,. .-..w.-.. ...... m... - .,. ...-.,... .. - -hg‘-.....“‘..mm v THE Ewe-mam STRUCTURE} or mam 9:0me . I. mum-comeumww SELF! CQNSiSTENT-FEELD CALcmmN 6F . THE LGw-L‘iifiGiELEfi-‘E‘RONQC STATES: . II. awww ENTERPREféTEQfi j ’ . ’ ' mssertatéon ior'thé Bégree ”of Ph.'D.-; ' MECHIGA?! STATE UNIVERSITY GREGORY DRVID GlLLlSPlE ' E375 ' ‘ LIBRARY WM Um i . .3 This is to certify that the ‘ thesis entitled 5 1e 5' THE ELE TRUCTURE 0F NITROGEN DIOXIDE GURATION SELF-CONSISTENT-FIELD LATION THE LON-LYING ELECTRONIC STATES: II. SPECTRAL INTERPRETATION presented by Gregory David Gillispie has been accepted towards fulfillment of the requirements for Ph.D. my” in Chemistry mm ’ Major profeRsor Date 2 ‘4 16175-1 0-7639 1 _:_L_ ABSTRACT THE ELECTRONIC STRUCTURE OF NITROGEN DIOXIDE I. MULTI-CONFIGURATION SELF-CONSISTENT-FIELD CALCULATION OF THE LOW-LYING ELECTRONIC STATES; II. SPECTRAL INTERPRETATION BY Gregory David Gillispie Multi-configuration self-consistent-field (MC-SCF) wave-functions have been computed for the low-lying RZAI, 2 ~2 ~2 4 4 2 + . 2, B 31' C A2, 32, A2, and Zg electronic states of N02. The minima of the A232, 8231, and EZAZ states have all been found to be within 2 eV of the minimum of the X2A1 ground state; for these states, C A B 2v potential surfaces have been constructed for purposes of a spectral interpre- tation. The 4B2, 4A2, and 22+ states are all more than 4 eV above the minimum of the ground state and have been examined in less detail. The study described here is an improvement on previous N02 ab initio calculations in three important areas: (1) The double-zeta-plus-polarization quality basis set is larger and more flexible; (2) The treatment of molecular correlation is more extensive; and (3) The Gregory David Gillispie electronic energies have been calculated for several different bond lengths and bond angles in each state. For the four lowest doublet states the following spectral data have been obtained: Te (eV) Re(A) Oe(degrees) 001(cm-1) 002 (cm-1) u (debyes) CZAZ 1.84 1.27 110 1360 798 0.05 8281 1.66 1.20 180 1192 960 0.00 A282 1.18 1.26 102 1461 739 0.46 Ezal 0.00 1.20 134 1351 758 0.37 (0.00) (1.1934) (134.1) (1358) (757) (0.32) The ground state experimental constants are included in parentheses. The estimated accuracy of the various param- eters is i 0.02 A for bond length, :.2° for bond angle, 1 10% for the vibrational frequencies, :_0.10 debye for dipole moments, and :_0.3 eV for the adiabatic excitation energies. Contrary to previous theoretical studies, but con- sistent with recent experimental evidence, the lowest excited state is of 2B2 electronic symmetry. Franck-Condom factors were calculated between the RZAI and A232 states; from these a theoretical A232 + i2 A1 absorption spectrum was generated. This spectrum, based solely on the ab initio data, is successful in accounting for many of the features in the low resolution NO2 absorption between 8500 and 0 6000 A. On the basis of experimental hot and cold band Gregory David Gillispie intensity ratios, along with the ratios predicted from the 232 + izAl origin is lower than the present experimental assign- ment of 1.48 eV. The A232 rotational constants as inferred Franck-Condon factors, a suggestion is made that the A from the accurate ab initio equilibrium geometry are vastly different from the results of experimental rotational analyses; this dissonance has been attributed to vibronic 2 coupling of the A 82 state with high vibrational levels of the ground electronic state. ~ It is also proposed that the origin of the B B1 + ”2 1 X Al absorption is about 925 cm- 1 lower than the present assignment of 14 743 cm- (1.83 eV); the analysis is based upon a consideration of isotopic band shifts and the impli- cation of these for the £231 asymmetrical stretching frequency. The fizBl state has a linear equilibrium geometry and its bond length is less than 0.01 A longer than in the izAl ground electronic state. Among the topics which are discussed in view of the computed potential energy surfaces are: predissociation for 1<3979 i and in the 2491 A band system, the N0(2H) + 0(3p) chemiluminescence, the photodissociation of NO2 by the 6943 A line of the ruby laser for which a single photon has an energy equal to only 57% of the N02 dissociation energy, a photodetachment experiment in N0; which may involve the 281 "ring state" of N02, the ability of level dilution to explain the anomalous fluorescence lifetime Gregory David Gillispie in N02, and the evidence supporting strong vibronic coupling of the A282 and RZAI states. The overlapping of the A282 + izAl and £231 + RZAI O absorptions in the 5200 to 3700 A region and the observation of Douglas and Huber that all bands become diffuse for O A<3979 A require that predissociation mechanisms be found ”2 for both the A B and B2131 states. In the latter state, 2 predissociation by vibration is the most likely possibility, ~ while the A B2 state could also undergo predissociation by vibration or be heterogeneously predissociated by high vibrational levels of the ground state. Possible mechanisms are suggested for predissociation of the upper state of the 2 2 ~ 0 — 2 B + X A 2491 A band system into either N(4S) + 02(329) 2 l or into NO(2H) + 0(3p). The visible and near-infrared NO(2H) + 0(3P) chemiluminescence must primarily be due to the electric- dipole allowed transitions A232 + izAl and 3231 + izAl, especially the former. The 3.7 um feature in the recom- bination has been tentatively assigned to a EZAZ + A282 electronic transition, rather than an izAl vibrational transition as has been suggested in the literature. The ruby laser photodissociation experiment has been interpreted by Gerstmayr, Harteck, and Reeves in terms of two consecutive single photon absorptions. If this be so, the likely sequence is A232 + izAl followed by EZAZ + ”‘2 A Bz. Gregory David Gillispie Herbst, Patterson, and Lineberger have found an N0; photodetachment process which is different from N0;(R1A1) + hv + N02(i2A1) + e. Although they have attri- buted their data to N0;* + hv + NO + O + e where N0;* is a [N‘O’Ol- peroxy isomer, we suggest that the process N0;(Ring state) + hv + N02(Ring state) + e be also considered. The "ring states" are the nearly equilateral triangle conformation states which have been found in ab initio calculations on both NO2 and 03, which is isoelectronic with NO}. It has been suggested several times in the liter- ature that the anomalously long fluorescence lifetime observed in N02 could be understood in terms of level dilution of the emitting state by a high density of izAl vibrational levels. However, the level density ratio of ~2 ~2 X Al to A B2 is much smaller than was inferred from the data of earlier, less accurate ab initio calculations. Consequently, this mechanism can only account for a small part of the anomalous fluorescence lifetime effect. An unusual feature has been found for the 22+ 9 state. The equilibrium geometry of this linear state has 0 0 two unequal bond lengths of 1.20 A and 1.42 A and the inversion barrier is approximately 800 cm-1. THE ELECTRONIC STRUCTURE OF NITROGEN DIOXIDE I. MULTI-CONFIGURATION SELF-CONSISTENT-FIELD CALCULATION OF THE LOW-LYING ELECTRONIC STATES: II. SPECTRAL INTERPRETATION BY Gregory David Gillispie A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree DOCTOR OF PHILOSOPHY Department of Chemistry 1975 AC KNOWLEDGMENT S I wish to thank Professor R. N. Hammer who first showed me the excitement of chemistry and who guided me through my undergraduate years. A debt of gratitude is also owed my teachers in physical chemistry and quantum mechanics: Professors J. L. Dye, G. E. Leroi, R. H. Schwendeman, J. F. Harrison, F. H. Horne, R. I. Cukier, W. Repko, and A. U. Khan. The financial support of the National Science Foundation, the Chemistry Department of Michigan State University, and the Argonne National Laboratory Center for Educational Affairs is also gratefully acknowledged. Special thanks are due Dr. A. C. Wahl for extending the opportunity for me to work in his group at Argonne National Laboratory, Dr. M. Krauss of the National Bureau of Standards for numerous insightful discussions and suggestions, and Mr. Al Hinds for helpful comments concerning the computer codes used in the calculations. Most of all, I would like to express my sincerest appreciation of the role that my advisor, Professor A. U. Khan, has played in my development. Not only has he been a ii friend as well as advisor, but has also always taught me to grow both as a person and a scientist. His encouragement and enthusiasm has inspired me throughout the five years of our association; without it, this study would not have been possible. iii TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . LIST OF FIGURES. . . . . . . . . . . . Chapter I. INTRODUCTION . . . . . . . . . . . II. MC-SCF CALCULATION OF THE LOW-LYING ELECTRONIC STATES . O C C O O O O O The OVC Approach to Triatomic MC-SCF Calculations and the Partitioning of the Orbital Basis . . . . . . . . Details of Configuration Selection . . . The Choice of the Basis Set and the Accuracy of the Computed Excitation Energies . . . . . . . . . . . An Analysis of Previous Theoretical Calculations and a Comparison With the Present Results . . . . . . . . . The Determination of the Potential Surfaces . . . . . . . . . . . The Accuracy of the Computed Spectro- scopic Constants. . . 2 An Unusual Feature of the ‘+ 29 Stat of N02. III. SPECTRAL INTERPRETATION. . . . . . . . A Qualitative Discussion of the Electronic Spectroscopy of N02. . . . . . . . Frangk-Condgn Factor Calculations. . . . The B281 + ngl Absorption System. . . . The B232 + XZAl Absorption System. . . . IV. CONCLUSIONS AND RESULTS. . . . . . . . iv Page vi ix 32 SO 60 65 69 74 74 106 116 125 148 Page REFERENCES 0 O O O O O O O O O O O O O 154 APPENDICES Appendix A. The Computer Programs and the Cost of the calculation. 0 O O O O O O O O O 160 B. The BaSis Set. 0 O O C O O O O O O 162 C. Molecular Orbitals of the OVC Wavefunctions for the Low-Lying Doublet States of NO2 . 164 D. Configuration Selection in "Well-Behaved" Polyatomic Molecules. . . . . . . . 173 Table II. III. IV. VI. VII. VIII. IX. XI. XII. XIII. LIST OF TABLES Zflu OVC Configurations and Mixing Coeffi- cients at R=2.25 Bohrs, 8=l80° . . . 2A1 OVC Configurations and Mixing Coeffi- cients at R=2.2552 Bohrs, 0=l34°, Energies of Various RZAl Wavefunctions in the [4s3pld] Basis . . . . . . . 232 OVC Configurations and Mixing Coeffi- cients at R=2.40 Bohrs, 0= 100° 2A2 OVC Configurations and Mixing Coeffi- cients at R=2.40 Bohrs, 9=110° . 4B2 OVC Configurations and Mixing Coeffi- cients at R=2.45 Bohrs, e=125° . . . 4A2 OVC Configurations and Mixing Coeffi- cients at R=2.45 Bohrs, 9=llS° . . . 22+ OVC Configurations and Mixing Coeffi— gients at R=2.45 Bohrs, 9=180° . . . Basis Set Dependence of N02 State Energies The 2A Electronic Energy Surface and Dipo e Moment . . . . . . . . . 2 The Bl (zflu) Electronic Energy Surface and Dipole Moment . . . . . . . . . The 2B Electronic Energy Surface and Dipo e Moment . . . . . . . . . The 2A2 Electronic Energy Surface and Dipole Moment . . . . . . . . . vi Page 25 30 32 33 34 35 36 37 40 41 42 43 44 Table XIV. XVI. XVII. XVIII. XIX. XX. XXI. XXII. XXIII. XXIV. XXV. XXVI. XXVII. The 482 Electronic Energy Surface and Dipole Moment . . . . . . . . The 4A2 Electronic Energy Surface and Dipole Moment . . . . . . . . The 22+ Electronic Energy Surface and Dipo e Moment . . . . . . . . Comparison of Important Aspects of N02 ab initio Calculations . . . . . ab initio Predictions of the Equilibrium Geometries of the Doublet Electronic States of NO2 . . . . . . . . ab initio Predictions of Excitation Energies in N02 . . . . . . . The OVC Equilibrium Geometries, Vibrational Frequencies, and Normal Coordinates for the Low-Lying Doublet States of N02. Comparison of the OVC and OVC-CI ab initio Spectroscopic Parameters for thE—X.A1 and 282 States of N02 . . . . . System Origins and Potential Constants of the 2B1 State of N02 as Determined by Hardwick and Brand . . . . . . Harmonic Frequencies of Isotopic Modifications in the 2A1 State of N02. . . . . Some Computed Franck-Condon Factors for A2B2 ++ XZAl Transitions in N02 . . Some A232 + izAl Vibrational Quantum Number Assignments of Brand gt El‘ and Proposed Reassignment . . . . . A Comparison of Experimental Band Intensity Ratios with a§_initio Franck-Condon Ratios for the Assignments of Brand 33 El. and the Proposed Reassignment . . . . . Various Determinations of the AZBZ Rotational Constants . . . . . . . . . vii Page 0 O 44 . . 45 . . 45 I O 52 O O 53 O O 54 . . 64 . . 66 . . 118 . . 120 O O 132 a . . 142 . . 142 . . 144 Table XXVIII. XXIX. XXX. XXXI. XXXII. XXXIII. XXXIV. XXXV. A Summary of the Theoretical Spectroscopic Parameters of the Low-Lying Doublet States Of N02 0 o o o o o o o o The RZAI OVC Orbitals for a Geometry of R=2.2552 BOHRS, 0=l34° . . . . . . The AZBZ OVC Orbitals for a Geometry of R=2040 BOHRS, 6:71.000. 0 o o o o o The B B1 OVC Orbitals for a Geometry of R=2.25 BOHRS, 0=180°. . . . . . . The CZAZ OVC Orbitals for a Geometry of R=2040 BOHRS, 6:11.00. 0 o o o o o OVC Configurations and Mixing Coefficients for H02 0 O O O O O O O O O O OVC Configurations and Mixing Coefficients for HONO. O O O O O O O O O O OVC Configurations and Mixing Coefficients for CH3NO O O O O O O O O O 0 viii Page 153 165 167 169 171 176 179 181 LIST OF FIGURES Figure Page 1. The nodal structure of the valence molecular orbitals in linear N02 . . . . . . . . l9 2. Flow chart of the OVC and OVC-CI configuration selection and orbital optimization procedures . . . . . . . . . . . . 23 3. Electronic energy as a function of bond angle for the low-lying electronic states of N02 . 46 4. Internal coordinate system for C2 geometries v in N02 0 o o o o o o o o o o o o 61 5. The Q3 potential of the 22; state of N02 as computed at the OVC-CI level . . . . . . 72 6. The adiabatic correlation diagram (ACD) for dissociation of NO2 into NO + O . . . . . 76 7. The ACD for dissociation of NO2 into N + 02. . 77 8. Schematic potential curves which could give rise to predissociation of the 2232 state of N02 into N(4S) + 02(323) . . . . . . 97 9. Schematic potential curves describing the ring states of N02 and N02 which could account for the unusual photodetachment process observed by Herbst, Patterson, and Lineberger . . . 101 10. Band origin isotope shifts in the BzBl + HZAI absorption system as calculated by Hardwick and Brand . . . . . . . . . . . . 122 ll. The'A2B2+X2Al absorption spectrum at 300°K generated from the gg_initio potential surfaces. . . . . . . . . . . . . 133 ix Figure Page 0 12. Superposition of a low resolution (~10 A) experimental N02 absorption spgctrum between 800 and 600 nm on the A23 + X Al spectrum generated from the ab initio potential surfaces . . . . . . . . . . . . 137 13. Superposition of a low resolution N02 absorption spectrum between 900 and 730 nm on the A 32 + X Al spectrum generated from the ab initio potential surfaces . . . . . 138 CHAPTER I INTRODUCTION The nitrogen dioxide (N02) molecule has long been of chemical interest because, although highly reactive, it is a stable free radical of non-zero spin with a high pro- pensity for dimerization. More recently, heightened ecological consciousness has focused attention on the role of NO2 in the photochemical smog cycle common to polluted urban atmospheres. In a similar vein, aeronomists have been attracted to the chemiluminescence known as the "air afterglow" or the "night afterglow" arising from emission from excited electronic states of N02, which are generated in the recombination of ground state nitric oxide molecules and oxygen atoms. Drawn to the molecule in many cases by the features mentioned above, the molecular spectroscopists have found additional challenges to be met. In absorption NO2 exhibits a discrete, nearly continuous spectrum from 8500 A to 3000 A. However, the spectrum is so complex and irregular that traditional methods of spectroscopic analysis have been less successful in elucidating the nature of the excited states involved than for many other molecules of comparable size. The diffuseness observed in the rotational structure at the first dissociation limit but the continuing vibronic structure is indicative of a predissociative process. Additionally, fluorescence lifetimes obtained from broadband excitation range from 40 to 90 usec, some two orders of magnitude longer than the value of 0.26 usec deduced from the integrated absorption coefficient. Related issues include whether or not the decay is exponential and its dependence on the wavelength of excitation. The common thread of these characteristics is that each involves the electronic structure of N02, especially that of the excited states. In 1965 Douglas and Huber [l] succeeded in analyzing the unperturbed K'=0 sub-bands in a bending progression of a 2B1 + X2 A1 transition. However, the observed bands in the 4600-3800 A spectral region are far from the extrapolated system origin between 8500 A and 6500 A predicted from a consideration of band isotopic shifts. No K'>O or upper state stretching features were found so the 281 geometry could not be inferred, although the state is expected to be linear, or nearly so, from the qualitative arguments of Walsh [2]. More recently, laser- induced fluorescence [3-7] and microwave-optical double resonance [8, 9] (MODR) experiments have identified a 2B2 excited state in the visible and near-infrared regions. The vibrational assignments of this state [5] are tentative, though, and the rotational analyses are for energetically isolated regions in which widely differing rotational constants have been reported. Consequently, although it is now firmly established that at least two electric-dipole allowed transitions, 2B2 e RZAI and 2 1' to the visible and near-infrared absorption of N02, the B1+ sz contribute excited state spectroscopic parameters and excitation energies are still subjects of intense research. The complexity of the spectra and the difficulty in analysis have been ascribed to various perturbations among the levels in this region. Attempts to identify the existence, nature, and mechanism of these perturbations have been greatly frustrated by the availability of several candidates. Non-empirical theoretical calculations [10-15] 4 2 have predicted low-lying 2A2, A2, 432, and 2 A1(2£;) states in addition to the experimentally observed RZAI, 2B2, and 2B1 states. However, the quantitative accuracy of these calculations is insufficient to justify using the theoretical results as the basis of a critical examination of the experimental data. An ab initio calculation which gives reliable values for bond lengths, angles, and excitation energies would provide the necessary framework for such an interpretation. In fact, in view of the complexity, accurate theoretical potential energy surfaces and descriptions of the electronic structure of the various electronic states involved are probably crucial in order to choose from competing perturbing mechanisms which disrupt the spectrum. In this study calculations are described which are significant improvements over the published literature. The basis set used for the trial variational wavefunction is larger and more flexible and the treatment of molecular correlation more extensive. In addition, the geometry has been varied for all the low-lying electronic states to cover a region sufficient to analyze the visible and near-infrared spectrum. The emphasis of this study is primarily on the description of the electronic states necessary for such an analysis; therefore, the reaction surfaces for dissociation and O + NO recombination and the region of very small bond angle have been given only a curSory examination. The electronic structure of such a variety of states can be described quantitatively by a multi-configuration variational trial wavefunction. In order to provide the most compact description, it is necessary that both the molecular orbitals and the configuration weights be simultaneously varied and optimized. This has been accomplished through the multi-configuration self—consistent- field (MC-SCF) formalism of Das and Wahl [16]. The experi- ence of application of the MC-SCF technique to diatomic molecules has shown that the dominant correlation effects are included in a set of Optimized Valence Configurations (OVC) which primarily determine the molecular extra correlation energy (MECE), that is, the additional corre— lational energy due to bond formation. The present study will discuss in detail the extension of the OVC method to , triatomic systems. After an analysis of the OVC approach to triatomic molecules is presented, the relative merits of MC-SCF calculations and the more traditional SCF and configuration interaction (CI) treatments are contrasted in Chapter II. The method of selection of configurations and the quality of the basis set are discussed and the present results are compared with those of previous theoretical studies. The details of fitting the ab initio energies at various geometries to potential surfaces are described; the resulting RZAI equilibrium geometry, vibrational frequencies, and dipole moment are compared with the experimentally determined values in an effort to gauge the accuracy and predictive capabilities of this study. Chapter III commences with a qualitative discussion of the electronic spectroscopy of NO2 in which a synthesis of relevant experimental data into the theoretical framework is attempted. Among the features which are treated are a characterization of the spectral regions in which the various electronic states manifest themselves, the observed predissociative processes, the visible and infrared NO(2H) + 0(3P) chemiluminescence, and the source of the anomalous fluorescence lifetime. Attention is then focused on some of the finer details of the two absorption systems which carry most, if not all, of the absorption oscillator strength in the visible and near-infrared. Toward this end, a technique is developed for evaluation of Franck- Condon overlap integrals from the ab initio potential surfaces. The 2B 2 1 + i A1 absorption is analyzed first with special consideration devoted to a prediction of the origin of this system, based on band isotope shifts. The apparently stronger 2B2 + RZAl absorption is examined next; a primary aid is an absorption spectrum constructed from the computed Pranck-Condon factors. Many of the long wavelength experimental absorption features can be corre- lated with those of this spectrum which is generated almost entirely by non-empirical means. A reassignment of the 2B2 origin is proposed which is supported by the relative intensities of hot and cold bands. Finally, it is suggested that vibronic interaction of the 282 state with high vibrational levels of the f2 Al ground state provides the only consistent explanation for the incompatibility of the experimentally determined 2B2 rotational constants and those inferred from the ER initio Born-Oppenheimer geometry of this state. Chapter IV summarizes the most important conclusions of this study. CHAPTER II MC-SCF CALCULATION OF THE LOW—LYING ELECTRONIC STATES The OVC Approach to Triatomic MC-SCF Calculations and the Partitioning of’the OrbItEIfBasis The MC-SCF calculations reported in this work were performed with the BISON-MC program deve10ped by Das and Wahl [17]. The details of the MC-SCF method and the computational procedures employed in this program have been discussed elsewhere [16] and only a brief review of the MC-SCF theory will be given here. The total electronic wavefunction is represented by a linear combination of configurations, 4k, + + + + w(r1, ....., rm) — : Ak¢k(rl, ....., rm) (1) where the Ei represent the spin and space coordinates of the ith electron. The configurations are appropriately projected linear combinations of antisymmetrized determinants over a set of molecular spin orbitals, {¢i}, + + A + + ¢k(r1,....., rm) = ga'smlul) ¢m(rm)I (2) where the symbolt§d 3 indicates that the determinant has I been projected to be an eigenfunction of the total spin and to possess the overall orbital symmetry of the electronic state in question. For a C geometry, the BISON-MC system 2v of codes only recognizes the symmetry operations of the identity and reflection through the molecular plane. This constraint, forced by programmatic considerations, can have serious implications for the orbital optimization of excited states. For example, if one is attempting to treat the 2B2 state of N02, "variational collapse" to the 2A1 ground state may occur since both of these transform as 2A' in the CS point group recognized by the program. This difficulty was circumvented by choosing the initial guesses of the mole- cular orbitals to transform as the irreducible representa- tions of the C2v point group; in all cases the orbitals maintained the desired symmetry throughout the orbital optimization and "variational collapse” was averted. The spatial portion of each molecular spin orbital is expanded in terms of a set of atom-centered basis functions, {xj}, ¢. = 2 cijx. (3) For NO2 and other polyatomic molecules, it is convenient to use Gaussian-type functions (GTF) for which fast mole- cular integral programs are available. In this study, the necessary integrals were evaluated with the integral generating code of the POLYATOM [18] system, suitably modified to provide an input into the BISON-MC system of codes. The MC-SCF procedure variationally determines the CI mixing coefficients, Ak, and simultaneously, the basis function expansion coefficients, cij' This added flexibility over an SCF + CI treatment permits a significant truncation of the length of the CI expansion while still providing a satisfactory description of the electronic structure. However, the adequacy of the description will depend some- what on the electronic property under consideration and the absolute lowering of the total energy is not necessarily the most desirable way to choose a configuration. The OVC method, in fact, attempts to distinguish between intra- atomic and inter-atomic types of correlation, specifically those portions of the correlation energy which vary signi- ficantly with geometry. In the case of diatomic molecules, quite compact MC-SCF wavefunctions have been found to quantitatively describe the binding in such molecules as F2 [19], CO [20], and OH [21]. Here the analysis used for the diatomics is extended to the triatomic systems and modified as required. The OVC model concentrates on the changes in the electronic structure which occur as the molecule forms. In the diatomic molecule case, these changes are referenced to an asymptote of Restricted Hartree-Fock (RHF) atoms. The base set of configurations is chosen to insure that the molecule formally dissociates into RHF atoms. Added to the 10 base set is a set which contributes to the molecular extra correlation energy (MECE) and whose correlation contribution vanishes at the dissociation limit. These two classes of configurations yield compact, physically meaningful wave— functions. Only for very high accuracy has it been found that the smaller variation in the intra-atomic correlation energy with the internuclear separation must be considered [19]. The localization of electronic charge in the molecule is determined by one or a few dominant con- figurations. Since we are concerned primarily with the lowest state of a given spatial symmetry and multiplicity, the state is usually valence in character, i.e., no principal quantum number promotion, relative to the constituent atomic orbitals, occurs in the molecular orbitals from which the dominant configurations are constructed. It is then desirable to define a non-virtual space composed of both core and valence orbitals. This space contains those orbitals which are linear combinations of the orbitals occupied in the constituent atoms of the molecule and those additional orbitals with the same principal quantum numbers as the occupied orbitals. For an A82 triatomic composed of first row atoms, each atom has five accessible orbitals, the ls, 23, 2px, 2p and 2pz. The non-virtual space y! therefore contains fifteen orbitals which are linear combinations of the fifteen accessible atomic orbitals. Of these, the core orbitals are those which are practically 11 the same in both the atoms and the molecule; the remaining non-virtual orbitals are the valence (i.e., bonding, non- bonding, and anti-bonding) orbitals. The virtual space contains all other orbitals. The experience from diatomic molecules would indicate that a relatively small part of the MECE contribution arises from excitations into virtual orbitals. Rather, such excitations mainly serve to describe atomic-like correlations, which are fairly constant over the entire range of inter-nuclear separations. The identification of the valence space for an MC-SCF calculation is desirable because the majority of critical molecular correlation effects reside within it. We further define valence configurations as those with all core orbitals fully occupied and all virtual orbitals unoccupied. The base configurations are usually those whose contribution varies most rapidly with internuclear geometry and are always valence configurations. Since the MC-SCF approach optimizes the molecular orbitals and configuration mixing coefficients simultaneously, a calculation based only on a careful selection of valence configurations still yields the important portion of the correlation energy. The remainder of the correlation energy is assumed not to vary with valence state or with the geometry of a given state in the neighborhood of its equilibrium geometry and away from near degeneracies with states of the same symmetry and multiplicity. 12 In all of the states of NO2 which we have con- sidered, there are five orbitals which are invariably of core character. In terms of symmetry orbitals, they are log, 20 , 30 , lou, and Zou for linear geometries; they 9 9 become lal, 2al, 3a1, lbz, and 2b2 for bent geometries. These correspond in a localized description to ls atomic orbitals on each atom and 29 atomic orbitals on the oxygen atoms. The role of the sixth lowest energy orbital, the 4a1 (409), merits further comment. Experience from MC-SCF calculations on first-row diatomics indicates that the core orbitals are the first four 0 orbitals, accounting for most of the ls and 2s character of the atoms. Correlation of these orbitals for purposes of consideration of differential atomic correlation affects the computed potential curves almost negligibly. If a similar situation prevails in N02, there will be six core orbitals. However, a preliminary calculation clearly demon- strated that such is not the case. Initially the 4a1 orbital was held doubly occupied in all configurations used 2 2 A1 and B2 states. in the orbital optimization of the i When excitations from this orbital were included, the adiabatic separation of the two states increased by 0.4 eV. Examination of basis set expansion coefficients of the 4a1 orbital in the two states reveals that in the 282 state it is a nitrogen 2s core-like orbital; however, in the 2A1 13 state, the nitrogen atom is strongly s-and-p hybridized and the 4a1 orbital is primarily describing N-O o-bonding. It might seem that this behavior could be ration- alized in terms of the very different bond angles in the 2Al (134°) and 2B2 (102°) states and subsequent changes in hybridization with bond angle. However, the 4a1 orbital retains its o-bonding character in the 2Al state even as the bond angle is reduced to 105°; similarly, in the 2B2 state, the 4a1 orbital is still mostly a nitrogen 25 atomic-like orbital for bond angles of 130° and greater. This behavior and the bonding in the various electronic states of NO2 will be more critically examined in the future. At the present, though, suffice it to say that it is speculative in AB2 (and ABC) triatomics to designate the first six (or more) molecular orbitals as core, and hence leave them uncorrelated in an MC—SCF calculation. For a CI calculation based on SCF orbitals it is even less advisable to make such assumptions. In an MC-SCF calculation, the core orbitals are quite free of DO con- tamination because it is energetically favorable for the Po character to be rotated from the core orbitals into the valence orbitals during the orbital optimization. In the single configuration SCF method, however, the core orbitals often contain a substantial amount of PO character because the total electronic energy is invariant to a unitary transformation of the doubly occupied orbitals. Conse- quently, unless a suitable unitary transformation is made 14 before the CI is performed, it is likely that substantial amounts of valence correlation will be missed if any but the lowest orbitals are designated as core. The previously described orbital partitioning has reduced the treatment of valence correlation in N02 to a 13 electron, lO orbital problem. In principle, one would form all possible configurations of the desired orbital symmetry and multiplicity for this number of electrons and orbitals; the molecular orbitals and mixing coefficients would then be optimized in a single computation. In practice, the large number of configurations makes such an approach unfeasible. The OVC method effects a substantial reduction in the number of configurations by focusing only on the MECE contributions and ignoring for the most part differential atomic correlation. Usually, a maximum of 10 configurations is sufficient for first row diatomic applications. If the extension of the OVC method to triatomics were straightforward, the only modification necessary would be an increase in the maximum number of configurations to allow for the additional bond. However, there are conceptual complexities in the treatment of poly— atomic, as opposed to diatomic, molecules and these are detailed in the next several paragraphs. The principal difficulty is the existence of four distinct asymptotes (AB+C, BC+A, CA+B, A+B+C) for a general ABC triatomic, in contrast to the single A+B asymptote for an AB diatomic. No longer is the designation of the base 15 set of configurations (i.e., those configurations which allow dissociation into RHF fragments) unique; rather, it is dependent on the dissociation process of interest. For example, consider the RHF configuration of the 2Al state of N02 2 2 2 2 2 2 1 ....(4al) (3b2) (Sal) (lbl) (4b2) (la2) (Gal) which cannot dissociate to RHF fragments for either the NO(2H) + 0(3P) or the N(4S) + 02(32;) asymptotes. In order for dissociation into NO(2n) + 0(3P), configurations with three open shell orbitals, one of a' and two of a" symmetry, must be included in the base set of configurations. On the other hand, the N(4S) + 02(32;) asymptote requires five open shell orbitals, and hence a different set of base configurations for dissociation in CS symmetry. (Dissoci- ation of the 2A1 state to this asymptote is symmetry for- bidden in CZV') Similar problems arise when one attempts to identify the MECE configurations. Consider the various roles of a double excitation 02 + 0'2 within the valence space as an AB2 triatomic dissociates into AB + B. At infinite separation of the two fragments, the possibilities are: a. ¢ ends up with the AB fragment and 0' belongs to the B fragment, or vice-versa. Then this excitation is a triatomic MECE contributor since its effect vanishes for infinite separation of the atom and diatom. 16 b. Both 0 and 0' belong to the atomic B fragment. The excitation in this case is a differential atomic (intra-atomic) correlation term. c. Both 0 and 0' belong to the AB fragment. Then the excitation is a differential diatomic correlation term, which has no counterpart for diatomic dissociation. The geometry dependence of the con— tribution of such terms to the correlation energy is quite large and its neglect does not appear to be justified. Moreover, if the excitation belongs to case (a), for example, for AB + B dissociation, it may belong to case (b) or (c) for A + B2 dissociation. Excitations from the valence space into the virtual space will also exhibit ambiguities in regard to classification into different types of correlation contributions. Although some of the advantages of the OVC approach are lost for polyatomic molecules, the consequences are not great for this study. Our interest is in the potential energy hypersurfaces relatively near their minima, since we are attempting to treat the electronic spectroscopy and structure of N02. Thus, our configuration selection is concerned with insuring that the dominant correlation contributions are included near the equilibrium geometry of each state. It is not necessary to 3_priori identify the base configurations or distinguish MECE from intra-atomic or differential diatomic correlation contributions. We 17 shall continue to characterize the approach we have taken here as the OVC method, although it is not being applied in strictly the same manner as for diatomic molecules. Details of Configuration Selection At the time of this study, the program constraints at Argonne National Laboratory for a basis set of the size we employed were a maximum of 20 configurations for MC-SCF orbital optimization and a maximum of 99 configurations in a CI calculation. The procedure we have followed for each electronic state of N02 is to generate an OVC configuration list and an OVC-CI configuration list, of which the OVC list is a subset. The OVC list contains what we shall refer to as the dominant configurations, i.e., those configurations which are most important in describing electron distribution and correlation in the molecule. The molecular orbitals are optimized in an MC-SCF calculation using the OVC configu— ration list, yielding what we term the OVC wavefunction. When higher accuracy is desired, a CI calculation is per- formed using the OVC-CI configuration list and the OVC orbitals. The resulting wavefunction is referred to as the OVC-CI wavefunction. The Restricted Hartree-Fock (RHF) configuration is by definition the most important configuration in the neighborhood of the equilibrium geometry; its normalized mixing coefficient is greater than 0.93 for all the states of NO2 which have been examined. Identification of 18 additional dominant configurations is first made in terms of valence charge transfer excitations, which are best demonstrated in a linear molecule. Such excitations are by far the largest MECE contributors in diatomics, and are similarly important in N02, although they have not been specifically classed as MECE terms. The Znu state of N02 is used as an example to illustrate the origin of the term "charge transfer." Exclusive of the five core orbitals, the valence orbital occupation of the zflu RHF configuration is 2 2 0 0 4 4 l (409) (3Gu) (509) Mon) (lnu) (11:9) (”0) . The 409 and 30u orbitals account for the two N-O 0 bonds and Sag and 40u are the corresponding anti-bonding 0 orbitals. The lnu orbital is a delocalized bonding w orbital and Zwu is the corresponding antibonding n orbital; lng is non-bonding, having density only on the oxygen atoms in the absence of polarization functions in the basis set. The nodal characteristics of the orbitals are schematically illustrated in Figure 1. Better than 80% of the valence correlation energy of this state is obtained within the following MC-SCF orbital occupancies: H6909 0-0 9 0-0 WONG—GOG 000 M 920 2 (DOOMQDOO Fig. 1. The nodal structure of the valence molecular orbitals in linear N02. J:- O 0' U.) U1 0 .1: O U lnu 111 S 21! L1 20 409 30u Sag 40u lnu lug Znu (l) 2 2 0 0 4 4 1 (2) 2 0 2 0 4 4 l (3) 2 0 0 2 4 4 1 (4) 0 2 2 0 4 4 l (5) 2 2 0 0 2 4 3 (6) 2 1 O 1 3 4 2 (7) l 2 1 0 3 4 2 (8) 2 2 0 0 4 2 3 (9) 2 l l 0 4 3 2 The first is the RHF configuration. The next three correlate the 0 bonds, the fifth correlates the n bond. The sixth and seventh simultaneously correlate a 0 and a w bond and the final two allow for redistribution on charge within the valence orbitals. In each case, electron density (charge) is transferred across the bonds in the molecule in contrast to excitations which are primarily localized on a single center. However, since the calculations were performed using (real) Gaussian basis functions, the n orbitals are two-fold degenerate rather than four-fold. Therefore, orbital occupancy 5, which is related to the RHF configu- ration by the double excitation lnuz + 2nu2, actually is spanned by four configurations: lflu2(x) + 2nu2(x), lnu2(y) + 2nu2(y), and lnu(x)lnu(y) + 2nu(x)2nu(y), which has two spin couplings. Also, any orbital occupancy with 21 more than one open shell can have the open shells coupled in various ways to give eigenfunctions of S2 and S2. The coupling is specified in vector addition fashion as the orbital occupancies are scanned from left to right. Orbital occupancy 7, for example, could give rise to as many as five open shells: l l l 1 1 (409) (Sag) (1nu(y)) (2"u(X)) (2nu(y)) and there are five possible ways to couple five open shells to give a doublet state. The MC-SCF wavefunction, computed for the valence charge transfer excitations, gives a quite satisfactory qualitative description of electronic structure. As discussed in Appendix D, selection of the valence charge transfer configurations is well-defined for the ground states of polyatomic molecules such as H02, HONO, and CH3NO, for which valence bond structures can be written. An exhaustive search of configuration lists is not necessary for an accurate description of electronic structure in these cases and this is a strong point in favor of the OVC approach, especially for large polyatomic molecules. Unfortunately, NO2 does not have a satisfactory valence bond description and a more rigorous approach to con- figuration selection must be taken if the accuracy necessary for a spectral interpretation of N02 is to be achieved. We detail below the selection of the OVC and OVC-CI lists which are vital for this improved representation. 22 For reasons of economy, the OVC and OVC-CI configu- ration lists were Optimized at a single geometry in each electronic state and then these lists were utilized for all the geometries considered in that state. Preliminary studies in a double-zeta quality basis set gave reasonable estimates of the equilibrium geometries in the states of interest and it was at these geometries that the configu- ration lists were determined. We further note that the configuration lists were optimized in this double-zeta basis set, rather than the extended basis set used for the actual construction of the potential energy surfaces. Figure 2 displays a flow chart of the complete con- figuration selection and orbital optimization procedure. Although several individual steps are involved as depicted in the figure, many of them could be combined, eliminating intermediate operator intervention. The basic philOSOphy of the approach is to successively expand the configuration lists by examining various classes of configurations and retaining the most important configurations, as gauged by the magnitude of their mixing coefficients. We identify four classes of configurations to be examined: 1. The valence charge transfer (VCT) configurations. These are specified in 3.2riori fashion; the experience from studies as described in Appendix D is used as a guide to their selection. —'—_—_| I | Construct initial set of m's L__-__J F TTTTT 7 I Construct initial 23 I aumumzcwc L _> 4W wave t‘mction I Calculate new OVC wave mnction YES Siould any higier excitations be in OVC list? _19_; Zonstmct final I OVC-CI list L____J Fig. 2. F-—‘fi Construct L VSD-CI list ___J F"-'”7 Construct higier excitation CI list L___ Calculate OVC-CI wavenmctions in sets wing OVC orbitals Calculate final OVC-C: raw-2‘: ation selection and orbital optimization precedures. Calculate VSD—CI wave Function Construct truncated l OVC-CI List ' I (A1>O .01) l l— _____ .1 Flow Chart of the OVC and OVC—CI configuration 24 2. Single and double excitations from the RHF con- figuration within the valence space. These comprise the "valence-singles-and-doubles" configuration interaction (VSD-CI) list and include the VCT configurations. 3. Triple and higher excitations within the valence 92229.- 4. Single and double excitations from the valence space into the virtual space. The initial step is to compute a CI wavefunction with the VSD-CI list using the orbitals from the original VCT MC-SCF calculation. On the basis of this wavefunction, the configurations with the largest mixing coefficients (usually Ak greater than 0.04) are collected into an initial OVC configuration list. A new MC-SCF wavefunction is computed for this list, followed by a VSD-CI on the new set of orbitals. In every case the OVC list was stable, i.e., those configurations with the largest mixing coefficients in the VSD-CI wavefunction are identical to the configura- tions which constitute the OVC list. The next class of configurations to be examined is that of triple and higher excitations from the RHF configu— ration within the valence space. Rather than consider all possible excitations of this type (~3000 for a doublet state in C symmetry), we have taken a modified approach. 2 2v From the OVC configuration list of the nu state of N02, given in Table I, we identify nine dominant o orbital 25 iHmoe. me 0 me o H m m H H m H o m H H886. me 0 me H H m m H m H o H m H Home? H m m o m m m o o m 396.? H m .e. o m m o m m o Hmmeoe me o m m H o m m o H m H 330.0. H m m o m m o m o m H8305 no 0 m H m m H H o e m m 393.? me o m m H o m m H o H m Romeo: me 0 me o H m m o m m H H H H 3.8.? me o m H m o m m o H H m 088.? me 0 me o H m m H m H o H m H 838.0 me 0 me o H m m H m H H o H m mHmGod me o N m H H m H e o m m H885: H m m m m o e o m m Seemed me 0 me o H m m H H m o H H m 93:5. H m m m o m o o m m mHmHHe me o m H m H H m o o m m mmmmme H m m o m m o o m N 6833868 9832 388.5% New WH ”I: “em m: “pH :6: mom 5% wee ll .3 oomauc .mmem mm . mum Ed. SCHEOQ GE 92. mZOHEchmzoo 96 a H 59E. 26 occupancies: (409)2(30u)2, (409)2(3ou)l(509)1, (409)2(30u)1(40u)1, etc. All possible electron rearrangements within the n orbitals for each of the dominant 0 orbital occupancies are formed; all which are single or double excitations from the RHF configuration are removed, since they have already been treated.in class 2. The rationalization of the restriction of class 3 configu- rations is that since the occupied o orbitals are much lower in energy than the n orbitals, any configuration involving extreme disruption of the o orbitals (e.g., (40g)l(509)1 (40“)2 is going to be so highly excited that its effect in a CI will be negligible. Effectively what is achieved is a full n electron CI for a well correlated 0 core. Slight modifications are necessary for the non- linear electronic states, as the distinction of a and w orbitals is no longer rigorous. These modifications are minimal for quasilinear cases, such as the 2A1 ground state of N02, where the genealogy of the orbitals for the bent geometries is easily traced to the linear precursors. For strongly bent states, such as the 2B2 and 2A2 states, the individual orbital structures must be more carefully examined in order to distinguish those responsible for c-bonding, but the task is not difficult. Despite the reduction in number of class 3 con- figurations to be considered, there are still too many (“150) to be examined in a single calculation with our existing codes. We therefore truncate the VSD-CI list to 27 those configurations with normalized mixing coefficients of 0.01 or greater (~40 configurations). To these are added various sets of the triple and higher excitations in a series of CI calculations based on the OVC orbitals. The final OVC-CI list is selected by scanning the various CI runs, including the VSD-CI calculation, and choosing those configurations predicted to have the largest mixing coefficients up to the limit of 99 configurations. In a few instances triple excitations were found to be important enough to warrant their inclusion in the OVC orbital optimization. The best example is in the 282 state where a triple excitation has a very substantial mixing coefficient of 0.07718. Finally, class 4 of configurations, single and double excitations from the valence space into the virtual space, must be addressed. The types of excitations which were examined are: (1) single excitations, ¢ + x: (2) diagonal double excitations, ¢2 + x2; and (3) double excitations, ¢¢' + t'x, in which one electron is excited into the virtual space, while another is rearranged within the valence space. Our investigation was limited to the lowest energy virtual orbitals, namely Son and 31:u for linear geometries; the former becomes 6b2 for bent geome— tries, while the latter splits into 8a1 and 3b1. For this particular investigation, an orbital basis was calculated in MC-SCF fashion for the most important 15 OVC configura- 2 2 2 2 tions, plus the excitations 3au + Sou , 409 + 50u , 28 1n 2 + 3n 2, and 1n 2 + 3n 2, which optimize the virtual u u g u orbitals. This was followed by a CI calculation using this orbital basis, including the most important valence con- figurations (again those with mixing coefficients greater than 0.01) and the class 4 excitations under investigation. For the 2nu state, no configurations involving the Scu orbital were significant contributors to the wavefunction. This is undoubtedly due to the high energy character of this orbital, which results from a complicated nodal structure along the internuclear axis. Consequently, the Sou orbital was deleted from further consideration. The magnitude of the energy lowering of the remaining virtual configurations was nearly 0.02 hartrees, 4 but constant to 5 x 10' hartrees over the range of bond lengths 2.25 to 2.45 bohrs. A similar study for the 2A1 ground state gave an equally constant energy lowering for bond length variation, and the magnitude of the virtual correlation energy was within 0.002 hartrees (0.05 eV) of that found for the 2 nu state. In the final calculations used to construct the potential surfaces, class 4 configurations were neglected. As support for this simplification we note that: l. The magnitude of virtual correlation appears nearly constant with geometry variation near the equi- librium geometry. Thus the characteristics of the potential surface of a given state are essentially independent of virtual correlation. 29 2. The magnitude of virtual correlation does not appear to vary substantially from state to state. Thus the electronic spectroscopy of the molecule, which reflects the energy separation of the various electronic states, is nearly independent of virtual correlation. In order to present a more quantitative illustration of the configuration selection process, we cite some details for the RZAI ground state. For this state only, the OVC-CI configuration list was actually determined in the extended double-zeta-plus-polarization quality basis set used for the generation of the potential surfaces. The optimization was performed for a bond length of 2.2552 bohrs and an angle of 134°, the experimental equilibrium geome- trical parameters; the OVC wavefunction at this geometry is given in Table II. The energy of the 2A1 RHF configuration is -204.0607 hartrees for the OVC orbitals. We note that this is 22E the 2A1 SCF energy in this basis since the orbitals were deter- mined for a set of configurations rather than the single RHF configuration; in the double-zeta basis, the SCF energy is 0.01 hartrees lower than the energy of the RHF configu- ration calculated for the OVC orbitals. The OVC energy is -204.2080 hartrees; the energy for the 99 most important VSD-CI configurations, based on the OVC orbitals, is lowered to -204.2309 hartrees. For the OVC-CI wavefunction, triple and higher excitation configurations were also 30 mmeme.ou o m m H m m e m o m mHmmo.ou m.o e m.o o H H m H m m o H H m moose.ou o m m H m m m o o m mmeso.e m.o H H m H m m H o o m m eHomo.ou o m m H m m o m m o Hmomo.o m.o H m.o o H m H H m m o H m H ssHmo.o m.e o o m m m m H H o H m mmmme.on m.o H m H H m H m e o m m emmmo.ou m.o o H m H m m H e o m m HHmmo.ou m.o o m.o o o m m H m m H H H H Hemme.o. m.o o o m N m H m e H H m mmmmo.o- m.o H m.o o H m H H m m H o H m momee.o m.o H m.o o H H m H m m o H H m mmseo.eu m m e H m m e o m m sMHme.o- m.o o H H m m H m o o m m emAOH.o m.o H H H m m H m e o m m HmmHH.o- m o m H m m o o m m msmmm.o o m m H m m o o m m mesmHeHeeeoo weds: 98:88.58 Hem NsH HsH Hem mes Hem mam as. mom Hes ozmane .mmmom mmmm.m& Be. WHZMHOHEOO ozUnE 32¢ mZOHBEUHmzoo o>o flaw HH mama. 31 included; the 99 most important lower the energy to -204.2422 hartrees. The total number of class 2 and class 3 valence configurations is 282, of which only 99 were included in the OVC—CI wavefunction. The effect of the truncation was examined in the following manner: the OVC-CI list was temporarily reduced to the 49 most important con- figurations (E=—204.2330 hartrees) and this list was com- bined with sets of the 183 omitted valence configurations in a series of CI calculations. Under the assumption of pairwise additivity of the neglected sets, the final energy would have been improved by only 0.004 hartrees for the entire 282 valence configurations rather than the more restricted set of 99 configurations. All of the information of this paragraph is summarized in Table III. On the basis of these numbers, we estimate for the izAl state of NO2 that: (l) the 18 configuration OVC wave- function accounts for about 80% of the valence correlation energy; (2) the 99 configuration OVC—CI wavefunction accounts for greater than 95% of the valence correlation energy; and (3) triple and higher excitations from.the RHF configuration contribute 6 to 8% of the valence correlation energy. It is important to note our definition of valence correlation energy as that portion of the correlation energy obtained within the set of orbitals designated as valence. Although only a fraction of the total correlation is accounted for, it is that portion of the correlation energy 32 TABLE III ENERGIES OF VARIOUS XZAl WAVEFUNCTIONS IN THE [4s3pld] BASIS Wavefunction Description Energy (In Hartrees) RHF Configuration from OVC Orbitals -204.0607 SCF (estimated) -204.07 OVC (18 Configurations) -204.2080 VSD-CI (99 Configurations) -204.2309 OVC-CI (49 Configurations) -204.2330 OVC-CI (99 Configurations) -204.2422 Full Valence OVC-CI -204.246 (282 Configurations, estimated) we have obtained which is crucial for a proper description of the electronic states. The OVC configuration lists and mixing coefficients 2 2 4 4 22+ are given for the B2, A2, B2, A2, and states in Tables IV through VIII. The Choice of the Basis Set and the Accuracy of the Computed Excitation Energies The double-zeta basis set used in the preliminary calculations was the Dunning [4s3p] contraction [22] of the (935p) primitive Gaussian basis set of Huzinaga [23]. In this basis the SCF energy of the ground state at-its experimental geometry is -203.9560 hartrees. A comparison 33 NHNNOS. me e N o N N N H o H H N smemee me H m5 H H H N H H N H o N N NsHsee me 0 me o H N H N H N H o H N mosses. me o N H H N N H o e N N Gommoe m5 H o N N H N H H o N N Hmmmoe me H me o H H N N H N o H H N sommoeu o N N N H N N e o N mmmmoel me H me H H N H H H N e H N N N885. o N N e H N o N N N 830.0 ms H me H e N N H H N H H H N GNSOS me o o N N N N H o H H N 380.0. ms H me e H N H N H N H e H N HESS me H N H H N N H e o N N 968.? N o N N H N o o N N sage: N N e N H N o o N N GNSHS me o H H N N N H o o N N iNmHe. me H H H N N N H o o N N Nmemms o N N N H N o o N N 888580 N552 $588.58 HnN NeH HsH Hem Nee Hem Nam Hep Nam Hes oooaum .mmmom oz.mum Be. S050 SS 9?. mZOHBéDmezoo o>o >H an”. N m m 34 omemo.e. m.o o m.o e H H H H N N e H N N mesmo.e m.e e m.e e H H H N N N H o H N moemo.e- e H N N N N e N o N Neemo.o m.o e e N H H N N H e N N Neeme.o- m.e o H e N H N N H o N N NNNse.e m.o e H e N N N N e H H N seeme.o- o H N N N N N o e N Haemo.e. m.e e e N H N N N o H H N Neemo.o. e H N o N N o N N N NNNGo.e. m.o H m.o e H H H H N N e H N N emmwo.o m.e H m.o e H H H N N N H e H N Nmmee.o- m.e o m.o H o H N H N N H H H N seGmH.ou N H e N N N e o N N mmHsH.o m.o H H H H N N N e o N N NGONH.o- m.e e H H H N N N e e N N 53.0 o H N N N N o e N N BeeHedeooo weds: 9588.58 HnN weH HeH Hem Nee Nam Nam Hes Nam Hes oeHHue .mmmom os.Nnm as mezmHoHsmmoo eZHHH: eze NZOHeNmseHszoo use NHN >393. 35 msose.ou m.H H H H N H N N o N o N Nsmso.ou m.H H H H N H N N N o o N emsme.ou m.H H H H N H N N o N N e oommo.ou m.H H H N H o N N H o N N eomme.on m.H H m.o o N H H H N N H o H N Hmmme.o m.H H m.o o N H H H N N o H N H smame.e m.H H m.o o H H N N N H H e H N smHeo.ou m.H H m.e o H H N N H N o H H N smmee.o| m.H H m.o o H N H H N N e H H N Nesmo.e m.H H N H H N N H e o N N mNGNH.e. m.H H H N H N H N o o N N Hmomm.e m.H H H H N H N N o e N N mesmHedeeoo weds: 8388-58 HeN NNH HeH H8 Nee Hem Nam Nee Nam Hes omNHue .mmmom ms.Num ee_mgzmHoHsmmco NZHHHz oze moneemseHszoo o>o Nms H> mama. 36 NHmmo.on m.H H H N N H H N o N e N msze.on m.H H m.o o H N N H N H o H H N Hmomo.on m.H H m.o e N N H H H N o H N H NNHmo.o m.H H . N N H N H H o o N N ssmmo.on m.H H H N N H H N N o o N NGHGo.on m.H H m.e o N H N H H N o H H N mONmo.ou m.H H H N N H H N o N N e NHmmo.o m.H H m.o e N N H H H N H e H N ONmGo.ou m.H H m.o e H N N N H H H o H N mmmwm.e m.H H H N N H H N o o N N meseHeHeNeoo NEE”: gaseoufiem HeN NNH HsH Hem Nee Hem Nam H3 NR Hes Oman G .958 mn. mum Ha EUEOU gang 9?. mZOHBEHano D>O NEH Egg. 37 HHH> EB mmHsoe me o H H N o N N o N H N messes- me H o N N H H N o N H N steed- me o o N N H N H o N N H mmmsoeu me o H N H o N N o N N H Romeo me o o N N H H N o N H N :23? o N N o N N N H o N mmmmoeu m5 H me H H N H o N N H H H N 380.? m5 H H H N o N N o N H N mNNSé me H o N N H H N H o N N 32.0.0 me 0 m5 H H H N H H N o H N N 0885 me H H H N o N N H o N N H385: me o o N N o N N H H H N $30.0; ms H ms H H N H H N H o H N N HNmNoen N N o o N N o H N N Hammad: me H me H H H N H H N o H N N msHmoS. o N N N N o o H N N Reece. N o N o N N o H N N SOHHS- o N N N o N o H N N OHHmmd o N N o N N o H N N 66633688 mans: 3:88.5em ”TN w: m: ”N m: “H see mom sem mes OSHA .meiom ms.Nnm .2 Eugene NE 93 mzoEEBES E6 WNN 38 with the results of Burnelle and Dressler [12] amply demonstrates the efficacy of the optimized Dunning con- traction. These authors also contracted the Huzinaga basis to [433p] but in an gd_hgg fashion; the energy of -203.8857 hartrees which they obtained is more than 0.07 hartrees above that for the optimized contraction and is indicative of the sensitivity of the energy to the con- traction scheme. The SCF energy obtained in the [4s3p] basis is some 0.164 hartrees from the Hartree-Fock limit of -204.12 hartrees as estimated by Schaefer and Rothenberg [24]. However, these authors demonstrated that allocation of a full set of single component Cartesian d-functions to each nuclear center lowers the SCF energy more than 0.12 hartrees over that obtained for the [432p] Dunning contraction [22]. Since the [432p] contraction is about 0.01 hartrees inferior to the [4s3p] contraction for N02, it is expected that the [433p] basis augmented with d-functions should be no further than 0.05 hartrees from the estimated Hartree-Fock limit. Such a basis, hereafter referred to as [4s3pld] or simply 431, is the one used here for all reported calculations. Due to the relative insensitivity of the energy to the choice of orbital exponent, no attempt was made to optimize the exponents of the d-functions; rather the eXponent was chosen as 1.0 for the nitrogen atom and 1.35 for the oxygen atoms. The former is close to the optimum Value found by Dunning in SCF calculations on the nitrogen 39 molecule [25] and the latter represents a reasonable scaling of the nitrogen exponent. A comparison of the OVC results in the [453p] and [4s3pld] basis sets in Table IX reveals that the computed transition energies between the ground state and the various excited states are increased in every case in going to the larger basis. The magnitude of this effect ranges from about 0.2 eV for the 2B2 - XzAl separation to nearly 0.8 eV in the case of 4B2 - XzAl. The energies that were obtained using the [453p1d] basis set at all geometries for the states considered here are given in Tables X through XVI for the XZ 2 4 2 + 82, 2A2, 82, 4 2 cases the iterative OVC process was continued until the energy had converged to 1 x 10"5 hartrees. Much of this 2 2 A1, 31‘ nu), A2, and states, respectively. In all information is summarized in Figure 3 where the energy variation with bond angle is depicted for the optimal bond length of each state. Computed bond angles are little different in the [4s3p] and [4s3pld] basis sets, generally agreeing to within 3°. However, the bond lengths are considerably shortened when the d-functions are added to the basis. For each state, the [4s3pld] basis gives a computed equilibrium bond length 0.10 i.0.02 bohrs shorter than for the [4s3p] basis. The same trend was noted for the ground state of 03 [26] in a calculation similar in approach to this one. nun-‘15 TABLEIX BASIS SET DEPENDENCE OF NO STATE ENERGIES 2 STATE BASIS SET* 8(80)fl E(OVC)” 080METRY*** 241 43 —203.9486 -204.1078 R=2.25, e=l34° 431 —204.0607 —204.2080 282 43 -203.8808 -204 .0729 R=2 .40, 0=105° 431 —203.9841 -204.1640 242 43 -203.9053 -204.0693 8:2.40, 9=110° 431 —203.9988 -204.1550 2B1 43 —203.9029 -204.0753 R:2.25, 0=180° 431 —204.0042 -204.1639 “82 43 -203.8961 -203.9924 R=2.40, 0=125° 431 -203.9701 -204.0634 ”42 43 -203.9226 -203.9832 Rs2.40, 0=110° 431 -204.0078 -204.0638 22* 43 -203.7326 -2o3.9370 =2.40, 0=180° 3 431 -203.8169 -204.0101 a: The [483p] basis set contraction of Duming [22] is desig- nated 43, while A31 denotes the '43 basis augmented with d-I‘unctions . u E(SC) is the energy of the RHF configuration in hartrees for the OVC wavefunction and is generally about 0.01 hartrees above the SCF energy of‘ that configuration for the same geometry. E(OVC) is the energy of the OVC wave function. m: 'lhe geometries were chosen to be near the equilibriun geom- etry for each state. 'lhe bond lengths are expressed in bohrs. I "‘ '21-“: ...-r- 141 TABLE x THE 24.1 ELECTRONIC ENERGY SURFACE AND DIPOLE MOMENT* R 0 E(SC) E(OVC) E(OVC-CI) 6(0v0) u(OVC—CI) 2.35 105 -203.97698 —204.14524 0.2775 2.40 110 —203.98714 -204.16501 0.2344 2.30 115 -204.02482 -204.18193 0.2543 2.35 115 -204.01486 —204.17918 0.2322 2.35 120 -204.02601 -204.18900 0.2107 2.3052 124 —204.04326 -204.19897 0.2024 2.25 125 -204.05424 -204.20196 0.2142 2.2552 129 -204.05816 -204.20597 —204.24033 0.1869 0.1852 2.3052 129 —204.04846 -204.20331 -204.23959 0.1732 0.1722 2.20 130 -204.06434 —204.20434 -204.23663 0.1931 0.1907 2.35 130 -204.03701 -204.19805 -204.23604 0.1554 0.1551 2.2052 134 -204.06649 -204.20674 -204.23907 0.1647 0.1617 2.2552 134 -204.06069 -204.20795 -204.24220 0.1519 2.3052 134 -204.05043 -204.20462 -204.24075 0.1433 0.1416 2.3552 134 -204.03636 —204.19756 -204.23560 0.1318 0.1309 2.2552 139 -204.06031 -204.20704 -204.24107 0.1206 0.1175 2.3052 139 -204.04962 -204.20329 -204.23918 0.1119 0.1094 2.3552 139 -204.03530 -204.19585. -204.23370 0.1039 0.1018 2 . 20 140 -204 .066 38 -204 .20550 -204 . 23737 0 .1202 0 . 116 8 2.2552 144’ -204.05716 -204.20359 -2o4.23734 0.0863 0.0826 2.3052 144 —204.04640 -2o4.19964 -204.23529 0.0803 0.0770 2.25 155 -204.04346 -204.18884 0.0143 2.25 165 -204.02440 -204.16961 -0.0351 a The dipole moment is emressed in of dipole accent is equal to 2.51) debye. atomic units; one atomic unit 42 TABIEXI THE 2131(211u) ELECTRONIC ENERGY SURFACE AND DIPOLE mMEN'I' R 0 E(SC) E(OVC) E(OVC-CI) u(0vc) Move—01) 2.2552 134 .203.95197 -204.11862 0.1742 2.3052 134 -203.94883 -204.12183 0.1573 2.3552 134 -203.94177 -204.12089 0.1402 2.2552 144 -203.97412 -204.13785 0.1464 2.25 155 -203.99066 -204.15170 0.1097 2 .25 165 -203 .99955 -204 . 15961 0 .0703 2.25 170 -204.00210 -204.16201 0.0459 2.30 170 -203.99453 —204.16078 0.0433 2.25 175 -204.00637 -204.16343 0.0231 2.30 175 -203.99592 -204.162o4 0.0217 2.25 179 -204.00414 -204.16389 0.0050 2.20 180 -204.00745 -204.16071 -204.17685 0.0 0.0 2.25 180 -204.00415 -204.16390 -204.18115 0.0 0.0 2.30 180 -203.99637 -204.16246 -204.18083 0.0 0.0 2.35 180 -203.98500 -204.15722 -204.17676 0.0 0.0 fl “1 R2 2.21 2.33 -203.99994 -204.16123 -204.17878 0.1048 0.1039 2.15 2.39 -203.99508 -204.15331 -204.17047 0.2129 0.2109 as 'Jhe bond angle is 180° for the final two entries of this table. ‘ ‘1-_-.r.-:Im 'I'HE2B2EIECTRONICEIWYSURFACEANDDIPOIENDVIEM‘ 143 TABIEHI R 0 E(SC) E(OVC) E(OVC)-CI) u(OVC) u(OVC-CI) 2 - 35 95 -203.99417 —204.16177 0.2302 2 - 40 95 -203.98715 -204.16245 -204.19456 0.2360 0.2271 2 - 145 95 -203.97661 -204.15971 -204.19326 0.2406 0.2299 2 - 30 100 20400032 -204.16273 0.1801 2 - 35 100 -203.99612 -204.16614 -204.19741 0.1861 0.1797 2 - 140 100 -203.98789 -204.16562 -204.19830 0 .1905 2 - 145 100 —203.97635 —204.16186 -204.19593 0.1966 0.1865 2 30 105 —203.99807 —204.16271 -204.19310 0.1319 0.1272 2 35 105 -203.99302 -204.16526 -204.19706 0.1373 0. 1314 2- 140 105 -203.98391 -204.16402 -204.19721 0.1435 0.1356 2 ’45 105 -203.97184 -204.15968 ~204.19424 0.1483 0.1388 2 . 50 105 -203.95689 -204.15278 0.1503 2 . 35 110 -203.98574 -204.16014 -204.19239 0.0849 0.0800 2 . £40 110 -203.97634 -204.15858 0.0897 2 ~45 110 -203.96386 -204.15400 -204.18895 0.0936 0.0867 2 30 115 -203.98036 -204.14948 0.0248 2 35 115 -203.97480 -204.15149 0.0299 . 140 115 -203 .96539 -204 .14991 0 .0342 2 35 120 -203.96070 -204.13983 -0.0269 2 3052 124 -203.95222 ~204.12654 -0.0759 2 - 3052 129 20393264 -2o4.10987 -0.1336 2 - 35 130 -203.92442 -204.10889 -0.1394 2 2552 134 -203.91085 -204.08347 -0. 1899 2 - 3052 134 -203.91074 20409094 -0. 1874 ~ 3552 134 -203.90638 -204.09425 -0. 1842 ’44 TABIEHII THEzAZEIEC'IRONICflIEMYSURFACEANDDIPOIEMMENT R 9 E(SC) E(OVC) E(OVC-CI) u(ovc) Move-01) 2 - no 100 -203.98961 -204.14520 0 .1166 . 2 - 35 105 -204.00088 .204.15010 0.0593 T 2 - 140 105 —203.99710 404.1529? 0.0682 2 - £15 105 -203.98944 -204.15239 —204.17267 0.0780 0.0835 ‘ 2 - 35 110 -204.00356 -204.15312 0.0081 ~. 2 - £40 110 -203.99882 -204.15500 -204.17446 0.0164 0.0219 2 . 45 110 -203.99027 —204.15358 -204.17437 0 .0269 0.0324 3 2 . 50 110 —203.97892 -204.14941 -204.17159 0.0372 0.0426 4 2 . 35 115 —2014.00075 -20'-I.15100 00443 J 2 . 140 115 -203.99531 -204.15231 -0.0349 2 . 145 115 —203.98631 -204.15042 -0.0245 2 . no 125 -203.97688 -204.13546 -0 .1408 2 . 35 130 -203.96829 -204.12090 -0.2022 2 . 3052 134 -203.95615 -204.10340 -0.2501 TABLE XIV THE “132 WCWMACEANDDIPOIEW L R 0 E(SC) E(OVC) E(OVC-CI) u(ovc) Move-01) r g. 1*S 120 -203.96357 -204 .06275 0 .0622 ' 2- “0 125 -203.97006 -204.06340 0.0451 2 - 45 125 -203.96538 -204.06368 0 .0380 2-50 125 -203.95775 -204.06110 0.0307 2- ’40 130 -203.96961 -204.06236 0.0225 2-45 130 -203.96473 -204.06236 0.0173 2 - 40 180 -203.88316 -203.98088 -203.99874 0 .0 0 .0 2- “5 180 -203.88090 -203.98275 -204.00134 0.0 0.0 50 180 -203.87593 -203.98181 -204.00n2 0.0 0.0 THE “A2 WC ENEHH SURFACE AND DIPOLE MCMENT 45 TABIEXV R e E(SC) E(OVC) E(OVC-CI) u(0vc) 11(OVC-CI) 2 - 45 105 -204.00754 —204.06388 0.0278 2 - no 110 -204.00780 -204.06378 0.0171 2 - £45 110 -204.00829 -204.06556 -0.0017 2 - so 110 -204.00696 -204.06485 -0.0262 2 - 145 115 -204.00695 —204.06479 -0.0358 TABLE XVI . mzzgmcmucmmmmnmmmm 5.1 a; E(SC) E(OVC) E(OVC-CI) Move) u(CVC-CI) 2 - 140 2.40 -203.81693 -204.01006 -204.07646 0.0 0.0 - I45 2.45 -2o3.81271 -204.00953 -204.07942 0.0. 0.0 ~50 2.50 -203.80041 -204.00621 -204.07815 0.0 0.0 2 - £00 2.52 -204.81304 -204.01040 -204.07987 0.1631 0.1594 - 34 2.58 -203.82215 -204.01385 -204.08127 0.3089 0.2990 2 - 28 2.64 -203.83535 -204.01896 -204.08267 0.4661 0.4481 -22 2.70 -203.85030 -204.02471 -204.08315 0.6072 0.5828 ~16 2.76 20386488 -204.02984 —204.08203 0.7449 0.7184 :5 Toe bond angle is 180° for all entries in this table. . M, “..,-.., ..; x 46 J 90 BOND 11101.5( degrees) Fig. 3. Electronic energy as a function of bond angle for the low-lying electronic states of 1402. For the four lowest dowlet states , the equilibrium bond length of eact12state has been used in the construction of the cuI'ves. The 2 B state was not included in this study; it§ position is known experimgntally. The sharp rise in energy as the 2 state 1'3 bent is based on SCF studies in a dowle-zeta quality basEs set. 1.43 47 The results we present here represent one of the first attempts to produce well-correlated wavelengths in a double-zeta-plus-polarization quality. basis set for several electronic states of a non-hydrogen containing polyatomic molecule. Unfortunately, excitation energies are not well characterized experimentally in N02 due to the spectroscopic complications. Consequently, not enough data is available T for a rigorous assessment of the accuracy of the computed ,‘ excitation energies. On the basis of the following '2 analysis, however, we do not expect that the computed j adiabatic transition energies for the 4 low-lying doublet States are more than 0.3 eV in error. We first assume that the magnitude of the corre— lation energy contributed by the configurations we have cOnsidered would remain constant as the basis set is expanded from its present quality to a Hartree-Fock limit quality has is set; this approximation must be very good once the Polarization functions have been included in the basis. Under this assumption, the error in the computed excitation ene:I':gies is partitioned into a differential basis set error I“ in)“ ' l 44 and a differential correlation error. The basis set error (the difference in the SCF energy of the RHF configuration computed in the [433p1d] basis and for a Hartree-Fock limit haSis) must be nearly the same for the izAl, 231, 2132, and 2A2 states. We base this statement on the fact that the Same orbitals are involved in forming the RHF configuration for each state. (Actually, a different orbital is involved 48 2 in the B1 RHF configuration, which is related to the 2A l RHF configuration by the single excitation 6a1 + 2b1. However, the Gal and 2bl orbitals are the degenerate components of the lwu orbital for linear geometries.) The single configuration (SC) energies are very good approximations to the true SCF energies; from Table Ix we see that the lowering of the SC energies for these excited states when the d-functions are added to the basis is equal 2 to the 5'! A1 SC lowering to within 8% for the 232 state, 10% for the 231 state, and 17% for the 2A of 2 state. The estimate Schaefer and Rothenberg places the [433pld] basis within 0 - O 5 hartrees of the Hartree-Fock limit. If in the approach to the HF limit from [dsBpld] there is the same percent Variation as in adding the d-functions to the [4s3p] basis, the maximum differential basis set error will be 0.17 x 0. o s = 0.0085 hartrees = 0.22 eV for the low-lying doublet States. The d—functions have a profound effect on the has is because they make possible the accumulation of Slactron density on the central nitrogen atom in certain orbitals (lug, laz) which is symmetry forbidden in an sE>~~basis. The symmetry of the d-functions on the nitrogen a“icnn is such that the la2 orbital will acquire some binding character as opposed to its essentially nonbonding character in the absence of polarization functions in the basis. Therefore, the prediction is for the SC energy of the 2A2 state to be improved less than for the other three c101.1blet states since the la2 orbital is only singly occupied .‘r'. _' "I 49 in its RHF configuration rather than doubly occupied as in the other states; the data of Table Ix confirm this behavior. It is our anticipation that changes in the relative quality of the basis set for the low-lying doublet states will be significantly smaller once the d-functions have been included in the basis. Consequently, we have reduced our estimate of the differential basis set error to 0 - 15 eV. The core orbitals are almost identical in each of the low-lying doublet states and we shall assume that di fferential core correlation is negligible. Still to be Considered are the differential valence and virtual corre- lation effects. We have only treated the virtual corre- lat ion in a cursory fashion, but the 0.05 eV differential Virtual correlation found for the 2A1 and 231 states seems to be a reasonable estimate and is the one we have chosen to use. Regarding the valence correlation, we previously ar g mad that we have considered all the important configura- tions by virtue of what is essentially a full 17 CI for well- Correlated a core. Further, we showed that our restriction to 99 configurations rather than the full set of 282 missed at most 0.002 hartrees (0.11 eV) for the ground state. The effect of the truncation did not seem any more serious for the other states, as measured by the mixing coefficients of the configurations which were deleted. Since we are again cOnsidering a differential effect, the valence correlation eI‘ror is likely to be much smaller than the 2A1 truncation 50 error of 0.11 eV. However, we have omitted some "highly o—excited" configurations. Although these are certainly \nery small contributors individually, there are a sizeable rnamber so we retain 0.10 eV as the estimate of the differ- eeritial valence correlation error. Should all these errors act in the same direction, a”? t1r1e total uncertainty in the computed (0-0) transition eerarergies would be 0.3 eV for transitions between the four J.c>Vvest doublet states of N02. These error bars should be ee>c12ended by 0.1 or 0.2 eV for the vertical excitation j eerieergies because the configuration lists were optimized at the equilibrium geometry of each state. The OVC configura- t:i.<:>ns for the excited states are still the most important iit: the equilibrium geometry of the ground state, for which ‘tf1152 vertical excitation energies are computed; however, the (3\7<::—CI lists could be slightly improved for this geometry b)’ reoptimization. Consequently, the computed vertical e3‘<<:=itation energies will tend to be a bit too high. The O\7<::-CI energy is at most 1 eV lower than the OVC energy; frllrthermore, we are considering the differential change of L2 tl"lis 1 eV contribution due to the reoptimization of the (DV7C2—CI list. To this we assign a maximum of 0.2 eV. kAnalysis of Previous Theoretical .JéQLCulations and a ComparISOn With 3% Present Results Since the first ab initio calculations on N02 eHPpeared in the literature in 1968, many of the succeeding 51 experimental studies have relied on the theoretical work for guidance in the interpretation of the data. The .results of our investigation, however, are in enough cases sufficiently different to warrant a reassessment of those interpretations. We therefore feel that a critical analysis cocE the previous theoretical treatments is necessary in c>1rder to justify our results. Tables XVII through XIX summarize details of 2113! initio studies of NO2 from the published literature and ifzrcom this work. Table XVII provides a quick comparison of tiklnree important aspects of the calculations, Table XVIII €15.53p1ays the computed equilibrium geometries of the low- lafierng doublet states, and Table XIX contains the calculated eXcitation energies. Both the work of Schaefer and Rothenberg [24] and that of Brundle 9.5.59: [27] are SCF studies of the ground 53tléate at its experimental geometry. Since this does not rfie‘sreal anything directly applicable to the spectroscopy of tr'1€_=_:molecule, no further comment will be made except to CCDIrmare the energy reported by Schaefer and Rothenberg and 'ttliat reported in this work. Though the basis set we have ‘JSBGed is actually slightly superior to theirs, our reported erlergy is higher. The energy we give is not an SCF energy; ra‘ther it is the energy of the RHF configuration, calculated EC): the OVC orbitals. Optimization of the orbitals for this Stingle configuration alone rather than as a member of a set .memm on» finance Ems on Hams 93:0 on» was how an own: use» mm 28m on» ma pom mamas on» .HmSmsom .8335 8.30QO 28 PHoomn no: v.8 mcmm H8 * 52 o .m 32.0 88.28- to: sue S a am 2. o 9.8 . 8m- 31 2m .1... lul 28.28- 888521 Ea $20.28 3 o .m In! ..ch H8 em -.il ......I 38.8? .mm mm 0885 ms o memos 38.8? 02855 new $28 3 o 23.0 828m- 252 822.9% mosfimwmé 8%? wommzm mom :1 m m mzoggéo OHEHZH Md moz mo mmbmmmé. gag mo zomgmzoo HE mama. 53 49.2” 26 05 um oopsosoo 0.33 gouge twosome one: . mowuwmo 5 we? econ can .12 5 ommmmaxm mam anemone ocoms O OHH em.H HOH mN.H owa om.H mma mH.H 3*suspm mass 3H mocwnmmmmv mad om.H mm om.H owe mm.a «ma =N.H scum ass .71. nan: moa nun: eea .unn ema nun: 20H moemnmmmmv seam AmH ooemsmammv NHH .... HOH .... omH ...: smH .... maao:ssm.cem amass mm mm mm mm m0 mm ma mm * a. m m HHHBn Ema. 02 mo Egan OHZOQHDE g8 E mo I go EDHmmHHHDQm E .8 monHbHQm—mm OHHHzH m< 54 .mmHUSpm once» :2 omoSHOcH mm: COHpompouca coapmpsmfimcoo 02*: .zwpmcm 2:202 opou HM2qu2mQMHU sou omuomppoo no: one ocm.>m :2 cm>2w mam mmfiwpmcm cOHQMpHOxo one: 22 m2 m2 22 02 m2 mosmpmmmm 2.2 ... ... 2.2 22.2 ... ... 22.2 N22 2.: nun nun v.2 om.m an: nun mm.m mm2 2m.2 2m.o nu. 2.m mo.m mm.2 22.m m2.2 mam 22.2 22.0 2.m 2.2 20.2 22.2 22.2 22.2 mmm 00.2 2.2 m.2 m.m m2.« :1: mm.m me.2 2mm 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 22m mHHmchsm maamcssm shoe m2s2 2*memm 28o sew 2wemo smog m2s2 2am 226cmm 2mm 2*2222 new 2wsme *AOuov *2202222> m XHX mumdfi oz 2H mMHummzm_ZOHB¢BHoxm mo mZOHBOHQmmm OHBHZH mfi. 55 of configurations would result in a further lowering of about 0.01 hartrees. For a somewhat less than double-zeta quality basis set, Pink [10] calculated SCF energies of several states at 4 different bond angles and a fixed bond length of 1.19 i. In a later study, he extended his basis to roughly double- zeta quality and recomputed the vertical excitation energies. Configuration interaction was also applied, but was limited to states of 2A1 symmetry and only a total of 29 configurations were included. The study of Del Bene [14] is noteworthy in that both the bond length and bond angle were varied to find the equilibrium geometry for the four lowest doublet states of N02. However, her basis set was the Pople £5 31. STO-BG minimal basis [28] and no configuration interaction was included. Nonetheless, such a treatment has been shown in the past to adequately reproduce geometrical trends even though calculated excitation energies must be viewed with caution. Del Bene's results are consistent with ours in that she predicts a low-lying 231 excited state which is linear and has a bond length only slightly longer than in the ground state. The prediction of 2B2 and 2A2 excited states with longer bond lengths and smaller bond angles than for the ground state are also in agreement with our findings. Even though her calculated excitation energies are not too reliable, they do indicate large differences between the vertical and (0-0) transition energies, ‘ifl‘fi- I {—..—2;, 56 especially for the 282 state. As will be discussed at the beginning of Chapter III, we feel that lack of recognition of this fact has greatly contributed to misunderstandings regarding the visible absorption of N02. The work of Gangi and Burnelle [13] is that most often cited by experimentalists in attempts to explain their data. There are numerous disagreements between their results and spectral interpretation and those which we present. Furthermore, these discrepancies are significant and we suggest that any analyses based on the data of Gangi and Burnelle should be carefully reexamined. Undoubtedly the most important factor affecting the accuracy of their results is the limited nature of their CI. We discussed previously the probable error of any SCF-CI approach which treats as core any orbitals other than those which correspond to ls orbitals in the atoms. Gangi and Burnelle, however, have arbitrarily designated the eight lowest molecular orbitals as core. Among these are the 5a1 and lb1 orbitals which should undoubtedly be treated as valence since they correlate with the lnu orbital for linear geometries. A preliminary calculation supports our reasoning. A minimal basis (STO-BG) SCF-CI calculation performed using the orbitals suggested by Gangi and Burnelle gave the same ordering of states and good agreement with their computed vertical excitation energies. The CI lists were then increased by including excitations from the lb1 orbital in 57 forming configurations. The energy of the 232 state relative to the ground state scarcely changed; however, the 231 state was found more than 0.5 eV higher than before, while the 2A2 state dropped nearly 1.5 eV relative to the ground state. The magnitude of these shifts clearly demon- strates the inadequate convergence of the CI expansions of Gangi and Burnelle. The recent study of Hay [15] includes the most extensive treatment of correlation up to the time of the work which we report here. For the STD-3G basis, he per- formed SCF-CI calculations for the low-lying electronic states; the CI wavefunctions included all single and double excitations from the RHF configuration in each state and only the ls-like molecular orbitals were designated as core. Although only vertical transitions were treated in this portion of his work, it is gratifying to us that he obtains the same ordering of states and rather close agreement with our calculated vertical transition energies. Because the basis set is minimal, the configuration treatment of Hay is necessarily valence (as we have defined “; the term) in nature and similar in size to the one which we have employed, although our basis set is much better. His CI is lower in energy by 0.1931 hartrees over his ground state SCF result, compared to a lowering of 0.1815 hartrees for our final wavefunction over the energy of the 2A1 RHF configuration computed with the OVC orbitals. As discussed previously, the actual SCF energy in our basis would be 58 about 0.01 hartrees lower if the orbitals were optimized only for the RHF configuration and our energy lowering due to introduction of correlation should be reduced by this amount. Nevertheless, it is likely that our treatment of correlation is more extensive since the CI of Hay is undoubtedly compensating for his limited basis as well as introducing correlation. Furthermore, there is a substantial oxygen 23 core correlation contribution in Hay's study which is not treated in the OVC-CI approach. As a check we also computed an OVC-CI energy for the ground state at its experimental geometry in the STO-BG basis and found an energy lowering over the SCF result of 0.1986 hartrees, slightly better than what Hay achieved. It is eXpected that the superiority of the OVC-CI procedure over SCF-CI will become more dominant as the flexibility of the basis set is increased. Despite the great range in the quality of the various 2Q initio treatments and our doubts about the validity of certain aspects of previous calculations, a consensus does emerge on several points: 1. The ground state bond angle is well reproduced, the 231 state is linear (the bond angle of 177° found by Pink undoubtedly results from his use of a quartic fit to 4 widely spaced points), and the 2B2 and 2A2 states are significantly more bent than is the XZAl state. 59 2. The 281 state is the first excited state for vertical excitation. 3. The vertical excitation energy of the 282 state is substantially lowered by the introduction of electron correlation. 4. The 432 and 4A2 states are very close in energy; also, the vertical excitation energies of these states are drastically underestimated at the SCF level. 5. The calculated excitation energies to the 2A2 state are not in good agreement. However, for the two studies in which correlation was carefully con— sidered (the work of Hay and our results), the 2A and 232 hypersurfaces are found to be very close 2 for vertical excitation. To emphasize the improvements of our results over previous theoretical treatments, we point out the following facts: (1) The only calculation which employed a basis set comparable in size to the one we used is that of Schaefer and Rothenberg. However, only an SCF calculation of the ground state at its experimental equilibrium geometry was performed. (2) The only calculation with a comparable treatment of correlation was that of Hay. However, his basis set was much smaller and the correlation was con- sidered only for the ground state equilibrium geometry. (3) The only calculation which also varied both bond angle and bond length was that of Del Bene. However, again the 60 basis set was quite small and in addition, no correlation effects were treated. The Determination of the Potential Surfaces The spectral analysis which we present in Chapter III requires a knowledge of the characteristics of the potential surfaces (i.e., equilibrium bond lengths and bond angles and vibrational frequencies) of the states of interest. In this section we describe the mathematical details of the deter- mination of the potential surfaces from our ab initio calculations. The discussion is necessarily limited to the space of C2v geometries. A convenient set of internal coordinates for describing the symmetric vibrations of the N02 molecule is shown in Figure 4, where R1 is one-half the separation of the two oxygen atoms and R2 is the distance from the nitrogen atom to the midpoint of the line joining the two oxygen atoms. The classical kinetic energy in terms of these coordinates is given by 2T 2 2.01; . :1;ng I}; (4) where the overdot indicates the derivative with respect to time and m0 and mN are the masses of the oxygen and nitrogen atoms, respectively. 0 O T R N i2 Fig. 4. Internal coordinate system for C vgeometries in NO 2 2' The generalized quadratic potential in R1 and R2 is _ _ e 2 _ e 2 _ e _ e 2(V—Ee) - f11(R1 R1) + f22(R2 R2) + 2f12(R1 R1)(R2 R2) (5) From the ab initio calculations we have the electronic energies for the several electronic states of NO2 at various values of R1 and R2. The electronic energies, which provide the potential for nuclear motion in the Born-Oppenheimer approximation, were least-squares fit to the functional form of equation (5). The six variable parameters, namely the three force constants, £11, £22, and £12, the two equilibrium geometrical parameters, R: and R3, and the energy at the equilibrium geometry, Be, were determined by least-squares optimization, using the STEPIT subroutine available from the Quantum Chemistry Program Exchange, 62 Indiana University. In general, those points with bond lengths within :.0.10 bohrs and angles within 1 10° of the equilibrium geometry for a given state were used in the determination. As described in Wilson, Decius, and Cross [29], equations (4) and (5) lead to the secular determinant fll-Zmox £12 -1 IF-G AI = = o (6) f12 £22-;22;E_ A o “N which yields two roots, A1 and x2, related to the harmonic frequencies of the symmetric vibrations by _ 2 Xi - (ZWCUi) (7) where mi is in cm-1 if Xi has units of sec-2. Successive substitution of the roots Xi into the matrix F-G-ll and solution of A11 1 = (8) A21 0 (F-G’ A 1) yields the amplitude ratios of Ali to A21. The internal coordinates are related to the normal coordinates, Q1 and 02, by e _ . = Rj-Rj - "1Aj101 + nZAjZQ2 3 1,2 (9) vb ll .I J: 63 where the "k are chosen to satisfy the relation 2 2 + 2f A + f22A2k 12 lkAZk] = ‘k ‘10) 2 "k‘fllAlk Inversion of equations (9) gives the normal coordi- nates in terms of displacements in the internal coordinates: _ _e _e Q1 ‘ b11(R1 R1) + b12‘R2 R2) (11) _ _e _e 92 ‘ b21(R1 R1) + bzz‘Rz R2) have the dimensionality massl/2 to give 1/2 The coefficients bij the normal coordinates in mass length. The normal coordinates thus defined cast the classical expressions for the kinetic and potential energy into the simple forms ’2 '2 2T a 01 + Q2 (12) 2(V-E ) = A 02 + x 02 (13) e l 1 2 2' Table xx summarizes the pertinent results for the low-lying doublet states of NO2 obtained with the methods of this section. We note that the OVC wavefunctions were used for this analysis; as will be shown in the following section, the higher accuracy OVC-CI wavefunctions change the characteristics of the potential surfaces to a negligible extent. 6" TABLE XX THE OVC EQUILIBRIUM GEDMEI‘RIES , VIBRATIONAL FREQUENCIES , AND NORMAL COORDINATFS FOR THE LOW-LYING DOUBIET STATES OF NO 2 2A1 2Bl 2132 2A2 Re (K) 1.186 1.197 1.252 1.273 9e e(degrees) 135.2 179.5 101.14 109.6 w:(cm "1) 1356 1192 1146i) 1360 61"(cm 1) 771 960' 781 798 niacin-s) 2.073 2.261 1.831 1.966 32(bohrs) 0.853 0.011 1.500 1.388 :211(harbrees/bohr2)" 1.585 1.708 1.076 1.116 r22( " ) 0.027 0.088 0.665 0.577 f12( " ) 0.41“ 0.070 0.h36 0.34“ baamn( am1/2) 8.181) 5.635 2.608 2.7141 b12(amul/2) 2.100 0.275 2.769 2.730 Wfi mull/2) 3.807 0.1198 5.020 _ 11.9148 b22(a1m1 am1/2) -2.308 -3.109 -1.l)39 -1.512 a Calculated.in.the bent molecule formalism” fl 0 1 hartree/bohr2 =- 15.57 mdyne/A Fv 65 The Accuracy o§;the Computed Spectroscopic Constants Because of the complex and severely perturbed absorption spectra of N02, not enough excited state spectro- scopic constants are known for a rigorous test of the accuracy of the theoretical constants. Comparison with ab initio studies on other, better understood molecules is not helpful either; to the best of our knowledge, there are no previous calculations of excited state spectroscoPic constants for a polyatomic molecule of this size at the level reported here. We must, therefore, rely on the szl constants for our analysis. We do note that tentative vibrational frequencies have been suggested from experiment for the 282 excited state and estimates of its equilibrium geometry have been given. However, there are great com- plexities involved with the data for this state and we defer a detailed discussion to Chapter III. In Table XXI are tabulated the ab initio spectro- scopic constants for the §2A1 and 232 states at both the OVC and OVC-CI level along with the experimentally known ground state parameters. The data amply support our con- tention that the OVC and OVC-CI wavefunctions give nearly identical descriptions of the potential surfaces. As is the case for diatomics, there is a clear implication that a small set of carefully selected configurations (i.e., the OVC configurations) yields the vast majority of the dominant coorelation effects; at present, though, the 66 . mm mocmnmhmm a: . mm mocgmmmm 2. Am Sufiamm. 3.0 on. 83 ~43” mew; 30:91.8 mm 5.0 :1. $3 :42 «mm; 258 «mm .335 zeme :39 .322 gamma; Aflucmsflmeav fmm. _ Rd we. 32 .32 83 5.88 seam mma at 82 meme 8H; 8.5 Heme“. 39303 a $183 m3 A7834”... Ammmnmmov mm 33 mm 0 m m H oz mo 82% mm 92 «mm mm. mom Em oEoomomBEm olulBHfi Q 8.26 ea 26 may mo 282380 dong 67 selection of these configurations in a priori fashion is less well-defined in polyatomics than for diatomics. (However, see Appendix D for a description of straight- forward configuration selection in some molecules other than N02.) Because the OVC and OVC-CI surfaces are so nearly the same for the szl and 232 states, no great effort was expended in computing OVC-CI surfaces for the 231 and 2A2 states; rather, the OVC-CI energies were calculated for only a few points near the minimum of these states for improved values of the (0-0) transition energies. Perhaps the most significant difference in the OVC and OVC-CI spectroscopic parameters is the longer bond lengths observed at the OVC—CI level. For both the XzAl and 232 states, as well as the 231 state (see Table XI), the computed equilibrium bond length is 0.01 to 0.015 A longer for the OVC-CI wavefunctions. This trend might well be expected from the experience with diatomic molecules. Hartree-Fock quality basis set SCF calculations on diatomics generally give bond lengths shorter than experiment [30]; the computed bond lengths then increase as correlation is introduced. At both the OVC and OVC-CI levels, the theoretical XzAl geometrical parameters are in very good agreement with the accurate microwave determinations [31], the errors being less than 1%. The harmonic symmetrical vibrational fre- quencies, as given by Arakawa and Nielsen [32], are also well reproduced by the gg_initio surfaces but in this case 68 the near coincidence must be regarded as somewhat fortuitous. The ab initio energies were computed on a relatively wide- spaced grid of geometries and the energy at each point was only converged to l x 10.5 hartrees. Thus, for the 14 OVC-CI points used in the XZAI surface determination, the standard deviation of the 39 initio and fitted energies was 1.3 x 10"4 hartrees (29 cm-l) and for one point as large as 2 x 10"4 hartrees (44 cm-l) with the sign and magnitude of the deviation at each point distributed in what appears to be random fashion. Although these residuals do not translate directly into an uncertainty for the vibrational frequencies, we estimate that a more compact grid of geometries and higher convergence of the wavefunctions would not cause a greater than 5% change in the values of ml and “2 which we report. The very small differences between the OVC and OVC-CI dipole moments indicate that the dominant molecular correlation configurations are also those which figure most prominently in determining the dipole moment, consistent with what was found in OVC studies of the diatomic hydrides and C0. This is in contrast to more standard CI treatments of dipole moments which often require careful selection of very lengthy configuration lists in order to obtain good results. The computed dipole moments were least-squares fit as a function of linear and quadratic displacements in the previously determined normal coordinates. The ground state 69 dipole moment at the computed equilibrium geometry is 0.37 debyes (polarity N-OE), only slightly larger than the most accurate experimental determination [33]. In contrast, the STO-BG and [4s3p] basis sets give moments of -0.06 debyes and 1.02 debyes, respectively, at the experimental equi- librium geometry. In the harmonic approximation, the intensity of a fundamental vibration is proportional to the square of the first derivative of the dipole moment with respect to the normal coordinate of that vibration, evaluated at zero displacement in the coordinates. Our fit indicates that on this basis, the bending fundamental for N02 should be roughly three orders of magnitude more intense than the symmetric stretch, qualitatively consistent with the experimental observation [34] that the rotational structure of the bend is easily analyzed under conditions in which the stretch is not seen at all. We suggest that the magnitude of the theoretical intensity ratio not be given too much emphasis, however, since the dipole moment expansiog contains large second derivative contributions (i.e.,(g—g) 012, etc.) which do not contribute to the fundamental intensities in first order. An Unusual Feature of the 22+ _§tate of NQ;_ g Several years ago, Mulliken [35] proposed that certain excited states of ostensibly symmetric A32 triatomic molecules might actually exhibit two unequal A-B bond 70 lengths at the minimum of the potential energy hypersurface. He further suggested that such behavior is most likely for states in which the 2b1 molecular orbital is occupied in the RHF configuration, his premise being that distortion from the equal bond length situation would reduce the anti- bonding character of this orbital. Prompted by these remarks, Coon and coworkers have examined band systems in N02 [36], $02 [37, 38], and C102 [38, 39] and have found evidence for the existence of unequal equilibrium bond length states. Their arguments are primarily based on the apparent necessity of a double-minimum potential along the Q3 normal coordinate in order to explain the unusual isotopic origin shifts and anomalously strong Av3=2 transitions in absorption. We have not theoretically studied the 2491 A band system of N02 discussed by Coon et al.; the upper state of this system is the second excited 232 state in N02 and there are conceptual difficulties involved in treating such cases by our approach. However, we do comment on this system in Chapter III in an analysis of predissociative mechanisms in N02. we did find evidence, though, for a non-symmetrical 22+ state, when examined at 9 the double-zeta basis set level. Further study in the equilibrium structure of the [4s3pld] basis has verified this behavior. Attention was first restricted to Dooh geometries; configuration lists ‘WQre optimized and a parabolic interpolation of three 71 equal bond length points gave a symmetric minimum bond length of 1.302 A. Further points were then obtained in which one bond length was contracted and the other lengthened by equal amounts. A lowering of the energy was found with a minimum for A03= :_0.11 A, and a barrier to inversion of approximately 800 cm-1 as depicted in Figure 5. Several comments are in order regarding the theoretical technique and results. 1. The configuration lists were optimized at a Dooh 22+ geometry and hence included only 9 contributors. Once the symmetry is broken, those configurations 22: symmetry in the equal bond length case will of also be involved and should be considered. However, they will only serve to magnify the asymmetry. 2. The wavefunction has not been analyzed in enough detail for an explanation of the asymmetry, although we do note that the RHF SC energies strongly favor unequal bond lengths. An avoided crossing mechanism [40] is improbable, since the first 2 excited state of 2: symmetry in N02 is most likely that analyzed by Ritchie and Walsh [41]. The 22: state is more than 7 eV above origin of this the ground state, and hence, better than 2.5 eV above the 22; state. 72 800 - O O O T 5 O O l RELATIVE ENERGY (cm71) 0’ 200 - .____l I. l l 1 l 1 | | l l 024 0-12 0 0-12 0-24 AQ3 (bohrs) Fig. 5. The Q potential of the 22+ state of N02 as computed at the OVC-CI evel. g The displacements are expressed relative to the equal bond length minimum for R = 2.116bohrs. 73 The 22; state is more than 1 eV above the NO(2H) + 0(3P) asymptote to which it adiabatically corre- lates. We therefore suggest that the apparent double-minimum character along the asymmetric stretch normal coordinate may be simply a mani— festation of the essentially dissociative nature of this state. CHAPTER III SPECTRAL INTERPRETATION A Qualitative Discussion of the fiIectronic Spectroscopy of NO, Despite a voluminous amount of experimental data, many of the details regarding the spectroscopy of NO2 remain uncertain. The best understood electronic transition is the 231 + XZAI absorption from 4600 to 3700 A, analyzed by Douglas and Huber [l], but it is not without its compleXi- ties. Less well characterized are the 282 9 XZAI absorption in the visible and near-infrared and the 2491 A band system to a different 282 excited state, hereafter referred to as the 2232 state. In addition, 2A2, 432, 4A2, 22A2 and 22A1(223) states have been suggested to be of importance for excitation energies less than 5 eV. None of these states has as yet been found experimentally, although some of them may be responsible for the two striking predissoci- ations observed at 3979 A [l] and in the 2491 A band system. It has been proposed in various analyses that one or more of 2A2, 482, or 4A2 contribute to the NO(ZH) + 0(3P) chemiluminescence; again, however, firm experimental confirmation is lacking. 74 75 The §g_initio calculations presented in Chapter II are a great aid in sorting out the possibilities. For the XZAI, 232, and 231 states, the extensive theoretical potential surfaces allow a detailed discussion of the 2B + 2 XZAI and 231 + XZAI absorption systems; these are described in the last two sections of this chapter. Here we confine our attention to some of the more qualitative features and attempt to formulate an interpretation which is consistent with as much experimental data as possible. A definitive resolution is not attainable at this time for every experi— ment, but a significant truncation in the number of possible explanations is achieved. Hopefully, this will enable an efficient approach to the planning of future experimental studies and will also expedite the analysis. An adiabatic correlation diagram (ACD) relating the molecular states with those of the various dissociative asymptotic products is a useful adjunct to our discussion. As the name implies, the ACD is valid within the regime of adiabatic potential surfaces and it is derivable entirely from group theoretic principles and a knowledge of the electronic energy levels of the states. Figure 6 contains the ACD for dissociation into N0 + 0 and Figure 7 displays the ACD for N + 02 products. The asymptotic energy levels are placed relative to the XZAI ground state of N02 from the thermodynamic dissociation energies and the spectro- scopic excitation energies of the atoms and diatomic fragments. All of the N02 levels up to the 22A2 state, 76 1400700000) 30010 r000) ~00“)er Nd‘n)*0(’r) 2A ".4, Dissociation Linear ~02 02" (0-0) C s Dissociation Vertical Transition Excitation Energies Fig. 6. The adiabatic correlation diagram (ACD) for dissociation of 1102 into N0 + 0. 'Jhe energies of thg asymptqges relative to the grognd stat II ofNO are3.llerorNO( II)+O( )and 5.08erorNO( )+O( ). 2 77 fl‘DMOJ’zg) 595) 10.02;) \\ N05) * 01 (‘AQ/ A. C?" Dissociation 02v (0-0) Vertical Transition Excitation Energies Fig. 7. The ACD for dissociation of ~02 into N + 02. The energies of the asynptotes relative to the ground state 3 _ of NO2 are 4.50 eV for N08) + 02(3)::5') and 6.88 eV for N(2D) + 02( £8) 78 plus the 4ng level, are from the OVC-CI calculations in the [4s3pld] basis reported in Chapter II. The (0-0) excitation energy of the 22 B2 state is known from experi- ment [36,42,43] and its vertical excitation energy has been estimated. No experimental information is available for the 22A2 state and it was not treated in our calculations; however, Hay [15] computed that it is 0.36 eV below the 2232 state for vertical excitation and we have used this separation for both the vertical and (0-0) excitation energies. It is quite possible that the order is actually reversed in one or both of these cases. Although neither 2 the 09 or 2H9 states were included in the calculations of Chapter II, the RHF configuration of each arises from the same (lnu)4(l1rg)3(21ru)2 orbital occupancy as the 4H9 RHF configuration. Hund's rules would predict the ordering E(4Hg)0 vibronic levels are determined by the Born-Oppenheimer potentials of both Renner components. This is of little consequence for low vibrational levels of the ground electronic state as there are no perturbing, isoergic 2B1 zeroth-order levels. However, the 8281 rovibronic manifold is severely affected as evidenced by the absence of any apparent K'>0 features in absorption. The same type of behavior is possible for the A2 2, CZAZ Renner-split pair of the linear 299 state. As shown in Figure 3, these two states are very close in energy over much of their hypersurfaces. For bond angles near 180°, this proximity is due to the p6 dependence of the Renner splitting of a 9 state, where p is the supplement of the bond angle. However, in the region of experimental observation, these states are so far from linearity that it is very doubtful that the proximity can still be attri- buted to Renner-Teller coupling [46,47]. The thrust of this argument is that for small bond angles, the electronic angular momentum of the 0 state is so effectively quenched that it is no longer valid to apply the Renner-Teller formalism. Instead, the closeness of the hypersurfaces is much more likely due to the similar bonding properties of the 4b2 and la2 molecular orbitals, which are respectively in—plane and out-of-plane non-bonding orbitals with 106 electron density almost entirely on the oxygen atoms. The RHF configurations of the A232 and C A2 states differ only in the single occupancy of these two orbitals; an analogous explanation holds for the closeness of the 4B2 and 4A2 states which arise from the linear 4H9 state. Franck-Condon Factor Calculations The large differences in the equilibrium geometries of the ground and excited electronic states suggest that rapid change of Franck-Condon factors with excitation wave- length may be responsible in part for the complexity of the absorption of N02. To examine this possibility, we have computed vibrational overlap integrals for various values of symmetric vibration quantum numbers in the upper and lower electronic states. Under the Born-Oppenheimer approximation, the wave- function separates into a product of an electronic wave- function and a vibrational wavefunction. was (q,Q) = ¢a (q,Q) Xas‘Q’ (14) where q indicates the totality of electronic coordinates, and Q symbolizes the nuclear coordinates. The electronic wavefunction 0a is determined for fixed nuclei and thus is parametrically dependent on the nuclear coordinates. The labels on the vibrational wavefunction denote the 8th vibrational level of the electronic state 0a; reference to explicit vibrational quantum numbers is deferred. 107 For each absorption band of an electronic tran- sition it is possible to define [83] a "partial oscillator strength." For the electric-dipole allowed transition Ebt +'Ias' this is given by 2 (15) was(q,Q)>q'Q s)/h, fl is the dipole f <¢bt(q.Q) u (Ebt-E bt,as = CVbt,as where C = 4nmec/3h, Vbt,as = moment operator, and the subscripts denote integration over a all electronic and nuclear coordinates. Substitution of equation (15) into equation (14) and integration over the electronic coordinates yields ~ 2 fbt,as - CVbt,as Q (16) where the electronic transition matrix element is It is usually assumed that the electronic transition matrix element varies slowly enough about some point QO in the region of significant overlap offbt andIas such that Pab(Q) may be replaced by Pab(Qo). This Simplifies equation (16) to ~ _ 2 bt,as _ CVbt,as 2 f {th(Q) xas(Q)>Q| (18) The squared overlap integral is the Franck-Condon factor Pab(QO) of the wavefunctionsifias and Ebt' Before describing the actual evaluation of the Franck-Condon factors, we note that the replacement of 108 Pab(Q) by Pab(QO) and its subsequent removal from the integration over the nuclear coordinates may introduce error in the case of very weakly overlapping vibrational wavefunctions, as has been discussed by Nicholls [84]. In the future we shall directly compute the electronic tran- sition matrix elements from the ab initio wavefunctions for N02 and examine the nuclear coordinate dependence. Our treatment is predicted upon the vibrations being harmonic which leads to great mathematical simplifi- cation. Although inclusion of anharmonicity would change the Franck-Condon factors somewhat, the effect will be small for low quantum numbers. Within this approximation, the vibrational wavefunction factors into a product of one- dimensional harmonic oscillators along the normal coordi- nates. For the particular case of a non—linear AB2 triatomic - 1/2 1/2 1/2 XaS(Q) - wvl(a1 Ql)¢v2(a2 Qz)wv3(c3 Q3) (19) where WV (ai/zQi) = Nv Hv (oi/201) exp(-ciQ§/2) (20) 1 i i and a. 1/2 N 2 = [‘55) 1 - (21) V. ‘n' v 109 The Hv. are Hermite polynomials of order vi and ai are 1 related to the xi of equation (7) by “i = Ai/Z/fi (22) Similar relations can be written for xbt with the Q. a 1, i' and vi replaced by Bi’ 0;, and vi. In general the normal coordinates will not be the same in the two differ— ent electronic states. As pointed out by Duschinsky [85], the two sets are related by Q. = z aika + A1 (23) k where the aik are the elements of a rotation matrix A in normal coordinate space, and the A1 are displacements determined by the different equilibrium geometries of the tWo states. The normal coordinates for each electronic state transform as the irreducible representations of an appro- priate point group. If both electronic states in queStion belong to the same (non-degenerate) point group, two general rules follow from the symmetry transformation properties of the normal coordinates: (l) aij is identically zero unless Qi and Qj belong to the same irreducible representation. (2) Ai is identically zero unless Qi transforms as the totally symmetric irreducible representation. The further requirement of invariance of the kinetic energy 110 .2 0. 2T = 201 = 201 (24) i 1 under the transformation of equation (23) forces the A matrix to be orthogonal. For the specific case of an AB2 triatomic, in which Q1 and Q2 transform as al in the C 2v point group and Q3 as b2, equation (23) expands to 01 cos 6 Sln 8 0 Qi A1 = - O ' Q2 Sin 8 cos 6 0 Q2 + A2 (25) I which leads to the following expression for the Franck- Condon overlap integral , . 1/2 1/2 1/2 . - . = Idoldgzwviml Ql)wvé(82 Q2)wv1[al ((205691 + SlneQz + A1)] 1/2 . . , B1/2 ml/Z x WV; [01.2 (-sin8§2l + cosBQ2 + A2)]}fdQ3wv3(B 3 Q3N1v30l3 Q3) (26) The integral over Q5 is straightforward and will be discussed later. Although the remaining two-dimensional integral can be evaluated in closed form, clearly the integrations will be extremely tedious for all but the very lowest vibrational quantum numbers. In a procedure which is in some sense the reverse of the one we are following, Coon, DeWames, and Loyd [86] approximated the two-dimensional integral by assuming that the symmetric normal coordinates in one state could be chosen parallel to those in the other state. Thus the rotation angle 6 of equation (25) was arbitrarily chosen 111 as 0°. Their goal was to extract the Ai from ratios of Franck-Condon factors determined by the experimental band intensities; the A1 were then combined with the known ground state geometrical parameters to give excited state geometries. In the examples they cited, the changes in geometry were small and good results were obtained. However, for large differences in the equilibrium geometries of the two states, the approximation 6 = 0° becomes inade- quate. The present needs are somewhat different since the normal coordinates, and hence the A1, are already known from the ab initio potential functions. Faced with a similar problem in calculating Franck-Condon factors ~1 between the 22A state of NO and the X A state of NO-, 1 2 l Herbst, Patterson, and Lineberger [73] resorted to numeri- cal integration for evaluation of the two-dimensional integral. We describe here an alternative method which is perhaps simpler, especially when only a few vibrational levels are of interest in one of the two states. Similar to Coon st 31. [86], we choose Coordinates for the ground state which are parallel to and displaced from the normal coordinates of the upper state, i.e., the bij of equation (11) are chosen to be those of the upper state, but R: and R3 are those appropriate to the ground state. These two coordinates satisfy the relations 112 _ v _e _e X1 - b11(R1 R1) + bi2(R2 R2) (27) II 0 H- I p H ___| _e l ...e '_ x2 b21(R1 R1) + b22(R2 R2) 02 A2 (28) Because X1 and X2 are not normal coordinates, the potential energy expression in terms of them contains cross terms _ 2 2 2(V-Ee) — kllxl + kzzx2 + 2k12x1X2 (29) although the kinetic energy expression is identical in form to equation (12) 2T = éi + é: = éiz + 652 = if + i: (30) Because of the cross term in the potential, the vibrational eigenfunctions for symmetric vibration are no longer the product of one-dimensional harmonic oscillators as in the normal coordinate system. Instead, the eigen- functions are linear combinations of such products along the coordinates X1 and X2 (v v ) 1/2 1/2 _ 1' 2 1/2 1/2 m,n (31) where _ 1/2 Yi - kii /h (32) The expansion coefficients are found by transfor- mation of equations (29) and (30) to the quantum mechanical hamiltonian, and construction and diagonalization of a truncated, infinite matrix whose elements are those of the 113 hamiltonian operator between the harmonic oscillator product basis functions along X1 and X2. The range of indices m and n is increased until the lower eigenvalues of interest have converged. It is important to note that this procedure gives the identical description as the normal coordinate treatment as was verified by comparison of the eigenvalues of the matrix diagonalization with those obtained from the ground state normal coordinate analysis. Naturally the basis set must be made larger to converge higher eigenvalues, but only the first few are necessary to treat the cold bands and prominent hot band absorptions in our study. If the interest is instead in Franck-Condon factors for emission from a few vibrational levels in the upper state to many levels in the ground state, it is more convenient to reverse the process and expand the upper state wavefunction using displaced ground state normal coordinates. The integral over 0i and 05 has now been reduced to 1/2 . 1/2 , 1/2 1/2 = (33) (VIIVZ) 1/2 , 1/2 , 1/2 , 1/2 mincmn The resulting one-dimensional integrals of displaced harmonic oscillators are easily evaluated with the aid of relations developed by Henderson, Muramoto, and Willett [87]. 114 For a given one-dimensional integral, these authors defined two dimensionless parameters (in our notation) _ 1/2 51 " (Bi/Y1) (34) _ 1/2 and gave the following expression Rm _ < (81/2 'lw I U2< '-A > - 26' 4 1/2 _piz/2 (36) R00 ‘ W0 9;) OIYi Q1 1)] ‘ 1+6 2 e i where _ 2 1/2 pi - Di/(l+5i ) (37) for the overlap integral when both quantum numbers are zero. Further, they developed the following two recursion relations which enable the rapid computation of the overlap for all other combinations of the quantum numbers. . (i) 1/2R (i) 1/2 R(i) ZOiDiRmn + (1+612)[2(n+1)]mn,n+1 + (2n) (512-1_R)mn_l (38) _ W/ - 261(2111)Rm_1’n (1) 1/2R (1) 1/2 2_ (i) 2D1R + 261(2n) Rm n- 1 + (2m) (6i .1)Rm-l,n (39) = (612+1)[2(m+1)11/2R$ii, n 115 The integral over Q5 could also be evaluated with equations (34) through (39). However, except for two linear states, only equal bond length geometries were studied and no information was obtained about the asymme- tric stretching frequencies of most of the excited states. In many cases, though, the Q5 integral is essentially unity if v3 = v3 and zero otherwise, as is shown by the following analysis. The symmetry properties of a harmonic, non-totally symmetric vibration require that its quantum number not change by an odd number for an electronic transitions Consider then the overlap integral over 05 for v3 = 0 and v5 = 0,2,4,etc. Since there is no displacement along Q5, D3 = 0, and equation (38) yields (3) 2 -1 R0 n+1 n. 63 "I37": " an: T“ (40’ R 63+l 0,n-l The ratio of R63) to R63) is therefore less than 0.1 for 0.75 i 63‘: 1.32. The Franck-Condon factor is the square of the overlap integral, so if 63 is in the specified range, the AV3 = 2 transition has less than 0.01 the intensity of the AV3 0 transition. Now 0, ll 3 (83/(13)2 = ws/w3 (41) 116 and unless the asymmetric stretching frequencies are greater than 30% different in the two states of the transition, the square of the 05 overlap integral is unity for v5 = 0 + v3 = 0 and zero for v3 # O + v3 = 0 to less than 1% error. The overlap is similarly diagonal for transitions from v3 # 0 levels. In summary of this section: we invoked the Born- Oppenheimer approximation and assumed that the electronic transition matrix element is independent of nuclear geo- metry in order to obtain an expression involving Franck- Condon factors and related to the transition oscillator strength. A set of (non-normal) coordinates were defined for the ground state which enabled the ground state vibra- tional wavefunction to be expanded in products of harmonic oscillators along the upper state normal coordinates. Consequently, the three—dimensional overlap integral was reduced to a linear combination of products of one- dimensional integrals which are easily evaluated. The BZB1+ K2A1_Absorption System The first published rotational analysis in the visible region of the absorption spectrum of N02 is that of Douglas and Huber [1], who assigned a series of bands in the region 4600 to 3700 A as perpendicular K'=0 + K°=l transitions to a 2B1 excited electronic state. The observed bands form a nearly harmonic progression in an upper state vibration, thought to be the bending mode, of 117 about 900 cm-1. From the qualitative arguments of Walsh 2 [2], the B1 excited state is expected to be low-lying and to have a substantially larger bond angle than that in the KZAl ground state. Both the 2B1 and KZAl states arise from Renner-Teller splitting of the lowest linear ZHu state; it was suggested by Douglas and Huber that the absence of any K'>0 features was likely due to perturbations created by the Renner-Teller interaction, which will disrupt all but the K'=0 upper state levels. The magnitude of the band isotope shifts of the 14N1602 and 15N1602 isotopic species indicate a system origin between 8500 and 6500 A (1.46 to 1.91 eV). Since the observed bands are far removed from the system origin, it is not possible to determine the 2B1 equilibrium geometry from the Douglas and Huber data. Hougen, Bunker, and Johns [49] have developed a hamiltonian, later extended by Bunker and Stone [88], which employs a curvilinear bending coordinate to account explicitly for the effects of large amplitude bending vibrations. After enlarging the Douglas and Huber absorp- tion study to include the 14N1802 species, HardWick and Brand [48] used this hamiltonian to mathematically extra- polate to the system origin of the 2B1 state. For each of the three isotopic species, the parameters of the hamil- tonian were varied to give the best least squares fit to the observed 2; vibronic levels of the excited state. The experimental band origins were best reproduced for a linear 2Bl state. The remaining three parameters, 118 quadratic and quartic force constants of the potential function and the system origin T0, were determined separately for each isotopic species by Hardwick and Brand and are given in Table XXII. TABLE XXII SYSTEM ORIGINS AND POTENTIAL CONSTANTS OF THE 2B1 STATE OF NO2 AS DETERMINED BY HARDWICK AND BRAND* 14 16 15 16 14 18 N 02 N 02 N 02 To (cm-1) 14743.5 14722.3 14717.9 0 K22 (md-A/radz) 0.46228 0.46595 0.46218 0 K2222 (md-A/rad4) 0.050054 0.051105 0.042047 *Reference 48. It was suggested by these authors that there is a possible uncertainty of one in their vibrational quantum number assignments. We have analyzed the results of Hardwick and Brand on the basis of origin isotOpic shifts and find strong evidence in support of a revisement of the system origin to one quantum of the bend further to the red. Our reasoning is detailed in the next several para- graphs. The form of the normal coordinates of an AB2 triatomic lead to the following relationships of the harmonic vibrational frequencies for the normal and isotopically substituted species [57] 119 (”1*92*)2 “02% ("16* + mo” Bent “’1‘”2 = “10.5%. “‘N + 2"‘0T (42) N02 “3*. 2 _ momu (mN* + 2mo* sin29> (1) - m ; * . 2 (43) 3 O mN (mN + 2m0 Sin a) “1* )2 = {‘19. <44) (1) m * Linear 1 0 N02 2 2 (“’2") = E) = mom“ (mN* + 2m0*) (45) w2 w3 m0;mN* TmN +2mo) where m0 and mN are the masses of the oxygen and nitrogen atoms, respectively, 2c is the equilibrium bond angle, and the asterisks denote the isotopically substituted species. Our analysis requires the harmonic vibrational ~ frequencies in the X Al ground state for the isotopic modifications of interest and these are listed in Table Table XXIII. The 14N1602 and 15N1602 data are from the infrared study of Arakawa and Nielsen [32] and ms for 14N18 14N160 0 results from the substitution of the 2 value of w3 into equation (43). In order to obtain ml and wz for 14N1802, we use the Coon, Cesani, and Huberman [36] measurement of v1 and v2, assume that the difference in v2 and ”2 is essentially the same as for the other isotopic species, and choose ml to satisfy equation (42). If anharmonicity is neglected, the origin of an electronic transition is given by 120 do = (Té-Tg) + %4mi+wi+w§) - %%wi+m§+wg) (46) TABLE XXIII HARMONIC FREQUENCIES OF ISOTOPIC MODIFICATIONS IN THE 2A1 STATE OF NO 2 14N16O a 15N160 a 14N18O b 2 2 2 ml (cm-1) 1357.8 1342.5 1308 :_2 02 (cm'l) 756.8 747.1 728 1,1 w3 (cm-1) 1665.5 1628.0 1633.7 aReference 32. bCoon, Cesani, and Huberman.[36] measured vi=1268.6 I 0.4 cm-l, vg=7zi.o :_0.4 cm-1 for 14N1802 The quantity (Té - Ta) is independent of isotopic substi- tution so that the origin isotopic shift is 3 l = _*=_. I._l*_lt_l:* A00 00 00 23:le3 ”j ) (wj wJ )] (47) For the isotopes 14N1602 and 15N1602, the ground state change in zero point energy is 31.2 cm-l. If the 2 excited B1 state is linear, then mi = wi*, and application of equation (45) reduces equation (47) to _ 1 A00 - 7(0.0234)(wé+w5) - 31.2 (48) The Hardwick and Brand analysis gives A00 = 21.2 cm.1 and predicts an wé in the neighborhood of 925 cm-1. The origin 121 shift then requires an m5 of 3500 cm-1, which is certainly much too large for such relatively heavy atoms. In Figure 10 we have plotted the isotope shifttrela- 14N16 tive to the o( 02) of the Hardwick and Brand calculated band origins versus 0(14N1602) and have extrapolated one quantum of the bend further to the red. A revised value of 1 14 16 15 16 A00 = 3 1.1 cm- for N 02 and N 02 results which reduces the estimate of mg to 2000 i 200 cm-1. From a l + comparison with the £9 ground state of C02, this seems to be reasonable. The asymmetric stretching fundamental of CO2 is 2349 cm.1 and addition of an electron into the Ztu antibonding orbital to give the electronic configuration 2 of the B B1(2Hu) state of NO2 will reduce this somewhat. The accurate OVC and OVC-CI theoretical data also strongly support the proposed revisement of the origin. As has been the case in all other ab initio studies, the B2Bl state is again found to be linear. Very encouraging 1 is the agreement of the ab initio mi = 960 cm- and 1 m3 frequency of 925 i’lo cm- = 2040 cm- with the experimental harmonic bending 1 and the asymmetric stretching frequency of 2000 :_200 cm.1 derived from the revised assignment of the system origin. The consistency seems good enough to warrant the use of these numbers in con- 1 junction with the theoretical mi = 1192 cm- to examine the origin isotope shift for 14N1602 and l4N1802. From the ab initio upper state vibrational frequencies and the experimental ground state frequencies the predicted origin 122 63835 ma 3932.8 ”CH5 cocoon.“ ..So an oopcfiompo mm Swapo omemammmmn can no coafiumoo mam. H .mmeu semen ten acetone: Eofiflafioaeflangfle8fi f%$%nfieaflfifim%83emflog$ Sada . m Hugs muoH x A cmazeeve ma NH 0H ma ea — 1 fl _ 4 n _ . \fH-M ocmum tam xuflzoumm mo ucoacmfimmm 5.3.3 soumxmllllullv \ .\\ \ \ . \ .\ \ \ I. .x \ ON 0 .\ \ \ 53.5 U. .\ \.O oocmwmmmom m. \ \ .\ . \ \ I. \. S \ \xxxx 1 es m N N \. so 0 o- o . m onazmio n onazeav .\\ x x. .m .\ \ S \X \\ m... .\ \ \ .I 00 M...” . \ o\ c . \ ‘\ m \ \ I \ .\x \\ \ “NomfizfiovooflzEVe P ...x .1 on \0\ \ \ \ \ sh! Eta Va I m! In th of vi 123 1 shift is 5.6 cm- to the red; the analysis of Hardwick and l l for Brand gives 25.6 cm- which is reduced to about 5 cm- the proposed reassignment of the origin. The OVC-CI adiabatic separation of the izAl and BzBl hypersurfaces is 1.66 eV. For the BzBl state, the zero point energy is calculated to be 2100 cm_1; when this is combined with the experimental izAl zero point energy of 1840 cm_1, a B2Bl + iZAl origin of 1.69 eV is obtained. The Hardwick and Brand 14N1602 origin is 1.83 eV which drops to 1.71 eV for the proposed vibrational quantum number reassignment. However, we do note that the OVC-CI origin has an estimated uncer— tainty of i 0.3 eV and is therefore consistent with either the Hardwick and Brand or revised origins. Three remarks concerning the validity of the isotope shift analysis are in order: (1) The least squares fit will be distorted if a local perturbation affects a given band origin more in one isotopic species than in the others and this distortion would be magnified in the long extrapolation to the system origin. (2) Although the force constants obtained in the reduction of the data for each isotopic species are very similar, they do vary slightly and hence must represent effective potential constants. A simultaneous fit of all three isotopic species is desirable, but requires knowledge of the anharmonicities associated with the stretching vibrations, which have as yet not been observed. 124 (3) We have assumed that the origin isotOpe shifts for the proposed reassignment of vibrational quantum numbers follow directly from an extrapolation of the calculated band origins obtained with the Hardwick and Brand quantum numbering. The possibility exists that recalculation of the least squares fit for the revised quantum numbering would substantially affect the computed band origins, necessitating changes in our analysis. Nevertheless, despite these caveats, there is a strong suggestion that the origin of the B2B1 + izAl 1 to the red of the value absorption in N02 is about 925 cm- given by Hardwick and Brand. Hardwick and Brand also employed the numerical vibrational wavefunctions which they obtained to correct the experimentally observed B$2 for the effects of large amplitude bending in order to yield extrapolated values of B6 and hence r6. They suggest that an equilibrium bond length of 1.23 A for the BzBl state, or a lengthening of nearly 0.04 A from the ground state, is indicated. How- ever, we favor the ab initio prediction of a less than 0.01 A increase in bond length and cite the following evidence: (1) The §b_initio bond length calculations are 0 expected to be accurate to 0.02 A and no worse than this for predicting trends. 125 (2) The izAl and B231 states are Renner-Teller split components of a linear 2Hu state and as such are expected to have very similar bond lengths. (3) No spectral features assignable to excitation of the symmetric stretching mode have been observed in the B231 + izAl system. Computed Franck-Condon factors, based on a change in the bond length of 0.01 A, predict that absorption to (l,v§,0) upper state levels should occur with intensities comparable to absorption to (0,vé,0) levels in the region of experimental observation. If the change in bond length between the two states is actually as large as 0.04 A, the stretching features should be quite prominent. (4) There is a trend in the data of Hardwick and Brand in that the lower the vibrational level from which the extrapolation from 3&2 to B6 is made, the shorter the predicted r6. Neglect of the effects due to bond stretching in the hamiltonian of Hougen EE.El° is the likely cause of this behavior. 2 2 The B 32,+ i A1 Absorption System Although the absorption features associated with the BZ Bl upper state are reasonably strong, they only account for a small number of the observed spectral lines. Polarization studies [45] established that most of the visible absorption intensity arises from a transition with moment parallel to the axis joining the two oxygen atoms, indicating a 2B2 upper state; the presence of the A232 126 low-lying excited state of N02 has since been verified by rotational analyses [3-9] obtained through several diverse experimental techniques. However, the analyzed bands are in widely separated regions of the spectrum and there is no clear relationship among the various sets of rotational constants which have been extracted. Consequently, there is no precise determination of the AZB equilibrium geometry 2 or vibrational frequencies. Brand and coworkers have suggested tentative vibrational quantum number assignments O [5] for the strongest cold bands in the 8500 to 6000 A spectral region and have associated the origin of the AZBZ + izAl transition with a feature near 8350 A [50]. However, as pointed out by these authors, separations of consecutive members of the assigned progressions show a several percent variation in the upper state vibrational frequencies. Their value for the system origin is based on the tempera- ture variation of the absorption; isotope substitution studies were inconclusive due to abnormally large and highly irregular band shifts. In view of the present uncertainties concerning the AZBZ state, the ab initio calculations which we have presented are of significant value because of the reliable zeroth description they provide. To be sure, the theoreti- cal vibrational frequencies and origin are not of a sufficient accuracy for a definitive assignment of the experimental spectral features. However, they are expected to be very useful for prediction of trends and evaluation 127 of possible interpretations. On the other hand, the ab initio equilibrium geometries should be quite representative of the 32 B2 Born-Oppenheimer state. Thus, if there are large deviations in the theoretical and observed rotational constants (as there are), these must be attributed to the influence of perturbations and not to any inherent deficien- cies of the theoretical approach. With the existence of these perturbations firmly established, the ab initio results further serve to assess the significance of various candidates for such a perturbing role. Although this aspect is only discussed in a qualitative fashion here, it does not seem unreasonable to anticipate direct evaluation of perturbation matrix elements from the ab initio wave- functions in the future. In this section we discuss the ab initio potential surface for the lowest 2B2 state of NO2 in the space of C2v geometries. From this surface and that of the izAl ground state, Franck-Condon factors (FCF) are computed using the techniques previously described. A theoretical absorption spectrum is then constructed from the PCP and a correlation with the experimental spectrum attempted. After the successes of the theoretical spectrum in reproducing experiment are examined in some detail, we conclude this section with an analysis of the vibronic perturbation most likely responsible for the spectral disruption. 128 The computed equilibrium geometry of the A B2 state 0 is Re = 1.26 A, 6e = 102° which represents an increase of ~ 0 0.06 A and a decrease of 32° from the respective X2A 2 1 B2 state at its minimum is calculated to parameters. The A be only 1.18 eV above the minimum of the ground state, but some 3.4 eV above izAl for vertical excitation. The theoretical prediction is therefore for the absorption to commence in the vicinity of 10 000 A, with progressions of increasing intensity in both the symmetric stretching mode and the bending mode, especially the latter, toward 4000 A and beyond. The experimental spectrum does in fact steadily increase in intensity to a maximum around 3900 A (3.2 eV) [89] from an assumed origin at 8350 X (1.48 eV). The agreement of the qualitative features of theory and experiment is pleasing but not a very stringent test of the reliability of the calculations. However, with the aid of the PCP from which an absorption spectrum is generated, more detailed comparisons are possible. The theoretical spectrum has been constructed under the assumption that only excitation of the symmetric vibrational modes in the upper state is important. Although this is largely dictated since only C2v geometries were considered in the ab initio calculations, any transitions with AV3 # 0 are expected to be quite weak, as discussed previously. Nevertheless, Brand, Hardwick, Pirkle, and Seliskar [5] found a strong v3 hot band in their absorption 129 study, which indicates the presence of vibronically allowed bands. This is supported in turn by the laser induced fluorescence study of Abe, Myers, McCubbin and Polo [3]. The fluorescence induced by the 4800 A Ar+ line consists of bands displaced to lower energy from the exciting radiation by frequencies corresponding to v1, v2, and 202 in the ground electronic state. However, additional lines were seen for which the displacement was close but not equal to 03 in the ground state. These somewhat unexpected lines were explained by Abe st 31. on the following basis: the laser excitation is from the izAl vibrationless level of vibrational symmetry Al to a level which has B2 (v5 odd) vibrational symmetry in the AZB excited electronic state. 2 This transition, which is vibronically Al + A1, obeys perpendicular rotational selection rules (AK = :1 in the limit of a prolate symmetric top) and hence is electroni— cally forbidden but vibronically allowed. The fluorescence, however, returns to the (001) vibrational level of the ground electronic state, and is fully allowed, corresponding to an A1 + B2 vibronic transition obeying parallel rota- tional selection rules (AK = 0). In a later microwave- optical double resonance (MODR) study, Solarz, Levy, Abe, and Curl [8] deduced the rotational quantum numbers involved in both the upper and lower states and confirmed this interpretation. The observance of Av = i.l transitions does not 3 . g_priori vitiate our neglect of asymmetric stretching 130 features in generating the theoretical absorption spectrum. As pointed out by Brand 25 31.[S], in a fixed frequency laser study, very weak transitions may appear with apparently appreciable intensity if the narrow laser line matches the bandwidth of the weak transition but does not significantly overlap the bandwidth of stronger, more allowed transitions. This appears to be the case for the 4880 A laser line; however, in emission the totally allowed parallel fluorescence is observed, but not the perpendicular, electronically forbidden, resonance fluorescence. Use of a tunable laser in this spectral region would reveal the relative intensities of the "unusual" transitions gig a gig the stronger, fully allowed transitions which are nearby in energy. On the other hand, the v3 hot band found by Brand 23 31. in a straightforward absorption experiment does indicate that such ostensibly weak transitions are fairly strong, and may in fact be competitive in intensity with the fully allowed transitions in certain regions of the spectrum. The appearance of these unexpected transitions could also be taken as evidence supporting the existence of vibronic perturbations in the excited state. The FCF necessary for construction of the theoreti- cal absorption spectrum were computed strictly on the basis of the ab initio equilibrium geometries, normal coordinates, 2 ~2 A1 and A B2 electronic states. The pertinent parameters have been and harmonic vibrational frequencies of the i given previously in Tables XX and XXI. Consistent with 131 the large differences in the equilibrium geometries of the two states, the FCF for low vibrational levels in each state are very small, as shown in Table XXIV. Once the PCP have been calculated, the theoretical absorption spectrum is easily generated. From the upper state Te' the vibrational frequencies of the two states, and the appropriate vibrational quantum numbers, the v frequency factor is calculated for each vibronic transition. Each FCF, multiplied by its frequency factor and a Boltz- mann factor for the chosen temperature, results in the partial oscillator strength of that transition. Plots of the partial oscillator strengths versus the transition frequencies comprise the theoretical absorption spectrum. The ab initio AZBZ + izAl spectrum for a temperature of 300°K is shown in Figure 11. In the computation of the transition frequencies, the ab initio value of mi was 1 1 reduced from 1461 cm- to 1410 cm- ; without this modifi- cation, many of the absorption features would be nearly indistinguishable at the resolution of Figure 11 since mi 1 is so close to Zmé (1461 cm- vs. 2 x 738 = l476cm-1). The choice of the modified mi smaller than Zmé was prompted by a comparison of the experimental and theoretical spectra. We also point out that the direction of the adjustment of mi is consistent with the probable effect of the intro- duction of anharmonicities. (The x11 anharmonicity constant is an order of magnitude larger than x for the izA 22 1 ground state [32].) 132 TABLE XXIV SOME COMPUTED FRANCK—CONDON FACTORS FOR K232 ee'igAl TRANSITIONS IN N02 (vi v5 vé) (v3 v3 vg FCF* Relative FCF 000 000 3.82 x 10'9 1.0 010 000 5.04 x 10‘8 13.2 020 000 3.30 x 10"7 86.4 100 000 1.80 x 10'8 4.7 200 000 4.37 x 10'8 11.4 110 000 2.43 x 10'7 63.6 120 000 1.63 x 10"6 427. 060 000 2.51 x 10'5 6570. 140 000 2.39 x 10‘5 6260. 000 010 7.41 x .10"8 19.4 010 010 8.68 x 10"7 227. 020 010 5.00 x 10"6 1310. 020 020 3.71 x 10‘5 9710. 000 100 5.45 x 10'9 1.4 * Computed from.the OVC—CI potential surfaces. 133 .mthBc mango Hmcoflpmhbfi. mpmum smog on» no @030an 98 omfiog who moqmo goo .mmommhsm .I 83568 63d: 8m m5 89C ompmpmcmw eooom pmagpmmam 8396QO e Tm e To. .3 .wE m... m: AECV Ehgs.» coo can oak IF+_F__ .Lfir_pd. 44er1..- _ : =~ _ .7 as ZOHUmm Mag/mm 38 .mO ZOHmdeunm D 41__r_4_:__ .r.1__p >— 4 __ i 13 _ A l j p 3m>3 N n 4 o k 8?: a v n 3m>$ T WWW 134 Some of the features of the theoretical spectrum which we feel are worthy of comment include: (1) The origin of the A B2 + izAl absorption is very weak relative to higher energy portions of the spectrum. The FCF for vertical excitation are probably on the order of 0.1, so that the origin is some seven orders of magnitude weaker. (2) The low energy region is going to be consider- ably complicated by unusually strong hot bands, even at room temperature. The transition (010) + (010) will occur at almost the same frequency as (000) + (000) since the bending frequency is nearly the same in the two electronic states. The computed FCF for this particular hot band is larger than that for the origin by a factor of 227 which outweighs the Boltzmann factor of 0.027 at a temperature of 300°K. At 500°K, for which the Boltzmann factor for a 02 hot band is increased to 0.11, the origin will be almost completely buried under this hot band. We have identified the cold bands in Figure 11 by labelling them by the upper state vibrational quantum numbers. We note that although the hot bands are relatively less prominent higher in the A B2 manifold, their effect is by no means negligible. (3) The intensity of the cold bands increases rapidly near the origin. On the basis of the PCP, the intensities of the cold bands (020) + (000), (010) + (000), and (000) + (000) are in the proportion 86:13:l. 135 (4) Higher in the AZBZ manifold, spectral features involving simultaneous excitation of both bending and symmetric stretching modes become relatively more intense than those in which only the bending mode is excited. Before proceeding to an actual comparison of the theoretical and experimental absorption spectra, we summarize some limitations of our approach. (1) The theoretical spectra are stick plots of intensity versus wavelength for the idealized case of a non-rotating molecule and the effects of rotational motion (K and N structure in the limit of a prolate symmetric top, Honl-London line strength factors, rotational selection rules, etc.) have been neglected. Thus in the normalization of the most intense line of the theoretical spectrum to the most intense "peak" of the experimental spectrum, some error is introduced since other background transitions are undoubtedly contributing some of the apparent intensity of the most intense peak. This error will be minimized if each dominant spectral feature is equally affected by the back- ground intensity on a percentage basis. (2) The variation of the electronic transition moment with nuclear geometry has been ignored in order to obtain an expression relating intensity to Franck-Condon factors. (3) The vibrational frequencies and normal modes have been found for infinitesimal amplitudes of the vibrations, yet the PCP of interest are primarily those for 136 transitions in which several quanta are excited, either in the upper or lower states, or both. Certainly as the domain of large amplitude vibrations is entered, the use of harmonic wavefunctions for computing the FCF becomes less satisfactory. Similarly, neglect of anharmonicity will affect the computed transition frequencies. (4) The final complication is more a peculiar property of NO2 than it is a limitation of our approach. As will be discussed, the evidence is indisputable that the A282 state is severely perturbed by high vibrational levels of the ground state. The consequences of this interaction are almost completely unknown in more than a qualitative sense and it is not possible to account for them in the generation of the ab initio absorption spectrum. Although these caveats are not inconsequential, the theoretical spectrum is quite successful in reproducing many of the qualitative features of the low resolution experimental absorption spectrum as shown in Figures 12 and 13. We would suggest that the experimental spectrum be viewed in terms of groups of bands, with the groups roughly centered at the wavelengths 892, 836, 791, 747, 710, 671, 647, and 615 cm-1. The existence of groups of bands arises from the near coincidence and overlapping of cold bands to the upper state levels (0,vé,0), (l,vé-2,0), (2,vé-4,0), etc. Complications are introduced by vibronic interaction with the ground state, the prominent hot bands at low energies, and the likely occurrence of Fermi 137 .pcmn Hmucmewummxm ummmcouum mnu ou pmufiameuoc ma Ecuuoomm HMUfluouoonu one .oooe cOauomumU pmumumca 0:» ca cmumummo 5H ammo m :0 Umusmmms cmmn mm: eauuommm Hmucmsaummxm one .mmomuusm Hoaucmuom Canada am on» Bonn pmumumcmm Eduuommm H¢~m + mmmm msu so an 000 new oom cmozumn fiauuommm :OAuQHOmnm ~02 Hmucosaummxm am oazv acausaommu 30H m mo IcOauamomummam .NH .mwm 35 5.02383, . C.“ O b P n .. a” Q m 8” 3-6%. I‘J—‘ _ __ . LLI SNSLNI EAIJN'ISH 138 Hmfiucmuom oflpflcfi mm may Eoum omumumcmm Esuuommm admx + mm cmm3umb Eduuomdm cowumuombm moz cofiusaommu 30H m mo cafluflmo AECV mBUzmqm><3 Gem F . b QMQ _ p . . 0%” p . L m 7‘." «u—d- uNI— J: .mmomwusm we» no an amp can com 665m .ma .mem LLISNHLNI HAILVTHH 139 resonance and significant anharmonicity. The very differ- ent contours of the various band groups can be attributed to the presence of strong hot bands at low energies and the increasing prominence of symmetric stretching features higher in the AZBZ manifold. We feel that the support for our qualitative inter- pretation is strong and that we have achieved a satisfactory zeroth-order description. Such a description is of prime importance if we are to hOpe to unravel the complexities of the electronic spectroscopy of N02. At this point we must await further studies by Smalley, Ramakrishna, Levy, and Wharton [82] for a more precise evaluation of our interpre- tation. These authors have achieved rotational temperatures of 3°K for a supersonic beam of argon seeded with N02. The N02 fluorescence excitation spectrum obtained by irradiating the beam with the output of a tunable dye laser is very close to the hypothetical case of a free, non-rotating molecule assumed in the generation of the E2 initio absorption spectrum. At present, their investigation has been limited to the 5800 to 6100 A region, but hOpefully further studies will be made in the red and infrared. For the construction of the theoretical spectrum, the ab initio AZBZ + izAl origin of 9700 cm-1 was used. Temperature studies by Brand and coworkers [50], however, have not revealed any cold bands at lower energies than one at 11 956 cm_1 and it is to this band that they assign the origin. A nearby hot band at 11 895 cm_1 is of nearly 140 equal intensity at 90°C and has been assigned as (010) t (010). At a more elevated temperature of 220°C, two hot 1 and 11 145 cm-1 are found with band 1 1 bands at 11 205 cm- contours very similar to the 11 956 cm- and 11 895 cm- bands, respectively. A simultaneous partial rotational 1 analysis of the 11 956 cm- and 11 205 cm.1 bands based on combination differences from ground state (000) and (010) 1 infrared studies [90] verified that the 11 205 cm- band 1 is a v2 hot band of the 11 956 cm- band. From the similar contour and 750 cm.1 separation, there is little doubt that 1 1 the 11 145 cm- band fits in with the 11 895 cm- band as a ground state v2 progression and it has been assigned as (010) + (020). The highest frequency member of this progression is a cold band at 12 646 cm"1 assigned as (010) " (000). Although the upper state bending frequency is not firmly established spectroscopically, on the basis of the theoretical spectrum we are confident that the 12 646 cm-1 1 cold band differs from the 11 956 cm- cold band in one additional upper state bending quantum as suggested by Brand gt 31. However, we would like to propose that the 11 956 cm—1 band may not be the origin of the AZBZ + XZAI absorption; for purposes of discussion we identify it as (030) + (000) which would be consistent with the ab initio origin of 9700 cm-1. Revised quantum number assignments for the other bands are given in Table XXV. 141 With the aid of the computed Franck-Condon factors, we have calculated various band intensity ratios for both the Brand 2E.El° vibrational quantum numbering and for our proposed reassignment. These have then been compared with two experimentally determined ratios from the spectra of Brand, Chan, and Hardwick [50] at the stated temperatures as is shown in Table XXVI. Although it is difficult to extract accurate intensity ratios from the experimental spectra and it would be desirable to have more comparisons, the intensity ratios are much better explained with the revised vibrational quantum number assignments. It is then necessary to explain the apparent absence of the (020) + (000) cold band if the assignment of (030) + (000) for the 11 956 cm.1 band is accepted. Again using the ab initio FCF, we find that at 220°C the intensity ratio of (020) + (000) to (030) + (010) is 0.16. The (020) + (000) band should occur at nearly the same frequency as the (030) + (010) band at 11 205 cm'l, in which case it may be buried underneath. As will be discussed momentarily, one must also consider the possibility that the (020) + (000) band is drastically shifted by vibronic interaction from its expected position. This shift could be as large 1 in which case the cold band might simply as perhaps 200 cm- be lost in the noise of the spectrum of Brand gt_gl. Vibronic interaction between the AZBZ state and high vibrational levels of the XZAl ground state has long been a popular feature in discussions of NO2 spectroscopy O ’.'.;‘.Y: ..— 142 TABLE XXV ’b 'b SOME A282 <- X2Al VIBRATIONAL QUANTUM NUMBER ASSIGNMENTS OF BRAND ET. A__l_.. AND A PROPOSED REASSIGNIVIENI‘ O,cm71 Assignment of* The Proposed Brand 22 a1. Reassignment 12646 (010)+(000) (040)+(000) 11956 (000)+(000) (030)+(000) 11895 (010)+(010) (OUO)+(010) 11205 (000)+(010) (O30)+(010) 11145 (010)+(020) (040)+(020) .— *Reference 50. TABLE XXVI A COMPARISON OF EXPERIMENTAL BAND INTENSITY RATIOS WITH AB I_NI_T_I_Q FRANCK—CONDON RATIOS FOR THE ASSIGNIVIENTS OF BRAND ELAL- AND THE PROPOSED REASSICNMENT Assignment of’ The Proposed Experimental Brand gt a1. Reassignment Ratio * 1(11956) 0.1 0.6 m1 1(11895) ** 1(11205) 0.01 0.6 m2 I(111555 X At a temperature of 90°C. ** At a temperature of 220°C. 143 [58-61], especially in regard to the anomalous fluorescence lifetime. Although the large spectral disruption is strongly supportive of such a perturbation, until quite recently the evidence was primarily of a qualitative nature. However, the several partial rotational analyses which have been lately achieved, in conjunction with the accurate ab initio calculations of this study, now provide compelling proof for the existence of strong AZBZ ++ XzAl vibronic coupling. In Table XXVII are given experimental A B2 rota- tional constants, the wavelengths at which they were determined, and the experimental techniques which were employed. The final set of constants is that inferred from the ab initio AZBZ equilibrium geometry. The dissonance of experiment and theory is striking and irreconcilable unless vibronic coupling is invoked. Even very conservative error bars of i_0.03 A in the theoretical bond length and i'3° in the bond angle do little to improve the agreement. It is also not possible to explain the differences in experiment and theory on the basis of vibrational-rotational inter- action, as the constants of such an interaction would have to be impossibly large. The recent model calculation of Brand, Chan, and Hardwick provides a most promising first step towards a resolution. These authors have roughly estimated the energies of B2 vibronic levels (v3 odd) of the XzAl ground 2 state in the vicinity of the A B + XZA1 cold band at 2 .om pew m moccamumm e: 144 * .m mocmpmmmm 3:. .N. mocgmmmm x. Epgomm $6 om.m esflsfldas 83682.8 335 8 83¢. 25. m9; Se. m. ommm .. .Q 0530 : O 8398QO 5 II II M memm 38.85 88338 2:48 es. scene . 898meth mm; m 33 884651883“ 28¢ mocmomeOSHH See mm: m mmmm 8685:6681“ *OHN use 883m coapmcfinmpmo $.85 A78: .8 2823883 835669 magmzoo qezofléom mmmm ems mo mzoflefizmmema mucus; HHBOA EB 145 11 956 cm‘l. Further, they estimated the Franck-Condon overlap of the AZBZ levels with the high vibrational levels of the ground state in which the AZBZ levels are embedded. Finally they deduce a representative vibronic coupling matrix element from the shift of one of the K sub-bands as indicated by their rotational analysis of the 11 956 cm-1 band. From these data, they construct an energy matrix, diagonal in N and K, which describes the vibronic inter- action and can be diagonalized to yield estimates of intensity variation due to the coupling and also approxi- mate rotational constants. When these constants are deperturbed to give the estimated rotational constants of the AZBZ Born-Oppenheimer state, a geometry of R = 1.26 A, 8 = 102° results which is coincident with the gb_initio geometry. The Brand, Chan, and Hardwick treatment is of a model nature and the agreement with the ab initio predictions is probably somewhat fortuitous. However, their analysis is highly significant because it provides a description of vibronic coupling which is not totally dependent on ag.hoc assumptions. There no longer appears to be any rational alternative to A282 + X2 A1 vibronic coupling which can satisfactorily account for the spectral disruption and highly perturbed AZBZ rotational constants in N02. Solarz 25.21: have suggested that their observations in a MODR study could be attributed to an unequal bond 2 length equilibrium geometry of the A B2 state [8]. although 146 1 alternative explanations are possible. Shortly thereafter Hinze, Solarz, and Levy [40] described schematic potential surfaces which could cause an ostensibly C2v electronic state to exhibit a double minimum potential along the asymmetric stretch normal coordinate. Relying on our preliminary ab initio calculations, they proposed that an avoided crossing of the AZBZ and 22A1 states over certain portions of their hypersurfaces and a subsequent double minimum potential was consistent with the MODR study. The calculations we have presented in Chapter II now exclude the 22Al state from such a role, as it is much too high in energy to affect the A2 B2 levels reached in the 0 4880 A optical transition in the MODR experiment. However, the Hinze 35 31. discussion is still relevant for the intersection of the AZBZ and XZA1 hypersurfaces. As shown in Figure 3, the X2 A1 surface intersects very near to the minimum of the A282 surface; for various fixed bond lengths, plots of potential energy as a function of bond angle show that the crossing point is on the large angle arm of the A282 curve and within ~0.3 eV of its minimum. Thus the stated conditions of Hinze gtflgl. for the existence of a C8 equilibrium geometry of the A28 state has likely to be 2 satisfied. As has been pointed out by Brand, Chan, and Hard- wick [50], though, it may not be meaningful to even ask whether or not the A282 state has a double minimum potential 1.1:“ WE.1 147 along 03. For Cs geometries near the intersection with the XZAl hypersurface, the Born-Oppenheimer approximation is likely not valid; if so, the concept of the potential function or surface is no longer well defined. U CHAPTER IV CONCLUSIONS AND RESULTS The major conclusions and results of this study are tabulated below in the sequence in which they have been discussed. (1) The orbital basis has been partitioned into core, valence, and virtual spaces and the dominant mole- cular correlation effects are found to reside in the valence space as expected. (2) The 4a1 orbital has been found to play a significantly different bonding role in the XZA1 and A232 states, which cannot be simply accounted for in terms of the different bond angles of the two states. (3) The designation of various configurations as describing molecular extra correlation energy or differ- ential atomic correlation is not as well defined for polyatomic molecules as it is in diatomics. (4) A technique was described in which various classes of configurations were successively examined; compact final configuration lists were formed on the basis of this procedure. 148 149 (5) The basis set used in these calculations is larger and more flexible than any previously employed in an NO2 ab_initio calculation. (6) Adiabatic excitation energies are estimated to be accurate to i 0.3 eV from an analysis of the quality of the basis set and the extent of the inclusion of molecular correlation. (7) The treatment of mOlecular correlation is more extensive than in any previous NO2 ab_igitig_calculation. Better than 95% of the valence (not total) correlation energy is obtained. 2 (8) The X A1 calculated equilibrium geometry is. 0 within 0.01 A and 1° of the experimental values. 2 (9) The X A1 calculated symmetric vibration harmonic 1 frequencies are within 10 cm- of the experimental m1 and m2. The estimated accuracy of the computed vibrational frequencies of the lowest doublet states is i 10%. 2 (10) The X A1 calculated dipole moment is 0.37 debyes which is only slightly larger than the experimental determination of 0.32 debyes. (11) The 22; (22A1) state exhibits a double minimum potential along Q3. 2 (12) The A B2 state is the first excited state in 0 N02. The long-wavelength absorption in N02 (4.: 7000 A) is only to the 3282 upper state and not to 8281. The AZBZ and .., 0 B2B1 states overlap at least in the 5200-3700 A spectral region. 150 2 ~2 (13) The A B2 and C A2 states adiabatically corre- late to the NO(2n) + 0(3P) asymptote along CS paths. (14) The 22; (22A1) state is not responsible for O 0 either the 3979 A or 2491 A predissociations in N02. (15) Both the AZBZ and 8281 states are predissociated o ... for A<3979 A. In the 8281 state, the likely mechanism is predissociation by vibration. Either predissociation by vibration or a heterogeneous predissociation by high vibrational levels of the X2 the A282 state. A1 ground state is possible for J. (16) The NO(2H) + 0(3P) chemiluminescence only involves the XzAl, A282, 82 (17) The 4B2 and 4A2 states do not contribute to B1, and CZA2 electronic states. either the NO+O chemiluminescence or visible absorption of N02. (18) The NO(2H) + 0(3P) chemiluminescence data regarding the peak at 3.7 um is consistent with a CZAZ + A 82 electronic transition. 0 (19) The predissociation in the 2491 A band system is likely due to either (a) a homogeneous predissociation into N(4S) + 02 (32;); or (b) predissociation by vibration into uo(2n) + 0(3p). O (20) The photodissociation of NO by the 6943 A ruby laser line probably involves a A2 followed by a subsequent CZAZ + AZBZ absorption. 2 B2 + XZA1 absorption 151 (21) The 1.8 eV photodetachment process in N02 may involve the ring states of N0; and N02. (22) Level dilution of the A232 state by high vibrational levels of the izAl ground state can only account for a small portion of the anomalous fluorescence lifetime. (23) The proximity of the A B2 and EZAZ potential w"! hypersurfaces is due more to the similar properties of the 4b2 and la2 molecular orbitals than it is due to Renner- Teller interaction in the 209 linear state. V 2 2A1 absorption '1 1 (24) The origin of the 3 B1 + i system is probably near 13 800 cm'l, about 900 cm' lower than the present assignment of Hardwick and Brand. 2 (25) The § B1 state is linear and has a bond length 0 an about 0.01 A longer than in the szl ground state. 2 (26) The A 32 equilibrium Born-Oppenheimer geometry 0 is R=1.26 A, 6=lOZ°. The Franck-Condon factor for the 2 ~2 vibrationless levels of the i A1 and A B2 states is of the order 10-8. (27) Near the origin of the A282 + i v2 hot bands will be more intense than cold bands, even at 2A1 transition, room temperature. ~ to o (28) The A232 + sz1 absorption between 8500 A and O 6000 A is qualitatively understood in terms of groups of 2 bands, with successive groups separated by the A B2 bending frequency. The members of each group correspond to cold 152 bands to the (O,v§,0), (1,vi-2,0), (2,vé-4,0), etc. upper state levels. (29) The relative intensities of hot and cold bands 2 indicate that the A B2 + izAl origin may be lower than the present assignment of Brand ggflgl. at 11963 cm-l. (30) The dissonance of the experimental A232 rotational constants with those inferred from the accurate All,“ ab initio geometry of this study can only be attributed to strong A232 - 22 A1 vibronic coupling. Table XXVIII summarizes the spectroscopic para- ,} meters of the low-lying doublet states and includes the most recent experimental determinations. 153 .mompm Hmmm. on» now coumgodmc omdm mmz an 80. oaom 1.. m3 no 33> < .... .HmSmH 0.5 on» on 83330.30 8m; 83.9, 333:8 .22. .mm< o no.“ on. cam Hmmm no.“ on. cam mm oflnm mam mm mcoflmgoamo 5'26 05 89C 3528 mam Rouge domapmnoocp mmm< cam ~41.ch one: .3Hmpcmnflo9no 58.908 no ownwoo fiE .m 0» :292 ohm 33m ogonw on» now 305 3:0 . mommnpcwnmo 5 ooosaofiu mam Eggnog 335on m5. a STE :3 8mg 3.11m: 38%.: 398 mm.mm .Hm Rd fig 32 in 84 8.0 new. so: 893 sec N om .m 2.0 mm. 3.: NS mm; mHA mmm Ammmv A83 Ammév A823 2 .H 8.0 8m 83 8H 84 84 2... Jam. mod was 82 0: SA .54 ~~ —__’Y \i and the x—direction is perpendicular to the plane of the molecule. —-->Y ——-9Y 164 TDAEEJZIXXJD( THE 1201 OVC ORBITALS FOR A (mommy OF R=2.2552 BOHRS, 0=l3u° Basis FUnction Orbital ‘9: 1% 13:1 231 3&1 2'02 “31 81 ...r- 04-00 000 . 0. 001 l 71-04.59.36£_JJ2QL);_0100_009__:L19993_ 32 0.00000 0.00154 0.44534 0.03014 0.00000 -0.13301 33 0.00000 -0.00314 0.00242 -0.03159 -0.00000 0.41942 _ .50 - 0. 00000 0. 00101 0. 00 1 34.4.0214 940-00 000__0.11341_ y 11 -0.00032 0.00000 -0.00000 0.00000 0.04233 0.00000 N’ Y2 -0.00203 -0.00000 0.00000 -0.00000 0.02321~:-0.00000 Y3 ...-0.00094 0.00000 -_0.oooon___o.ooooo --0.00444_1_0109000. 21 0.00000 0.00004 0.00144 ' 0.001557 -0.00000 0.15010. 22 '0400000 0.00I14 -0400059 0.01672. 0.00000 0. 20391) g} . -0.00000. 0.00094- 0.00049..-0-02433._=0-00000.110.05290 32 0.32454 0.31311 0.00152 0.11344 0.13334 02002411 33 11 -0.00144 -o.02319 -0.00505--20139120_le0;40130.1:01001122 30 0.00044 -0.02114 -0.00544 -0.31949 -o.34511 -0.14211 11 0.00213 0.00430 0.00113 - 0.05433 0.05411 0.11339 (31 12 l.-0.00003. 0.00292..-0.00144.__0.04112___0.04524__.0121293; 13 0.00105 0.00374 0.00040 0.04434 0.05101 0.03131 z1 -o.00015 -0.00035 -0.00021 0.00321 -0.01094 0.01914 22 1.....0.0001.3... 0.00092..t0.000£1___04014&1._=0.01140_;_D.031214 23 1-0.00035 -0.00013 -0.00014 0.00213 -0.01034 0.02103! - O O“ O 32 —-0732654 1. — . 0. 31377—m-04-00-l-51—0-4-1-1-06 b—AHJO-J 33 0.00144 «0.02319 -0.00505 o0.39120 0.40130. -0.001121 30 -0.00044 -0.0z114 -0.00544 -0.31949 0.34511 -0.14211, 02 11 _..0.00213 -0.00430---0..00113-4—0r0543-3—0.05411—-04+1-334- 12 -0.00003 -0.00292 -0.00144 -0.04112 0.04524 '0-212931 ‘13 0.00105 -0.00374 -0.00040 -0.04434 0.05101 4.03137. 21 ..-—0.00015 -0.00035~—-0400021———0eoo321-——0.01093—-—0701913— 22 -0.00023 0.00092 -o.00021 0.01441 0.01123 - 0.03121_ z3 0.00035 -o.00013 -0.0oo14 0.00213 0.01034 0.02103 . 00 67660tIT5=6T666II:==6t6T6511==6766666===6:66666= 11 0.00000 -0.00019 -0.00022 -0.01943 - 0.00000 0.011132 1) 22 0.00000 0.00019 0.00009 0.01149 0.00000 - 0.01954; YZ -—-0.000‘391— -04 00000—-0.-00000-——0r0000-0—-0-:01320-‘—0f00000‘-1 YY -0.00012 0.00009 0.00043 0.00256 , 0.00150 0.02914 ' (31 zz .-..0.00002 0.00014._.0.00001___0.00119.__0.001134—-0.0012- 12 0.00009 0.00033 ~0.00013 0.00214 0.00120 -0.00553 u 0.0(500r—O. o o " o I " .00 .1 (32 yy .” .0.00012 0.00009 -.0.00043___0.00254_m.0.00150___0.029 zz -0.00002 0.00014 0.00001 0.00119 -0.00113 -0.00125 yz 0.00009 -0.00013 0.00013 -0.00214 0.00120 0.00553 166 TABLE XXIX. 2W ' Basis nmctle'n 1 Orbital 3b2 731 l31’2 531 "b2 531 ’81 3.1.0000Lewo910---.0. 00000—14110“. -0.00000 -o.04363 82 --0.00000 -0.)5992 0.00000 0.10024 -o.ooooo -o.04244 S3 - 0.00000 ‘1 . 0. 7395 S -0. 00000 -0. 33503 4.0.00000—._0.-1.7138—'- 84 ...0..00000__.0. 29M 7 .0. 00001140146121. - 0.00000 0.44994 Y1 0.95346 0.00000 «1.59751 -0.00000 -o.on¢5 0.00000 N 12 0.31311 -0. 00000 -0.42144 0. 00000 '.--0.10935..-0..00000_. Y3 ..-_o.03445__0.00000 0.04342 -..-91.90.9111. -0.04305 0.00000 21 -0.00000 0.27214 -0.00000 0.10597 -o,ooooo 1.0.355"; . 22 ' 0.00000 0. 36310 0. 00000 0. 25007 _. -0. 00000....(163554... -- ZE 1. 0.00000 . .0.0I665 0.00000 04.10571. .- -0.00000 ~0420135 .. 0.0T2'U—0'Z’036’4'8—"07'037' - . o, . . 82 0.01460 0.05349 0.05427 -0.02446 L__o.00999-_o,.01514—.— S3 11.10.03223__:0. 19190 --0. 22885 -_ -OJTSSI- -o,oz1u -0,05139 .1 3‘! 0.03624 -0.24 72.0 -0.18393 0. 10163 -0. {0973 -o,o¢”9 Y1 : “0421068 '0. 34339 '0. 30312 0. 01530 —-0.-0640-3--0r01-60'0-* 0]. Y2 20.26.76640. 36883 ._ -0..3 1606.____0._0L§J.3_ -0.094§¢ -0.0£521 Y3 . - 0.14056 '0. 05 06 7 '0. 00664 0. 02 430 1 -o,09029 -0.01499 -- z1 ' 0.01525 0.01999 0.09210 0.15424 1-0.1.79; ---0.—21999— V 22 Winn—00491.1 ..0.14243...-o..20.1m_ -0.32404 0.23493 23 --0.01444 0.01433 0.11143 0.10410 . -0.21331 0.25135 - o 40 " o I " .U ...‘0 . ... '0 -4 82 ;--0.01.430—-o.115 34.9----0..05421——-0r02-444—- 4.00999 0.01516 33 - 0.03 223 --0. 19193 0.22335 0.01553 0.02711 «3.05139 3“ 3 -0.03424 -0.24120 0.13393 - 0.10143 4091340449343... Y1 ——0721343-'—o.34339---0-.—303H-0—.-01+30—' -0.06W3 0-01600 02 Y2 -0.24144 0.34333 -0.31404 -0.0:413 9 '0-09‘“ 0-045“ Y3 ... - 0.14 054 0. 05 04 1 -0. 00444 -0. 02 430 5—104’019-—°~N‘93— Zl —-0:01-525---0.01999—-0-.09230—o-.-15 . 0-2‘79‘ °-“”9 22 g-u.0104z 0.04911 -o.14z43 0.20201I 0-32‘0‘ 0-13‘93' 23 . 0.01444 0.01433 -o.11143. 0.10410. 41-21“? 0-451'5. 71';— -- ° N. ° -.- ‘ _. to“ l -0'00000 0.04930 k 0.00000 -0.0411.9 0.00000 -0.03114 . ' - N 22 f-0.00000 0.03451 0.00000 0.02340 -04Q0090.__9...01.8§J_~_ YZ —0.04419--0.00000--0.-09454—0;-0 . 0-03103 ‘ °-°°°°° T . T‘Tw‘h‘ - . 4 . . - . 2 . 01 W -o.02221 -0.0o310 -0.00251 0.00413 L—o.00134__-_o_._o_0u.3_ zz —0.00591—-0.00102--0.0051-1—-0.0090 ' 0.01209 0.00091. 12 3 0.01404 0.01394 0.00015 0.01411 ' -0.01432 0.00409 ,1 n " 0 do ”13 o o ....04.. ...,—....OLOQZOZ.-. YY .._0.01.21.1._-0.00310....0..00251—.0..00313_ 0.00234 -0.00133 . y 02 zz .-0.00591 0.00102 0.00511 -0.00909 ,-0.01209 0.00092- Y2 0.01404 -0.01394 0.00015 -0.01411 _.-0..o.1031_‘_-:9.9_040L, 1bl la.2 2b1 i "111 ”" 40.24245 ' 0.00000 0.32005 N x2 _ -0.34042 _ 0.00000 . 0.4_2,34_9_ x13: -0.14423 -0.00000- 0.19431 - . . - . l 01 X2 .. '0-2260? ..,04-3423.3..;:.9.e.3.2.47.7__ x; -0.11514 0.24131 -0.03540 - 4 b” - o I-OOZ ll; 02 x2 ......-0.22402,....-0.34?33_.-.0.32.411._ 11 -0.11514 -0.24131 -0.03540 + o - 0 ‘0400000 N xz _,_--0.oz433 -0.00000.__.0.01439 "IT—"'7. 21 . 0. 7 01 x7. __g._00391 -o.00124_;_0_._g_1_14_44_ 02 10: 167 TABLE XXX THE 212132 ovc ORBITALS FOR A GEOMETRY OF R=2.uo BOHRS, e=100° Basis flmction Orbital lb‘2 lal 23.1 33.1 2b2 ital s1 -o.ooo1;o 10.00100 -0.4o232 0.01521'~ o.ooooo""=0‘.‘073'T{' $2 0.00000 -0.00129 -0.4S970 0.01384 ~0.00000 -0.13002. 83 1... -- 0.00000 . 0.00192 0.04 433...20..0302.5__s.0..00000___0..52011— 30 -0.00000 0.0.1035 0.04909 0.01153 0.00000 0.55335 Y1 0.001113 -0.121100() -U.00000 0.00000 -0.0?710 '0.00000 . N 12 P- - 0.0003 1 o. 0.10:: 0 0. 00 00114040000411.01312—0200000— 13 -0.00014 «1.04000 0.00000 0.00000 0.02043 0.00000 21 0.00000 0.00054 -0.o1321 -o.o1435 -0.00000 -o.14155 ' 22 ... -0..ooouo ~0. 0005 3 -u .014511....-.0..0095.4 _4..0oooo_.-0.20431._ 23 41.00040 41.101211 -0.01044 0.02304 0.00000 -o 31 . 4 - .4 €33“—'—0'0T0. 0'8—‘0—0'7'1. FT—-0"'.o1413 0.00914 32 --.-0-33455 :0. 3344 1 0. 00 15 2_o.11430__-.0..1.21.91——0.o441.4_ 33 -0.02443 0.02421 -0.oo450 -o.391 1 0.41514 -o.04554 80 -0.02241 0.02291 -0.01033 -o.373 3 0.33242 -0.11444 11 .-_.0.00352 «1.00339 4.00041--_0.0322-1_.-0.03431__0.0103-1— 01 ya 0.00191 «1.00134 -0.00012 0.03533 -0.o3505 0.00301 13 0.00244 «1.00215 -0.oo133 0.02491 ~0.o3211 -o.o151o 21 ,. 10.00304 ..0.0u404 -0.00215__=0..0431_1__0.01320.__-21.023z.4_ 22 -0.00290 0.00333 -0.00239 -0.05434 0.04552 -0.03434 z3 ~o.oona 0.00335 0.00242 -o.o4545 0.03315 0.02415 - 31 41.41441 -0.4I413 0W . .- . ———0.0 4113—0—00 4_ 52 -0.33455 41.3.1441 0.00152 0.11430 0.12191 0.01414 33 0.02443 0.02411 -o.oo450 -0.39131 -0.41514 -0.04554 s11 ,-0.oz241 . 0.02291. .-0..o1.033._.0.31.33.3_s0..3 ‘ 11444— 11 0.00342 0.0.1339 -o.ooo41 -o.o3221 -0.03431 -o.01031 - 02 y; 0.00191 0.00133 0.00012 -o.o3533 -o.o3505 -o.0030.1 Y3 _u.00244 ....0.0021s___0.001_33_-.n_n249 ‘ 21 0.00344 0.00404 -0.00215 -0.04311 -0.o3320 -0.02324 ° 22 0.00290 0.00333 -o.00239 -0.05434 -o.o4552- -0.03434 I 23 . ~0.00213 o. .1013)” ...0.0024L__-0.0454 440.113.315—042415. XX . - .12 . . - o .0 yy 0.00000 «1.00030 0.00039 —o.oo314 -o.ooooo 0.00941 1 N 22 0.00040 -o.0uo19 4.00122. -o.004o3 -0.ooooo -o.o1093 . 92 0.00043 «1.00000 0.00000 0.00000 -0.01999 0.00000 - T O - o - 4 o -03 - 3°00 ' YY .._9.-.°.°°"" '0-‘10009 -‘0.e°°9§§__Q-.901 1&6 ...0400.13‘_19_1_Q9U5_J 01 22 0.00023 -0.00029 0.00041 0.00243 -0.00235 0.00352 yz 0.00012 -o.00001 -0.00042 -o.00054 -0.oo104 -o.00341 u ... 70.0001. -IL 00 .- 0 . -..,. -0. .. ......_....9. . 02 .1111 -0.00009 -o.0oon9 -0.00043 0.00124 0.00134 -0.oo134 ' zz -0.ooo23 -0.00029 0.00041 0.00243 0.00235 0.00352 12 _ 0.00012 0.00001 0,00042____0.00054_--o,.oo1o4___0_._0034_1_ 168 TABLE XXX , continued Basis function Orbital 3b2 781 5b2 531 ”b2 6&11 81 I 0.00000 "‘“Ji‘ofi‘T'S’T‘b'b'm-‘a - . 30:01:02' 0.00000— 40.00110 i 0.00000 -0. 11718 0. 00000 -0. 015 71 “.0, 0000 o_-o,_gsoo s__.__ ...0.00 000. . 0. 55 709- .- 0.-00000—---0.-0070 7-.— -0.00000 0. 17705 0.00000 0. 30598 -0. 00000 0.15092 0.00000 -0.16059 Y1 0.20105 0.00000 -0.09017 -o.00000 :10303110___0.90009__. IQ Y2 ——0.32571u-oo.ucono-m-0.s0207——-0.00000—h-0.03898 -o.00000 Y3 0.02530 0.00000 -0.0sr71 0.00000 .0.ooe7s -0.00000 21 w—0.00000 0.02130 -0.00000 -0.00702' 0.99090 0.23200__ 22 .‘-—0.0oooo--mo.40137——-0.00000———0.00000-1 0.00000 0.30022 23 0.00000 'n.10213 -0.00000 -0.00003! 0.00000 0.11730 31 0.01005 0.1-37.79 0.733535 0. 11039 '.s-0._007_1,3__-;0_._gga1t__ S2 .—0.012030«~o.05129——~0.00959———0r02020--0.00970 -0.00357. 83 -0.03009 -0.1uzn1 -0.10909 .-0.00900a 0.031591; 0.02533 _SH 0.04008 -0.a109a -0.19sza -0.07893;_olg‘oal__gq‘g]bgj__ Y1 .--0.12203mw-n.23350~—-0v10500———0r+0«05-' 0.23273 0.12071 -(31 Y2 -0.15037 -0..a031 -0.?1050 0.20000 0.30912 0.15053 Y3 -0.00032 -0.10782 -0.03010 0.10335 __0.20010 Q.05303___ 21 ..0.10090 n.0.20239_-0~21280———0v+9+5+- 0.17200 -0.13930 22 0.71012 0.21000 0.25391 0.25050 0.22219 -0.17005 Z3 . 0.10530 -0.00/05 0.00515 0.19010 __o._1067_l._;:_a‘m5_a; ”In --0.01005 . 0.03075---0.01 - . . - .0 l 82 -0.01£03 0.05119 -0.0£959 0.01026 0.00976 -0.00857; S3 0.03069 -0.18'fll 0.18969 ‘0'”69“.:fl;93269.__fl‘01811_; SH -0.000as----0. 21898—0H9W-0.0¢087 ‘ -0.07663 02 Y1 -0.12283 0.23356 -0.18550 -0.16005 0.23273 -o.1u71 Y2 ‘0.15037 0.!8‘31 “0.11650 -0.20_“6b J.‘3‘091_3_:_n‘lb0u_ Y3 “0.00432—0.107H——-0.-03010——0r+0430- 0.20070 '0.05363 21 -0.16093 0.20719 -0.23280 0.19151 -o,17205 -o,13930 22 -0.11012 0.21088 -0.25$9l 0.25690 -29J2219_:.Q0JJ605;_ Z3 4:0.10520__.0.00200__.0.00045___0.48010_-0.10070 -0.11750 TWO—"WEI?! . 0 . . - . N YY - 0.00000 -0.«.0 703 0. 00000 -0.03777. _0.._0_00_0_0 3 22 -0.00000 0. 1.64027 0.00000 0.03171. 0.00000 0.03672 YZ 0.06911 0.00000 -0.06553 0. 00000 -0.02257 -0. 00000 XX 0. 00656 0. (m -0. 0 - . :9,-QOOL3__1Q&9§( lq__ YY 59,0001?“ 0-00165_:Q._®_¢_OJ__9.._QL ‘2}. 0.01180 0.02090 (IL 22 -0.01001 0.00303 0.00020 --0.01310 -0.0o097 . 0.00307 - YZ 0.01861 0.00004 0.00198 0.000924:0.00§66 ‘0 01036 H -'0.00656 . 0.UUJS9__0.9QM._Q9_36I 00 - 0 Y! 0.00017 0.00100 0.00207 0.01293 ~0.01104 0.02090 (32 zz 0.01001 0.00300 -0.00020 -0.01310 0.00097 ; 0.00307 ' Y2 1.9.01.0“. —o.0009_0_0.90110_:_0200090.70730063' 'F‘.unéa‘""0 _ -0, 21 9., ... can‘t?! 7'39 5". N X1 ., _. . X2 -0.31091 0.00000 0.43575 x3 -0.10z10 -0.00000 . 0.21979 ‘0. , 0‘ .. .... Q. . . o x3 -0.13903 0.20100 -0.00051 XI ” ...U' ----... 0. . ...." 2 .... 02 x2 -0.23501 -o.35239 4.31207. x3 -0.13953 -0.20100 -o.00001 “XY 0.00000_,-o.03733“_20700000 __£%_h.-.'°-02235 -n.00000 0.03172 XY -0.01741 0.01132 0.01220 01 xz 0.01209 -o.u1001__-0.00090 ..... .: “XT‘T““' 0.01701 0.01132 -o.012z0 02 X7. 0.012710 0.01M! -0.00090 1159 s~fl6v.61 emo~).61 monno.a ~6u~6.6 nau66.6.1 omh66.6 m0666.61 n~666.61 «6666.6 «0N66.6 mmN66.6 om~_6.o1 N6666.61 nau66.6 6~n66.6 n6666.6 o~666.6 -666.61 [ .FNN66.6 ¢N6H6.61 m6nn6.61 ~6—~6.6 nnuoc.6 owm66.6 66666.61. n~666.61 69666.61 ¢ON66.61 mn~66.6 ¢m-6.6 86666.61 n0u66.61 6~n66.6 m6666.6 ou636.6 ~«666.6 66666.61 mmnn6.c1 <66c6.6 nno¢o.6 66666.61 ann~6.61 16666.6 66663.61 “0666.6 66666.61 nqcn6.61 66666.61. on—66.61 66666.6 onn~6.6 n~666.6 60633.0 66666.6 omNm6.61 hom06.6 6mo-.61 oo~06.61 uenm6.61 mo—n6.61 m0666.6 no~n6.c1 $6666.o ownhn.61 6h66n.6 660m~.61 N—~6u.61 ONum6.61 hum06.o1 ~6N66.o ~h~66.61 6N666.61 . . a..(. ~ooon.6 nN6n~.61 6~666.61 umn-.61 666mm.41 60w6fi.61 06066.6 606~3.c1 oO~66.31 .emh~.6 oom¢~.61 ~NN66.61 6Ns¢6.61 Cnmov.61 Q~h66.61 mcmoooo JauNnoul mg666.6 .anMpé- ELSA. 39.5.3.1 nmnood 32F... omumo.o1 ~0moo.o1 one—u..- oohoo.o ~0nmo.o- mo.mo.o moooo.o- oo~93.o .uuoo.0 daWHflAu119dxa1d1IAa0dNJj1Id4a31qIIAfl4wqJ11IAad3fld111zadd1quIJHAGQA1114§SZWduI m~0~n.ou ocean.oaq mduo~.ou acmm~.o onueo.o1 00.no.o douoo.ou. .meoo.o nn~oo.o . nooo~.o1 muon~.on ogooo.o amn-.o- 00oan.° no.0».ou 00000.0 condo.c- oonoo.o 44wH1A1113nJA1d1IJ4adqA1IIddHx1d1IJaAd1JHIIdAud9A111d4021d1141309411lddxawq11 «no.0.o n.m.o.° 0..oo.° omogo.o ~00n~.ou oon~..o oodoo.o-. .flnnn.o uuo~n.o 1pmmnc.o mo~mo.o mamop.o nnmapuo oeoo~.on . . .3 J . r. . a a; 70" -N¢6.61 66666.61 n6nNn.6 66666.6 -mn6.61 66666.6 66666.61. 66666.6 . mN~66.61 666~o.61 66666.6 6666~.6 66666.61 nmemo.61 ¢6n66.61 66666.6 66666.61 .n—~66.61 66966.61 ¢~m¢6.6 66666.61. n66~m.6 66666.6 n6n~6.61. 09066.61 66~66.61 66666.6 69666.61 flomnq.61 66666.61 ~6-~.61 66666.61 ~vo~6.6 6~0v6.61 -~66.6 66666.6 m 1 11.1. 1. .. . . . 111111111. .1111 -1 f 1 --- 1-2.1.111..- 11 '11 om cm 0: cm om om OH OH Hmpfipno mo Ho mo Ho coapocsm mammm oomauo .mmmom mm.mum wmemzomw < mom mqo Hmma mme Hxxx mqm<9 170 00000.01 nom~0.0 o~u~6.01 .IJ9J14142313|433J11 ooonn.o 60006.: enun~.0 s w 3 0N 0H F A00 A00 A00 H . . ,--. - - ‘ oe-6.6 0m040.0 hme~6. ¢¢m~n.01 anonn.61 o-n~.0 an¢o~.01 o_0mn.01 noNs~.6 --qnuuacal aflnmuddlllnoudem cemun.01 6n6nn.0 0-n~.0 :.1 . llldflddAIQIIIQdGGQ1un-N w0nn9.0 00606.0 ~n9an.n nms~n.6 06660.0 0N0nN.n QQQLQSLfiDLfiB QQQLQQLQQ fifi NO Ho NO HO : w : Axv.m Ax0.H 0.0.0 Hmufiphu coauocpu mammm [0005008 . dog Ema. 1371 flyflflJEIXXXII THE 021-12 ovc ORBITALS FOR A GEOMETRY OF R=2.uo BOHRS, 0=110° Basis function 01 02 01 O2 Orbital lb 28.1 381 2b 2 Mal ‘W 2 1a1 .1' ~o;oqgoo -g,00001__.-o.59.191 - $1 0.0167.\__0..90000__0...1219L 52 3; 0.00000 -o...0005 -o.11132 0. 01111 -0. 00000 ...-0 17151;:_; 33 " 0.00000 ~n.00021 -0.00072 -0.03170 *-0.00000 0. 52201 1 Sta _-.Q.Qooua__.0. 0011 0..- 0. 00191 --o-0193.0_.0..00 0004.04.19”; .11 -0.00011 0.0..01‘0 -0.00000 -o.ooooo -0.02011 -0.00000'. 12 ‘_-o.00127 -o.0..0'0 0.00000 0.00000- -o.00159 0.000002; ;y3 .....o.00011_-,0.00000.- .0.00000.1-.-0..00000..:..o.o11714000000011: .21 . 0.00000 «1.00013 -o.00150 -o.0115-1 -o.00000 '-o.1755.3 . lzz .g-Z'-0.00000 -0.0ul1'l 0. 00017 -0. 007111 ' 11. 00000 '-0. 2153911 _ z3 ..-.n.oaou1___0-0..o1,1_-o.oo131—042321wooww12ufi ‘31 0.11010 -0.11011 0. 00019 0.159901 --0. 10112 0.01050 . 32 ”.hto.32111-.‘-0 32121 0. 00017 - 0.13130. -0. 11151 n10.01513 s3 1.2.0.000sz 0.01 1__- 0. oo 03 2...eo-1oo$5_2_.o.1 1439—4001911... .31 '1 0. 00117 -0. 00170 -0. 00023 --0.37770 . 0.37511. -o.11301.. .111 . .0. 001111 -0. 00132 -0. 00017 ~ 0.03177 .-'-0. 03993 :1 0. 0153111 Y2 .;-‘.-.0.00051_._0. 0003 3..-000112_0.012151—-0101222—-'--0.-01101—'—- Y3 0.00015 1-0. 01051 -0. 00021 - 0. 03023- I-o. 03190 : -0 01515;. .21 .'-o.oo102- 0.0.017 . 0.00017 -o.01153 '- 0.01111‘-°-0.02112‘.': 22 _.0.00.011..-0.0001 I__.0.00071_'_-o.-0511—1———o.05101-=—o.0291-3- , 23 -o.00051 0.00011 - 0. 00011 -0. 01325 - 0. 03759 .- 0.03291.“ ' 31 «1.710110 «1.110111 0. 000?! 0. 0m 32 S3 .;.0. 00022. 0. 0.011 o0. 00032 -o.1005'5 .-o.11139..--o. 01931 311 .fi- -.0 00117 . -0. 00120 -0. 00023 ”41.37770 '-0.37911"’r-o. 12301; Y1 ...2-0600111._°_0. 00132- -.0. 00017._-G+0337 {—00.03993—0-01-0-1-53-04 1'2 -0 00051 ' -0 0. 033 0. 00112 --0. 01251 -0. 01222 .-0. 01101 1 Y3 '1.'...0.00015 - 0. 00056 (1. 00021 '-0. 0302.3 '-0. 031190 0.01515" 21 ..'_"_'.0.00402_0 00017....0. 000174000005-‘3‘4-0. o1111-’:-0.02114.1 22 -0. 00011 «1.00017 0. 00071 -0. 05111 -0. 051213-0. 02913 23 5'. 0.00051 1 0.00011 0.00011 -c.o1325 “0.0375911 0.0329“ XX U. . 1) ...... . ...—... ' .. ‘ _. YY ..o.ooooo «1.00051 0.00009 '-o.oos1.9 --0.00000. 0.00169 ; 22 ' "0.00000 -0. 00015 -0. 00030 --0. 00121 -0. 00000. -0. 01205 YZ ‘ 111.100.0000u._0.00000'_.0.00000_-..0.01-792__0..00000. T - Y! W 0.00001 '-0. 00012 -0. 00011 1 o. 00129.1-0. 0007713 0. .000203 ooou_:0..0000115_9.1102.011._-:.11.00211'.__n.nnm. 0.00011 «1.00013 0. 00027 --0. 00031 0.00003 -0. -00912 WONT—WGWW . z .oo L...—.0-00001..-0. 00012. ..—o.oo o11..-o.001z9--_-.0 000‘"'——-0r0002.-0-§ -o.00011 -0.00021 -0.00005 0. 00200 0.002711 0. 410721; .o.ooo11 0.00013 -0.00027 : 0.00031 0.00013?- , 0.00917.g . . ' 1 1372 GHUELE XDOCLI, (XMTtirnmxi “* ‘?:=: Basis fUnction Orbital 302 7a1 502 5111 111.2 5.1 31 1.0. ‘00"000‘...:0.09000 42.00000 ‘0..01600. -o;ooo(,-o -o.o2.179 82 0.00000 ~0.13215 0.00000 «1.02373. -0.00000 —0.05112. 33 -0.00000 ' 0.511110 0.00000 0.05715_0-00000___0.u91.1. le 1.0100000...0.-21111 -0,000..0 0.19911 -0.00000 -0.15151 Y1 0. 211 13? -0.00000 -0.1J'2voz 41.001100 0.02179 -0.00000 bl 12 0.32391 1-0.00000 -0.59101 0.00000___0.03111 0.00000.. Y3 .0.001s7..-.o.00000 -0.01 7,13 0. 00000 0.03517 0-00000 21 1-0.00000 0.12151 0.00000 -0.01015 0.00000 0.21301 22 0.00000 - 0.11735 -0.0o.00 0.00301_.=0.00000.._0.319L0. 23 0.00000-—.0.02110 —0_00000 -0.01211- -0.00000 0.12933 31 0.01300 0.03173 0.03013 0.01313 -0.00051. -0.00117‘ 32 0.01110 0.05155 0.05211 0.01850.40.oon¢.-.-o.4.0930—= S3 »-0.01003—:-o.20320 -0.20021 -0.0s951 0.00510 0.03152 8“ 0.02319 -o.10923 -0.17911 -0.01525 0.01159 -0.01157 Y1 -0.11011 1-0.21112 -o.22310 0.13025._.0.17012~—.0.11111— <31 Y2 v-0222117——-0.30730 -0.23257 0.11191 0.23026 0.10211 Y3 -o.12911 -0.00111 -0.02191 1.10990 0.17011 0.01771‘ . 21 0.10111 ' 0.15311 0.10317 0.21009.__0.21311__-0.11930— . 22 ; L-0.13313e-0.15513 0.22037 0.20152 0.27711 -0.15291_ 1 23 1 0.05195 . 0.01111 0.01217 0.21121 0.11070 1-0.11109 1 31 -0.01300 ' 0.03173 -0.03533 0.01313___0r00051——-0.00004—s S2 L~0101711—L—0.0s115 -0.nsz12 0.01050 0.00111 -o.0o930 S3 ' 0.01103 -—0.20370 0.20121 -0..5951‘ -0.00500 0.03152. 5" -0.02 369 -0. 199! 3 0. 171111 -0. 00 52 5....M41.59_.o,.97.151_ Y1 M.-10010—'-—0.-21007 «1.22310 --0.-'1-3125 0.11032 41.11155 <32 22 -0.22117 - 0.30730 -o.23257 ~-0.11791 0.23021 -0.10211 ‘13 -0.12901 ' 0.00611 ‘0.02190 '0-10990—0..1.1000——-010077-1- 21 .-0.10011-4—0.15300 -0.19337-u-0.21009 -0.21311 '-0.11939 .22 -0.13363 0.15513 -0.22931 0.20857 -o,z'n2.1 1-0-.1529; Z3 -0.05095 0.01111 -0.01237 0.21021__=0-11020_Ln0.1 109- "1I"5..0. 000 —L- .03151.- . 000 -«- r00 2 0.00000 -0.01171, YY ' -0.00000 0.00551 0.00000 -0.03710 -0.00000 0.02710“ N 2% -0.00000 ' 0.01510 0.00000 0.02909_.o.ooooo__o,.93095_1 Y "XX"‘ YY ..0.05317...0-00000 -0.09905 “10.00000 -0.01519 -0.00000 Ff 0 0 U o " o o 6 ‘ 0 . 3|- 0]. -0.01156 '-0.00610 0.00237 0.0096£-_0.01.000_'__0.02391. 1} ZZ .10.0Q5£1_L.0-01096 u.003£9..-0-01061: -0.0106‘ 0.00023 3 Y2 0.01862 0.00545_fi_u.00527 0.005811 0.00393 1-0.01530 ; xx -0.00706 '0.00511 -0:“0117 -0Ti0T732:IE:35331:::32583§EZ : YY _~0.01151—;-0.00111 -0.00237 - 0.00912. -0.01011 0.02397' 02 22 0.001171 0.01000 -0.0031'9 -0. 0104113 0.01051 0.00003 YZ 0.01012 ‘-0.00515 0.00127 -0.00501;__o.oo;o._._o.g+;gog ' x1 V’;o.21919 -0.00000 ' 0.13153 3 DJ x2 --0.31310 - 0.00000 -’-0.415529---i X' -0.11221 -n.00000 0.17135 ' "1d?"'70TTTI70"'0717015"=077r130" 01 x2 -0. 23711 -- 0. 31.111 —-0.—30v710-v x3 -0.13110 0.23710 -0.09555 x1 -07 - . - . x3 -0.13150 -0.23710 --c.09555 N Xi ...—:boonbt- - o - 0 X2 '"-9L2229’ --0.00000 9203027-' 01 X7. ”“1141! [07 41.1.0909 9,00 09') XY 0.01026 0.01370 '0100100-‘ X7. 0.01107 0.0-.989 0.00065 (D2 APPENDIX D CONFIGURATION SELECTION IN "WELL-BEHAVED" POLYATOMIC MOLECULES ! a .41....— .._-_ APPENDIX D CONFIGURATION SELECTION IN "WELLPBEHAVED" POLYATOIVUIC MOLECULES In those molecules for which a "good" valence-bond type wave- function may be written, the dominant molecular correlation configura- tions are easily identified in a E13311. fashion. A valence-bond type wavefunction is understood to denote a case in which the molecular binding is well described by localized o and n bonding and anti-bonding orbitals, in addition to core and lone-pair orbitals; this is in con- trast to the necessarily delocalized orbital structure of N02. In these "well—behaved" molecules, the most important corre- lating configurations relative to the RHF configuration fall into three classes: (1) Diagonal bond correlation configurations . Such configura- it tions are generated by 02 2 excitations from local- +0 and m2 +132 A—B A-B A-B A—B ized bonding orbitals into the corresponding anti-bonding orbitals. (2) $plit—shell multiple bond correlations. This class is of n a the form OA-BflA-B+°A-B"A-B where the o and tr orbitals are the compon- ents of a double (or triple) bond. For states with closed shell RHF configurations, these excitations lead to two open a shells . Con- figurations with the unpaired o electrons singlet coupled are much more important than for triplet coupling. 173 U 174 (3) rIWo bond gut—shell excitations. Classes (1) and (2) are identical to the molecular extra correlation energy (MECE) configura- tions, which are vital for an accurate description in diatomic mole- cules. However, unique to polyatomic molecules are configurations which correlate two orbitals which are not localized on the same pair of nuclear centers. The most important of these are * i m 5‘) OA-B °B—C‘*°A—B C's—c ‘ * * and b ) °A-B"B-c*°A—B "B-c in which there is a nuclear center common to the two localized bonds being correlated. Again, as in class (2), overall singlet coupling of b the 0 open shells contributes much more than triplet coupling. Other excitations such as n2+oZEB or n2+1r:33, where "n" is a lone-pair orbital localized on a single center, or single excitations are an order of magnitude less important for introduction of correla- tion than the configurations of classes (1), (2), and (3). As illustrations, the ground electronic states of HO HONO, 2, and CH3NO will be discussed. I. H02. 'Ihe hydroperoxyl radical ground state has the electronic 4. symmetry 2A" and as a free radical does not quite conform to our defi- b nition of a molecule with a "good valence-bond" structure. However, the deviation is not large and some extensive CI calculations exist for the H02 ground state which provide some interesting comparisons. The RHF configuration of H02 in its 2A" ground state corre- sponds in a localized description to <1s01)2(1s02>2<2s01)2<2s02>22(n;_0>1 175 where pz,02 is an in-plane lone-pair orbital perpendicular to the 0—0 bond. Due to the influence of the unpaired electron, the bonding 1r orbital electron density is predominately on the central oxygen atom, while the Hit orbital is located more on atom 02. The qX‘ZA" bond angle in H02 is approximately 105° and the central oxygen atom is essen- tially unhybridized. 8"“ The basis set used in this study was the [1433p] contraction of Dunning, augmented with single component d-functions on oxygen (exponent = 1.35) and p—functions on hydrogen (exponent = 1.00). For v, ' a nuclear geometry of RH_Ol=l.80 bohrs, R01a02=255 bohrs, 9=100°, J 8 configurations were included in the OVC orbital optimization. 'Ihese configurations are included along with their mixing coefficients in Table XXJCLII; the four core orbitals which are doubly occupied in all configurations have been omitted from the table. 'lhe single configuration (SC) energy of the RHF configuration for the OVC orbitals is -150.20112 hartrees; this is probably less than 0.01 hartrees higher than the true SCF energy in this basis. For the 8 OVC configurations, the energy is -l50.288’-l7 hartrees. A CI calculation which includes all single and double excitations from the RHF configuration within the space of valence orbitals (those given in Table )OOCEII) using the OVC orbitals gave an energy of 450.2906” hartrees. From the SCF estimate of -lSO.21 hartrees, we obtain an energy lowering of 0.078 hartrees for the 8 OVC configurations, with an additional improvement of only 0.002 hartrees for the extension to the valence-singles-and doubles (VSD-CI) level. Thus, a smell 176 ivy mmmun-. .II r1; moamo.o m.o o m H o H m m H mmmmo.o m.o o m.o H H m H H m H H s:mso.o m.o o m.o o H m H H m H H Hmomo.o m.o H m H H o m H m HHmHH.o m.o o m H H o m H m sHmmo.ou H m o m m m o msme.o- H m m o m o m mmmem.o H m o o m m m mpcmHOHccmoo wcHXHz mcHHasooucHam mouHmF mouHoF mo.Hmo Helmo mo.Na No.8D Hon:b mom mom mezmHonmmoo QZHXHZ 92d mZOHEO HHHXNN mqm¢9 177 subset of configurations selected in a; pr_'i_o_r;i_ fashion yields «97% of the valence correlation energy of the REA" ground state of H02. Cole and Hayes [91] have performed a VSD-CI calculation in a smaller basis set and for a somewhat shorter O-O bond length (RH-01: 1.8114 bohrs, R01 _O2=2.32L| bohrs, 0=100°, ESCF=-150.0068 hartrees) ; however, the CI employed SCF orbitals as compared to our use of the multi-configurationally optimized OVC orbitals . IIheir energy lowering upon the introduction of correlation is 0.026 hartrees, or one-third that of our calculation. Clearly evident is the superiority of the OVC orbitals over the SCF orbitals for purposes of introduction of correlation into the wavefunction. rmere is a study by Liskow, Schaeffer, and Bender [92] in a double-zeta basis set (ESCF=-150.1562l hartrees for RH_01=1.80 bohrs, 801 _O2=2.60 bohrs, O=100°) which employed [500 configurations in an iterative natural orbital (INC) treatment. 'lhe final energy of -l50.24138 hartrees gives a correlation energy lowering of 0.085 har— trees, an improvement of about 10% over our 8 configuration OVC wave- flmction. For the low-lying doublet states of N02, thougu, the CI energy lowering decreased by about 10% when d-functions were added to the double-zeta basis. Thus, our 8 configuration OVC wavefunction produces the same amount of molecular correlation as the 500 con- figuration INO wavefunction of Liskow et al. II. HONO. No all M calculations of nitrous acid electronic structure have appeared to date in the published literature [93] and we have not examined all valence configurations; thus it is not pos- sible to make detailed comparisons such as were done for H02. J “£39....”— I Q ‘._ I 178 'Ihe calculations were done in the Dunning [1433p] basis at the experimental geometry of the trans isomer as determined by Cox _e_t _a_l_. [9“] H ’/ o,)\: 4 O 7 0 Exclusive of the six core orbitals, which correspond to the ls and 2s atomic orbitals on the heavy atoms, the RHF configuration in the localized description is )2 2 2 2 2 2 (OH-01 (OOl-N) (ON—02) (po,O2) ("N—o2) (pm,01) where po,02 is an in—plane lone-pair orbital perpendicular to the N—02 a bond and p",Ol is a lone-pair orbital perpendicular to the molecular plane. The OVC orbital optimization was performed for the configurations given in Table XXXIV. The SC and OVC energies are respectively 404.5773 and -2014.73Ol hartrees and the energy improvement due to correlation for the 13 a m selected configurations is 0.1528 hartrees; this com- pares favorable to the lowering of 0.1592 hartrees for 18 carefully selected configurations in the BCZAI ground state of N0 and in the 2 same basis set . We note that configurations 6 and 7 have very small mixing coefficients; this is as expected since there is no nuclear center common to the two bonds which are being correlated by these con- figurations . 'J _ 179 Hmomo.ou o m.o H H H H o o m m H m m 0mmOH.ou o m.o o H H H o o m m H m m mammo.o o m.o H o m o H H m m m H H mmeso.o o m.o o o m o H H m m m H H mmHHo.ou o m.o H o m H H o m m H H m mmmmo.ou o m.o o o m H H o m m H H m mmsoo.o o m.o H o m H o H m m H m H mHmoo.o o m.o o o m H o H m m H m H mmmeH.ou m o o o o m m m m m mHmmH.o- o m o m o m m m o m mommo.o. o m o o m m m m m o mmumo.ou o m m o o m m o m m omem.o o m o o o m m m m m pcmHoCumoo wfiHeuooucHam No.2F No.2F No.2D zuHoo Helpo 8:5 moan .8120 2:50 Hone0 SHE a. .... * ... 020m mom EOEOO UZUQE Q72 WZOHBEUHEZOO 96 Egg. 180 III. CH3NO. For nitrosomethane we have only considered the con- figurations of classes (1) and (2). In addition to five domly occu- pied core orbitals, 13(2), 151%, 1520, 2sg, and 281%, and an in-plane lone- pair orbital, pi O’ which is perpendicular to the N—O a bond, the ’ CH3NO RHF configuration contains 6 bonding orbitals 2 2 2 2 2 2 (CC-H1) (00412) (Go-H3) (CC-N) (ON-O) ("N-o) For this calculation the Dunning [1152p] contraction was used and the geometry was identical to that chosen by Ha and Wild [95], who per- formed an SCF-CI computation in a basis set which is about 0.05 har— trees inferior'to the Dunning basis at the SCF level. ‘lhe 9 class (1) and (2) OVC configurations, expressed in an excitation formalism notation, are included in Table XXXV along with the mixing coefficients. The OVC energy of -168.9292 hartrees is an improvement of 0.1572 hartrees over the SC energy. In contrast, Ha and Wild ob- tained a lowering of only 0.1037 hartrees over their SCF result for a "'-.-u CI calculation of 169 configurations; this set was selected from a r very much larger set by application of an energy criterion desigled i”. to identify those configurations most efficient for the introduc- tion of correlation. In none of the examples cited have we rigorously examined the effect of various classes of configurations in introducing correlation into the wavefunction; also, only ground electronic states were considered. Nevertheless, it is evident that at least for some molecules, the majority of the valence correlation energy 181 TABLE XXXV OVC CONFIGURATIONS AND MIXING COEFFICIENTS FOR CH3NO Configuration Mixing Coefficient RHF 0.95853 (OC_H1)2+(0;_H1)2 -0.07908 (OC_H2)2+(O;_H2)2 -0.07908 (CC-H3)2+(°C-H3)2 '0’07906 (0N4)2+(o;_0)2 —0.07386 (OC_N)2+(o;_N)2 -0.09811 (mN_O)2->(n;_0)2 -0.1826u (ON—0)("N-0)‘* 0.086214“ (“;o)(";—0) " 0.07988## #Spin-coupling 0 0.5 0. fl WSpin-coupling l 0.5 0. “a: can be accounted for within a small set of configurations chosen in a 31.15; P231931 fashion. This is of great significance for the extension of v a_b_ initio techniques to larger polyatomic molecules, for which the huge total number of valence configurations precludes a "brute force" approach to introduction of molecular correlation. IIHflllfllfllmlH ! B Ill: 1 "6 .W. ”I” N m U Io fun