THESIS This is to certify that the dissertation entitled The Electronic Structure of First-Row Negative Ions and Transition Metal Atoms presented by Beatrice Helen Botch has been accepted towards fulfillment of the requirements for FAD «mum P7944” I 111 .-‘ A".‘L t- a Major professor Date M MS U is an Affirmative Action/Eq ual Opportunity Institution 0- 12771 MSU LIBRARIES 1—;— RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. THE ELECTRONIC STRUCTURE OF FIRST-ROW NEGATIVE IONS AND TRANSITION METAL ATOMS By Beatrice Helen Botch A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1981 ABSTRACT THE ELECTRONIC STRUCTURE OF FIRST-ROW NEGATIVE IONS AND TRANSITION METAL ATOMS BY Beatrice Helen Botch Negative Ions The electron affinities of carbon, oxygen and fluorine have been calculated using a compact MCSCF wave- function with a [4s,4p,3d] Gaussian basis set. This wave- function describes radial correlation of the 2p electrons which is found to have a large differential effect between atom and anion, and also includes the (232->2p2) near-. degeneracy effect. Radial correlation of the 2p electrons increases the calculated electron affinity by as much as 1.2eV over the Hartree-Fock value. An orbital model is discussed which ascribes this effect to the diffuse nature of the orbital occupied by the (Z+1)st electron of the anion. Configuration interaction calculations based upon the MCSCF wavefunction, result in electron affinities comparable in magnitude to the large basis set calculations of Yoshimine and Sasaki.2 The importance of higher-order angular functions and higher-order excitations is also examined. Transition Metal Atoms The major differential valence correlation ef- fects of the lowest lying states arising from the szdn, n+1 2 configurations of the first-row transition sd , and an+ metal atoms have been characterized using MCSCF and CI pro- cedures. The important correlation effects are found to be first, angular correlation of the 432 pair arising because of the near degeneracy of the 4s and 4p orbitals, and second, radial correlation of the 3d electron pairs. This large differential radial correlation of the 3d electrons can be interpreted as being due to nonequivalent n+1 2 excited states. Both of d-orbitals in the sd and dn+ these effects can be incorporated into a simple MCSCF wavefunction which reduces the error in the excited state atomic dissociation limits (~0.2eV in Sc-Cr, and No.5ev in Mn-Cu for the sdn+l-szdn excitation energy), yet still is of a form which lends itself easily to molecular calcula- tions. This dissertation is dedicated to my whole family Mom and Dad, Laurie, Paul, Dennis and Barbara, Ruth and George, and Steve and Carol. I thank each of you for your patience and love which have helped me to see my very slow steps as progress. ii ACKNOWLEDGMENTS First, I would like to thank my research director, Thom Dunning, whose understanding and encouragement have helped me to grow from scientific infancy to a much higher level of maturity. More important than the many valuable lessons is the friendship I take with me of a person I respect very deeply. To my major professor, Jim Harrison, who on the day I asked to join his group told me: ' The last think this world needs is another quantum chemist' , thanks for giving me the opportunity to succeed and fail as I would. To my good friends Larry Klein, Trish Maxson, Emily Rose, Carl Gagliardi, and Mark Beno who have shared with me the highs and lows of graduate school and have kept me from losing touch with myself, thanks. I extend a special thank you to Larry Harding for the many evenings over ice cream and wine and for lightening considerably the burden of 'finishing'. And finally, I wish to thank my teacher, my mentor and my friend, Pat Faber, who set the hook back in freshman chemistry and has kept me from drowning ever since; may I have the chance to do the same someday. iii TABLE LIST OF TABLES. . . . . LIST OF FIGURES . . . . INTRODUCTION . . . . . Part A: METHODS I. INTRODUCTION . II. HARTREE-FOCK . III. IV. V. GENERAL PROCEDURE. VI. MOLECULAR CODES. LIST OF REFERENCES Part B: NEGATIVE IONS Theoretical characterization of negative ions. lation of the electron affinities of carbon, oxygen, OF CONTENTS CONFIGURATION INTERACTION. MULTI-CONFIGURATION SCF. and fluorine. I. II. III. INTRODUCTION . A. EXperimental Background. B. Theoretical Background . ORBITAL MODELS FOR NEGATIVE IONS CALCULATIONAL DETAILS. iv Page vii ll 14 15 16 17 Calcu- 19 21 23 27 35 n+2 IV. CALCULATION OF THE ELECTRON AFFINITIES OF CARBON, OXYGEN AND FLUORINE. . . A. Carbon . . . . . . . . . . . B. Oxygen . . . . . . . . . . . C. Fluorine . . . . . . . . . . V. DISCUSSION . . . . . . . . . . . VI. CONCLUSIONS. . . . . . . . . . . LIST OF REFERENCES . . . . . . . Part C: TRANSITION METAL ATOMS Valence Correlation in the szdn, sdn+l, states of the First-Row Transition Metal I. II. III. INTRODUCTION . . . . . . . . . . A. Background . . . . . . . . . B. Basis Set. . . . . . . . . . and d Atoms. VALENCE CORRELATION EFFECTS IN THE EARLY TRANSITION METAL ATOMS, Sc-Cr. . A. Scandium and Titanium. . . . B. Differential Trends. . . . . C. Orbital Interpretation . . . D. CI Calculations . . . . . . E. Experimental Excitation Energies . . VALENCE CORRELATION EFFECTS IN THE LATE TRANSITION METAL ATOMS, Mn-Cu . A. Differential Trends. . . . . B. CI Calculations on the Nickel Atom . C. Experimental Excitation Energies . . Page 38 38 41 44 46 50 52 55 56 59 60 63 67 70 78 87 87 88 89 91 IV. OTHER CONSIDERATIONS. . . . . . A. Core Correlations . . . . . B. Relativistic Effects . . . V. CONCLUSIONS . . . . . . . . . . LIST OF REFERENCES. . . . . . . REPRINT: On the orbital description of the 433dn+1 states of the transition metal atoms. . . vi Page 94 94 96 97 99 102 Table Part B II. III. IV. VI. VII. LIST OF TABLES Page Electron affinities and ionization energies of the first-row atoms. . . . . . 20 A comparison of calculated electron affinities of hydrogen, lithium and boron through fluorine. . . . . . . . . .. 25 A comparison of the REF, UHF and GVB energies for H, H', and He. . .. . . . . . . .|. .. .30 Optimized 3d and 4f Exponents for carbon, oxygen and fluorine . . . . . . . . . . . 37 Hartree-Fock, MCSCF and CI calculations on the 3P state of the carbon atom and the 4S state of the carbon anion . . . . . . . 39 Hartree-Fock, MCSCF and CI calculations on the 3P state of the oxygen atom and the 2P state of the oxygen anion . . . . . 42 Hartree-Fock, MCSCF and CI calculations on the 2p state of the fluorine atom and the 1S state of the fluorine anion . . . . 45 vii Table VIII. IX. Part C II. III. IV. VI. VII. Page MCSCF and aCI correlation energies relative to Hartree-Fock for the iso- electronic series: 4S, 2P, and 1S. . . . . . 47 Errors in the calculated electron affinities of carbon, oxygen and fluorine . . . . . A comparison of numerical and Gaussian basis set of calculations of the Hartree- n+1 2 n n+2_32 Fock sd -8 a , A(n+l), and d n, d A(n+2), excitation energies of scandium to c0pper. . . . . . . . . . . . . . . . . . 61 Calculated and experimental energies for the szdl, ad2 and d3 states of the scandim atom. I O I O O O O O O O .- O O O O, 65 Calculated and experimental energies for the szdz, sd3, and d4 states of the titanium atom O O O O O O O O I O O O O O O O O O O O 6 6 2 Valence MCSCF (482,4p2), (3d ,4d2) correla- tion energy differences from Hartree-Fock. . 68 Calculated and experimental energies for the szd3, sd4 and d5 states of the vanadium atom O O O I O O O O O O O O O O O I O O O O 80 Calculated and experimental energies for the szd4,sd5, and d6 states of the chromium atom 81 1 4s3dn+ -4323dn calculated and experimental ex- citation energies for scandium to chromium. . 82 viii Table Page VIII. 432, 3d2 and 433d pair correlation energy differences from Hartree-Fock for the chromium atom. . . . . . . . . . . . . . 83 IX. 3dn+2-4sz3dn calculated and experimental excitation energies for scandium to chromium. . . . . . . . . . . . . . . . . 86 X. Calculated and experimental energies for the szd8,,sd9 states of the nickel atom. . 92 XI. 4sBdn+l-4sz3dn calculated and experimental excitation energies for manganese to c0pper. . . . . . . . . . . . . . . . . . 93 ix Figure Part B 1. Part C 1. LIST OF FIGURES A comparison of the RHF and GVB radial amplitudes for the Is orbitals of H, H- and He. n+1-4523dn excitation energies of n+1 433d scandium to c0pper, [E(sd )-E(szdn)]. . Calculated (3d2,4d2) energy lowering relative to HP for the 32dn and sdn+1 states of scandium to chromium, [EHF—EMCSCF]'° . . . . . . . . . Radial plots of the Hartree-Fock and non- orthogonal d-orbitals for the 3F(4823d2) state of titanium . . . . . . . . . . . . Radial plots of the Hartree-Fock and non- orthogonal d-orbitals for the 4F(4s3d2) state of scandium Differential (3d2,4d2) energy lowerings 2 n n+1 for the s d and sd states of scandium to copper O O O O O O O O O O O O O O O O Page 33 58 71 76 77 90 INTRODUCTION There are many seemingly simple problems of chem- ical interest which have presented theoreticians with dif- ficulties, both computationally and conceptually, for many years. This work is concerned with two such problems: the determination of accurate atomic electron affinities; and the calculation of the electronic excitation energies of transition metal atoms. Each is of theoretical interest because the procedures developed for accurately describing atomic anions and metal excited states provide the frame- work in which concepts about the structure and reactivity of molecular species are developed. Experimentally, while the atomic systems are in general well-characterized, molecular species have been difficult to generate and Spectra, once obtained, must lean heavily on theory for in- terpretation. Thus, accurate experimental values aid theory in calibrating the methods which are employed, while reliable theoretical values aid experiment in inter- pretating data from the more complex molecular systems. A general development of the methods used in calculation of these quantities is in Part A of this thesis. Part B is concerned specifically with the calculation of the electron affinities of carbon, oxygen and fluorine; 1 while Part C is concerned with the calculation of the 52dn n+ n+2 . . . . . l or d electronic exCitation energies of scandium to sd to COpper. In both of the atomic systems above the Hartree- Fock wavefunction inadequauay represents the states with n+1 the 'extra' electron, i.e., the 2322p anion, or the n+1 and 3dn+2 483d states of the metal atoms. Previously, the electron affinities and the excitation energies have only been recovered with high-order configuration inter- action (CI) calculations. These types of calculations are inappropriate for molecular systems and can, in fact, obscure important physical information regarding the dif- ferential correlation effects. We have found that these states are more prOperly represented when the 'extra' electron is allowed to be in a radially inequivalent or- bital (szpnp' for the anions, sdnd' or dnd'd" for the metal atoms), which is more diffuse in nature than the other 2p or 3d orbitals. A more balanced zero-order de- scription of the ground/excited state, or atom/anion systems can be obtained from a simple multi-configuration (MCSCF) wavefunction which includes two valence correlation effects: 1. The differential radial correlation effect due to the loosely-bound electron in the excited states and anions, 3 2. The (sz,p2) near degeneracy effect due to the near degeneracy of the ns and un- occupied np orbitals. This MCSCF wavefunction removes many of the inconsistencies found at the Hartree level resulting in more accurate energy differences. CI calculations based upon the MCSCF wavefunction give electron affinities and excitation energies which are now in good agreement with experiment and provide a simple and consistent method for treating correlation effects in molecular systems containing nega- tive ions or transition metal atoms. Part A: METHODS I. INTRODUCTION The wavefunctions and energies which are used to characterize the electronic prOperties of atoms and mole- cules result from solution of the time-independent Schrodinger equation within the Born-Oppenheimer approx- imation. The nonrelativistic electronic Hamiltonian, in 1 atomic units , has the form N n z 2 A 1 2 AB H = Zn. + .2 r.. + A (4) While the choice of we is arbitrary, it is best chosen to approximate.as closely as possible, the exact wavefunction. For a given functional form, the 'best' wavefunction is one for which the energy is a minimum, i.e., for which the first order change in the energy with respect to any vari- ation in the wavefunction is zero 5E(wo)=0. Variations in the wavefunction can be introduced by means of the second fundamental principle, the expan- sion theorem. This theorem states that any normalizable function may be represented as an expansion in terms of a complete set of functions4. Thus, the single particle orbitals above may be represented as an expansion in a- known basis (pi =21; aiu X11 (5) and the n-particle wavefunction may in turn be expanded in terms of the orbital basisS W(l n) j J J (6) where each configuration, ¢j, is a symmetry-adapted linear combination of Slater determinants possessing the spatial and Spin symmetry of the electronic state of interest. Variations in the wavefunction translate into variations in the expansion coefficients {aiu}’ {Cj}, which are determined to minimize the energy, the variational me- thod. A wavefunction is, thus, defined by the configura- tions included in the n-particle expansion and the eXpan- sion bases from which the configurations and orbitals are constructed. II . HARTREE-FOCK In the Hartree-Fock (HF) procedure, the simplest form of wavefunction is assumed by truncating the configura- tion expansion at a single term wHFtl.....n) = ¢HF (7) thus, as in Equation (3a), the HF wavefunction is a simple antisymmetrized product of orbitals. Minimization of the total energy with reSpect to variations in the orbitals results in the HF equations for each orbital. For a closed-shell HF wavefunction with n/2 doubly-occupied or- bitals these equations are of the form ‘1’ 34¢ 13(1) ¢lB(2)oo. n8 (n) (8a) 7 HF _ h oi - Si oi (8b) hHF=h+ 2 (2J. -K.) (8c) 3 J J The Jj and K3. are the familiar Coulomb and exchange Operators l J. = < . ——— . 8d 3 ¢Jlrlzl¢3> ( ) - .2. Kj ¢i - <¢jlr12l¢i>¢j (8e) Since these operators depend upon all of the occupied or- bitals, the equations must be solved self-consistently. Expanding each orbital in terms of a known-set of basis functions7'8 results in the matrix eigenvalue equation . HI? I Z: < Xulh IXV> ai\) = Si;ai\) (93) v HP iii O.‘= 6133i. (9b) where the energy is determined from the secular equation HF Ill-z -ei$|=o (10) If {xu} is complete, this is the exact HF solution. Since it is not possible to use a complete basis, only an ap- proximate solution is usually obtained, referred to as the SCF solution. Since most of the electron density of a chemical system is concentrated around the nuclei, it is reasonable that the expansion basis, or basis set, be comprised of functions which are centered on each atom and which de- crease exponentially from their centers, linear combination of atomic orbitals (LCAO). The two types of functions most commonly used are Gaussian and Slater functions which are of the form 2 xi Ym Zn e Er (11a) rn-l e-gr ng‘ei‘i’) (11b) The range of radial and angular functions required to re- present different chemical processes has been thoroughly investigated and is summarized in Reference 9. For the two systems of concern here, negative ions and transition metal atoms, it was necessary to augment the atomic bases with diffuse functions to describe ionic and excited—state behavior. This is discussed in more detail in later chapters. The physical significance of the HF wavefunction is best understood by examining the potential terms in Equation 8b. Note that an electron in orbital i is not exposed to the full l/rij potential due to the other electrons, rather it sees a potential of 2(2Jj "Kj )1 J averaged over all of the electrons in the system. The na- ture of the HF potential has been discussed by Sinanoglulo where he compared the full Coulomb potential to that of the HF. He showed that the HF potential accounts for the long- range effects of the Coulomb repulsions between electrons. The difference between the two potentials, the interactions neglected in the HF picture, are short range in nature falling off rapidly as the distance between the electrons increases. This implies that there are two effects which determine the nature of the interactions between electrons in atoms and molecules 10 - The average interactions of each electron with the (n-l) others and, - The instantaneous interactions as pairs of electrons closely approach one another. The HP wavefunction, by describing the average interactions among the electrons, accounts for mainly the long-range part of the Coulomb repulsions. Thus, the single-particle picture where each electron moves in an individual orbital, is maintained by the HF method, in the sense that the orbitals are now determined in the average field of the (n-l) other electrons as well as the attractive field of the nuclei. While this neglects the short-range instant- aneous interactions, those which are dependent upon the in- dividual motions of the different particles, it still provides a well-defined reference point for more sephisti- cated approaches. The term 'correlation energy' is defined in terms of the HF model as the difference between the exact nonrelativistic energy of a system and the HF energyll Ecorr I Eexact EHF (12) referring to the interactions neglected in the HF model. This is an appropriate term provided that the HF wave- function describes the major physical features of the system. For closed-shell species at equilibrium this is the case but, in general, a single configuration descrip- tion is too restrictive. For example, orbital degeneracy effects or molecular dissociation cannot be represented 11 11’12 and consequently, other con- by a single—configuration figurations are needed to give a prOper 'zero order' de— scription. Since this deficiency is, not strictly, a cor- relation effect in the dynamic sense of short-range in- stantaneous interactions, there are two types of correc- tions that need to be considered beyond the HF model: those which arise from imprOper representation because of the single-configuration nature of the HF wavefunction; and the 'true' instantaneous correlation effects arising from short-range Coulomb repulsions between electrons in the same Spatial region. Both types of interactions can be represented by wavefunctions that do not truncate the ex- pansion in the orbital basis at a single configuration. III. CONFIGURATION INTERACTION In the configuration interaction (CI) procedure the wavefunction is written as a linear combination of many 13 orbital configurations , that are, in general, taken to be orthonormal, Equation (6). Variation of the configura- tion coefficients to minimize the energy leads to the matrix eigenvalue equation (H-E’b)¢.=o (13) where the elements of H are defined between configurations (1H),). = <¢ilH|¢j> (14) and the eigenvalues are determined from the secularequation lll-l-EAI =0 (15) 12 If the expansion basis is complete this results, in prin— ciple, in the exact nonrelativistic energy of the system. This energy is independent of the orbital basis, although the convergence of the expansion, the number and types of configurations necessary to achieve a particular level of accuracy can be accelerated significantly if the orbitals reflect the general characteristics of the wavefunction. In practice, it is not possible to expand the wavefunction in terms of a complete set of functions and both the expansion basis and the configuration set must be truncated. The energy of this approximate wavefunction is no longer independent of the orbital basis. The SCF procedures are used to define a physically relevant set of occupied orbitals. In addition, an apprOpriate set of cor- relating orbitals are needed. There are two major require- ments which these 'virtual' orbitals should fulfill in order to adequately represent the correlation effects among the occupied orbitals. They should be concentrated in the same region of Space as the occupied SCF orbitals; and they should have additional nodal surfaces which can allow for effects such as in/out, left/right, or up/down correla- tion. Thus, the virtual basis is principally comprised of higher angular functions with radial extents similar to those of the occupied orbitals. These requirements are well understood and are discussed in more detail in Refer- ence l3. 13 The extent of the configuration expansion is limited by the finite size of the orbital basis. A full CI is one in which all of the possible configurations constructed from a finite basis are used in the expansion. The energy of this wavefunction is dependent only upon the space Spanned by the orbital basis not upon the individual orbitals, and is the best energy which can be obtained within the given basis. In general, full CI's are not possible for all but the smallest orbital sets and the configuration list must be further reduced. Configurations can be classified ac— cording to the number of replacements, or excitations, which occur relative to a given set of reference configura- tions, single, double, triple, quadruple, etc. Within each level the resulting energy is independent of the in- dividual virtual orbitals, again dependent only upon the Space which they Span. If the zero-order wavefunction is a good representation of the system, the dominant cor- relation effects enter the CI expansion in terms of double excitation configurationslo'IB-ls. Thus, most CI calculations consist of single and double excitation con— figurations relative to the HF configuration or a more general zero-order wavefunction. The importance of the higher order triple and quadruple excitations is a topic 16,17 of current interest and is Specifically addressed in this thesis for the two systems noted previously. 14 IV. MULTI-CONFIGURATION SCF In the multi-configuration self-consistent field (MCSCF) procedure, as in the CI procedure, the n-particle wavefunction is constructed from a linear combination of configurations (Equation 6), but now both the orbitals, {oi}, and the CI coefficients, {Ci}, are optimized simulta- 18'19. Thus, the matrix eigenvalue equations for flu: neously orbitals (Equation 9b), and for the CI coefficients (Equa- tion 13), must be solved, though the form of the orbital equations are not as simple as the closed-shell HF equations, requiring solution of more than one pseudo—eigenvalue equa- tion. In this way, the correlating orbitals of the CI expansion discussed previously are well-defined, increas- ing the occupied SCF space to include a set of active orbitals having variable occupancy to describe correlation effects among the valence orbitals, in addition to the core orbitals which remain doubly-occupied throughout the cal- Culation. The generalized valence-bond (GVB) wavefunction is an example of such a wavefunCtion which accounts essen— tially for proper molecular dissociation and orbital degeneracy effectszo. While this procedure could, in principle, be used to define the entire correlating space, in practice this has not been found to be beneficiall7. Large MCSCF wavefunctions suffer from problems with con- vergence and interpretability, but perhaps more importantly, not all of the correlating orbitals need be defined 15 self—consistently. The orbital set can be divided into two groups, the primary and secondary orbital setsZI. The en- ergy of the chemical system is critically-dependent upon the orbitals of the primary set since it is this set that represents the major physical features of the system. The energy is only weakly-dependent upon the orbitals of the secondary set, that are necessary for providing minor cor— rections to the wavefunction. V. GENERAL PROCEDURE The general procedure used in the following cal- culations has been to determine a set of occupied orbitals self-consistently, that define the zero-order wavefunction. For many systems, the single-configuration HF wavefunction is apprOpriate, for other systems, more than one configura- tion is needed. These orbitals along with a proper set of correlating virtual orbitals are then used in CI calcula— tions to account for other correlation effects. We have investigated alternative forms of zero— order wavefunctions for negative ions and transition metal excited states since it is found that HF inadequately re- presents these systems. Consequently, the correlation ef— fects in these systems have been particularly confusing when cast in the HF framework. When cast instead in terms of an MCSCF framework these effects become very consistent and lend insight into the physical nature of these systems. 16 VI. MOLECULAR CODES The integrals were calculated using the BIGGMOLI integral program of R. C. Raffenetti22 as well as his in- tegral transformation programs. The MCSCF calculations were done using the ALIS MCSCF program from Ames Laboratory, Iowa State, by K. Ruedenberg, S. T. Elbert and coworkersle. The CI calculations were carried out using the CITWO pro- gram from the California Institute of Technoloqy by F. W. Bobrowic223. LI ST OF REFERENCES 17 LIST OF REFERENCES 1The atomic unit of length is the Bohr (1a0=0.52918A). The atomic unit of energy is the hartree (lh-27.121eV=627.51Kca1/m). 2See, for example: L. Pauling and E. B. Wilson, "Introduction to Quantum Mechanics"(McGraw-Hill, New York, 1935), chapter 7. 3An upper bound to a higher eigenvalue can be obtained when it is possible to insure that the approximate wave- function is orthogonal to all of the exact eigenfunctions of H corresponding to lower eigenvalues. 4For a general discussion, see for example: R. McWeeney and B. T. Sutcliff, "Methods of Molecular Quantum Mechanics" (Academic, New York, 1969), pp. 21-25. Sp. 0. L6wdin, Rev. Mod. Phys. 32, 328 (1960). 6See, for example: F. L. Pilar, "Elementary Quantum Chemistry" (McGraw-Hill, New York, 1968) chapter 13. 7c. c. J. Roothaan, Rev. Mod. Phys. 23, 69 (1951). 8C. C. J. Roothaan, Rev. Mod. Phys. 32, 179 (1960). 9T. H. Dunning, Jr., and P. J. Hay, in "Methods of Electronic Structure Theory", edited by H. F. Schaefer, III, (Plenum, New York, 1977), pp. 1-27 100.Sinanoglu, J. Chem. Phys. 36, 706 (1962). llp.<0. Lawdin, Adv. Chem. Phys. 2, 207 (1959). 122:. Clementi and A. Veillard, J. Chem. Phys. 44, 3050 (1966). 18 131. Shavitt, in "Methods of Electronic Structure Theory" edited by H. F. Schaefer, III (Plenum, New York, 1977), pp. 189-275. 14R. K. Nesbet, Adv. Chem. Phys. 14, l (1969). 153. P. Kelly, Adv. Chem. Phys. 14, 129 (1969). 16R. Krishnan, M. J. Frisch and J. A. Pople, J. Chem. Phys. 72, 4244 (1980). 17W. D. Laidig, P. Saxe and H. F. Schaefer, J. Chem. Phys. 73, 1765 (1980). 18K. Ruedenberg, L. M. Cheung, S. T. Elbert, Int. J. Quantum Chem. 16, 1069 (1979). 19A. C. Wahl and G. Das, in ”Methods of Electronic Struc- ture Theory" edited by H. F. Schaefer, III (Plenum, New York, 1977), pp. 51-78. 20W. J. Hunt, P. J. Hay and W. A. Goddard, III, J. Chem. Phys. 57, 738 (1972). 21T. H. Dunning, Jr., "Determination of orbitals for use in configuration interaction calculations", in Post Hartree-Fock: Configuration Interaction, NRCC Report, Lawrence Berkeley Laboratory, 1978. 22R. c. Raffenetti, J. Chem. Phys. 58, 4452 (1973). 23F. W. Bobrowicz, ”Investigations of Spin-Eigenfunction Correlated Wavefunctions", Ph.D. thesis, California Institute of Technology, 1974. Part B: NEGATIVE IONS Theoretical characterization of negative ions. Calculation of the electron affinities of carbon, oxygen, and fluorine. 19 I . INTRODUCTION The electron affinities of atoms have been notor- iously difficult to calculate from first principles. While experimentally most atoms are found to have a bound negative ion, Hartree-Fock (HF) theory is unable to predict the sta- bility of all but a few of these.1 For 2 < 10, HF finds only C- (43) and F-(IS) to be bound, the errors in the cal- culated electron affinities being 0.7eV for carbon and over 2eV for fluorine (Table I). By including electron cor- relation beyond HF, in particular through the configuration interaction (CI) method, the stability of anions can be cor- rectly predicted, but it has not been clear what level is necessary to obtain this result. Extensive CI calculations have been reported by Yoshimine and Sasaki2 (Y&S) of the correlation energies of the first-row atoms and their an- ions. Despite the use of large basis sets and the inclu- sion of up to quadruple excitations relative to the HF wave- function the calculated EA's are still in error by as much as 0.3-O.4eV. Since the error in the HF electron affinity, the neglected differential correlation energy, varies con- siderably from atom to atom, HF does not provide an appro- priate zero-order description of negative ions. Yoshimine and Sasaki's results indicate that CI's based upon the HF configuration are poorly convergent, requiring many confi- gurations for very little improvement in the overall EA. Their approach would be impractical for molecular anions in 20 TABLE I. Electron affinities and ionization energies of the first-row atoms. All quantities are in ev. Electron Affinity ionization Energy aExperimental bNumerical Hartree—Fock cExperimental Hydrogen 0.754 -0.33 13.60 Helium <0 - 24.58 Lithium 0.620(7) -0.122 5.39 Beryllium <0 - 9.32 Boron o.27au:.010)d -0.268 8.30 Carbon 1.268(5) 0.549 11.26 Nitrogen -0.07(8) -2.150 14.54 Oxygen 1.462(3) -0.541 13.61 Fluorine 3.399(3) 1.363 17.42 6These are the 'recommended' values taken from reference 1. bFrom reference 17. CR. 8. Leighton, ”Principles of Modern Physics“, (McGraw Hill, 1959), pp. 727-729. dFrom reference 10. 21 which much less extensive treatments are unavoidable, yet electron correlation must be included before accurate re- sults can be obtained from the calculations. Clearly, a more general ab initio procedure needs to be develOped which results in a more consistent descrip- tion of negative ions, though not necessarily giving the exact electron affinity. We have approached this problem by redefining the zero-order wavefunction using a.multi— configuration self-consistent field (MCSCF) function, which provides a more balanced desoription of both the neutral and the anion and can be used conveniently in singles and doubles CI calculations (MCSCF+1+2). This approach yields consistent errors at the MCSCF level, provides a convenient method for introducing triple and quadruple excitations (relative to the HF function) into the CI wavefunction, and EA's comparable to Y&S using a much smaller basis set and CI expansion. A. Experimental Background There are many excellent reviews of the empirical, semi-empirical and theoretical procedures which have been 1'3-7 Work rele- used to determine the electron affinities. vant to ours is presented below, the reader being referred to the above reviews for more thorough discussions. Experimental determination of electron affinities is a difficult task so that until recently the EA's of many elements could only be determined by extrapolation 22 procedures based upon trends in isoelectronic series (horizontal analysisa). For those systems which could be studied eXperimentally, there are two standard techniques which have been employed for the direct determination of electron affinities.1'3 The first is photodetachment threshold spectroscOpy where the long wavelength threshold for detachment of the electron is measured, giving the EA of the anion. The second is negative ion photoelectron Spectroscopy where a fixed frequency of light, larger than the EA of the neutral, is used to detach the electron, whose kinetic energy is then measured. The EA is determined with reSpect to a known reference based upon the conservation of energy. While each technique has its own inherent limita- tions (which are very thoroughly discussed in reference 3) they Share two technological problems, the availability of a suitable photon source and a suitable ion source. Ad-I vances in laser technology have helped to reduce some of the limitations of the photon source, but lack of ion sources has slowed progress in the measurement of EA's. Recently, Sputter-type ion sources have been used to gene- rate beams of atomic anions for many transition metal 10 whose EA's were then atoms , 9 and some main group elements , determined using photoelectron Spectroscopy. While the range of elements able to be studied experimentally has broadened, molecular EA's are still largely undetermined.4 Yet accurate EA's can be critical for determining the pr0perties of many neutral species; for example, the 23 controversy over the singlet-triplet separation of methylene, as measured from the photoelectron spectra, hinges on a know- 11-14 ledge of the electron affinity of the 3B1 state. B. Theoretical Background Theoretical determination of electron affinities is also a difficult task because a balance must be struck between the description of a neutral Species and its nega- tive ion, a system having one more electron. The electron affinity is found by subtracting the calculated ground state energy of the anion from the neutral, a positive value in- dicating that the anion is stable. Because of the extra correlation energy associated with the (z+l)st electron in the negative ion, calculations will favor the neutral, re- sulting in EA's which are in general too low relative to experiment15 or even negative as is found from most HF cal- culations. If the total nonrelativistic energy is expressed as the sum of the HF energy plus the correlation energy,16 the EA is Simply the difference between these quantities: EA = AEHF + AEcorr (1) For atoms the HF differences can be calculated very accu- rately using the near HF-limit energiesl7 obtained from 18 numerical procedures and therefore, the problem of 24 calculating electron affinities has been seen as one of obtaining the differential correlation energy between the atom and anion. One approach to the calculation of electron af- finities has been the pair-correlation schemes of 19,20 21,22 Nesbet or Weiss. In these, the total energy is written as: E=EHF+Zei+Zeij (2) EHF being the total HF energy, 8i the single-particle cor- relations, and eij' the correlation energy of the ij-pair. It is assumed that the 5's can be calculated independently by separate CI calculations for each ij-pair and summed to obtain the total correlation energy. These methods were compared by Y&S2 to the more complete CI+1+2 calculations (Table II), where they were able to Show that although the EA's obtained are often in excellent agreement with ex- periment, this is due to cancellations between truncation of the basis set and neglect of higher-order terms, and at the limit of a complete basis set, the pair methods over- estimate electron affinities, being 0.2eV too high for both oxygen and fluorine. Schaefer, et al?3(l969) approximated the EA's of the first-row atoms based upon a first-order CI wave- function (FOCI) which includes the internal (near- degeneracy), and the semi-internal (near-degeneracy and 255 TABLE II. A comparison of calculated electron affinides of hydrogen, lithium and boron through fluorine. All quantities are in eV. H Li 8 c N o r 88 : aNumerical -0.33 -0.12 -0.27 0.55 -2.15 -0.54 1.36 ova : bGoddard 0.38 0.28 rocr : CSchaefer -0.61 0.11 -2.45 -1.12 0.53 CI(pairs): dMoser 1 and 2 particle terms 0.39 1.46 0.19 2.06 4.18 + 3 particle terms 0.22 1.29 -0.12 1.43 3.37 °w61ss 1.47 3.47 fYoshimine and Sasaki 1.71 3.62 CI : gYoshimine and Sasaki xr+1+2 0.15 1.11 -o.s7 1.04 3.00 sr+1+2+3+4 0.17 1.13 -0.46 1.17 3.15 hExperiment 0.75 0.62 0.28 1.27 -0.07 1.46 3.40 l‘From Reference 17. bFrom reference 24. cFrom reference 23. a Prom references 19 and 20. O I From references 21 and 22. From reference 2. 9Energies obtained from the frozen K-shell values, reference 2. h'Remausnended‘ values taken from reference 1. 26 polarization) effects. The effects represented by the FOCI wavefunction were found to favor the neutral atom, and rather than improving the calculated electron affinity, re- sulted in poorer agreement with experiment. As mentioned previously, Yoshimine and Sasaki2 (1974).have published the most extensive CI calculations to date on the first-row atoms and their negative ions. Their intention was to eliminate any basis set error and obtain the exact correlation energy of each by using a very large Slater basis which, upon reduction by an approximate natur- al orbital analysis, consisted of an (8s,7p,6d,5f,4g,3h,21) orbital set. Singles and doubles CI calculations were car— ried out based on the HF reference configuration (HF+1+2), and then the importance of higher-order terms was examined by including selected triple and quadruple excitations (HF+l+2+3+4). Their results are also summarized in Table II where only the L-shell correlation energies have been used Since it was shown by their calculations that K—shell and KL-intershell correlations contribute less than lO-ZeV to the EA, and, these are the energies most comparable to our ls(HF) frozen core CI calculations. Note that the singles and doubles CI wavefunction does correctly predict B- and 0- to be bound whereas the FOCI23 does not. Although they were able to calculate approximately 95% of the estimated correlation energy of the atom and anion, only approximate- ly 85% of the EA's were obtained. The HF+1+2 EA's are in error by 0.l-O.4eV. Triple and quadruple excitations 26:3 improve the energies by only m0.02eV for boron and carbon, and m0.lSeV for nitroqen, oxygen and fluorine which reduces the error in the calculated EA's to 0.1-0.3eV. While all of the previously mentioned calculations have examined the overall correlation energies in the atom and anion, none have addressed some key questions which can lend insight into the description of negative ions: --What are the minimum correlation effects requir- ed to account for the stability of the anion? --How is the electronic structure of the ion dif- ferent from that of its isoelectronic neutral? --What other types of correlation effects are differential between atom and anion and what is the minimum level required to describe each (particularly with reSpect to the higher an- gular momentum functions used in all of the previous calculations)? The first and second of these questions were ad- dressed by Goddard24 (1968) in his paper on the stability of the negative ions of hydrogen and lithhm.and will be discussed in detail below. We have attempted to address the last question through MCSCF and CI calculations which isolate various differential contributions to the EA's of carbon, oxygen and fluorine. 27 II. ORBITAL MODELS FOR NEGATIVE IONS A negative ion is characterized by a very diffuse charge distribution relative to the neutral atom. If the energy to remove the 'last' electron in an atom, which is on the order of 10eV, is compared to that of the anion, which is on the order of leV, it is clear that the (z+l)st elec- tron is much more loosely bound than the other z electrons (Table I). The failure of the HF wavefunction to predict the stability of negative ions can be related to its inabil- ity to allow for this diffuse nature of the orbital for the (Z+l)st electron. Using Goddard's example,24 this point is illustrated by contrasting three different zero-order repre- sentations of the l5(132) state of H-: the restricted Hartree-Fock (RHF) wavefunction; the unrestricted HF (UHF) wavefunction; and the generalized valence bond (GVB) wave- function;25 in terms of the physical model represented by each, and their ability to predict the stability of the anion. The RHF wavefunction is of the form: = 2 dlsHF 018 ‘1’th (3) Physically, this wavefunction represents two electrons Sing- let-coupled in the same Spatial lsHF orbital. The UHF wave- function is given by: ‘PUHF =dllsalss “8 (4) 28 This wavefunction relaxes the restriction that the electrons be in the same Spatial orbital, but at the ex- pense of the Spin symmetry,\pUHF, being a mixture of the singlet and triplet couplings of the electrons. An alternate way to relax the Spatial restric- tions on the Is electrons is to use the GVB wavefunction for H- which allows the two electrons to be in nonequiva- lent ls orbitals while maintaining the prOper spin sym— metry: qJGVB =J(lsls' + ls'ls)aB (5) The overlap between ls and 15' is nonzero. Equation 5 can be expressed as an equivalent two configuration MCSCF wavefunction in terms of an orthOgonal basis26 which, computationally, is more convenient to use: 2 2 )pMC “((CllsMC - c228MC )08 (6) = 0 2 2_ c1 + c2 - l 29 The relationship between the nonorthogonal GVB orbitals and the orthogonal MCSCF orbitals is given by: ls = (2.)"5 fine)” lsMC + (l-s)Li 25140] (7a, 15' = (2)45 [(l+S);5 lsMC - (1-S);’ ZSMC] ‘ (7b) c -c 1 2 S = = cl+c2 All of these wavefunctions maintain the single-particle in- terpretation of an electron moving in an average potential due to the other electron and the nucleus. The energies and EA's calculated with each of these wavefunctions (Equations 3,4,6) are summarized in Table III. The (58/38) hydrOgen basis of Huzinaga27 was used for these calculations but, in order to obtain a negative orbital energy for H-, it was neceSsary to augment this set with two diffuse functions (determined by an even- tempered expansion), to give a (7s/Ss) basis set. The RHF wavefunction, which places two electrons in equivalent ls orbitals, results in the energy of the HF anion being 0.33eV above the energy of the hydrogen atom, predicting the anion to be unstable relative to the atom. The UHF wavefunction, not being restricted to keeping the electrons in the same Spatial orbital, places the extra electron at infinity, also predicting an unbound negative 3(3 TABLE III. A comparison of the RHF, UHF and GVB energies for H, H-, and He. Units are as indicated. an 2 a - I , b l g S) H g S) SHE EA He( 8) AHF hartree hartree ev ev hartree eV RHF -.4998 -0.4877 0.0 -0.33 -2.8612 0.0 UHF -0.4998c - 0.0 - GVB —0.5250 1.01 0.37 -2.8774 0.44 Experimental 0.75 aCalculated using an augmented (Se/3s) r (7s/Ss) basis. See text. bCalculated using the (63/43) cSet to atomic value. basis set from reference 29. See reference 24. 31 ion. The GVB wavefunction, on the other hand, correctly predicts the electron to be bound with an EA of 0.37eV. Thus, by simply allowing the electrons to be in radially nonequivalent ls orbitals while maintaining the singlet Spin symmetry, a prOper zero-order physical description of H- is obtained. Goddard showed this to be true for the 15(282) state of L1’ as well.24 Radial plots of the ls and the ls and 13' GVB HF' orbitals help to illustrate the difference between the RHF and GVB models. The RHF orbitals for H and H- are pre- sented in Figure la. Since the ls electrons partially shield one another from the nuclear charge, forcing both electrons to be in the same orbital results in a decrease in the attraction felt by each, and the anion orbital (dashed line) expands relative to the atom (solid). A1- ternatively, the 2nd electron could be placed in a more diffuse orbital which does not shield the other electron as effectively (GVB), and thus does not decrease the net at- traction as significantly.28 These ls and 13' orbitals (dashed lines) are presented in Figure lb along with the atomic ls orbital (solid). The diffuse nature of the IS' orbital is very apparent, the overlap with the tight ls, being only 0.57. Note that the tight ls orbital has es- sentially the same radial form as the atomic orbital. For comparison Figure 1c shows the lsH and the FI ls and 13' GVB orbitals for the l8(152) state of helium using the same two configuration wavefunction as for H- Figure Figure Figure Figure la. 1b. 1c. 32 A comparison of the RHF and GVB radial am- plitudes for the ls orbitals of H, H- and He. Distance is in bohr; plots are all on the same scale. H (solid) and 8‘ (dashed) RHF orbitals. = 1.50 and 2.51ao for s and H“, reSpective- ly. H RHF (solid) and H’ GVB (dashed) orbitals. = 1.50, 1.44 and 5.07ao for ls 1 ls GVB, reSpectlvely. RHF' sGVB He RHF (solid) and GVB (dashed) orbitals. = 0.93, 0.69 and 1.22aO for lSRHF.15GVB ls GVB' respectively. AMPLITUDE AMPLITUDE l 0%.0 LG 2.0 DISTANCE (0,) v, I 2.0 ms”: H’Is GVB '-° H'Is 33 AMPLITUDE .1. 0%.0 I0 2.0 DISTANCE (00) Figure l. 2.0 3.0 4.0 DISTANCE (6,) 34 (Equation 6). Note now, the higher overlap between ls and 15', S=0.88, and the small deviation from the HF orbital. Further note that the (lsZ->252) radial correlation energy is much smaller for He, 0.44eV, than for H-, l.OleV (Table III). The helium orbitals, with their high overlap, are typical of radial correlation within a singlet-coupled electron pair, indicating that the electrons are bound nearly equivalently. The H- orbitals, on the other hand, vary much more from one another and the HF orbital, indicat- ing that the electrons are not bound equivalently. Rather, one electron is bound as in the atom while the second is only loosely bound, occupying a much more diffuse orbital. This suggestsa general physical model for negative ions where an anion can be thought of as having 2 electrons bound as in the neutral atom with the (Z+l)st electron oc- cupying a more diffuse orbital. A negative ion differs from its isoelectronic neutral because in the neutral all (Z+l) electrons are bound nearly equivalently. Thus F- is physi- cally very different than Ne, C- very different than N. We have applied this model of a loosely-bound electron in the negative ion to developing a balanced zero- order MCSCF wavefunction for the atom and anion of carbon, oxygen, and fluorine. When viewed in terms of this model, many of the apparent inconsistencies in the HF representa— tion are clarified, revealing a very consistent, physical- ly reasonable description of the negative ion. 35 III. CALCULATIONAL DETAILS The basis set used in the MCSCF calculations was the (lls,6p) Gaussian basis of Duijneveldt29 augmented with an additional p function determined by an even-tempered ex- pansion of the last two functions. This was contracted to [4S,4p] using the general contraction scheme of Raffenettin), based upon the HF orbitals of the ground state of the atom. This contraction resulted in HF EA's which are about 0.01eV less than the numerical results.17 The configurations used in the MCSCF calculations were chosen to describe the differential p2 pair energies between the atom and anion due to the diffuse nature of the (Z+l)st electron in the anion. This requires configura- tions similar to those used in H- and Li-; double excita- tions from 2p2 into 3p2. This type of radial correlation describes not only the tight/diffuse nature of the (2+1) electrons in an anion, as demonstrated previously for H-, but also simple in/out type correlation arising from pair repulsions, as seen in He. It must, therefore, be included in the zero-order wavefunction of both the atom and the anion, although as will be seen, the effect is much more dramatic in the anion. Thus the wavefunctions used in the following MCSCF calculations are: Atom: «I! c 2322pn - C. 1 ZS22ph-23p2] (8a) 2 Anion: «11 [c12522pn+1 - C 2322pn-13p‘] 2 (8b) 36 In addition, for the 3p(2$22p2) state of carbon, the (252->2p2) near-degeneracy effect,31 arising because of the unoccupied 2p orbital, was included in the zero-order wave- function by adding the configuration: c32p4a8ao (8c) For the CI calculations the [4S,4p] basis was augmented with 3 sets of d-funCtions. The first two functions were taken from a two-term Gaussian expansion of a Slater 3d function.32 These exponents were then scaled together to minimize the energy for the neutral in terms of an MCSCF+1+2 wavefunc- tion. Then a more diffuse exponent was added and optimized for oxygen and fluorine to describe the anion. This was done by allowing all single and double excitations out of one of the doubly occupied 2p orbitals, based upon the. MCSCF orbital set of the anion. Since for carbon there is not an equivalent procedure to follow, the third exponent was chosen in the same ratio as Optimized for fluorine. These exponents are given in Table IV. An f-function was also added to the fluorine basis and optimized for both the atom and the anion in HF+1+2 and MCSCF+1+2 calculations. The exponent was found not to be a strong function of the state or orbital set and, there- fore, the optimum values for 2P and 1S were averaged (cf 3 1.9167). For the most part the CI wavefunctions consisted of all single and double excitations from either the HF 37 TABLE IV. Optimized 3d and 4f Exponents for carbon. oxygen and fluorine Carbon Oxygen Fluorine £36 1.0886 2.2887 3.0238 0.3223 0.6777 0.8954 0.0954 0.2651 0.2651 1 . 9167 38 reference configuration (HF+l+2) or from the MCSCF reference configurations (MCSCF+1+2). These calculations were parti- tioned to include first the (Sp) Space, then the effect of adding the d and f-functions was examined. For all three systems full-CI's within just the (p) Space were performed to test the adequacy of the (2p2->3p2) MCSCF description, and for carbon and oxygen selected full-CI calculations are also reported. The lS(HF) orbital was kept frozen in all of the calculations, and symmetric orbital sets were used. IV. CALCULATION OF THE EA'S OF CARBON, OXYGEN AND FLUORINE A. Carbon The results for carbon are summarized in Table V. Carbon is somewhat anomalous because of the near-degeneracy 31 effect found in the 3P(235p2) state of the atom but not in the negative ion. Since this effect is ignored at the HF level, the energy of the 3P state is artificially high,“ resulting in a larger EA than would be the case if HF treated both the atom and anion equivalently. If the (282->2p2) near-degeneracy effect is included in the 3p 33 this state wavefunction, the GVB description of carbon, is lowered by 0.47eV giving an EA of only 0.06eV. Allowing the 2p orbitals to radially correlate by including 2p2 to 3p2 excitations, lowers the 3p state by 0.12eV. This correlation lowers the 4S state by 0.49eV, larger than expected based on the effect in the 3p state. Full CI's within the (p) Space improve the MCSCF energy of 359 TABLE V. Hartree-Fock, MCSCF and 3CI calculations on the 3P state of the carbon atom and the ‘5 state of the carbon anion. Basis set: [4s,4p,3d), units are as indicated. 3PI232222) ahr 45(2s2293) our g5 artree eV hartree eV eV Hartree-rock ‘ -37.6880 0.0 -37.7076 0.0 0.33 ncscr (282+292) -37.7054 0.47 0.06 (2p2r3p2) -37.6924 0.12 -37.7256 0.49 bc1 (9) Full -37.6924 0.12 -37.7269 0.52 scscr (282*2p2)+(2p2*3p2) -37.709s 0.58 0.44 cc1 (sp) sr+1+2 -37.7293 1.12 ' -37.7588 1.39 0.80 ncscr+1+2 '37.7296 1.13 -37.7607 1.45 0.84 Full -37.7297 1.13 ~37.7611 1.46 0.85 dCI (Spd) HF+1+2 -37.7740 2.34 -37.8114 2.83 1.02 ncscr+1+2 -37.7747 2.36 -37.8144 2.91 1.08 Full -37.7755 2.38 a'eCI Yoshimine & Sasaki ar+1+2 1.11 ar+1+2+3+4 1.13 r . Experiment 1.27 ‘The ls orbital remained doubly-occupied in all calculations. bFull CI within the (p) space. cExcitations allowed only in the (sp) space. 4 .From reference 2. f'Recommended' value from reference 1. Excitations allowed in the (spd) space. 40 the 4S state only slightly, having no effect on the 3P state, demonstrating the adequacy of the (2p2->3p2) MCSCF descrip- tion for carbon. The result of incorporating both effects, near-degeneracy and radial 2p2 correlation in the 3P wave- function is a lowering Of 0.58eV, just the sum of the in- dividual effects, and results in an MCSCF EA of 0.44eV for carbon. The orbitals obtained from the HF and MCSCF cal- culations were used in CI+1+2 and full CI calculations within the (Sp) space. The total lowerings (relative to HF) obtained from the full-CI calculations are 1.13eV for the 3P state and 1.46eV for the 4S state, predicting an EA of 0.85ev.. The MCSCF+1+2 calculations lower both states by essentially the full-CI result, 1.13eV for C and 1.45eV for C-. HF+1+2 calculations within (sp) give a similar 3P- lowering but are 0.07eV higher for the 4S state, predicting the EA to be 0.80eV. Thus, about 57% of the EA of carbon is obtained by full correlation within the (sp) space. En- largement of the (Sp) basis would improve this to perhaps 65-70%, leaving about 0.4eV which can be ascribed to higher angular momentum terms. CI calculations within the (Spd) Space are also reported in Table V. A full-CI was only possible for the 3P state, lowering it by 2.38eV relative to HF. The MCSCF+1+2 and HF+1+2 lowerings for this state are Similar, being only 0.02 and 0.04eV higher than the full-CI result. The energy of the anion is affected more than the neutral 41 by the higher-order excitations in the MCSCF+1+2 calcula- tion, picking up 0.08eV over the HF+1+2 energy and giving an EA of 1.08eV. The HF+1+2 EA is 1.02 eV. While about 0.2eV of the EA is Still unaccounted for at this level, a com- parison to the CI calculations of Y&S2 show that our HF+1+2 EA is only 0.09eV less than theirs, thus providing an estimate of the basis set limitations within our calcula- tion. Since the MCSCF+1+2 wavefunction contains quadruple excitations relative to the HF configuration, the EA can be compared to the HF+1+2+3+4 result, being only 0.05eV less than this value. If the basis set error is on the order of at least 0.1eV, a more complete basis would yield an MCSCF+1+2 EA larger than obtained by Y&S and about 0.1eV smaller than experiment. B. Oxygen Calculations on the oxygen 3P(2322p4) neutral atom and the 2P(2322p5) negative ion are summarized in Table VI. HF finds the anion to be unbound (as was also 23). The (2p2->3p2) 3 the case for the FOCI description radial correlation effect lowers the P state by 0.9leV, and the anion by 1.92eV. Now the energy Of the anion lies below the atom giving the correct physical description at zero-order with an EA of 0.46eV. Full-CI's within the (p) space are more important in oxygen than in carbon resulting in an additional lower- ing Of 0.l7eV for the atom and 0.47eV for the anion over the 42 TABLE VI. Hartree-Fock, MCSCF and 8CI calculations on the 3P state of the oxygen atom and the 2? state of the oxygen anion. Basis set: [4s,4p,3d), units are as indicated. 3 2 4 2 S P(2s 22 ) as? 9(23222 ) 388 55 ar ree ‘EV artree ev ' eV Hartree-Foch -74.8077 0.0 -74.7875 0.0 -0.55 MCSCF (2p2-3p2) -74.8411 0.91 -74.8580 1.92 0.46 b CI (9) Full ~74.8474 1.08 -74.8754 2.39 0.76 cc1 (sp) HF+1+2 -74.8774 1.90 -74.9057 3.22 0.77 MCSCF+1+2 -71.8792 1.94 -74.9132 3.42 0.92 Full -74.8794 1.95 - dCI (spd) HF+1+2 -74.9S44 3.99 -74.9871 5.43 0.89 MCSCF+1+2 -74.9S7S 4.07 -74.9976 5.72 1.09 7'.CI Yoshimine s Sasaki HF+1+2 1.04 HP+1+2+3+4 1.17 {Experimentk 1.46 aThe 1s orbital remained doubly-occupied in all calculations. bFull CI within the (p) Space. cExcitations allowed only in the (sp) space. dExcitations allowed in the (Spd) Space. ‘trom reference 2. 1’Recommended' value from reference 1. 43 MCSCF (2p2->3p2) energies. This is because more than one effect must be described by the 2p and 3p MCSCF orbitals once the 2p orbitals become doubly occupied. For the neutral, correlation of the intra-p2 pair as well as the inter-p2 pairs must be represented by the two orbitals. For the anion, in addition to these, the effect of one diffuse electron correlating with four tighter electrons is also present. Allof these effects are averaged in the MCSCF procedure but are more fully represented by addi- tional p-functions, as demonstrated by the full-CI (p) energies.34 Full-CI calculations within the (Sp) space could only be done for the neutral because of the size of the configuration list, lowering the 3P state by 1.95eV rel- ative to HP. As was found for carbon, the MCSCF+1+2 and HF+1+2 correlation energies for this state are only slightly higher, 1.94 and 1.90eV respectively. The energy of the anion is lowered by 3.42eV for MCSCF+1+2, while it is 0.2eV smaller for HF+1+2. Thus, the effect of the triple and quadruple excitations upon the energy of the anion is becoming pronounced. The HF+1+2 EA is 0.77eV. The MCSCF+1+2 EA is 0.92eV, 63% of the experimental value. For the CI calculations within the entire (Spd) virtual Space, the higher-order excitations begin to show an effect on the energy of the neutral, HF+1+2 being 0.12eV above the MCSCF+1+2 energy. For the anion this 44 effect is even larger, 0.29eV. The HF+1+2 EA is 0.89eV, 0.15eV less than that of Y&S, again an indication of basis set limitations. The MCSCF+1+2 EA is 1.09eV, only 0.08eV less than that of Y&S, but still in error by 0.37eV with experiment. Correcting for limitations in the basis set, we estimate the MCSCF+1+2 limit for the EA of oxygen to be more than 1.24eV. C. Fluorine Calculations on the 2P(2$2p5) atom and 18(23226% anion of fluorine are summarized in Table VII. HF, while properly predicting the stability of F-, accounts for less than half of the experimental EA of 3.40eV. Radial (292 l by 1.16eV, resulting in an MCSCF EA of 2.51eV. Full-CI ->3p2) correlation differentially lowers the S anion calculations within the (p) Space Show the same trend as for oxygen reducing the energy of the anion more than that of the neutral, giving an EA of 2.78eV. CI's within the (sp) and (spd) Spaces also Show the trend that the higher- order terms included in the MCSCF+1+2 wavefunction are more important for the anion than the neutral; a differen- tial effect of 0.l8eV for (sp) and 0.25eV for (Spd). Within the (sp) Space the MCSCF+1+2 calculations account for 87% of the EA of fluorine. Within the (Spd) Space, the MCSCF+1+2 EA is 3.16eV, HF+1+2 being 2.9leV. Thus, in the [4s,4p,3d] frozen core basis the HF+1+2 EA is only 0.09eV less than that of Y&S, while the MCSCF+1+2 EA is 45 TABLE \II. Hartree-rock, MCSCF and aCI calculations on the 2p state of the fluorine atom and the 1S state of the fluorine anion. Basis set: [4s,4p,3d,1f], units are as indicated. 2P(2522252 AH? 15(232226) AHF §§ hartree eV hartree eV eV Hartree-Foch —99.4067 ' o.o -99.4563 0.0 1.35 e MCSCF (2p‘t392) ~99.4646 1.58 -99.5570 2.74 2.51 b CI (9) Full -99.472¢ 1.79 ~99.S746 3.22 2.78 cc: (5;) HF+1+2 -99.5043 2.66 -99.6068 4.09 2.79 HCSCF+1+2 -99.5070 2.73 ~99.6163 4.35 2.97 dCI (spd) HF+1+2 -99.5948 5.12 -99.7016 6.68 2.91 MCSCF+1+2 -99.5992 5.24 -99.7153 7.05 3.16 CI (spdf) HF+1+2 ~99.6159 5.69 -99.7229 7.25 2.91 MCSCF+1+2 -99.6202 5.81 -99.7369 7.64 3.18 "°c1 Yoshimine 5 Sasaki HF+1+2 ’3.00 HF+1+2+3+4 3.15 ¢ . ‘Experiment 3.40 3The ls orbital remained doubly-occupied in all calculations. bFull CI within the (p) space. cBxcitations allowed only in the (sp) space. dExcitations allowed in the (spd) space. eProm reference 2. f‘Recommended' value from reference 1. 46 actually 0.01eV better. Addition of an f-function shows no differential effect for the HF+1+2 EA, while the MCSCF+1+2 EA is only improved by 0.02eV. If the basis set error is about 0.1eV, the EA obtained with the MCSCF+1+2 wavefunc- tion should be about 0.1eV less than the experimental value. V. DISCUSSION The importance of the differential p2 correla- tion energy in determining the electron affinity of an atom is best illustrated by comparing isoelectronic sys- tems. These energies relative to HP, are given in Table VIII for the systems: C-, N(4S); 0-, F (2P); and F-, Ne(18), at various levels of calculation (oxygen 3P is also included for completeness). Comparing the anion and isoelectronic neutral, it is clear that correlation of the p-electrons is larger in the anion than the neutral, even though the p-orbitals contract as Z increases and so an Opposite trend might be expected. For carbon the 3P state shows a p2 repulsion energy of 0.12eV/pair. If the effect in the anion were the same, the lowering in the 48 state would be 0.36eV (0.12 x 3 pairs), while it is, in fact, 0.l3eV larger. This can be associated with the extra correlation energy due to the diffuse electron. Note though, that the effect for nitrogen atom is 0.34eV, as predicted from the pair energy for carbon. The im- portance of this differential radial correlation increases 447 a , . . . . TABLE VIII. MC‘CF and Cl correlation energies relative to Hartree-Fock for the isoelectronic 0 series: 45(2322p3) of C and N; 2P(2522p5) of O- and F; and 18(232p6) of F- and Ne. All units are in eV. ucscr Vbc1(p) cCI(sp) (2 2,3 2, Full HF+1+2 MCSCF+1+2 Full 43(2822p3) c' 0.49 0.52 1.39 1.45 1.46 d'eu 0.34 0.35 1.12 1.14 1.14 A 0.15 0.17 0.27 0.31 0.32 3P(2s22p4) o 0.91 1.08 1.90 1.94 1.95 2P(2522p5) 0' 1.92 2.39 3.22 3.42 F 1.58 1.79 2.66 2.73 3 0.34 0.60 0.56 0.69 15(2s22p6) 9‘ 2.74 3.22 4.09 4.35 d'fxe 2.33 2.55 3.50 3.61, A 0.41 0.67 0.59 0.74 The ls orbital remained doubly occupied in all CI calculations. Full CI within the p-space. Excitations allowed only in the (sp) space. Calculated using the (11s,7p/4s,4p) basis from reference 29. OD-OO'OI EHF--54.4000 hartree. f .- EH? 128.5431 hartree. 48 with the number of p electrons, where for F- the effect accounts for 0.4leV more correlation energy than found in neon. Table Ix summarizes the errors in the calculated EA'S at various levels for HF-based and MCSCF-based wave- functions. Since in every case the difference between atom and anion is the addition of one more electron, it is rea- sonable that the differential correlation energy be similar from atom to atom. This is consistent with the concept that the addition of an electron to a system should cost about leV in correlation energy. Thus, the error in the uncorrelated EA should be about leV. But for HF, as pre- viously mentioned, these errors range from little more than 0.5eV to over 2eV as the p-orbitals become doubly occupied. The FOCI wavefunction, by removing the degeneracy and po- larization effects but not the differential p2 correlation increases this error to 2.01:0.8SeV. The MCSCF description which includes the differential (2p2->3p2) radial correla- tion (and the near-degeneracy effect in the 39 state of carbon), results in EA's which do reflect this trend, being in error by 0.91:0.09ev, even though the p-orbitals are doubly occupied in oxygen and fluorine. Thus, the nonrelativistic electron affinity (Equation 1) may be more consistently represented by: EA = ABC + AEcorr (9) Carbon Oxygen Fluorine Mean Error Hartree-Peck 0.74 2.01 2.05 1.40 t 0.66 aSchaefer FOCI 1.16 2.58 2.87 2.01 1 0.85 HF+1+2 (sp) 0.47 0.69 0.61 0.58 t 0.11 (Ipd) 0.25 0.57 0.49 0.41 t 0.16 b (spdf) 0.49 YSS HF+1+2 0.16 0.42 0.40 0.29 i 0.13 cacscr 0.83 1.00 0.89 0.91 1 0.09 MCSCT+1+2 (sp) 0.43 0.54 0.43 0.49 t 0.05 (spd) 0.19 0.37 0.24 0.20 f 0.09 b (Spdf) 0.22 Y&S HF+1+2+3+4 0.14 0.29 0.25 0.21 t 0.08 419 TABLE IX. Errors in the calculated electron affinities of carbon, EA(expc)-EA(1), All units are in eV. oxygen and fluorine, aFrom reference 23. b cCalculated using equations 8a, b and c. Frozen K-shell values taken from reference 2. 50 where ABC is the difference between the MCSCF energies of the neutral and anion, and AEcorr is on the order of leV.. Comparing the CI calculations based upon the HF wavefunction to those based upon the (2p2->3p2) MCSCF wave- function also points up the consistency of the MCSCF ap- proach. While the CI+1+2 error within the (sp) Space is 0.5810.llev for the HF-based calculations, this is re- duced to 0.49:0.05eV for those based upon the MCSCF wave- function. CI's within the (Spd) space result in HF+1+2 errors of 0.4110.16ev, which is reduced to 0.2810.09eV if the MCSCF reference wavefunction is used.35 These results can be compared to the results of Y&S, 0.2910.l3ev for the HF+1+2 errors. Including higher-order excitations, HF+1+2+3+4, which are most comparable to our MCSCF+1+2 calculations, Y&S have an error of 0.21:0.09eV, only 0.07eV less than ours deSpite the far larger basis sets and configuration lists. Note that the importance of the higher-order terms for the description of the anion in- creases with the number of p—electrons. VI. CONCLUSIONS The zero-order MCSCF wavefunction which includes the differential radial correlation due to the diffuse nature of the (z+l)st electron in the negative ion, prOp— erly predicts the stability of H-, Li-, and 0-. This wave- function gives a consistent error of approximately leV in the calculated EA. At higher levels, CI+1+2 calculations 51 based upon these orbitals and configurations provide a con— sistent method for introducing triple and quadruple excita- tions (away from the HF configuration) into the wavefunc- tion. With only a [4s,4p,3d] basis we have calculated the HF+1+2 EA's of carbon, oxygen and fluorine to within 0.12eV of the large basis set calculations of Y&S.2 If the configuration list is based on the MCSCF generating configurations and orbital set, this difference is reduced to 0.07eV, where for fluorine the resulting EA is slightly better than obtained in the near basis set limit HF+1+2+3+4 calculations of Y&S. The MCSCF+1+2 EA'S differ from those determined experimentally by about 0.2eV. LIST OF REFERENCES 52 LIST OF REFERENCES lH. Hotop and W. C. Lineberger, J. Phys. Chem. Ref. Data 4, 539 (1975). 2p, Sasaki and M. Yoshimine, Phys. Rev. A 9, 17 (1974); F. Sasaki and M. Yoshimine, Phys. Rev. A 9, 26 (1974). 3R. R. Cordermen and W. C. Lineberger, Ann. Rev. Phys. Chem. 30, 347 (1979). 4B. K. Janousek and J. I. Brauman, in "Gas Phase Ion Chemistry", Vol. II, edited by Michael T. Bowers (Academic, New York, 1979), pp. 53-86. 5H. S. W. Massey, "Negative Ions" (Cambridge University, London, 1976), 3rd edition. 6H. S. W. Massey, in "Advances in Atomic and Molecular Physics", edited by Sir. David R. Bates (Academic, New York, 1979), pp. 1-36. 7C. F. Bunge and A. V. Bunge, Int. J. Quantum Chem. Symp. 12, 34s (1978). 8R. J. Zollweg, J. Chem. Phys. 50, 4251 (1969). 9C. S. Feigerle, R. R. Corderman, S. V. Bobashev, and W. C. Lineberger, J. Chem. Phys. 74, 1580 (1981). 10C. S Feigerle, R. R. Corderman, and W. C. Lineberger, J. Chem. Phys. 74, 1513 (1981). 11P. F. Zittel, G. B. Ellison, S. V. O'Neil, E. Herbst, W. C. Lineberger and W. P. Reinhardt, J. Am. Chem. Soc. 98, 3731 (1976). 12L. B. Harding and w. A. Goddard III, Chem. Phy. Lett 55, 217 (1978). 53 13S. K. Shih, S. D. Peyerimhoff, R. J. Buenker, and M. Peric, Chem. Phy. Lett. 55, 206 (1978). 14For a recent review of this problem see: W. T. Borden and E. R. Davidson, Ann. Rev. Phys. Chem. 30, 125 (1979). 15For transition metal atoms calculated EA'S may actually overestimate the experimental value due to neglect of differential relativistic effects. See reference 7. 16For relativistic discussion see reference 7. 1‘75. Fraga, "Handbook of Atomic Data", (Elsevier, Amsterdam, 1976). 18C. Froese-Fischer, "The Hartree-Fock Method for Atoms" (Wiley, New York, 1977). 19C. M. Moser and R. K. Nesbet, Phys. Rev. A 4, 1336 (1971). 20C. M. Moser and R. K. Nesbet, Phys. Rev. A 6, 1710 (1972). 21A. W. Weiss, Phys. Rev. A 3, 126 (1971). 22M. A. Marchetti, M. Krauss, and A. W. Weiss, Phys. Rev. A 5, 2387 (1972). 23H. F. Schaefer III, R. A. Klemm, and F. E. Harris, J. Chem. Phys. 51, 4643 (1969). 24w. A. Goddard III, Phys. Rev. 172, 7 (1968). 25This is referred to as the GF wavefunction in reference 24 26W. J. Hunt, P. J. Hay, W. A. Goddard III, J. Chem. Phys. 57, 738 (1972). 27s. Huzinaga, J. Chem. Phy. 42, 1293 (1965). 28This effect is very Similar to that found in the sdm'l states of the transition metal atoms, except that the 54 diffuse nature of the orbital occupied by the 'extra' electron is much more dramatic for the (z+l)st electron in an anion than for the (n+1)st electron in a metal atom; B. H. Botch, T. H. Dunning, Jr., and J. F. Harrison, J. Chem. Phys. in press. 29F. B. van Duijneveldt, IBM Technical Report No. RJ945 (1971). 30R. C. Raffenetti, J. Chem. Phys. 58, 4452 (1973). 31E. Clementi, A. Veillard, J. Chem. Phys. 44, 3050 (1966). 32m. H. Dunning, Jr., J. Chem. Phys. 55, 3958 (1971). 33W. A. Goddard III, T. H. Dunning, Jr., W. J. Hunt, and P. J. Hay, Acc. Chem. Res. 6, 368 (1973). 34Other forms of MCSCF wavefunctions were examined where, for example, (232p->353p) or (232->3sz) type configura- tions were included. The orbital characters were af- fected very little by these additions and CI' 5 based upon these orbitals gave essentially the same energies and EA' 3 as the (2p2->3p2 ) wavefunction. 35The results for carbon and fluorine are in slightly better agreement than for oxygen. This may be due to the stability of the 1/2- filled and filled 2p shells of C and F . 34Other forms of MCSCF wavefunctions were examined where, for example, (232p->333p) or (232->332) type configura- tions were included. The orbital characters were af— fected very little by these additions and CI's based upon these orbitals gave essentially the same energies and EA'S as the (2p2->3p2) wavefunction. 35The results for carbon and fluorine are in slightly better agreement than for oxygen. This may be due to the stability of the 1/2- filled and filled 2p shells of C and F . Part C: TRANSITION METAL ATOMS 55 I. INTRODUCTION There is much interest currently in obtaining a consistent theoretical description of the electronic states arising from the szdn, Sdn+1, and dn+2 1-4 configurations of the transition metal atoms. While Hartree—Fock (HF) calculations are known to inadequately represent these low- lying states, it has not been well-understood what level of description is required. This inability of the HF model to reproduce the atomic separation has important conse- quences for the description of the bonding between transi- tion metal atoms and other atoms and molecules. In part- icular, for manganese through c0pper, the atomic dissocia- tion limits for small molecular Species are biased in 2dn state in the HF picture by as much as favor of the s 1.3eV compared to experiment, raising serious doubts as to the validity of interpretations based on molecular calcu— lations which do not go beyond the HF model. We have examined the differential correlation effects within the lowest lying states corresponding to the Szdn, sdn+l, and dn+2 configurations of the first row transition metal atoms using MCSCF and CI approaches (nonrelativistic) with the intention of: 1. Characterizing the major valence cor- relation effects in these states, and, 2. DevelOping a compact yet accurate multi- configuration description for each of the states. 56 A. Background The metal atoms typically have a szdnground state n+2 n+1 as the first excited state, and the d state sever— sd a1 eVs higher. Experimentally5 the excitation energies 1 5 (AE) follow two trends, decreasing with Z for d to d (half-filled shell), increasing abruptly at d6, then de- 10 n+1 2 n creasing again to d (filled shell). The Sd —s d AE'S denoted by A(n+l), are plotted in Figure la. Note that at Cr(d5) and Cu(dlo) the two states invert, sdn+l becoming more stable, i.e., the ground state. Numerical HF results6 are also presented in Figure lb as the error with reSpect to experiment (solid lines). While the general trends are reproduced by HF, the AB in Sc-Cr is underestimated by ~0.3eV favoring the sdn+1 state, and overestimated in Mn-Cu by ~1.0eV favoring the szdn state. These trends are more consistent if view- ed in terms of the number of singlet-coupled electron pairs in each state which are not expected to be represent- ed well in the HF description. In Mn-Cu the two states have the same number of singlet-coupled pairs, i.e., the same multiplicity, and so the HF level of description should be comparable for both states. In Sc-Cr on the 2 other hand, the s dn state always has the one 452 Singlet- coupled pair whereas the sdm'l state, being high Spin coupled, has no Singlet pairs. Thus, a description for the first half of the row comparable with that of the second half would require correlation of the 452 pair. The dashed 57 Figure l. l-4sz3dn excitation energies of scandium to c0pper, 4s3dn+ [E(Sdn+l)-E(szdn)]. All units are in eV. a) Experimental values (reference 5). b) Error in the numerical HF excitation energye, [AHF - AEEXPER] (solid line). The dashed line is the HF error cor- rected for the (452,4p2) near degeneracy effect, AEH + 0.78eV(an). F Figure l. ENERGY (eV) 58 Mn Fe 59 line gives the AE's for Sc-Cr including this correlation (to be discussed in detail later). Now the calculated trend is clear; the AE'S are consistently overestimated, 2dn state by m0.5eV for Sc-Cr, and ml.0eV for Mn-Cu. For the dn+2-szdn AB, A(n+2), the same type favoring the s of HF trends are observed, the error being ~1.4eV for Sc-V and ~3.5eV for Cr-Cu. Clearly, the inclusion of electron correlation is necessary to prOperly represent the excitation energies of the low-lying states of the transition metal atoms. n+1 A most puzzling point is that is the sd and dn+2 states which are less accurately described, even though for dz—d5 the electrons all occupy different orbitals and all are high Spin coupled. It is this problem which our paper addresses, first by examining in detail the differential correlation effects in terms of a valence MCSCF wavefunc- tion, and then by comparing the AE'S obtained with this approach to valence CI and experimental values. B. Basis Set The primitive Gaussian basis set used in the following calculations, (l4s,llp,6d), is that of Wachters7 augmented with two additional p functions to describe the 4p orbital and one additional, diffuse d function to l and dn+2 states.6 describe the 3d orbital of the sdn+ This set was contracted to [55,4p,3d] using the general contraction scheme of Raffenetti.8 Since the contraction 60 was based on the atomic orbitals for the 52dn state we ex- pect a slight bias toward this state.. The HF AE'S, A(n+l) and A(n+2), for this basis are compared to numerical re- sults in Table I. In general, the basis set error is on the order of m0.1eV for A(n+l) and m0.lSeV for A(n+2). For Sc-Cr CI calculations were carried out in which a full set of single-component f-functions was added to the basis and the exponent Optimized for each state of the atom based upon a CI wavefunction which included all single and double excitations with reSpect to the MCSCF configurations. The Optimal f-exponents changed signifi- cantly from one atom to another but not between states of the same atom; consequently an average f-exponent was used for each atom: SC(0.27), Ti(0.45), V(0.77), Cr(l.l4). The [53,4p,3d,1f] basis was examined further. for titanium where an additional function Of each sym- metry type was added and the exponent was Optimized in CI calculations with the ls-3p core orbitals frozen. The additional functions were found to have no significant 2 2 3 effect on the s d to sd excitation energy.9 II. VALENCE CORRELATION EFFECTS IN THE EARLY TRANSITION METAL ATOMS, Sc-Cr Since energy differences are the quantities re- lated to experimental observables, the important correla- tions are those which are differential between the states 61 TABLE I. A comparison Of numerical and Gaussian basis set of calcula- 1 2 n 2 2 n tions of the Hartree-Fock sdn+ -s d , A(n+1), and dn+ -s d , A(n+2), exci- tation energies of scandium to COpper. All quantities are in eV. A(n+l) A(n+2) numericala Gaussian _A__ numericala Gaussian _A__ Sc 1.00 1.10 0.10 4.47 4.63 0.16 Ti 0.54 0.63 0.09 4.25 4.41 0.16 V 0.12 0.21 0.09 3.27 3.44 0.17 Cr -1.27 -1.17 0.10 5.75 5.90 0.15 Mn 3.32 3.38 0.06 9.15 Fe 1.80 1.86 0.06 7.46 CO 1.53 1.55 0.02 7.05 Ni 1.28 1.30 0.02 5.47 Cu -0.37 -0.34 0.03 aFrom reference 6 62 of the atoms. There are three types of pair correlations which arise in the (4s,3d) valence shell: 1. 32 correlation of the 4szpair (largely the near degeneracy effect) 2. d2 correlation of the 3d electrons 3. sd correlation between the 4S and 3d electrons We have used an MCSCF wavefunction to describe the 52 and d2 correlation effects. The sd correlation effect was not represented at this level but was, however, included in subsequent CI calculations (discussed in more detail later). Correlation of the 432 pair is clearly a dif- ferential effect, occuring only in the den state. The most important correlating configurations are those which allow angular correlation of the 43 pair represented by a double excitation from 43 into the 4p orbitals. This is the third-row analog Of the well-know near degeneracy effect found in first-row atoms.lo While the magnitude Of this effect decreases with z in the first row as the 2p orbitals become occupied, the 4p orbitals for the states of interest in the metal atoms are unoccupied, so the effect would be expected to remain relatively constant across the row. Inclusion of this effect favors, i.e., differential- 2 1y lowers the energy of the s dn state. The d2 pair correlation effect, on the other hand, occurs in all States Of the atoms. We have found that radial correlation of the 3d electrons is by far the 63 most important contribution to the energy11 and thus, have used correlating configurations which are double excita- tions from 3d2 into 4d2, for all 3d pairslz. This effect would be expected to be proportional to the number of pairs Of d-electrons, i(i-l)/2 for d1, favoring the states in the following order: dn+2 > sdn+1 > SZdn. Since the d-orbitals are known to contract with increasing Z, the d2 correlation effects would also be expected to in- crease with Z. Thus, the valence MCSCF wavefunctions used in the following calculations include angular correlation of the 452 pair and radial correlation of the 3d2 pairs: 32d“ : c14323dn + c24p23dn + c34323dn‘24d2 (la) sdn+1 : c14s3dn+1 + c24s3dn‘14d2 (lb) dn+2 : cl3dn+2 + c23dn4d2 (lc) A. Scandium and Titanium Scandium and titanium, being the simplest transi~ tion metal atoms, serve as a useful point to begin discus- sion Of the above effects without the added complication which more d-electrons present. Scandium has a 2D(4523d) ground state with the 4F(453d2) first excited state 1.43eV higher. The present HF calculations predict an excitation 64 energy of 1.10eV, 0.lOeV above the HF limit and 0.33eV be- low the experimental value (Table II). The valence MCSCF wavefunction for the 32d1 state contains no d2 correlation term since there is only one d-electron. Addition of the 4p2 configuration lowers the energy by 0.75 eV. The MCSCF wavefunction for the sd2 state has one configuration in ad- dition to the HF configuration to describe the correlation between the two d-electrons. The d2 correlation energy for this state is 0.21eV. The resulting excitation energy, 1.64eV, now favors the 82d1 state by 0.21eV. Improvement of the core basis as judged from numerical HF calculations could reduce the difference with eXperiment to about 0.1eV. 23d2) ground state with the Titanium has a 3F(4s 5F(4s3d3) first excited state 0.8leV above it (Table III). The HF AB is 0.63eV, again 0.09eV above the HF limit, but underestimating the experimental value by 0.18eV. The 82d2 state has both 52 and d2 correlation terms. The 32 correlation accounts for a 0.77eV lowering relative to HF, slightly larger than for scandium. The d2 correlation in 32d2 is only 0.09eV. The combination of both 32 and d2 correlation is essentially additive, 0.85. In the sd3 state of Ti the d2 correlation energy is 0.49eV. The valence MCSCF gives an AB of 0.99, 0.l9eV above experi- ment.13 While the d2 state of titanium (szdz) shows a lowering of 0.09eV for one 3d2 pair, the effect in scandium (sdz) is 0.21ev, 2.5 times larger. Further, the €55 TABLE II. Calculated and experimental energies for the s d , sd2 states of the scandium atom. Basis set: (Se, 4p, 1f]. Units are as indicated. ?0(4s‘3d) “8(4s3d‘) “8(3d3) 2 2’ Energy AH? Energy AHF A(s d) Energy £§§_ A(s d) hartree e .artree eV 9 ar ree eV eV HF -759.7251 0.0 -759.6847 0.0 1.10 -759.5551 0.0 4.63 MCSCF 3d2-4d2 -- -7S9.6924 0.21 -759.5833 0.77 8011‘ ~759.7527 0.75 -- 1.64 -- 4.61 CI HF+1+2 -759.7625 1.02 -759.7005' 0.43 1.69 -759.5929 1.03 4.62 MCSCF+1+2 -759.7636 1.05 -759.7007 0.44 1.71 -759.5987 1.19 4.49 Pullb -759.7638 1.05 -759.7007 0.44 1.72 -759.5991 1.20 4.48 Experimentalc 1.43 4.18 aMCSCF wavefunctions defined by equations la—c b cFrom reference 5 Full-CI for 3 valence electrons 66 2 TABLE III. Calculated and experimental energies for the s d , sd , and d4 states of the titanium atom. Basis set: (5s, 4p, 3d, 1f]. are as indicated. 5 4’ jF(4s27 ) SF(4s3d3) D(3d ) Energy 4H? Energy LHF lgszdzz Energy AHF 0(s d ) hartree ev hartree eV V hartree eV eV HF -848.3927 0.0 -848.3697 0.0 0.63 -848.2306 0.0 4.41 MCSCF 3d2-4d2 -848.3961 0.09 -848.3877 0.49 -848.2761 1.24 n 4s‘-4p2 -848.4210 0.77 -- -- Ful‘a -848.4241 0.85 -- 0.99 -- 4.03 CI HF+1+2 -848.4377 1.22 -848.4009 0.85 1.00 -848.2909 1.64 4.00 MCSCF+1+2 -848.4396 1.27 -848.4019 0.88 1.03 -848.2974 1.82 3.87 Fullb -848.4399 1.28 -848.4020 0.88 1.03 -848.2982 1.84 3.86 Experimentalc 0. 81 3 . 55 aMCSCF wavefunctions defined by equations 1a-c. bFull-CI for 4 valence electrons cFrom reference 5 67 d2 correlation energy in the sd3 state of Ti, 0.49eV, is much larger than the 0.27eV (0.09eV/pair x three pairs) that would be predicted on the basis of a simple pair de- pendence of the correlation energy. B. Differential Trends . The results of the 32 and d2 correlation effects for the early transition metal atoms are summarized in Table IV. Here we use the symbol €232+4p2 to denote the energy lowering, relative to the HF energy, Obtained from 2 the (432,4p2) correlations of the 4S 3dn states. Similar- 2 ly, e 2 2 represents the (3d ,4d2) energy lowering of n 3d +4d those states with n d-electrons. Columns 1-3 Show the effect of including 4p2 and 2 4d2 configurations in the 4S 3dn wavefunction. As mention- ed earlier, €282+4p2 is relatively constant being 0.78 t 2 n n . 0.03eV. For the 4s 3d states €3d2+4d2 increases approx- imately quadratically with the number Of d-electrons fol- lowing the simple pair formula: n - e3d2+4d2 ‘ Aedd n n-l) (2) 2 where Aedd = 0.087eV, and corresponds to the energy lower- ing associated with correlation of a parallel Spin 3d electron pair. Note that it is also approximately the pair energy calculated for the 52d2 state of titanium. The 68 2 TABLE IV. Valence MCSCF (4s ,4p2), (3d2,4d2) correlation energy differences from Hartree-Fock. All quantities are in eV. __ 43230“ 433dn+1 3dn+2 Fulla 4s2+422 3d2+4d2 3d2+4d2 3d2+4d2 Sc 0.75 0.75 0.0 0.21 0.77 Ti 0.85 0.77 0.09 0.49 1.24 v 1.03 0.79 0.27 0.84 1.69 Cr 1.28 0.80 0.52 1.21 2.48 Mn 1.55 0.81 0.79 1.85 Fe 2.03 0.82 1.29 2.48 CO 2.53 0.83 1.81 3.16 Ni 3.07 0.84 2.37 3.86 Cu 3.67 0.84 2.99 4.59 aMCSCF wavefunction defined by equation la.’ 69 first column shows the results of incorporating both ef- fects into an MCSCF wavefunction (Equation 1a). As can be 2 2 seen, the d and s energies are essentially additive as would be expected if the orbitals are indeed concentrated in different regions of Space and the correlation effects noninterfering. If the same type Of d2 correlation energy were 1 associated with the (n+1) electrons Of the S.dn+ states of the atoms, €n+§ 2 would simply be equal to en 2 2 of 3d +4d 3d +4d Z+1. Comparing columns 3 and 4 shows that this is not the case. Consistent with the results given above for Sc and 1 Ti, the d2 correlation energy for the sdn+ state in all cases is larger than for the 32dn state with the same number of d-electrons. Apparently a different type of correlation is involved with the'(n+l) electrons Of the dn+1 3 state than the simple pair repulsions between the n electrons in the den state. We have found the d2 correlation energy for the n+1 state to follow a form which is the sum of two terms. sd The first term represents the correlations between the n d-electrons as in the ground state; the second describes the extra energy of the (n+1)8t electron correlating with each of the other n: n+1 e _ n(n-l) 3d2+4d2 ‘ Aedd "'2“— + A’de' n ‘3) 70 The difference between the d2 correlation in the two states should, therefore, be linear in n: n (4) where the SlOpe of the line gives this extra energy associat- ed with the (n+1)St electron. Figure 2 shows a plot of the d2 correlation energies for the two states. The energy low- ering in the szdn state is represented by the diamonds. The dotted line is a plot of equation (2) with AS = 0.087eV. dd The observed good agreement between the calculated energy lowerings and dotted line, rms = 0.005eV, validates the sim- ple pair model of this correlation effect. The d2 correla- tion energy for the sdn+1 state, €n+l , is represented by 3d2+4d2 the circles. The difference between these values (squares) is seen to be linear in n, where the value of the SlOpe, is 0.18ev, with rms=0.04 (solid line). The dashed . n+1 Aedd’ line, associated with the sd state, is simply the sum Of the solid and dotted lines. C. Orbital Interpretation From the above discussion, it is clear that the d2 correlation effects in the Sdn+1 states are unusually large. An understanding of the unique nature of this extra cor- relation energy is given by contrasting these states tO the analogous Spn+1 states of the first row atoms. In carbon, for example, the excitation 32p2+sp3 places a ZS electron into a 2p orbital. The Shielding of the nuclear charge 71 O T \ \ l ENERGY (0V) 9 (I 0.0 Figure 2. Calculated (3d2,4d2) energy lowering relative to HF fOr the 2 1 s dn (diamonds) and sdn+ (circles) states of scandium to chromium, [E 2 HF-EMCSCF]. The dotted line through the s dn points correSponds to the formula, 0.087 x n(n-l)/2 (rms=0.005eV). The squares correspond to the calculated differential energy lowering between the two states, while the solid line corresponds to the equation 0.18 x n (rms=0.04eV). The dashed line is the sum of the solid and dotted lines. All units are in eV. 72 by the 25 and 2p electrons is essentially equivalent accord- 14 SO that this excitation is not ex- ing to Slater's rules, pected to greatly change the net potential felt by the 2p electrons. For carbon the p2 radial correlation energy for 22p2) state is calculated to be 0.12eV using an the 3P(Zs MCSCF wavefunction which includes the 3p2 configurations. If the pair effects were transferable, the p2 correlation energy for the 58(232p3) state would simply be three times this amount, or 0.36eV. The calculated energy lowering of 0.30eV is only slightly less. In the transition metals, on the other hand, the excitation den to sdn+1 removes an 'outer' 4s electron, placing it into an 'inner' 3d orbital. In this case, Slater's rules state that the 4s electrons do not shield electrons in the 3d shell, however, 3d electrons do par- tially shield one another. If the (n+1)St d-electron were put into an orbital equivalent to the other n there would be a decrease in the net potential felt by all Of the d-electrons, resulting in an expansion of the entire 3d shell. This is, indeed, what happens in a HF wavefunction.)5 Alternatively, the (n+1)St electron can be placed in a somewhat more diffuse orbital, 3d', which does not shield the other n 3d electrons as effectively, and, thus, does not decrease the net potential which they feel. It is this later model that explains the anomalous d2 correlation n+1 effects in the sd states noted above. 73 This type of description for the dn+1 states Of the transition metal atoms was first prOposed by Froese Fischer15 3d10(lS)nl, where she suggested that the last 3d electron in discussing the ionization potential of COpper should be treated differently from the previous nine, 9 I 3d 3d (lS)nl. This was further discussed for the 7P(3d54p) state Of chromium11 where, by allowing the fifth 3d elec- tron tO be nonequivalent to the other four, the imprOper HF ordering of the two septet P states arising from 3d54p and 3d44s4p is corrected. In support of this picture of the 3d correlation effects we compare the calculated d orbitals of the 4sz3d2 state of Ti to those of the 433d2 state of Sc. In order to meaningfully interpret orbitals from an MCSCF calculation it is simplest to first transform them into an equivalent independent particle wavefunction involving non- 16 orthogonal orbitals. This is analogous to the natural orbital to nonorthogonal pair orbital transformation used to interpret generalized valence bond wavefunctions.l7 In the following, 3di and 4di denote the various components of the orthogonal 3d and 4d natural orbitals Ob- tained from the MCSCF calculations, while di andcl; denote the nonorthoqonal orbitals of the equivalent independent particle wavefunction. The necessary orbital transforma- tion can then be written: 4di4d ] ad 2 j l 12 3 = - ' [2(1+S )] [[did'j + didjlaa 74 where, d [(1+5)/21l5 3d + ((1-s)/2]* 4d d' ((1+s)/21is 3d - [(1-s)/21‘5 4d 8 = (cl-c2)/(cl+c2) To make the independent particle nature of this wavefunction more evident we define a projection Operator .3? which eliminates all but the F component of a many elec- tron wavefunction. With this definition, the wavefunction for the 3F state of Ti may be written as: ypddidsaa where now the physical interpretation Of these orbitals is clear; one electron is bound in orbital di and the second electron is bound in a nonequivalent orbital, d3. The projection Operator then assures that the wavefunction will have the correct Spatial symmetry. The overlap between the nonorthogonal orbitals, d and d' is high when the contribution of the 4d correlat- ing orbital is low, resulting in two orbitals very similar 18 to the HF orbital (if S=l, d=d'=3dHF). Comparing these orbitals to 3dH is then a measure of the breakdown of the F single configuration representation of the 3d orbitals inherent to the HF wavefunction. In Figures 3 and 4 the radial amplitudes of the d, d' and 3dHF orbitals are plotted as a function of the distance from the nucleus, R. The Ti 32d2 nonorthogonal d orbitals (dashed lines) and the HF orbital (solid line) are Shown in 75 Figure 3. The essential equivalence of these orbitals is reflected in their high overlap, S=0.91, and in the small deviation of the expectation value of R, , from that of the HF orbital: 1.20, 1.82, and 1.46 aO for d, d' and 3dHF' reSpectively. Contrast this with the Sd2 orbitals Of Sc plotted in Figure 4. Here the nonequivalence of the d and d' orbitals is reflected in the lower overlap, S=0.81, and in thalarger deviation of from HF: 1.52 and 3.06ao compared to 2.13ao for 3dHF' The nonorthogonal d orbital in the sd2 excited state of scandium is actually very similar in radial extent to the 3d orbital in the 52d1 ground state (=l.68ao). Thus,an orbital picture emerges which ascribes a different type of binding to the 3d electrons in the Szdn 1 states of the transition metal atoms:- - The 32dn states have n 3d-electrons in and ad n+ essentially equivalent orbitals with Aedd = 0.087eV correlation energy for each pair of d-electrons. - The Sdn+1 states have n 3d-electrons bound approximately the same as in the ground state with the same correlation energy, Aedd for the —2%2:$L pairs; and one electron, the (n+l)St, in a more diffuse orbital with an additional Asdd' = 0.18eV correlation energy for 76 AMPLITUDE -05 l l l l 0.0 (.0 2.0 3.0 4.0 DISTANCE (00) Figure 3. Radial plots of the Hartree-Fock (solid) and nonorthogonal 2 d-Orbitals (dashed curves) for the 3EMS 3d2) state of titanium. 77 | 5 f T I l — A -( LG [I ‘\ ’ \ I \ \ I \ u.) I \ C) I \ a 1 x _ —‘ ~ III! _’ 0.5 ' ’t “‘ \ Q. I ‘s‘ 2 I “~ <1 I, ‘~ \\‘:. _____ 0.0 I """"" ()5 1 l L L 0.0 |.0 2.0 3.0 4.0 DISTANCE (00) Figure 4. Radial plots of the Hartree-Fock (solid) and nonorthogonal d-orbitals (dashed curves) for the 4F(4S3d2) state of scandium. 78 each of the n other d-electrons with which it correlates. The HF description, restricted to a Single confi- guration wavefunction, is unable to allow for the diffuse 3d' orbital in the sdn+1 state, resulting instead in equi- valent 3d orbitals, all of which expand relative to those of the 32dn state. The valence MCSCF wavefunction, which lifts this restriction, finds that n electrons are bound as in the 32dn ground state, with only the (n+1)St electron occupying a more diffuse orbital. D. CI Calculations While the valence MCSCF wavefunction presents a consistent picture of the differential s and d correlations within the szdn and sdn+1 states of the atoms, its utility as a practical method for use in electronic structure cal- culations depends upon its ability to track the energy dif- ferences obtainable in a full valence CI. In the following section, we compare the excitation energies obtained at each level in order to determine the additional correlation effects which have been neglected or underestimated in the valence MCSCF wavefunction. Valence CI calculations within the entire virtual Space were carried out for scandium to chromium which in- cluded all single and double excitations with reSpect to the HF reference configuration (HF+1+2), or the MCSCF reference configurations (MCSCF+1+2), constructed from the 79 MCSCF orbitals for each state (Tables II, III, V, VI). For both Sc and Ti full valence CI calculations are reported as well. As previously noted, for the CI calculations the basis set was expanded to include an f-function. 2 n+1 states it is Discussing first the s dn and sd seen from Table VII that the MCSCF calculations faithfully represent the AE'S obtained from the CI calculations, the differences ranging from 0.04 to 0.07eV. Note that HF+1+2 gives AE'S closer to experiment than either the MCSCF+1+2 or the full-CI,19 but all three are quite com- parable and well-represented by the valence MCSCF wave- function. In order to verify that the 82 and d2 differential pair correlation energies are well-represented by the valence MCSCF wavefunction, pair CI calculations were car- ried out consisting of Single and double excitations from the 43 or 3d orbitals into the virtual Space for the 5D and 7S states of chromium. In addition, the Sd correlation energy was estimated from a Similar CI calculation by a1— 1owing simultaneous single excitations from the 4s and 3d orbitals into the virtuals. These results are summarized in Table VIII where the difference with EH is given for F MCSCF, and CI calculations in which excitations were al— lowed within the (S,p,d) and (S,p,d,f) virtual Space. Comparing the energies of the MCSCF wavefunction with the CI within the (S,p,d) Space, it is noted that there is less than 0.1eV improvement in either the 32 or d2 8() TABLE V. Calculated and experimental energies for the szd3, sd4 and d5 states of the vandium atom. Basis set: [5s,4p,3d,1f). Units are as indicated. 4“45239) 60(433d4) 6suds) 3 5 ; Energy AHF Energy AHF 3(32d ) Energy AHF A(s“d ) hartree 5V hartree 8V 6V hartree 6V eV HF -942.8678 0.0 -942.8602 0.0 0.21 -942.7414 0.0 3.44 MCSCF 3d2-4d2 -942.8777 0.27 -942.8912 0.84 -942.8037 1.69 482-4p2 -942.8966 0.79 -- -- 8011‘ -942.9058 1.03 -- 0.40 -- 2.78 CI HF+1+2 -942.9255 1.57 -942.9117 1.40 0.38 -942.8245 2.26 2.75 MCSCF+1+2 -942.9286 1.65 -942.9l37 1.45 0.41 -942.8322 2.46 2 62 Experimentalb 0.24 2.46 aMCSCF wavefunctions defined by equations la-c. bFrom reference 5 811 TABLE VI. Calculated and experimental energies for the szd4,sds, and d6 states of the chromium atom. Basis set: (5s,4p,3d,1f]- Units are as indicated. 50(4-23df)_ 75(4s3d5) 58(306) Ener AHF Energy AHF agsjd‘) Energy AHF 0(82d42 hartree eV hartree eV eV hartree SV SV HF -1043.2891 0.0 -1043.3323 0.0 -1.17 -1043.0724 0.0 5.90 MCSCF 3d2-4d2 -1043.3082 0.52 -1043.3766 1.21 -1043.1637 2.48 482-4p2 -1043.3184 0.80 -- -- €011a -1043.3360 1.28 -- -1.10 -- 4.69 CI HF+1+2 ~1043.3648 2.06 -1043.4046 1.97 -1.08 -1043.209lb 3.72 4.24 MCSCF+1+2 -1043.3695 2.19 -1043.4077 2.05 -1.04 -1043.2258b 4.17 3.91 Experimentalc -1.00 3.40 aMCSCF wavefunctions defined by equations 1a-c bConverged only by deleting the 4s orbital from the cFrom reference 5 virtual Space 1 TABLE v11. 453dn+ energies for scandium to chromium. 82 All units are in eV. -4523dn calculated and experimental excitation Sc Ti V Cr HF 1.10 0.63 0.21 -1.17 MCSCFa 1.64 0.99 0.40 -1.10 CI: HF+1+2 1.69 1.00 0.38 -1.08 MCSCF+1+2 1.71 1.03 0.41 -1.04 Fullb 1.72 1.03 Experimental: Relativisticc 1.43 0.81 0.24 -1.00 'Non-Relativistic'd 1.31 0.67 0.07 -1.21 aMCSCF wavefunctions defined by equations la and lb. bFull-CI for (n+2) valence electrons c From reference 5 dCorrected for differential relativistic effects taken from numerical relativistic HF calculations (reference 27) for each state. 2 83 TABLE VIII. 4S , 3d2 and 4s3d pair correlation energy differences from Hartree-Fock for the chromium atom. All units are in eV. LCSEF. S0(4sz3d4) 4s2 0.80 3d2 0.52 453d -- 75(4s3d5) 3d2 1.21 433d -- 5D(3d6) 3d2 2.48 HF+1+2 a fl 0.87 0.54 0.26 0.09 3.11 AMCSCF 0.07 0.02 0.26 0.09 (spgf)6 AMCSCF 0.88 0.08 0.90 0.38 0.48 0.48 1.79 0.58 0.21 '0.21 3.72 1 24 aSingle and double excitations restricted to the (Spd) virtual Space. bSingle and double excitations allowed within the entire (spdf) virtual Space.. 84 correlation energy for either state. SO inclusion of other S,p or d functions does not affect the description of these correlation effects which are indeed well-represented by the valence MCSCF wavefunction. The sd correlation effect is seen to favor the szdn state, the state with the most s-electrons. This implies that all three pair effects in- cluded in a single and double CI within the (S,p,d) virtual space Should increase the AE relative to the MCSCF energy, 20 which, in fact, it does by 0.2eV. If f—excitations are 2 allowed in the CI, the 3 description is still unaffected, 2 but now the energy lowerings from d and sd excitations are significantly larger. Relative to the Spdf-CI then, the MCSCF calculation for the 82d4 state neglects correlation effects worth 0.94eV while for the SdS state the difference is 0.79eV. Thus, the differenCe between the MCSCF and' Spdf-CI AB is only 0.15eV. Thus, without f-functions the Sdn+l-Szdn excita- tion energy will increase at the CI+1+2 level over the valence MCSCF energy because the differential Sd correla- tion, favoring the 82dn state, is now included. Including f-functions has little effect on the $2 correlation energy but increases both the d2 and the sd pair correlation energies, leading to essentially the same AE as obtained at the valence MCSCF level. While the lowering from the Sd excitations shows a differential effect in the pair-CI calculation, it Should be noted that the three effects, taken together, are not completely additive, implying 85 that the results in Table VIII overestimate this effect. Secondly, this correlation description is not expected to change the qualitative features of the orbitals signif- icantly, thus supporting its exclusion at the MCSCF level. It is appropriate at this point to discuss the calculations on the dn+2 states of Sc-Cr. In general, 2 the dn+ states are expected to be more difficult to describe than the sdn+1 states using a valence MCSCF wavefunction which makes use of only 3d and 4d orbitals. Now three nonorthogonal orbitals, d, d', and d", may well be required to represent the diffuse nature of the (n+1)St and (n+2)nd d-Orbitals.21 Although the d2 valence MCSCF energy lowerings relative to HF, €n+§ 2, are larger than 3d +4d n+ seen for Sd 1 (Table IV), CI'S within the (3d,4d,5d) Space indicate that the 5d orbital also plays an important role in describing the d2 correlation effect, i.e., the 3d and 4d MCSCF orbitals alone are not sufficient. Comparing these results to other levels of description (Table IX), it is seen that the valence MCSCF wavefunction is a much bet- n+2 state than is the HF wave- ter description of the d function, leading in all cases to AE'S less than 0.2eV above those obtained from CI's in which the entire virtual Space is included. We note that a large part of the dif- ference between the MCSCF and full virtual CI+1+2 results can be accounted for by allowing excitations from the MCSCF wavefunction into a third set of d-functions (5d). n+2 TABLE IX. 3d -4sz3dn calculated and experimental excitation ' energies for scandium to chromium. All units are in eV. Sc Ti V Cr HF 4.63 4.41 3.44 5.90 MCSCFa 4.61 4.03 2.78 4.69 CI: b ar+1+2 4.62 4.00 2.75 4.24 MCSCF+1+2 (3d,4d,5d)c 4.53 3.86 2.59 3.92 all virtualsd 4.49 3.87 2.62 3.91b Full-CIe 4.48 3.86 -- -- Experimentalz' Relativisticf 4.18 3.55 2.46 3.40 'Non-Relativistic'g 4.00 3.33 2.18 3.09 aMCSCF wavefunctions defined by equations la and 1c- bThe d6 state converged only by deleting the 4s orbital from the virtual space. cSingle and double excitations of the 3d electrons restricted to the (3d,4d,5d) space. dSingle and double excitations Of all (n+2) valence electrons allowed into the entire virtual Space. eFull-CI for the (n+2) valence electrons. fFrom reference 5. 9Corrected for differential relativistic effects taken from numerical relativistic HF calculations (reference 27) for each state. 87 As indicated by the CI calculations, the valence MCSCF wavefunction provides a compact representation of the differential 82 and d2 correlation effects between the 82 n+1 d“, , and dn+2 states of Sc-Cr. While this form of the wavefunction is not as accurate for the dn+2 state, it is sd still a far better description than the HF wavefunction provides. For all of these states, the valence MCSCF des- cription condenses the major valence correlation effects into a form which lends itself to molecular calculations. E. Experimental Excitation Energies Comparing the calculated AE's A(n+l) and A(n+2), to the experimental values for Sc-Cr (Tables VII and IX), indicates that even with full valence correlation of the (4s,3d) electrons the AE's are overestimated, favoring the 82dn state by N0.2eV for A(n+l), and ~0.3eV for A(n+2). Note that the error is much larger, 0.52eV, for the d6-szd4 AE of Cr due to the presence of the doubly-occupied d-orbital in the upper state. III. VALENCE CORRELATION EFFECTS IN THE LATE TRANSITION METAL ATOMS, Mn-Cu We have examined the differential valence correla- tion effects between the 32dn and sdn+1 states of manganese through COpper in terms of the valence MCSCF wavefunction (Equations la-c) discussed previously. The same type of 32 and d2 correlations were incorporated into the MCSCF 88 orbitals as in the first half of the row except now, Since the d-orbitals are doubly-occupied, there are four types of d-d interactions: (aB)ii, (aB)ij, and (88)ij, as well as the (0101)ij term from before. Thus, configurations which allow iii—emf and 3d? ad?-—»38.4a.3a.4a., as well as l J 1 1 j j 3dinT>4di4dj, were included in the calculations. A. Differential Trends The results of the S2 and d2 differential correla- tion effects for the late transition metal atoms are sum- marized in Table IV. The S2 correlation in the 32dn state was again found to be essentially constant and approxflmate- 1y equal to that for the early transition metal atoms, 0.82 i 0.02eV. As in the first half of the row, the 52 and d2 correlation energies are found to be nearly additive. Because the d2 correlation effects are now complicated by the additional interactions due to the presence of the B-electrons, they are best understood by considering all of the 0,8 interactions within the n and (n+1) d-electrons. Since both states have five d-electrons, there will always be 5(4)/2 aa-interactions. In addition, for the szdn state, there are (n-S) doubly-occupied orbitals so there will be: (n-S) (018)ii interactions, 4(n-5) (dB)ij interactions, and, (n-S)(n-6) /2 (BB)ij interactions. 89 For the Sdn+1 state there are (n-4) doubly-occupied orbitals, with the correSponding number of interactions, resulting in the differential energy expression: 0E3d2+4d2 = 611(08) + 4eij(08) + (n-5)eij(88) This eXpression is linear in n as with the first half of the row (Equation 4), but has an additional constant term, due st to the correlations of the (n+1) B-electron with the five a-electrons. In Figure 5, AE3d2+4d2 is plotted as a func- tion of n for Sc-Cr as well as Mn-Cu. As predicted, these differences are linear in n, the line for Mn-Cu shifted upward relative to Sc-Cr because of the additional constant term. Note that the SlOpeS of each are Similar, being 0.18 and 0.14 eV (rms=0.02) for the first and second halves~ reSpectively, indicating that the correlation energy of the (n+1)St electron with the other n electrons is similar for Sc-Cr and Mn-Cu. This d-d' interaction energy for the second half is also ~0.leV higher than that found in the 2 s dn state, implying again that the (n+1)St electron is loosely bound. B. CI Calculations on the Nickel Atom Because of its experimental and theoretical im- portance,22 we have carried out valence CI calculations 2 8 9 (HF+1+2) for the S d and sd states of the nickel atom, within the [55,4p,3d] basis, using the orbitals obtained 90 ( r I I r I l I I LS" -' 3 3 > loob — O C ‘i’ “I 0.5- .1 00 l L l A J J l l L ScT) VCrMnFeCoNiCu Figure 5. Differential (3d2, 4d2) energy lowerings for the 52dn and n+1 sd states of scandium to COpper as calculated in this work. All units are in eV. 91 from the valence MCSCF calculations. These results are sum— marized in Table X. The valence MCSCF wavefunction pre- dicts an sdg-szd8 AE of 0.51eV while HF+1+2 gives a value _of 0.42. Similar valence HF+1+2 calculations by Martin,2 in which the HF orbitals were used as the expansion basis, resulted in an energy separation of 0.32eV. This separa- tion increased to 0.46eV upon uncontracting the 3s and 3p core orbitals, whereupon, addition of an f-function to describe (4s,3d) correlation, was found to lower this AE to 0.30eV. These results are consistent with the conclu- sions for the first half of the row that the valence MCSCF wavefunction does present a reasonable description of 2 n+1 the valence correlation in the S dn and sd states even when doubly occupied d-orbitals are involved. C. Experimental Excitation Energies The excitation energies calculated with the valence MCSCF wavefunction are compared to the experimental values in Table XI. While the error for HF ranges from 1.0 to 1.3eV, this has been reduced using the valence 23 This error is MCSCF approach to 0.5-0.2eV for Fe-Cu. larger than seen in Sc-Cr but is still reasonable since it is estimated that single and double valence CI calcula- tions would give Similar results. 92 TABLE X. Calculated and experimental energies for the 32d8 and ad9 states of the nickel atom. Basis set: [Ss,4p,3d]. Units are as indicated. 38(4sz3d8) 30(4s3d9) A Energy AHF Energy AHF A(s[d8) hartree eV hartree eV eV HF -1506.8214 0.0 -1506.7736 0.0 1.30 MCSCF 3d2-4d2 -1506.9084 2.37 -1506.9155 3.86 432-4352 -1506.8522 0.84 -- ‘ Fulla -1506.9341 3.07 -- 0.51 CI ar+1+2 -1506.9533 3.59 -1506.9377 4.47 0.42 Experimentalb -0.03 ( _— aMCSCF wavefunctions defined by equations 1a and b. bFrom reference 5 93 1 TABLE XI. 4s3dn+ -4sz3dn calculated and experimental excitation energies for manganese to copper. All units are in eV. Mn Fe Co Ni Cu at 3.38 1.86 1.55 1.30 -0.34 MCSCFa 3.07 1.40 0.93 0.51 -1.26 Experimental: Relativisticb 2.14 0.88 0.42 -0.03 -1.49 'Non-Relativistic'c -0.38 aMCSCF wavefunctions defined by equations la and lb. b From reference 5. cCorrected for differential relativistic effects taken from numerical relativistic HF calculations (reference 2) for each state. 94 IV. OTHER CONSIDERATIONS 2 Since the valence correlated energies of the s dn n+1 states of the transition metal atoms are still in and sd error by N0.2eV for Sc—Cr and 60.5 for Mn-Cu compared to the experimental AE'S, there must be other differential ef- fects which are of importance. The two major effects which have been neglected are: 7- Correlation effects involving the 'core' (33,3p) electrons -- Relativistic effects A. Core Correlations While true core electrons would be eXpected to be unaffected by changes within the valence electron occupancy, the (33,3p,3d) orbitals are all concentrated in similar regions of space so that changes in the 3d occupancy could induce a differential effect in correlation of the 3S and 3p electrons. There are three types of correlation effects which could arise: l. The (3s,3p,3d) near degeneracy effect 2. Core/valence dynamic correlation 3. Correlation of the 3s and 3p electrons, including Space and Spin polarization effects. The near degeneracy effect observed in the N=3 shell is analogous to the effect in (4s,4p) discussed previously only now excitations into the empty 3d orbital 95 are involved. It can be represented by a CI wavefunction which allows double excitations from (3s,3p) into 3d. As with the first row, this effect will decrease as the d-orbitals become occupied. It will favor the state with the fewest d-electrons, Szdn, Since the more d 'holes' there are, the more orbitals available for correlation. This effect is evident in the paper by Guse, et al.3 (labeled as a differential core-core correlation effect). The core/valence dynamic correlation effect describes the instantaneous correlation of the core and valence electrons. It is represented by simultaneous single excitations of (3s,3p) and Gd,4s) into the virtual Space. This effect would be largest for the state with the most d-electrons, Since it is the 3d's which Should interact most strongly with the (3s,3p) electrons de- n+l 2 n creasing the Sd -s d excitation energy. Preliminary calculations24 Show that a second tight f-function is needed to describe this correlation prOperly25 as the dominant configurations are those which involve 3p,3d+nd,4& Spatial and Spin polarization effects among the 3s and 3p electrons can be induced by the asymmetric charge distribution of the 3d and 4S electrons. Munch and 4 Davidson, in calculations on the F(szd3) state of vana- dium,26 noted the importance of the Single excitation 3p+f, which describes the polarization of the 3p shell by the asymmetric d shell. Thus, a second f-function is also needed to describe Spatial polarization of the 3p electrons 96 as well as the dynamic correlation between core and valence. These effects will be state-dependent since the orbital occupancies vary considerably from Sc-Cu. For example, the Cr 7S(sd5) state Should Show no Spatial ef- fects but Spin polarization due to the Six a—electrons may be important; whereas for Ni15(d10) both effects will be zero. It is not clear what kinds of differential trends will emerge due to these effects. Thus, two competing effects occur in the correla- tion of the n=3 Shell: the (33,3p,3d) near degeneracy ef- fect favoring the Szdn state; and the dynamic correlation of the 35 and 3p electrons with the valence 4s and 3d, n+1 2 favoring the sd and dn+ states. A third effect, Spa— tial and Spin polarization within the (35,3p) electrons may also be important but not clearly favoring either the ground or excited states. Data taken from Guse, et al.3 imply that correlation of the core electrons increases in importance from Sc-Cu. B. Relativistic Effects Relativistic HF calculations have been carried 27 for Sc-Cr to determine the dif- 2 n+1 and dn+2 states out by Martin and Hay ferential effects between the s dn, sd of the transition metal atoms. Their results indicate that in all cases the relativistic energies lower the Szdn states. The differential effect for A(n+l) is 0.1eV for Sc, increasing to 0.2eV for Cr. The differential effect 97 for A(n+2) is 0.2eV for Sc, increasing to 0.3eV for Cr. It appears that this trend prevails across the row so that for nickel the relativistic effect has increased the sdg-szd8 Splitting by 0.35eV2. Because this effect is large, the calculated nonrelativistic AE'S Should actually be com- pared tO 'nonrelativistic experimental' values, given in Tables VII, IX and XI, where the experimental values have been corrected for the differential relativistic effects taken from Martin and Hay's numerical HF calculations. 2 With the s dn state being differentially lowered by these effects, the AE'S which are comparable to the calculated values are all smaller in magnitude than the reported values and consequently, the error in the calculated values is even larger. V. CONCLUSIONS A reasonable description of the differential 2 n+1 valence correlation effects within the S d“, sd , and n+2 d states can be obtained at the MCSCF level by using 2 a wavefunction which incorporates (3d ,4d2) radial cor- 2,4p2) angular correlation effects. This relation, and (43 simple wavefunction reproduces well the results of a single and double excitation valence CI calculation including f-functions in the basis. The AE'S obtained from the simple MCSCF wavefunction are in error by 0.2eV for Sc-Cr and 0.5eV for Mn—Cu for the Sdn+l_32dn excitation energies. 98 An interpretation which is consistent with n+1 and dn+2 these results is that in the Sd states of the transition metal atoms the (n+1)St and (n+2)nd 3d-electrons are not bound as tightly to the nucleus as the other n electrons. These States are more apprOpriately described as: n+1 4s 3d —————o 4s 3dn3d' n+2 3d -———>3dn3d' 3d" signifying that n electrons are bound as in the 32dn ground state with the additional d-electrons occupying more diffuse orbitals, 3d' and 3d", nonorthogonal to the other 3d orbitals. If relativistic effects are included in the description of each state, the error in the excitation energy appears to increase. This implies that a highly accurate description of the splittings of the low-lying states of the transition metal atoms requires that differ- ential correlation involving the (3s,3p) 'core' electrons also be taken into account. LI ST OF REFERENCES 99 LI ST OF REFERENCES 1T. H. Dunning, Jr., B. H. Botch, J. F. Harrison, J. Chem. Phys. 72, 3419 (1980). 2R. L. Martin, Chem. Phys. Letters 75, 290 (1980). 3M. Guse, N. S. Ostlund, G. D. Blyholder, Chem. Phys. Letters 61, 526 (1979). 4C. w. Bauschlicher, Jr., J. Chem. Phys. 73, 2510 (1980). SAveraged over M. values; energies from C. E. Moore, Atomic Energy Levels, Nat'l. Bur. Stand. (U.S.) Circ. 467, Vols. 1 and 2 (1949). 6P. J. Hay, J. Chem. Phys. 66, 4377 (1977). 7A. J. H. Wachters, J. Chem. Phys. 52, 1033 (1970). 8R. C. Raffenetti, J. Chem. Phys. 58, 4452 (1973). 9Similar calculations on the nickel atom by Martin2 support this conclusion. l°:8.C:1ementi, A. Veillard, J. Chem. Phys. 44, 3050 (1966). 11This effect was noted earlier by Froese Fischer; C. F. Fischer,'The Sixth International Conference on Atomic Physics Proceedings' , edited by R. Damburg, Plenum Press, NewYork (1978), p. 77. 12In a previous letter1 we wrote the MCSCF wavefunction in a single excitation form. While for a Single pair the single and double excitation representations are equiva- lent, once more than one pair is involved this is not the case. We have found the double excitation form to yield more consistent results and to be more convenient to use. 100 13Uncontraction of the basis to [6s,6p,3d] resulted in an energy of 0.94eV. 14J. C. Slater, 'Quantum Theory of Atomic Structure’, Vol. 1, McGraw-Hill Book Company, Inc., New York, 1960. 15C. Froese Fischer, J. Phys. B: Atom Molec. Phys. 10, 1241 (1977). 16w. A. Goddard, 111, Phys. Rev. 157, 81 (1967). 17W. J. Hunt, P. J. Hay, W. A. Goddard, III, J. Chem. Phys. 57, 738 (1972). 18The HF orbital is approximately an occupation number weighted average of d and d'. 19If the HF+1+2 calculations are based on the HF orbitals rather than the MCSCF orbitals, this difference appears to be more pronounced resulting in AE'S which are lower than obtained from the MCSCF+1+2 calculation, but which are found to increase upon inclusion of higher order excitations (see nickel discussion). 20This increase in the AE(eV) for the rest of the row is: Sc(0.13), Ti(0.ll), V(0.19). 21The d' and d" orbitals may be Similar in radial extent. 22J. O. Noell, M. D. Newton, P. J. Hay, R. L. Martin, F. W. Bobrowicz, J. Chem. Phys. 73, 2360 (1980), and references therein. 53The error is larger for Mn,0.9eV, since pairing of the d electron in the Sd6 state increases the AE at this level. 24From unpublished calculations on the 52d2 and Sd3 states of the titanium atom. 25Without the additional tight f-function Martin2 saw only 0.1eV improvement in the sdg-szda AE of nickel by includ- ing single and double excitations from the 33 and 3p orbitals. 101 260. Munch, E. R. Davidson, J. Chem. Phys. 63, 980 (1975). 27R. L. Martin, P. J. Hay, private communication REPRINT 102 On the orbital description of the 433w“ states of the transition metal atoms" Thom. H. Dunning. Jr. Theoretical C hemutry Group. Chemmry Division. Argonne National Laboratory. Argonne. "(than 60439 Beatrice H. Botch'” and James F. Harrison Department of Chemistry. Michigan State University. East Lansing. Michigan 48824 (Received 7 December I979. accepied (8 December N79) The diverse and complex chemistry of the first row transition metal atoms is in large part due to the near degeneracy of the 4s and 3d orbitals. Thus, depending on the molecular environment, the formal configuration of the transition metal atom may be 4.42371", 4s3d'”‘, 01' 3d'". Clearly then, it is important that the theoretical methods used to study tranSition metal compounds be able to accurately predict the relative energies of these atomic states. Despite its widespread use in calcula- tions on molecules containing transition metal atoms, the Hartree-Foe): method does not satisfy this criterion, e.g. , for the nickel atom Hartree-Fool: calculations‘ place the ’D(4s3d°) state i. 28 eV and the ‘S(3d‘°) state 5. 47 eV above the 3I“(4.«i’3d‘) state, whereas the experi- mental separations are -0.03 eV and 1.71 eV.‘ In calculations on the ground and low-lying excited states of the titanium atom, we have found that the Hartree-Foch descriptions of the 433dJ and 34‘ states are inadequate. These calculations indicate that the “(he list" 3d orbital of the 4534!" states and the “(n +1)st" and “(n +2lnd” 3d orbitals of the 3d'" states of the transition metal atoms are functionally inequiva- lent to the other 3d orbitals, being much more diffuse. Thus, the proper orbital configurations of these states are 453d'34‘ and 3d'8d'3d". The primitive basis set used in the calculations on the titanium atom, (14sllp6d), is that of Wachters’ aug- mented with two additional 11 functions to describe the 4p orbital and one additioml, diffuse d function to de- J. Chem. Phw. 72(5), 1 Mar. 1980 0321 -96m/80/053419-02$Ol .(X) scribe the 3d orbital of the 453(1’ state. ‘ Hartree-Fock (HF) calculations with this basis set predict a sF-‘F splitting of 0. 55 eV; numerical HF calculations‘ give a nearly identical result, namely, 0.54 eV. For the sub- sequent atomic calculations the primitive set was con- tracted to [6561534] using the general contraction scheme of Raffenetti.s if one of the 3d orbitals in the sI-‘(4s3d°) state of tita- nium is functionally different from the remaining two 3d orbitals, the resulting projected Hartree-Fact: (PHF) wave function has the form (with the core orbitals de- leted) TABLE 1. Summary of Hartree-Foe): (HF) and multi- cont‘iguration Hartree-Foe): (MCHF) calculations on the 51714.434”) and 'Fih'lld‘) states of the titanium atom. ‘F(4ssd") 3F(4s2342) E," (hartree) -848.3715 -848.3918 isnfirdri (eV) 0.55 Eu," (hartree) - 848. 3882 - 848. 4202 c. 0. 90 0.96 c; 0. 43 0.29 5”,-En' (CV) -0.45 -0.77 AimyistF) (9V) 0.87 "° O 1983 American institute of Physics 103 3420 THE TITANIUM ATOM 13¢ r r r T V I T r 3d 0 3 _i-__1___.1___J malTAL AMPLITUDE ii l 00 l 1” ---“~-- Af I F )4: I l \ ,1 I " 1 .0" . .l 1 L 1 L 1 l 00 LOO 3.20 41.0 640 MM) FIG. 1. Radial plots of the 4s. 34, and 34’ orbitals of the ’inner’) state of the titanium atom obtained from multicontigurao tion Hartree-Fool: calculations. inf!) snowman, (1a) where 5 is a projection operator which insures that the wave function has the proper spatial symmetry. Noting that 3d' scu3d+c.4d , (1:) can be rewritten as a mulliconfiguraft'on Hartree- Focl: (MCHF) wave function Wucnél-‘l-c,d433d'aaao4c,v!4s3d'4d-aoaa . (lb) ft is in this form that calculations were carried out with the BlSON-MC program.“ There is, of course, a simple relationship between the two sets of coefficients (cu, c.) and (c,, c.) and, correspondingly, between the two wave functions (is) and (1h). The energy computed with (1), along with the HF re- sults, are given in Table l. The size of the energy lowering, 0.45 eV, and the large value of c” 0.43, re- flects the unusual strength of this correlation effect even though the electrons involved are in spatially different 34 orbitals and are high spin coupled. The 43, 3d, and 3d' orbitals obtained from the MCI-1F calculations are plotted in Fig. 1. The difference in the spatial extension of the 3d and 3d' orbitals is clearly evident. The 34' orbital is much more diffuse than the ad orbital (the overlap being only 0. 70) and is more like the 4s orbital in radial extent. hi earlier HF calculations on the transition metal atoms Hay‘ found that the 3d orbitals of the 43341" ‘ states were more diffuse than those of the 4s‘3d“ states. In fact, as noted above, basis sets appropriate for the 4034' states must be augmented with an additional, more Letters to the Editor diffuse d function to properly describe the 453!" states. From the present calculations we see that this uniform expansion of the 3d orbitals is a consequence of the equivalence restriction in the HF calculations and, when projection effects are properly taken into account, only one of the 3d orbitals becomes diffuse. To a good ap- proximation the 3d orbital of the 4531!“ state of titanium obtained from the HF calculation is an occupation num- ber weighted average of the 3d and 3d' orbitals. Turning now to the ground state, 31“(4s'3d'), of the titanium atom, we use the MCI-1F wave function, than?“ :- c1 .445’3d'afloa +c,544p’3d'oflao , (2) where the second configuration accounts for the 4s-4p near degeneracy effect. The results of the HF and MCHF calculations on the ’1’ state of titanium are also summarized in Table 1. Use of (2) decreases the energy of the ’1? state by 0. 77 eV over that obtained with a single configuration. Using (1) for the '5' state of titanium and (2) for the ’F state, we obtain a ‘F—‘F splitting of 0.87 eV, in good agreement with the experimental separation of 0. 81 eV.’ In her MCHF calculations on the copper atom, Froese- Fischer' noted that inclusion of the 4334'“ configuration in the calculations on the ’S(433d‘°) state led to a dra- matic decrease in the energy of this state (> 2 eV). Further, she noted that the resulting wave function was equivalent to a PHF wave function of the form 4s3a‘3d’. Thus, both the present titanium calculations and the copper calculations of Froese-Fischer" argue for the importance of a PHF description of the 4334'" states of the transition metal atoms. . "Work performed under the auspices of the Office of Basic Energy Sciences. Division of Chemical Sciences, U. S. De- partment of Energy. 'lpartictpont in the Laboratory-Graduate Partlcipantship Pro- gram administered by the Argonne Center for Educational Affairs. lS. Frags, Handboob of Atomic Data (Elsevier, Amsterdam. 1976). 2A veraged over M, values: energies from C. E. Moore. Atomic Energy Levels. Natl. Bur. Stand. (0.8.) Circ. 407. Vols. 1 and 2 (1949). ’A. J. ii. Wachters. J. Chem. Phys. 52. 103:) (1970). ‘P. J. Hay. J. Chem. Phys. 64. 4377 (1977). See also, a. R. Brooks and H. F. Schaefer In. Moi. Phys. 84. 193 (1977); w. C. Swope and H. F. Schaefer [I]. (bid. 84. 1037 (1977); and T. Smedley and W. A. Goddard m (unpublished). ’R. C. Raffenetti. J. Chem. Phys. 50. 4452 (1973). “'0. Des andA. c. with). as modified by a. C. Raffenettl. ’C. Froese-Fischer, J. Phys. B 10. 1241(1977). J. Chem. Phys., Vol. 72. No.5, 1 March l9m Part C: TRANSITION METAL ATOMS