ABSTRACT ABSTRACTION VS. INSERTION IN THE C + H2 AND N+ + H2 REACTIONS, A2 INITIO POTENTIAL ENERGY SURFACES; THE ELECTRONIC STRUCTURE OF THE 3A2 AND 382 STATES OF METHYLENE. BY David Arnold Wernette The first part of this thesis is concerned with the reaction of A(3P), where A = C or N+, with H2(12;) to form AH2 in its ground 331 state (i;gL, insertion) and AH in its lowest 2 and H states plus H(ZS) (iLEL, abstraction). A minimal basis set was employed, and with the constraint that the ls function on the heavy atom was always doubly occupied, a full CI wave function was used. Symmetry orbitals were constructed with the Gram-Schmidt technique, starting with the ls function on A. A minimum energy re- action path (MERP) for each potential energy surface was determined by locating the minimum energy H-H separation as a function of the A-H separation, where A-H2 is the 2 distance from A to (1) the center of mass of H2 for C2v geometries and (2) the nearest H in C00V geometries. David Arnold Wernette Since C(3P) + H2(l£;) correlates with an excited 331 state of CH2 and C(3P) + H2(3Zz) correlates with the ground 331 state, the lowest 381 surface for the CH2 in- sertion reaction is characterized by a double trough, where the maximum or barrier in between is the result of an avoided crossing of two SCF surfaces whose lineages are l + 3 + . ~ H2(.Zg) and H2( Eu). At a C-H2 separation of 2.8 bohrs, 1 + the 29 trough merges with the barrier leaving a single trough which leads to CH2(3B1) at the global minimum. 4. 2 its ground 3B1 state traces its lineage to N(4S) + H;(2£:) Except for the fact that the trough which leads to NH in rather than N+(3P) + H2(3£:), the basic topology of the lowest 331 surface for the NH; insertion reaction is the same as the CH2 surface. At infinite A-H2 separation, the H-H separation is 1.667 bohrs, E(C; 3P) = -37.4701 a.u., E(N+; 3P) = -53.5347 a.u., E(N; 4S) = -54.0629 a.u., and E(H2; 12;) = -l.1131 a.u., while at the global minimum the A-H2 separation is 1.050 bohrs (CH2) and .395 bohrs (NHE), the H-H separation has increased to 4.142 bohrs (CH2) and 4.468 bohrs (NHE), +. 3 2! a.u. The calculated AH is -39 and -108 kcal/mole for CH and E(CH2; 3Bl)= 48.64563 a.u. and E(NH B1) = -54.8200 2 and NH: respectively. The barrier to reaction or activa- tion energy is calculated to be 82 and 41 kcal/mole for CH2 and NH3, respectively. David Arnold Wernette The energy along the CHH(3£—) abstraction MERP increases monotonically from a C-H2 separation of ~4.5 bohrs to ~2.2 bohrs (where the H—H minimum disappears and there is no barrier to dissociation into CH(42') + H(ZS), while the H-H separation goes from 1.7 bohrs to infinity. The CHH(3H) abstraction MERP behaves similarly and its barrier to dissociation into CH(2H) + H(ZS) is zero at a C-H2 separation of ~2.55 bohrs. We obtain E = -38.0214 a.u. at Re = 2.335 bohrs for CH(4Z-), E = -38.0346 a.u. at R3 = 2.552 bohrs for cnczn), and E = -.4970 a.u. for H(ZS). The calculated AH is 41 and 32 kcal/mole for the Z and H abstraction reactions, respectively. The barrier to reaction is 42 and 35 kcal/mole for the 2 and H reac- tions, respectively. Since both of the NHH+ abstraction MERP's.(3Z- and 3H) indicate rather large relative increases in energy would be required to stay on them until they be- come equal in energy to NH+ + H (the NH+(2H) has Re = 2.477 bohrs and E = -54.1383 a.u., and the NH+(4Z-) has Re = 2.481 bohrs and E = -54.1569 a.u.), we did not ex- tend the Z or H MERP to N-H2 separations less than 1.9 and 2.0 bohrs, respectively. The energy of the H MERP increases monotonically as the N-H2 separation decreases. Since there is a deep minimum in the 2 MERP {-37.7 and -34 kcal/mole with respect to N+ + H and NH+ + H, 2 David Arnold Wernette respectively) at an N-rH2 separation of 2.665 bohrs and an H-H separation of 2.13 bohrs, the global minimum on the 2 surface corresponds to the bound linear [NHH]+ com- plex in a 32- state. Since the energy of both MERP's in- crease rapidly for N-H2 separations less than 2.5 bohrs, the barrier to reaction at an N—H2 separation of 2.5 bohrs is 0 and 8 kcal/mole for the Z and H abstraction reactions, respectively. The corresponding values of AH are -4 and 8 kcal/mole, respectively. The barriers to reaction predict that (l) the ab- straction reactions proceed with greater ease than the + 2 with greater ease than the analogous CH insertion reactions and (2) the NH reactions proceed 2 reactions. A comparison with eXperimental results suggests that the rotation of the Hz to form a system with CS symmetry may be important in the dynamics of these reactions. The second part of this thesis is concerned with existence of a bent 3A2 state with an apprOpriate energy (~8.75 eV above the ground state) and angle (~125°), so as to be consistent with the interpretation of the elec- tronic spectrum of CH2 which predicts a ground 381 state (~136°). This interpretation also predicts that the 3A2 state is heterogeneously predissociated by a 332 state. A minimal STD-3G basis was augmented by a set of 3p and 3d functions, and §3P = .47 and C3d = .34 were determined by Optimizing the energy of excited states of David Arnold Wernette the carbon atom. The energy and angular dependence of the non-Rydberg or valence state and three Rydberg states for each symmetry and the ground 3B1 state were deter- mined by constructing solutions of the HFR restricted Open shell equations. The existence of three highly bent 3A states 2 with angles of 127°, 120°, and 113° corresponding to 0—0 transition energies of 8.86, 8.30, and 7.53 eV, re- spectively, was obtained. Several 332 states, which may be reSponsible for predissociation, were obtained. ABSTRACTION VS. INSERTION IN THE C + H2 AND N+ + H2 REACTIONS, AB INITIO POTENTIAL ENERGY SURFACES; THE ELECTRONIC STRUCTURE OF THE 3A2 AND 382 STATES OF METHYLENE. BY David Arnold Wernette A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1975 ACKNOWLEDGMENTS The support of my wife, Suzanne, to whom this dissertation is dedicated, is gratefully acknowledged. Many helpful discussions with Richard Liedtke and Gregory Gillispie are gratefully acknowledged. Finally, I wish to express my deep appreciation to Dr. James F. HarriSon, who has played the major role in my development as a scientist. ii TABLE OF CONTENTS LIST OF TMLES O O O O O O O 0 O O O 0 LIST OF FIGURES . . . . . . . . . . . INTRODUCTION. . . . . . . . . . . . . Chapter I. THE c + H2 AND N+ + H REACTIONS 2 Potential energy surface preliminaries. The C + H2 insertion reaction . . . . . The N+ + H2 insertion reaction. The abstraction reactions . The C + 82 reactions . . The N+ + H reactions. . Discussion. .2. . . . . . . II. THE 3A2 AND 332 STATES OF CH2. The states considered . . The basis set . . . . . The method of calculation The results . . . . . . . Discussion. . . . . . . . O O O O 0 Appendix I. THE THEORY . . . . . . . . . . The Schradinger equation. . Configuration interaction . Open shell self-consistent field theory II. COMPUTATIONAL TECHNIQUES FOR THE POTENTIAL ENERGY SURFACES. . . . . . . The basis set . . . . . . The determinants and configurations Spin eigenfunctions . . . . products . . . . . . . . “FEENCES O O O O O O O O O O O O 0 O The quality of our representation . . Characterization of the reactants and Page iv vi 71 71 82 92 92 101 129 130 142 Table 1. 2. LIST OF TABLES Orbital symmetries for CH2 . . . . . . . . Electron configurations for the eight 3A2 and 332 states of CH2 generated by single excitations from the ground 3B 1 state. The major atomic function component of the orbitals involved in the excitations . . . The linear states of CH and their electron configurations. . . . . . . . . . The orthogonal Gram-Schmidt C atom basis functions . . . . . . . . . . The coefficients (dj) and exponents (aj) of the gaussian functions which represent the C, N, and H atomic orbitals. . . . . . Orthogonal Gram-Schmidt molecular orbitals in sz geometries . . . . . . . . Orthogonal Gram-Schmidt molecular orbitals in C00V geometries . . . . . . . . The fifty-one 381 configurations for AH2 . The forty-eight 3H configurations for AHH X The fifty-one 32- configurations for AHH . The eighteen 3H configurations for AH . . Y The ten 42- configurations for AH. . . . . The twelve 32- configurations for NH . . . The thirty-six 3P, 1D and 1S configurations for A . . . . . . . . . . . Determinants and configurations arising from four unpaired electrons (with MS a 0) . iv Page 60 61 62 65 93 96 96 102 107 110 115 116 118 119 125 Table Page 16. Determinants and configurations arising from five unpaired electrons (with MS==fl/2) . 125 17. Determinants and configurations arising from six unpaired electrons (with MS a 0). . . 127 18. C, N+ and N energy differences in electron volts. . . . . . . . . . . . . . . 132 19. Diatomic R.a values in bohrs. . . . . . . . . . 139 20. Diatomic De values in electron volts . . . . . 139 Figure 1. 2. 3. 4 9. 10. 11. 12. 13. 14. LIST OF FIGURES The internal degrees of freedom of AH2 . . Schematic H3 equipotential curves. . . . . Schematic energy profile along the H MERP 3 Cross sectional cuts of the two lowest 3 Bl surfaces Of CH at R = w . . . . . . . 2 Cross sectional cuts of the two lowest 331 surfaces of CH at R = 4.0 bohrs . . . 2 Cross sectional cuts of the two lowest 331 surfaces of CH at R = 3.4 bohrs . . . 2 Cross sectional cuts of the two lowest 331 surfaces of CH2 at R = 2.95 bohrs. . . Cross sectional cuts of the two lowest 331 surfaces of CH at R = 2.8 bohrs . . . 2 C + H2 correlation diagram (sz) . . . . . The "MERP's" and barrier for the lowest 3B1 surface of CH2. . . . . . . . . The MERP energy profiles and barrier for the lowest 381 surface of CH2. . . . . Schematic equipotential curves for the lowest 381 surface of CH2. . . . . . . . . Cross sectional cuts of the lowest lying 331 surfaces of NH; at R = 1 x 1012 bohrs. Cross sectional cuts of the two lowest 331 surfaces of NH; at R = 3.5 bohrs . . . vi Page 10 11 12 16 17 18 20 23 25 Figure 15. 16. 17. 18. 19. 20. 21. 23. 24. 25. 26. 27. 28. 29. Cross sectional cuts of the two lowest 331 surfaces of NH; at R = 3.0 bohrs . . . Cross sectional cuts Of the four lowest 3B1 surfaces of NH; at R = 2.5 bohrs . . . The "MERP's" and barrier for the lowest 381 surface of NH3. . . . . . . . . The MERP energy profiles and barrier for the lowest 381 surface of NH3. . . . . Coordinates for the abstraction reactions. MERP energy profiles for the lowest 2 and H surfaces of CHH and the H barrier. . . . Cross sectional cuts of the lowest 2 sur- face of CHH at R = 2.5, 2.4, 2.3 and 2.2 bOhrs. O O O O I O O O O C O O O O O 0 Cross sectional cuts of the lowest H sur- face Of CHH at R a 2.8, 2.6, 2.58 and 2.5 bOhrs. O O O O O O O O O O O O O O O O The "MERP's" for the lowest 2 and H surfaces of CHH and the H barrier. . . . MERP energy profiles for the lowest 2 and H surfaces of NHH+ . . . . . . . . . Cross sectional cuts of the lowest 2 surface of NHH+ at R = 2.0, 2.25 and 2.5 bOhrB. O O O O I O O O O 0- O O O 0 Cross sectional cuts of the lowest H surface of NHH+ at R = 2.0, 2.25 and 2.5 bOhrsO O O O I O O O O O O O O O O The "MERP's" for the lowest 2 and H surfaces Of NHH+ . . . . . . . . . . . . N+ + H2 correlation diagram (sz). . . . . Molecular orientations for linear and non-linear CH2 0 o s o s O o s o o o 0 vii Page 26 27 29 31 33 35 37 38 40 42 45 46 48 54 58 Figure Page 30. The 3A2, 332 and 331 energy curves . . . . . 68 31. CH2 and NH; orientations in sz and va geometries . . . . . . . . . . . . . . . 94 32. The H; 22; and 22: energy curves . . . . . . 133 33. The CH (2H) energy curve . . . . . . . . . . 134 34. The CH (42-) energy curve. . . . . . . . . . 135 35. The NH+ (2H) energy curve. . . . . . . . . . 136 36. The NH+ (42-) energy curve . . . . . . . . . 137 37. The NH (32-) energy curve. . . . . . . . . . 138 viii INTRODUCTION In the last 15 to 20 years there has been a rapid development in computer technology which has had a dra- matic effect on science. The approximate solution from ’first principles of the SchrOdinger equation for a poly- atomic molecule with more than a few electrons is char- acterized by heavy computational requirements. Although the fundamental equations for the electronic structure of polyatomic systems have been known for oyer 40 years, the develOpment of high-speed computers was crucial for the exploration and development Of techniques for solving the required equations at an 5g initio level. we have studied the electronic structure of several chemical reactions and several excited states of methylene at an ab initio level. CHAPTER I THE c + H2 AND N+ + H2 REACTIONS The first part Of this thesis is concerned with the potential energy surfaces for the reactions of the carbon atom and the nitrogen cation with molecular hydro- gen. The initial impetus for this study was an interest in the ease with which CH2 in its ground 381 state might be formed from C(3P) and H2 (12;). Two possible approaches were considered to be important. The first is the inser- tion reaction C + I = C1: . (1) H H In this reaction, the C atom is assumed to insert itself into an H2 molecule via an approach characterized by sz symmetry at all internuclear distances. The second ap- proach is the abstraction reaction c + H - H a c - H + H . (2) In this reaction, the C atom is assumed to abstract a hydrogen atom from the H2 molecule via an approach characterized by Ccov symmetry at all distances. The abstraction is part Of an overall reaction scheme whose second part is CH-i-HHCH2 or CH+H2=CH2+H. We considered only the initial abstraction represented by (2). This problem is easily extended to include the + reactive nitrenium ion, NH2 As in CH2, we considered the formation of NH; in its ground 3E state from N+(3P) l and H2(12;). The same insertion and abstraction reactions were considered for NHE. Although NHS is isoelectronic with CH2, the different nuclear charges in the two systems would lead one to anticipate the possibility of fundamen- tally different results. Since equations (1) and (2) represent only two of many possible reactions, some of the other reactions that may be written may be equally important. However, we anticipate that a detailed study of the potential energy surfaces of the reaction Of C(3P) and N+(3P) with H2(12;) will indicate whether abstraction (Gav) or in- sertion (sz) is preferred. Since NH; is isoelectronic with CH2, we hope to determine the role the positive charge plays in differentiating between the two approaches and the two systems. Potential Energy Surface Preliminaries There are 3 internal or relative degrees of free- dom in AH2 systems. These are illustrated in Figure 1 and are represented by R, the distance from the A nucleus to the center of mass H2, r, the H - H separation, and e, the angle between E and R. F“ 2“ o A:$A center Of mass of H2 Figure l.--The internal degrees of freedom of AHZ’ Although our calculations were restricted to 6 = 90° (sz) and 6 - 0° (Cmv), we shall have occasion to dis- cuss CS geometries where 0° < 6 < 90°. We computed approximate eigenvalues and eigen- functions of the SchrOdinger equation for a large number of geometries. The eigenfunctions and their variations were used to give insight into the electronic changes as the nuclei assumed different configurations. From the eigenvalues, we extracted a minimum energy reaction path (MERP). That point on the MERP which corresponds to the maximum.energy in the MERP is the saddle point. The saddle point may be viewed as separating the reactants and products. Since the MERP is usually defined [21] as the path of steepest descent from the saddle point of the potential energy surface, V, it may be generated by following the vector -VV ==" (EV/3R) i - (EV/3r) 3 from the saddle point (where -$V == 0). These definitions are illustrated in Figures 2 and 3. In this work, we constructed the MERP by using a more pedestrian technique which allowed for the possibility Of multiple local "MERP's." At’a given value of R, the r distance was varied. The r distance which resulted in a minimum in the energy defined a point on the MERP. Since more than one minimum may be found, there may be more than one MERP. This process was repeated over the whole range of relevant R values. For those surfaces with more than one MERP, the overall MERP was defined by the set of r values, detenmined at various values of R, which correspond to the lowest energy minima. Our goal was to determine the overall MERP, not full potential energy surfaces for the reactions. The barriers to reaction we calculated were defined as the energy at the highest maximum in the overall MERP minus the energy of the reactants. The barrier to reaction, Eb, is illustrated in Figure 3 for the H3 surface. The C + H2 Insertion Reaction The lowest 381 surface Of CH2 is characterized by an avoided crossing with the second lowest 331 surface. The two (SCF) surfaces which would have resulted if there were no avoided crossing are denoted by their lineage at [ * SADDLE POINT . -— - — MERP Hl-Hz SEPARATION- __., *’ 1 \ {—W-g—g 4—Hl-H2+H3 HZ-HB SEPARATION --> Figure 2.--Schematic H3 equipotential curves. REACTANTS I PRODUCTS SADDLE POINT Figure 3.--Schematic energy profile along the H3 MERP. R = w. One surface originates from C(3P) + H2(1£;), while the other surface originates from C(3P) + H2(3ZE), Or equivalently from C(3P) + 2H(2S). A particularly lucid view of the avoided crossing and the two resulting surfaces is given by cross sectional views of the surfaces at various values of R. Figures 4 through 8 show cuts at R = w, 4.0, 3.4, 2.95, and 2.8 bohrs, respectively. When comparing Figures 4 through 8, note that different energy scales are used. As shown in Figure 4, for R = m we Obtained the 12+ 9 a uniform energy lowering by an amount equal to the energy 12; curve is characterized by a minimum at r a 1.667 bohrs and a monotonically increasing energy as r increases from 1.667 bohrs. The 32: curve is repulsive usual potential energy curves of H2( ) and H2(32:) with of C(3P). The and its energy decreases monotonically with increasing r. l + For all values of r, the 2g curve is always equal to or below the 32: curve. Both curves merge at r = m resulting in a doubly degenerate pair of surfaces. As the H2 molecule is brought closer to the C atom, the above description remains essentially unchanged until R ~ 4.8 bohrs. At R ~ 4.8 bohrs, a maximum in the 12+ 9 3 + and a minimum in the Eu curves are found at relatively large r values Of ” 5.5 to ~ 6.5 bohrs. This is the initial stage of the metamorphosis of each state into a resultant 331 state. Although when the H2 system is non- interacting with C(3P), the 12; and 32: states are of -38.20 - + 32+ u -38.30 — e s a: . 5 a -38.40 '- I E. . z m I C(3P) 2 . +2H( S) -38.50 F . l + O . 2:g e e ' e ‘38'58 I 0'] 1 l 1 1+ 2 4 5 6 m 1.667 r (H-H) BOHRS Figure 4.--Cross sectional cuts Of the two loweSt 331 surfaces of CH2 at R = w. -38.45 -38.50 I I I I I I I I I g I - I 5 I a I n: I I!) z I "‘ I -38.55 ' I I I | I I I 3' '1' “i: '4: v] In. 3859i- . 1 J I! I I 11L 1‘ 2 3 41‘ 5 6 on H2 CH2 r(H-H) BOHRS Figure 5.--Cross sectional cuts of the two lowest 331 sur- faces Of CH2 at R = 4.0 bohrs. 10 -38.43 I- .2 C(3P) .-'+2H(ZS) -38.47 I- ° ENERGY (A.U.) JJ 3 II I II 1 1‘ 2 4f 5 1L5 a: H2 CH2 r(H-H) BOHRS Figure 6.--Cross sectional cuts of the two lowest 381 surfaces of CH2 at R = 3.4 bohrs. 11 -38.32 L- % -38.35 P 5-38.40" 51 >1 3:” I53 z I31 -38.45 - I . : C(3P)+2H(23) I l : l I , ' l I ,‘ l I {3 3 2 48'5” 4| SI .1 I I I I 1 4 .. H2 CH2 r (H-H) BOHRS Figure 7.--Cross sectional cuts of the two lowest 381 sur- faces of CH2 at R = 2.95 bohrs. 12 -38.33 P g] -33.35 I- -33.40 I'- ‘ 'I l ’7 “’ D . I 5 I I S I a: [11 Z I!) C(3P)+2H(ZS) -38.50 - '3 l l I -38.52 If I I I L 1 1‘ 2 3 4f” 5 1’ .. Figure 8.--Cross sectional cuts of the two lowest 381 sur- faces of CH2 at R = 2.8 bohrs. 13 strictly different spatial and spin symmetry, in the presence of C(3P), both states contribute to 3B1 states of CH2. The correlations are 3. 1‘1 1+.2_ 3 CI P. 2px ZPZI + H2 ( zg. cg) — CHZI Bl) 3.11 3+.11_ 3 and C( P, 2px ZpY) + H2 ( Zu' Cg on) - CH2( Bl) . At R ~ 4.8 bohrs, the two curves are almost degenerate at large values of r. The result is a mixing of the two curves (or surfaces) and an avoided crossing. In Figures 5 through 8, the solid lines are cross sectional cuts of the two lowest 381 surfaces. The two surfaces whose avoided crossing resulted in these sur- faces are represented by dashed lines in the region of the avoided crossing. We define the (12;) minimum at smaller r values as the "first minimum" or "first trough." The (32:) minimum at larger r values is defined to be the "second minimum" or "second trough." In Figure 5, where R = 4.0 bohrs, the lowest 381 surface is characterized by the first minimum at r = 1.642 bohrs, a slight maximum at the avoided crossing at r = 4.10 bohrs, and the second minimum at r 5.6 bohrs. From R “ 4.8 to R ~ 2.8 bohrs, the lowest 331 sur- face is characterized by a double trough and a maximum or barrier in between. These extrema become more evident as R decreases from 4.8 bohrs, as may be seen in Figures 6 l4 and 7. In this region, the character Of these extrema change significantly. The energy at the bottom of the first trough and the energy at the maximum increase as R decreases. Since the energy of the barrier is not in- creasing as fast as the energy of the first trough, the first trough is disappearing. The energy Of the second trough is decreasing very rapidly as R decreases. While the location of the first trough remains essentially unchanged at r ~ 1.65 bohrs, the location of the barrier and second trough are moving rapidly toward smaller r values. At R ~ 2.8 bohrs, the maximum and thus the avoided crossing have moved to such small values of r so as to coalese with the first minimum, as shown in Figure 8. The lowest 381 surface is now characterized by a single minimum, This single minimum is the second minimum whose 3 + lineage may be traced to C(3P) and 112( Sn). It is this minimum that then leads directly to CH2 in its ground 331 state as R decreases further. It is worthwhile to point out that this assign— ment must be tempered with the knowledge that it is im- prOper to speak of C(3P) and H2(32:) in this region of the surface. We are describing the 331 state of CH2 with a linear combination of 51 configurations, and the char- acterization of the surface as having 3:: lineage is a result of detailed analysis of this function. 15 The nature of the avoided crossing is easily seen by considering the correlation diagram shown in Figure 9 for the two 331 states. Note that the other possible 381 state which arises from C(10) and H2(323) is not energetically relevant. Since C(3P) + H2(IZ;) correlates with an excited 3B state with an orbital 1 occupation lai Zai 1b: 3a: lbi 4a: and C(3P) + H2(32:) correlates with the ground 381 state with the orbitals occupied as lai 2ai lb; 3a: lbi, the result is the in- dicated avoided crossing. In Figure 10, the first MERP which corresponds to the first trough, the second MERP, and the values of r at the maximum energy in the barrier are plotted for the lowest 381 surface. The rate of decrease of the second MERP, as R decreases, slows down for R values less than ~ 2.8 bohrs, where the first MERP and barrier have merged. There is a minimum in the second MERP at R ~ 2.1 bohrs. The merging Of the second MERP with the barrier is at R “ 4.8 bohrs. The energy profile along the two MERP's and the maximum energy of the barrier are plotted in Figure 11. The global minimum corresponding to CH2 in its ground 331 state with an energy Of - 38.64563 a.u. is at R = 1.050 bohrs and r - 4.142 bohrs. With reSpect to decreasing R, we see the first MERP merging with the barrier at R ~ 2.8 bohrs. With respect to increasing R, we see the second MERP merging with the barrier at R ~ 4.8 bohrs. 16 3 2 2 0 2 1 Bl (la1 2a1 lb2 3a1 lbl 3 4ai) 3 + C( P)+H2( Eu) I _ \I V \ I 32 m H 3 1 + C( P)+H2( £9) 3 2 2 2 1 1 B1 (la1 2a1 1b2 3a1 lbl) DECREASING R R.- w —5 Figure 9.--C + H2 correlation diagram (C 2V)' 17 6.0 b 5-5 b SECOND MERP 5.0 '- GLOBAL MINIMUM 4.5 - 0'1 é 4’0 ' BARRIER 8 ES A 3.5 L g 3.0 b 2.5 _ FIRST MERP 2.0 _ 1.5 csFflI — P J, J l_ 1 1 2 i; 3 4 5 w R(C-H2) BOHRS Figure 10.--The "MERP's" and barrier for the lowest 381 surface of CH2. 18 I I . 3 , C( P)+ - ‘ :,_e e e 2 2H( S) I E- . I I -38.50 -- ' I ' FIRST MINIMUM I :2; I I 5' - h- ' I w 38.55 g . I?!) E . . I - ., C(39) ' l + SECOND MINIMUM +H2I 29) I GLOBAL .. MINIMUM I I . ' I I I.- -38.65L I l 1 l I 0 1 2 3 4 5 l w RIC-H2) BOHRS Figure ll.--The MERP energy profiles and barrier for the lowest 3B1 surface of CH2. 19 A rough hand-drawn set of equipotential curves for the lowest 331 surface are presented in Figure 12. All available points were used to indicate the approximate tepology of the surface. Although the curves are not quantitatively correct, we feel that they do provide a good qualitative overview of the surface. The dashed, dash-dot, and dotted lines represent the second MERP, the first MERP, and barrier, respectively. The first MERP cuts across and into the side of the barrier which may be likened to a mountain. Eventually, the first MERP dis- appears leaving a steep descent down the side of the mountain into the valley below (i;§;, the second MERP) and thus eventually to the global minimum at 3» . Our barrier to reaction, Eb, change in enthalpy for the reaction, AH, and overall MERP will be based on the formation of CH2 from C(3P) + H2(IZ;l. As may be seen in Figure 11, the energy of each MERP is equal to - 38.484 a.u. at R a 3.125 bohrs. From R a m to R = 3.125 bohrs the overall MERP is the first MERP. At R = 3.125 bohrs, the overall MERP proceeds from the first MERP at r a 1.65 bohrs to the second MERP at r = 4.45 bohrs while rising over the barrier, 6b. For R < 3.125 bohrs, the overall MERP is the second MERP. For the overall MERP, the barrier to reaction is 3 .14- Eb s eh - E(c. P) — E(H2, £9) = .130 a.u. = 3.54 eV = 81.6 kcal/mole . 20 ~ -38.64 ~ -38.62 ~ -38.60 ~ —38.58 -38.56 ~ -38.54 ~ -38.52 ~ -38.50 ~ -38.48 —38.46 etc. \OmNIO‘U'IIhWNH I H O l r(H-H) BOHRS R (C-Hz) BOHRS Figure 12.--Schematic equipotential curves for the lowest 381 surface of CH2. 21 Since there is a very slight minimum in the bar- rier at R = 4.2 bohrs, the lowest possible value of Eb is Eb = .1176 a.u. = 3.20 eV = 73.8 kcal/mole . This energy corresponds to the approximate energy required to break the H2 bond. Perhaps a more realistic prediction of the bar- rier to reaction would result if the Hz were to remain bound in the first MERP until it became unbound at R ~ 2.8 bohrs. This results in Eb = .155 a.u. = 4.22 eV = 97.3 kcal/mole. Our reaction surface yields an exthothermic reaction with o = . 3 - o 3 _ o 1 + AH(O K) E(CH2, B1) E(C, P) E012, £9) = - .0624 a.u. = - 1.70 eV = - 39.15 kcal/mole . This result is 50.6% of the eXperimental result [22], AH(0°K) = - 77.33 kcal/mole. The N+ + H2 Insertion Reaction The basic topology of the lowest 381 surface of NH; is the same as the lowest 381 surface of CH2. There is a first minimum, trough or MERP at smaller r values, a second minimum, trough, or MERP at larger r values, and 22 a maximum or barrier at intermediate r values. Figure 12 also describes the qualitative features of the NH; surface. The first minimum traces its lineage to N+(3P) and H2(12;). The second minimum does not trace its lineage to N+(3P) + H2(32:) (as suggested by the C + H2 results) but to N(4S) + H:(2£:), or equivalently N(4S) + H(ZS) + H+. The N+(3P) + H2(3£:) surface as well as the N(2D) + H;(22;) do interact significantly with these lower surfaces. The complexity of the situation is il- lustrated in Figure 13, where the relevant manifold of 331 states are shown by a cross sectional cut with R fixed at 1 x 1012 bohrs. The potential energy curves of the H2(1£;), H2(3ZE), H;(2£;) and H;(ZZ:) states may be easily seen. They are, of course, lowered uniformly by the energy of the appropriate state of N or N+. Since R = 1 x 1012 bohrs, the various states of H2 and H; are non—interacting and there are no avoided crossings. As the two systems approach each other and the metamorphosis into an NH; 381 description begins, each curve or surface crossing in- dicated in Figure 13 becomes avoided. We note that at very large values of R there will be a barrier or maximum due to the avoided crossing of the two lowest surfaces at r ~ 5.0 bohrs. The corresponding barrier in CH2 merged with the second trough at R ~ 4.8 bohrs and was non- existent at very large values of R. -54.30 - ' + 2 + I H2( 2:11) I I I . I I I "54340 I- . I ' N(ZD) ' +H(2S) I +H+ I = :a': :3 I . N+(1D) é . +2H(2S) v I I I V -54.50 - I E] I I ' . N+(3P) ». . 2 2 _ +2H( S) + + H2( 29) I u . 3 + H2‘ 2u) N<4s) . +H<25) + -54.60 r- - 1 + +H H ( Z . g) I I I I I -54.65 1 l l l I ‘1, l 2 3 4 5 6 0° r(H-H) BOHRS 3 Figure l3.--Cross sectional cuts of the lowest lying B1 surfaces of NH; at R = l X 1012 bohrs. 24 Figures 14 through 16 show cross sectional cuts of the two lowest 331 surfaces with R fixed at 3.5, 3.0, and 2.5 bohrs, respectively. As we moved the two hydrogen nuclei toward the N nucleus, a minimum in the lowest 331 surface starts at R " 5.0 bohrs and r ~ 8.0 bohrs. This is the second minimum which is the NH; analog of the second minimum in CH2. In CH2, the second minimum was associated with its lineage from H2(32:). In NEE, the second minimum is associated with its lineage from H;(2£:). There is no corresponding maximum in the 12; curve, as was the case in CH2. The lowest 331 surface of NH; is now character- ized by the first minimum at r = 1.7 bohrs, the second minimum and the maximum or barrier in between as a result of the avoided surface crossing. The location of the barrier for R > 3.5 bohrs was not determined. Since it was at r ~ 5.0 bohrs at R = 1 x 1012 bohrs, it seems reasonable to assume it lies between r = 1.7 and 5.0 bohrs. The basic features of the lowest 331 surface of NH+ from R ~ 5.0 to R ~ 2.6 bohrs are the same as those 2 for the CH surface from R ~ 4.8 to R ~ 2.8 bohrs. The 2 energy of the first trough is rising and the energy of the second trough is decreasing as R decreases. Neither of these changes are as dramatic as in the CH2 surface. As R decreases, the r values which define the first and 25 -54.47 -54.50 -54.55 ENERGY (A . U .) r(H-H) BOHRS Figure l4.--Cross sectional cuts of the two lowest 331 sur- faces of NH; at R = 3.5 bohrs. ENERGY (A.U ) -54.52 “54.54 -54.56 -54.58 26 b N+<3P) J +2H(ZS) y— o N(4S)+H+ f +H(zs) \1’ 2“ . _. I . I l 5 I . " 2.. 9:! flls'q: I i“! 1. ltr 1.§7f 2.0 2.5 .3.0 w H2 r(H—H) BOHRS Figure 15.--Cross sectional cuts of the two lowest 381 sur- faces of NH+ at R = 3.0 bohrs. 2 -54.45 -54.50 -54.55 ENERGY (a.u.) -54.67 N+(3P)+2H(ZS) N(4s)+H+ I . +H(§S) L— .' h : J, J_ LL 2 3 4 5 w H2 NH; r(H-H} BOHRS Figure 16.--Cross sectional cuts of the four lowest 331 sur- faces of NH; at R = 2.5 bohrs. 28 second MERP's remain essentially unchanged and decrease rapidly, respectively. The r values which characterize the maximum in the barrier decrease rapidly from R ” 3.5 to R “ 2.6 bohrs.- Eventually, as on the CH2 surface, the barrier and the first minimum merge as R approaches ~ 2.6 bohrs. The result is a single minimum or trough for R less than ~ 2.6 bohrs in the lowest 331 surface. This is the second minimum, whose lineage may be traced to N(4S) + 11382:) , and it leads directly to NH: in its ground 331 state at the global minimum. Cross sectional cuts of the third and fourth lowest 331 surfaces are also plotted in Figure 16. The first minimum at R a 2.5 bohrs resides totally in the second lowest 331 surface, which also has a maximum. The shape of the second lowest surface whose energy is rising appears to be affected by strong interactions with the third and fourth lowest 331 surfaces. The first MERP, second MERP, and the values of r at the maximum energy in the barrier are plotted for the lowest 331 surface in Figure 17. The barrier was not determined for R > 3.5 bohrs. We suggest that it seems reasonable to assume that the barrier will monotonically increase from r = 3.38 bohrs at R = 3.5 bohrs to r ~ 5.0 bohrs at R = w. Except for the fact that the barrier in the NH; surface is present for all R > 2.6 bohrs, this 29 8.4 - 8.0 - SECOND MERP 7.0 h 6.0 #- GLOBAL 5'0 T' MINIMUM , r(H-H) BOHRS a. To BARRIER 3.0 F" FIRST MERP 2.0 "' 1.6 k l 1 L 1 1 1 o 2 4 5 .. R(N-Hz) BOHRS Figure 17.--The ”MERP's" and barrier for the lowest 381 sur- face of NH3. 30 plot is remarkably similar to the corresponding plot ob- tained for CH2 in Figure 10. The energy profile along the two MERP's and the maximum energy of the barrier for lowest 3Bl surface of NH; are plotted in Figure 18. The global minimum corre- sponding to NH; in its ground 331 state with an energy of - 54.81995 a.u. is at R = .395 bohrs and r = 4.468 bohrs. The global minimum is substantially lower with respect to the reactants than the CH2 global minimum. Except for the depth of the global minimum, the energy changes in Figure 18, though paralleling those for CH2, are not as large or dramatic. The merging of the first trough and the barrier at R ~ 2.6 bohrs may be inferred from Figure 18. The global minimum is indeed very shallow. The bar- rier to linearity is so small so as to not be detectable on the scale of Figure 18. There is a minimum in the energy of the barrier (see Figure 18) at R = 3.1 bohrs which also coincides with the value of R at which the two energy profiles for the two MERP's are equal. Thus, E for the overall MERP b is equal to the lowest possible value of Eb for the reac- tion. The overall MERP (1) is the first MERP from R = w to R = 3.1 bohrs, (2) proceeds from r = 1.8 bohrs at the first MERP to r = 4.5 bohrs at the second MERP, and (3) is the second MERP for R < 3.1 bohrs. The energy of the -54.50 -54.55 -54.60 -54.65 -54.70 ENERGY (A.U.) -54.75 -54.80 -54.84 31 p— BARRIER P N(4S)+H+ _ +H(ZS) L . F . N+(3P) ' + SECOND +H2(l£g) MINIMUM _ FIRST MINIMUM b GLOBAL . MINIMUM If 1 I V 1 4 1 1 o 1 2 3 4 5 m R(N-H2) BOHRS Figure 18.--The MERP energy profiles and barrier for the lowest 381 surface of NH3. 32 barrier at R = 3.1 bohrs is Eb = - 54.582 a.u. Thus, the barrier to reaction is - - +0 3 - I 1 + Eb — eb .E(N , P) E(H2, 2g) = .066 a.u. = 1.80 eV = 41.4 kcal/mole . The NH; barrier is about one-half of the corre- sponding CH2 barrier, a significant difference. Although the true barrier in the NH; reaction might easily be less than 50% of the calculated barrier, our results suggest that it is non-trivial. Although this is contrary to the hypothesis that reactions involving ions tend to pro- ceed with little or no barrier, we caution that our re- sult is based on an examination of only the C approach. 2V If we consider the H2 as remaining bound in the first trough until it becomes unbound when merging with the barrier, we estimate from Figure 18 that Eb is ap- proximately .1 a.u. = 2.7 eV = 63 kcal/mole. Our reaction surface predicts an exthothermic reaction with 3 + 3 1 + AH(0°K) B1) - E(N . P) - E(H2. lg) + - .172 a.u. = - 4.68 eV = - 108.0 kcal/mole , which is 75% of the experimental result [23], AH(298°K) = - 144.0 kcal/mole, and represents a significantly higher 33 percentage of the experimental result than the calculated AH for CH (50.6%). 2 .The Abstraction Reactions The coordinates used in this study were R, the distance between A and the closest hydrogen, and r, the H - H separation. These are illustrated in Figure 19. X A _...___.._ I I I K R flt—v r‘-—* a: :I: v N Figure l9.--Coordinates for the Abstraction Reactions we fixed R and moved the far right proton by varying r until a minimum in the energy was found. At r = w, the energy of the system was that of the AH molecule at a separation of R plus the energy of H(ZS). Neither CH- + H+ nor NH + H+ were obtained as the lowest asymptotic limit for the abstraction reactions considered. The bar- rier for the far right hydrogen to leave was just the dissociation energy of A - H - H into AH + H. At R = m, the barrier is De for H2(1Z;). As H2 approaches A, there will always be a bar- rier to dissociation as long as the A - H - H energy is below the energy of AH + H at the same A - H separation. An important question is whether the energy of this 34 barrier increases monotonically from the minimum as r in- creases or whether the energy barrier has a maximum. If A - H - H is bound with respect to AH + H and its energy has risen above energy of AH + H, then there must be a maximum in the barrier to dissociation. The C + H2 Reaction We have studied the C + H2 abstraction reactions for the lowest lying Z and H surfaces, 32' and 3H. At R = w, the r value at the minimum energy is 1.67 bohrs which is characteristic of H2(12;). Since both the Z and H surfaces correlate with C(3P) and H2(12;), the two surfaces are degenerate at R = w. Their barrier to dissociation is De(H2; 12;) = .1192 a.u. There is no maximum in the barrier to dissociation, except at r = w. This description remains essentially unchanged as R decreases until R ~ 4.5 bohrs. The minimum energy with respect to r for both the 2 and n troughs begins to in- crease as R descreases from ~ 4.5 bohrs. The energy at the minimum, which is the energy profile along the MERP, is plotted for both surfaces as a function of R in Figure 20. Also plotted as a function of R are (l) the CH(4Z-) + H(ZS) and CH(2H) + H(ZS) curves for r = m; and (2) part of the barrier which corresponds to the maximum energies in the dissociation of CHH(3H) into CH(2n) + H(ZS). From R ~ 4.5 bohrs, the minimum energies increase very rapidly 35 -38.49'-' -38.50- CH(42 >+H<2s) H BARRIER TO DISSOCIATION -38.52 - . I :- I . . ' CH(2H)+H(ZS) E; -38.54 # . I 5 >4 U M m I Z [:1 CHH(3H) -38.56 _ I CHH(32‘) I -38.58 - ' ‘z. 3 1 + .. C( P)+H2( 2g) I I I I L_ 2.0 2.5 3.0 3.5 4.0 w R(C-H) BOHRS Figure 20.--MERP energy profiles for the lowest 2 and H sur— faces of CHH and the H barrier. 36 on the 2 surface until R ~ 2.2 bohrs and on the H surface until R ~ 2.55 bohrs. At R ~ 2.2 bohrs, the 2 surface is characterized by no barrier to dissociation into CH(4Z-) + H(ZS). On the H surface, the barrier to dissociation into CH(2H) + H(28) disappears at R ” 2.55 bohrs. Figures 21 and 22 follow these disappearances by displaying the r dependence of the energy for the 2 surface at R = 2.5, 2.4, 2.3, and 2.2 bohrs and for the H surface at R = 2.8, 2.6, 2.58, and 2.5 bohrs. From Figure 21, we see that on the 2 surface, the H - H potential well broadens and the energy at minimum rises as R decreases. The Z MERP moves to larger values of r as R decreases. There is no maximum in the barrier to dissociation for R < 3.0 bohrs for the 2 surface. The 3251 slight barrier at R = 2.2 bohrs in Figure 21 is energetically insignificant, especially at our level of description. At R ~ 2.2 bohrs, the H - H minimum dis- appears and the system can drOp into CH(4£-) + H(ZS). From Figure 22, we see that the H surface behaves similarly, except that it is characterized by a maximum in the barrier to dissociation for the range of R values considered. The location of this maximum moves to smaller values of r as R decreases andat R ~ 2.55 bohrs, coaleses with the H - H minimum. For R less than ~ 2.55 bohrs, the energy of CHH(3H) decreases monotonically to the energy of CH(2n) + H(ZS) as r increases. -38.51 -38.52 ENERGY (A.U.) -38.53 37 CH(4Z-)+H(zs) R=2.2 RCH=2°5 . '° RCH=2°2 I ‘__._e — .-,- - ~”—' RCH-2'3":::> R=2.3 . RCH'2.4 ° .- - R=2.4 ' - R=2.5 .1 J I I 2.0 2.5 3.0 3.5 ‘1’ w r(H-H) BOHRS Figure 21.--Cross sectional cuts of the lowest 2 surface of CHH at R = 2.5, 2.4, 2.3 and 2.2 bohrs. 38 -38.522 L CH(2H)+H(ZS) -38.525 - . '. '. RCH=2'8 .1 R=2.6 °. :5 -38.530 - ° ‘ : <3 2 .°2 RCH=2’5 s -_ g RCH=2.6 2 I23 -38.535 b ° R=2.8 -38.540 r . . l 1 I I 1.5 2.0 2.5 3.0 3.5 ‘1r w H2 r(H-H) BOHRS Figure 22.--Cross sectional cuts of the lowest H surface of CHH at R = 2.8, 2.6, 2.58 and 2.5 bohrs. 39 The 2 MERP, H MERP and the values of r at the maxima in the H barrier are plotted as a function of R in Figure 23. Both MERP's show a rapid increase toward larger values of r as they approach the R values where they have no barrier to dissociation. When the MERP's reach these R values, they rise vertically to r = w since there is no barrier to dissociation. The overall 2 MERP is the MERP plotted in Figure 23. The barrier in the Z abstraction reaction is 3 [11 ll E(CHH at R 2‘) 2.2 bohrs; 3 l + - E(C, P) - E(H2, 2g) .0673 a.u. 1.83 eV = 42.4 kcal/mole . For the overall reaction, 4 2 AH(O°K) E(CH at Re’ 2‘) + E(H; S) 3 1 + " E(Co P) - E(H21 29) .0649 a.u. = 1.77 eV = 40.7 kcal/mole . Since the energy of CH(2n) + H(ZS) equals the CHH(3H) energy at R ~ 2.65 bohrs, the overall H MERP will follow the n MERP in Figure 23 from R = m to R = 2.65 bohrs. At R a 2.65 bohrs, the overall MERP will go from r a 1.98 bohrs to r = m. Since the energies of the maxima in the barrier are essentially equal from R ~ 2.55 to 4O 3.2 - 3.0 - HBARRIER ‘2 m 2.5 b’ O m "I: J CHH(32-) MERP :1: 7: “— 2.0 — CHH(3H) MERP )- 1'5 I J I I L 2.0 2.5 3.0 3.5 4.0 m RCC-H) BOHRS Figure 23.--The "MERP's" for the lowest 2 and H surfaces of CHH and the H barrier. 41 R ~ 2.7 bohrs, we calculated only one barrier to re- action: I!) ll _ _ 3 _ .3 _ . 1 + b eb(CHH at R e 2.65 bohrs, H) E(C, P) E(HZ’ 29) .0563 a.u. = 1.53 eV = 35.4 kcal/mole . For the overall reaction, AH(0°K) E(CH at Re; 2n) + E(H; 25) 3 l + E(C, P) - E(Hz, £9) .05165 a.u. = 1.405 eV = 32.4 kcal/mole . This result is 39.5% larger than the experimental value [22], AH(0°K) = 23.2 kcal/mole. The N+ + H2 Reaction As in the C + H2 abstraction reactions, all the nuclei were on the Z axis with the N - H and H - H separa- tions represented by R and r, respectively. When N+(3P) is infinitely far from H2, we have R = w and r = 1.67 bohrs (the H2(1X;) Re value) at the minimum energy. When NH+(4Z- or 2H) + H(2S) is formed, we have R = 2.5 bohrs for both states and r = m. In Figure 24, we display the minimum energy as a function of R and r for the 2 and H surfaces. There is a relatively deep global minimum (1.63 eV with respect N+(3P) and H2(IX;)) occurs in the Z +, surface corresponding to the linear complex [NHH] in a 42 -54.60 b NH+(42‘)+H(25) -54.62 — NH+(2H)+H(ZS) -54.64 — + 3 l + N ( P)+H2( 29) D 5 NHH+(3H) >4 U M g m -54.68 - NHH ( 2‘) —54.71 — . m I 3.} N I I I I I I I A 2 3 4 5 6 y .. R(N-H) BOHRS Figure 24.—-MERP energy profiles for the lowest 2 and H sur- faces of NHH . 43 3... 2 state. This may be compared with the monotonically increasing energy of the 32- state of CHH shown in Figure 20. The binding energy of 1.63 eV for the [N - H - H]+ 32- state is about one-third of the correSponding computed value for NH;(3B1), 4.68 eV. In the complex, the N - H distance is 2.665 bohrs while the H - H separation has increased to 2.13 bohrs; the energy is - 54.7079 a.u. Also plotted as a function of R, in Figure 24, are the NH+(4Z-) + H(ZS) and the NH+(2H) + H(2S) curves at r = w. At R = 2.0 bohrs, well past the NH+(2H) mini- mum at R = 2.477 bohrs, the energy of the H MERP still lies below the energy of NH+(2H) + H(ZS). At R = 1.9 bohrs, well past the NH+(4Z-) minimum at R a 2.481 bohrs, the energy of the 2 MERP still lies below the energy of NH+(4£-) + H(ZS). Both states are characterized by a potential which is less favorable to dissociation than their CH2 counterparts. Since both surfaces indicate that rather large relative increases in energy will be required to stay on the NHH+ MERP's until they become equal in energy to the NH+ +H curves, we did not extend the 2 or H MERP to R values less than 1.9 and 2.0 bohrs, respectively. Since the NH+ 42- and 2H minima are both close to R - 2.5 bohrs, we focused our attention on the section of both surfaces at R = 2.5 bohrs. This decision was also influenced by the facts that the [N - H - H]+ complex has 44 its minimum energy near R = 2.5 bohrs, and the energy along the H MERP increased quite rapidly for R < 2.5 bohrs. Cross sectional cuts, as a function of r with R = 2.0, 2.25 and 2.5 bohrs, are plotted in Figures 25 and 26 for the Z and H surfaces, respectively. At R = 2.5 bohrs, the energy increases monotonically as r + w on both surfaces to NH+ + H. We have assumed that the curves for the other values of R behave similarly. In Figure 25, the energies at the Z minima rise rapidly, while there is only a small shift to larger r values at the minima, as R decreases. If we assume that the interpolated portions of Figure 25 are correct, then the barrier to dissociation decreases rather slowly from .0612 a.u. at R = 3.0 bohrs to .0528 a.u. at R = 2.5 bohrs to .0390 a.u. at R = 2.0 bohrs. At R = 1.9 bohrs, the barrier is ~ .03 a.u., and the rate of decrease of the barrier seems to be slowing. When the barrier does go to zero, it will be at a high energy compared to the scale of Figure 25. In Figure 26, the energy and the H - H separation at the H minima rise rapidly as R decreases. If we assume that the interpolated portions of Figure 26 are correct, then the barrier is decreasing rapidly from .0253 a.u. at R = 3.0 bohrs to .0073 a.u. at R = 2.5 bohrs to .0022 a.u. at R = 2.0 bohrs and vanishes at R ~ 1.8 bohrs and a relatively high energy. 45 -MI RNH=2'O RNH=2.25 ,.-"' RNH=2.5 -54.66)— ' :5 ' + 4 + 2 5 NH ( 2 )+H ( S) G If] -54068— . - E: I R=2025 ' R=2.5 -54I70P . “54.71— I l l I 1 l1 Tb 2 3 4 5 6 7 w H2 r(H—H) BOHRS Figure 25.--Cross sectional cuts of the lowest 2 surface of NHH+ at R = 2.0, 2.25 and 2.5 bohrs. 46 NH+(2n)+H(ZS) -54.53- A -54.60 h- RINK-4.0 D: . . I III. ooooooo a. SE S a: [I] 2 III -54.62I. R=2.25 RNH=2.25 RNH=2.5 -54.64 '- [ 2 3 4 s a 741r-1:- H 2 r(H-H) BOHRS Figure 26.--Cross sectional cuts of the lowest H surface of NHH+ at R = 2.0, 2.25 and 2.5 bohrs. 47 In Figure 27 the Z and H MERP's are plotted as a function of R. The rate of increase in the 2 MERP is fairly constant from R ~ 5.0 to R ~ 1.9 bohrs. The Z MERP shows no tendency to increase rapidly as the CH2 MERP's did. The H MERP increases rapidly and linearly for 2.0 < R < 3.0 bohrs. For the R values considered, neither MERP shows a very rapid increase to r = w as the CH2 MERP's did just prior to their barriers to dissociation going to zero. The barrier to reaction is zero for the 2 surface. For the Z abstraction reaction, we calculate AH(O°K) = E(NH+ at R = 2.5; 42-) + E(H; 2S) + 3 l + - E(N , P) - E(HZ’ 2g) = - .0061 a.u. = - .165 eV = - 3.8 kcal/mole . This result is only 27.3% of the experimental value [23], AH(298°K) a - 13.9 kcal/mole. In our reaction scheme, NHH+(3£-) is a reaction intermediate, and NH+(4£') + H(25) are the products. Since the global minimum on the 2 surface corresponds to the bound [N - H - H]+ complex, the [N - H - H]+ complex.might be described as the product where Eb = 0 and AH(0°K) = - .0589 a.u. = 1.605 eV 2 - 36.8 kcal/mole. The barrier to dissociation of NHH+(3Z-) into NH+(4Z-) + H(ZS) at R = 2.5 bohrs is 33 kcal/mole. Since the prediction of 48 307- 3.5 ‘ NHH+(3H)‘MERP 3.0 b (D m :13 O m a; 2.5 ' Si . H o GLOBAL MINIMUM .I + NHH (3 - 2.0 - Z ) MERP 1.5 J l I s 1' . J 4 U1— R(N-H) BOHRS Figure 27.--The "MERP's" for the lowest 2 and H surfaces of NHH+. 49 the products and their relative probabilities of being realized requires a detailed consideration of the dynamics of the system, we can say little more. Since there is no maximum (except at r = co) in the barrier to reaction on the H surface, Eb = AH(0°K) = E(NR+ at R = 2.5; 2n) + E(H;ZS) + 3 1 + - E(N I P) " E(Hzl £9.) and E = AH(0°K) b .0125 a.u. = .34 ev = 7.85 kcal/mole . Discussion The experimental values for the AH of reaction for the lowest surfaces which correSpond to NH;(3BI), CH2(331), CH(2H) + H(2S) and NH+(4Z-) + H(ZS), are - 144.0, - 77.33, 23.2 and - 13.9 kcal/moles, respec- tively, while our corresponding computed values are - 108.0, — 39.15, 32.4 and - 3.8 kcal/mole. There seems to be correlation between the magnitude of the experi- mental values and our ability to predict them. As the magnitude of AH increases, the quantity [(AH expt expt - AH )/AH also increases. The smaller the here exptl energy difference between the reactants and products, the less sensitive our predictive capabilities become at our level of description. 50 The effect of increased nuclear charge in the in- sertion reactions, as discussed earlier, could be seen 4. 2 relative to CH2. -However, a sizable barrier was still by a large decrease in the barrier to reaction for NH inferred for the true NH; surface. The barrier to re- action for the C + H2 2 and H abstraction reactions were 42.25 and-35.4 kcal/mole, while the NH; analogs were 0 and 7.85 kcal/mole, respectively. Even though the dif- ferences in these barriers are about the same or less than the difference in the insertion reactions, the ef- fect of increased nuclear charge appears to be more significant in the abstraction reactions. Since the H must dissociate from an NH+, the charge in the NHH+ system seemed to have a profound effect on the NHH+ ab- straction reactions. The Z and H NHH+ barriers do not reflect this, since we computed them at the lowest pos- sible barrier to reaction. The nature of the NHH+ 2 and H "MERP's" do reflect this effect. Nevertheless, we sug- gest there will be barriers in the true C + H abstraction 2 surfaces and little or no barriers in the true N+ + H2 abstraction surfaces. In order to compare abstraction vs. insertion, we will use the results for the 2 abstraction reactions, since they correlate directly with CH2 in its ground 331 state. Both abstraction reactions have barriers to reaction which are about 40 kcal/mole less than the 51 corresponding barriers in the insertion reactions. A straightforward comparison using the barriers to reaction predicts the abstraction mechanisms to be preferred in both systems. There is not very much eXperimental information on these reactions. There are two previous eXperimental studies which tend to suggest by comparison with our re- sults that the description of these reactions is more complex than the constrained systems we considered. Braun, Bass, Davis and Simmons [24] considered the reaction of C(3P) with H2 to form CH in an inert 2 argon medium. The C(3P) was generated by the flash jphotolysis of C203. At high total pressures, where the 4 2 argon pressure was 7 x 10 Nm- and the H 3 2 pressure was of the order of l x 10 Nm-z, they described the reaction as being highly efficient and characterized by having a high collision efficiency estimated to be within 1 and .1 They concluded that the reaction had a very low activation energy. Since our results suggest that the barrier to reaction on lowest true 381 surface describing C + H2+CH2 is substantial, there may well be a path to CH in its 2 ground 3B1 state from C(3P) and H2(1£;) that avoids the predicted barrier. Fair and Mahan [25] have studied the N+(3P) + H2(12;) + NH+ + H reaction using crossed beams. At relative energies near and below 1 eV, they interpret 52 their results as predicting a long-lived collision com- plex. At relative energies above 2 eV, they find no evidence of a long-lived complex and describe the reac- tion as proceeding by a direct interaction mechanism. They associated the long-lived complex with the system visiting the deep ground state 3B1 potential well of NH3. The correlation diagram they presented predicted only a slight minimum of ~ .5 eV with respect to N+(3P) + H2(IX;) for NHH+(32') in CGov geometry. Since our sur- face predicts the existence of a bound (1.63 eV) [N--H--H]+ complex, this 32‘ state may also result in a long-lived complex. The NHH+(3£-) may also provide a low energy path to NH;(381). In the NH; insertion reaction (CZV), the barrier is encountered until R decreases to ~ 3 bohrs. From R at “ 3 bohrs to the global minimum, the energy is :monotonically decreasing. In va geometries, the energy decreases monOtonically as R decreases to ~ 2.7 bohrs and then increases as R decreases from ~ 2.7 bohrs. The NH; system.may find a considerable potential energy con- straint to be in a va geometry until R has decreased to this range and thereby avoid the large barrier in a sz geometry. At R from “ 2.7 bohrs or less, the potential energy may dictate a rotation of the H to yield NH;(331) 2 whose energy then decreases rapidly. 53 The rotation of H2 to yield a system with a CS symmetry could be important in explaining the discrepancy between our prediction and the results of Braun gt_al. The essence of this theory was discussed by Fair and Mahan. In order to find a direct interaction mechanism for the higher relative energies leading to NH+ + H, they noted that if the system is initially in C geometry, a 2V rotation of the H2 gives the system CS symmetry. The 381 and 3A2 states arising from N+(3P) and H2(1£;) become 3A" states which are degenerate and should exhibit an avoided crossing. This is shown in the schematic C correlation 2V diagram in Figure 28. The avoided crossing of the 331 sur- faces we characterized is indicated. The avoided crossing of the 3 A" surfaces is inferred by the dotted lines. Fair and Mahan proposed a diadiabatic surface jump so as to avoid the long-lived NH;(3BI) complex at higher relative energies. The results of Braun et a1. may well result if (1) the C(3P) and H2(12;) initially form CH2(3A2) before 3 the A" avoided crossing, (2) rotate into C geometry, 8 and-(3) adiabatically via the avoided crossing end up on the portion of the lowest 3A" surface which leads to CH2(331) when rotation of the H2 occurs again. This path should provide a much lower barrier than staying in a sz geometry when forming CH2(3Bl). 54 3 B1 4 +2+ N( s)+H2< Zn) I \l IK\ >+ I U m 3 m A 5: +3 1+ 2 N ( P)+H2( 2g) 3 Bl DECREASING R R: 00 _:. Figure 28.--N+ + H2 correlation diagram (sz). 55 Since the 3A2 state also correlates directly with 3 2 the lowest H of AHH and therefore with the lowest H state of AH, the rotation of H into C geometries pro- 2 S vides a number of possible paths to AH + H and AH2(381). The determination of the CH and NH; potential 2 energy surfaces for CS geometries would be very important in exploring the hypotheses presented. The apparent com- plex nature of chemical reactions for even these rela- tively simple systems presents a formidable problem. Our increasing ability to understand chemical reactions has brought a very important dimension of chemistry into better focus. CHAPTER II THE 3A2 AND 3B2 STATES OF CH2 In 1961, Herzberg [1] published an interpretation of the UV spectrum of CH2 based on a linear or nearly linear ground state. In this interpretation, the diffuse" band at 1415 A.was assigned to the 32; + 32; transition. In 1971, Herzberg and Johns [2] prOposed a new interpretation of the 1415 A band which was consistent with more recent experimental and theoretical predictions of a ground 3 B1 (32; in Duh) state. As Herzberg noted in his earlier paper, the K' = 0 + K" = 0 subbands could be the result of a bent-bent type transition. However, the absence of the higher subbands K' a l + K" = l, K' = 2 + K" a 2, etc. favored a linear-linear type transition. On the other hand, the absence of these higher subbands could be the result of heterogeneous predissociation of an ex- 3A2(3£; in Duh) state by a 332 state. The 1415 A band may then be ascribed to a 3A2 + 3 cited B1 transition. The predissociation mechanism would involve a radiationless transition from the discrete vibrational levels of the 3A2 state to the continuous levels of the 32 state. This implies that the continuous levels be 56 57 energetically appropriate so as to overlap with the dis- crete levels and lead to dissociation. Since an electronic state of different symmetry (32) is responsible for the predissociation, it will be heterogeneous. The total symmetry of the wave function will be the direct product of the symmetry of the elec- tronic and rotational states. For non-linear CH2 rotating about the Y' axis (see Figure 29), the rotational wave function is of B1 symmetry. The total symmetry of the rotating excited 3A2 state for K 3‘ 0 is A31 0! .Bic’t = B 2' while that of the non-rotating state (i,e,, K = 0) is el rot A2 ’3 A1 = A2. In view of the ad-hoc nature of this interpretae tion and its significance in determining the geometry of the lowest 381 state of CH2, we attempted to confirm it theoretically by constructing representations of the ex- cited states of CH2 of 3A2 and 332 symmetry. The States Considered The SCF function describing the ground 331 state of CH2 is characterized by the orbital occupancy lai Zai lb: 3a: lb: (3). The 2al and lbé orbitals are responsible for virtually all of the bonding in CH On the basis of 2. a preliminary CI calculation, we found that a basis set with orbitals up to principal quantum number N = 2 on the carbon atom resulted in CI wave functions whose 58 Y Y j + NON-LINEAR OR // C GEO TRY 2v ME (: ——>Z ‘\ \\\\\ CENTER OF F1 MASS LINEAR OR Dooh GEOMETRY I——0——1—9-< Figure 29.--Molecular orientations for linear and non-linear CH2. 59 energies predicted only one significant 3A2 state. The predominant configuration in the wave function describing this 3A.2 state was lei Zai lb; Bai 1b: (4). Harrison and Allen [26], by using valence-bond (CI) wave functions, found that the energy of this lowest 3A2 state decreased HCH from eHCH a 180°. This non- Rydberg state is predicted to be unbound with respect to monotonically with e C(3P) and H2(12;); it could not be the state responsible for the observed 1415 A band. Harrison and Allen also predicted a minimum in the energy (vs. eHCH) for the lowest 382 state, making it unlikely that it is the state required for hetero- geneous predissociation. Since the main thrust of our work was concerned with the existence of the appropriate 3A2 state, we did not attempt to find a mechanism for predissociation. Since there are no energetically appropriate non- Rydberg 3A2 and 332 states, we decided to consider the N = 3 Rydberg states of 3A2 and 382 CH2. For comparison, we also considered the non-Rydberg 3A2 state (4) and the non-Rydberg 382 state 2 2 1 l 1a1 2a1 lb 3a 2 1 lb1 . (5) In order to leave the CH2 bonds intact, we con- sidered states arising from excitations of an electron from the 3a1 and 1b1 orbitals in (3). Figure 29 gives 60 the molecular orientations for linear and non-linear CH2. Table 1 lists the atomic functions and their symmetries. The 6 gauSSian d functions dx2, dXY' dXZ' dYZ, de, and dzz have been transformed into a 35 function dx2 + dY2 + dZZ and five 3d functions: 2dY2 - dX2 - dzz, dxz - dZZ, C1XY' dxz' and de° Table l.--Orbital symmetries for CH2- C2v symmetry th symmetry Orbital type ' a1 O9 3, H1 + H2 b2 Ou py, Hl — H2 b1 1Tu pX al Tru p2 a1 09 2dY2-dX2-dzz a2 Hg dXY b2 Hg de bl Ag dxz al 69 dx2 - dzz 3 3 Four A2 and four B2 states were studied, one non-Rydberg and 3 Rydberg states of each symmetry. These 8 states are listed in Table 2 with their electron con— figurations and the excitations which generated them from 3 the ground Bl electron configuration. Also included is 61 Table 2.—-Electron configurations for the eight 3A and 2 332 states of CH2 generated by single excita- tions from the ground Bl state. The major atomic function component of the orbitals in- volved in the excitations. State Electron configuration Excitation ll 3A2> 1ai 2ai lb; 3a: la; 1bl + la2 |2 3A2> lai 2ai lb: lb: 2b; 3a1 + 21:2 3 2 2 2 l 1 l3 A2> 1al 2a1 1b2 1b1 3b2 3al + 3b2 3 2 2 1 2 1 I4 A2> 1a1 2a1 lb2 3al lb1 1b2 + 3a1 3 2 2 2 1 1 ll 82> lal 2a1 1b2 3a1 2b2 1bl + 2b2 I2 332> lai 2ai lb; lb: la; 3al + la2 3 2 2 2 1 1 l3 32> lal 2al 1oz 3al 3b2 1B1 + 3b2 3 2 2 1 1 2 |4 32> 1a1 2al 1b2 3a1 1bl 1oz + 1b1 1b2 ~ 2pY 2b2 ~ 3pY 3al ~ 2pz 3b2 ~ 3de 11:1 ~ 2px 1a2 ~ 3dXY a list of the atomic functions which are the major com- ponents in the orbitals involved in the excitations. The I4 3A2> and I4 3B2> states are the non-Rydberg states. The corresponding linear states and their electron con- figurations are listed in Table 3. 62 Table 3.--The linear states of CH2 and their electron configurations. Non-linear state Linear state Configuration Il 3A2> 3Au' 32 a; a; 0: vi 3d“; I2 3A2> 3mg a; a; a: w: 3pc: I3 3A2> 3Au, 32 a; a; 0: vi 3dw; .. 3.2) 3mg a; a; .3 .3 I1 332> 3mg a; a; a: w: 3pc; I2 332> 3Au, 32 a; a; 0: wt 3dn; I3 332> 3Au, 32 a; a; 03 w: 3dr; I4 382> 3H9 0; a; at «3 Our wave functions for these states were linear combinations of two Slater determinants which resulted in spin eigenfunctions. Since our basis functions did not have spherical harmonics as angular functions, our wave functions for the linear geometry were not eigenfunctions of the £Y Operator (see Figure 29). The appropriate linear combinations which do have the correct rotational symmetry about the Y axis are 63 3Au = I2 332> - |3 332> 3An = II 3A2> + I3 3A2> 3:: = I2 382> + I3 332> 32; = |1 3A2> - |3 3A2> . (6) The following pairs of states were degenerate at 6 HCH 3 3 3 3 3 180°: |1 A2> and I3 A2>, I2 32> and I3 32>, |2 A2> and Il 332>, and I4 3A2> and I4 382>. If we had used the linear combinations in equation (6), the A states would have been degenerate at BHC a 180°, while the 2 states H would not have been. Finally, as a reference from which we may cal- culate excitation energies, the ground 331 state (3) was characterized. The Basis Set A minimal basis set of atomic functions was sup- plemented by a set of 3p and 3d atomic functions on the carbon atom. The ls, 23 and 2p atomic functions on the carbon atom and the ls function on the hydrogen atom were linear combinations of 3 gaussian functions. The coef- ficients were determined by Hehre, Stewart and Pople [27] by a least squares fit to Slater-type functions. The 3p and 3d atomic function expansions were characterized by Stewart [28] in the same fashion. 64 The expansions were determined with the Slater exponent, c, set equal to l. The exponents in the ex- pansion must be scaled by the c appropriate for the atom involved. The values of C for the N = l and N = 2 atomic functions on carbon and the N = 1 atomic function on hydrogen were the standard molecular values for an average molecular environment given by Hehre, Stewart and Pople [27]. There are no reported values of C3p and C3d for the carbon atom. We determined C3p and C3d by minimizing the energy of excited states of the carbon atom. There is no particular excited state of carbon which would have allowed us to calculate the one best value for C3p and C3d for all of the 3A2 and 382 states involved. Since all of our Rydberg states were triplets with an unpaired N = 2 electron and an unpaired N = 3 electron, we decided to consider the 3P, 3D and 3F states arising from the carbon atom electron configuration ls2 2 2p1 3d1 3 28 3 in the determination of C3d‘ Similarly, the P and D states which arise from the configuration ls2 2s2 2p1 3p1 were used in the determination of C3p' The energies were determined by the CI method. Orthogonal atomic functions, which were generated by the Gram-Schmidt technique by means of a FORTRAN program [7], are listed in Table 4. Of the 36 determinants which 2 l l arise from ls2 2s 2p 3p , only two were required to 65 Table 4.--The orthogonal Gram-Schmidt C atom basis func- tions. ¢l = xls ¢2 = X25 ’ axle ¢3 a XZpX ¢4 = XZPY ¢5 = X2pz ¢6 = X3px ’ bxsz ¢7 ‘ x3pY ' bxsz @- m ll X3pz - bepz ¢9 3 x35 - CXls - eXZs ¢1o a 2x3d22 ‘ x3dx2 ' X3dY2 (1,11 = X3dx2 ’ x3dY2 ¢12 = x3dxy ¢13 = x3dxz ¢14 = X3de Where X33 = x3dx2 + X3dY2 + x3dZZ a = = < > b XZPXIX3PX — 3C1 "' C2 c -'____77— l-a 66 Table 4.--Continued ac "C e=_2___2_!-. -l-a c1 = 3 C2 = 3 represent the ML = 0, MS = 0 components of the 3P and 3D states. Of the 60 determinants which arise from 132 232 2p1 3dl, only five were required to represent the MS = 0, ML = 0 components of the 3P, 3D and 3F states. The minimum energy for the 3P state was at C3P = .48, and for the 3D state it was at §3p = .46. We chose C3p = .47. The minimum energy for the 3P, 3D and 3F states were at C3d = .33, .35 and .34, respectively. We chose 53d = .34, which Coulson and Stamper [31] also found in their study of the Rydberg levels of linear CH2. The Method of Calculation The wave functions and energies of the four 3A2, four 3B and ground 331 states were constructed by solving 2 the HFR - OS - SCF equations outlined in Appendix I. The required one- and two-electron integrals over atomic functions were generated by the IBMOL program [6]. The HFR - OS -SCF routine was part of the POLYATOM system [29]. 67 Our intention was to qualitatively represent the features of these states and determine which are bound. To this end, we fixed the CH bond length in the 3A 3B 2 and 2 states at RCH = 2.2 bohrs, the value used by O'Neil, et al. [30]. The HCH angle was varied from 90 to 180°. Although we did not determine the global energy minimum for the surfaces, by varying the angle we were able to predict whether a minimum in the energy existed. A pre- liminary investigation of the first, second, and fourth 3A2 states by means of a CI wave function revealed that they were bound with respect to a variation of R from CH ~ ~ = o 2.1 to 2.3 bohrs, at BHCH 120 . Since previous calculations on CH2 (331) predicted ~ 0 ~ ' eHCH 130 and RCH 2.1 bohrs, we fixed RCH at 2.1 bohrs and calculated the energy of the 331 state at BHCH==120, 130 and 140°. The Results The calculated energies of the 3A2 and 332 states 3 ... O at eHCH — 90, 120, 150 and 180 and the B1 state at eHCH = 120, 130 and 140° are plotted in Figure 30. The 381 minimum energy was found by interpolation to be at eHCH = 124° with an energy of - 38.4430 a.u. Since the curves in Figure 30 are for SCF states, curves for the same symmetry will cross. The true curves will be those which result when avoided crossings (dotted lines) are included. 68 3 -38.01 - . '3 A2> -38.05 -38.10 ~38.15 ENERGY (A. U . ) -38.21 -38.43 —3a.45 L 1 I l 1 90 120 150 180 HCH ANGLE (DEGREES) Figure 30.--The 3A2, 3B2 and 331 energy curves. 69 Herzberg's prediction [2] resulting in the bent triplet ground state requires that the excited triplet is bent (~ 125°) and ~ 8.75 eV above the ground state. Our curves, when avoided crossings are used, indicate three 3A2 states with minimum energies at ~ 125°. These 3 A states have angles of 113, 120 and 127° with 0 - 0 2 transition energies of 7.53, 8.30 and 8.86 eV, respec- tively. The quality of our representation precludes any quantitative predictions or assignments; however, we do predict three likely candidates for the required bound 3 A state. The existence of these bent 3A states lends 2 support to Herzberg's hypothesis. 2 The nature of our results precludes any definite statement concerning an appropriate 332 state to cause predissociation. There is a bound 3B2 state at eHCH = 180° which has a maximum energy at eHCH ~ 120°. Since this 332 state is unbound for BHCH ~ 120° and has a favorable location energetically, it may be the required 332 state. Discussion The existence of 3 bound 3A2 states with appro- priate energies and angles so as to be consistent with Herzberg's prediction has been demonstrated. The exist— ence of an appropriate 332 state was neither demonstrated nor precluded. 70 An extension of this work could (1) use a better quality basis set, (2) consider the full surfaces around the global minimum and include C geometries, and (3) in- S clude correlation effects by means of a CI calculation. The first and second points would be controlled by simple economic and time factors. The third aspect is, however, more complicated. If an SCF calculation were used to generate the molecular orbitals for the CI, the choice of which state would be appropriate for generating the SCF orbitals would be difficult since there are a number of avoided crossings and the 3A2 states are very close to each other. Even for a minimal basis set calculation, the CI eXpansion would have to be truncated. The selec- tion of a single set of configurations to represent the various 3A2 states would be difficult. An iterative natural orbital scheme or a multi-configuration SCF ap- proach should provide a costly but realistic solution. APPENDIX I THE THEORY The Schrodinger Equation This thesis deals with the approximate solution of the Schrodinger equation for the CH2 and NH: systems and is based on three general approximations: (1) the individual system is not subject to any external forces, (2) the Born-Oppenheimer approximation is used, and (3) a non-relativistic formalism is used where all particles are assumed to be point masses. The first approximation implies that there are no applied external fields such as magnetic or electric fields. Further, it implies that there are no inter- actions between similar systems. This approximation is tantamount to saying that we are considering an isolated system. The potential energy of our system may be written with no time dependence. For a general M particle system, the non-relativistic time-dependent Schrodinger equation is .5. _ . 3 STI'P - 1R ‘5? ‘Y I (7) + + -> Where ‘1’ = W(rl' 1:2,...prM't) I 71 72 CT = T + v , A .Kz M vi T = (‘Tfo Z (574 = kinetic energy operator , i=1 i G = G (:1, $2,...,rM,t) = potential energy Operator .. Unless otherwise stated, we will use atomic units. The following conversions are used: (1) l a.u. = l Hartree = 27.21165 eV/molecule = 627.524 kcal/mole , (2) 1 bohr = .529177 x 10"8 cm, and (3) mass of the electron = electronic charge a H = 1 . Since 9 is not a function of time, equation (7) results in the time-independent non-relativistic Schrodinger equation, 'GTw = ETw . (3) where w = w(§l,}2,...,§M) I and W = We-iETt The energy of the system is postulated to be ET' The Born-Oppenheimer approximation allows us to solve for the electronic motion with the assumption that the nuclei are in fixed positions. Let the subscripts a,8,... denote nuclei and i,j,... denote electrons. 73 The Hamiltonian for a 2N electron system may be written as “T = 2E + VN + TN ' A 2N 2N 2N M—2N z where he = («:4 2 vi + 2 (FL) - z 2 (EL) , i i w 2 WE 0N = wE(ri;ra)wN(ra) where 0E depends parametrically on the nuclear coordinates. Equation (8) breaks up into a purely electronic Schradinger equation, A IQEWE = EEwE ' (9) where EE = purely electronic energy and a Schrodinger equation for the nuclear motion, A 'erN = ETwN , (10) where fiN=T +13, and E = EE + VN = electronic energy. 74 The electrons move much faster than the more massive nuclei and adjust rapidly to any nuclear changes. Thus, the electronic energy is a smoothly varying function of the nuclear parameters and provides the potential which governs the nuclear motions in equation (10). Our work will be concerned with the approximate solution of the purely electronic Schr6dinger equation (9) which will henceforth be called the Schrfidinger equation and be written as where ‘t m d> E. €- III me By the word energy, we will mean the electronic energy, E. Equation (11) is solved by fixing the nuclear co- ordinates and solving for w and EE for 2N electrons in the field of the fixed nuclei. Since 6N is a constant, we simply add it onto EE to find E. Configuration Interaction In general, the exact solution of the Schr6dinger equation (11), for a system with one electron is w(§) = z ai°i‘;’ (12) i=1 75 where the set of functions 0; is complete. In order to account for electron spin, we redefine our expansion (12) in terms of a complete set of spin functions or spin-orbitals, 01, which are products of spatial and spin functions. We write 0 = X a.¢. (13) . i=1 1 1 with 01 = @ia, 02 = Ois, 43 = ¢éo, etc. Equation (13), which defines a configuration interation (CI) wave func- tion, can be extended to a 2N electron system. The re- sult is w(; I? '00.]; )3 z 2 on. 2 Co Co once. P o o 1 2 2N i1=l 12=1 12N=1 l1 12 J'2N i112°”12N ' where P. . . _ + + + 1112’°‘12N - 0i (r1)¢iz(r2)... . . . ¢No(2> . . . . (ZNIY'I/2 3¢l(2m 761mm $No(2N)I . If azis the permutation Operator which interchanges the l/2 Z I\ 6? all0 is for normalization and e = + 1 if 7‘ coordinates of electrons, then a = (2N!)- where (2Nl)-]'/2 is equivalent to an even number of transpositions and - 1 iff? is equivalent to an odd number. Equation (14) then becomes‘ c D . (15) w(1'2'ooo'2N) 3 I I I "b18 O 77 Practical considerations dictate that we truncate expansion (15), by truncating our set of spatial functions, 0;, to obtain cIDI . (16) M 1p: I=0 The choice of the set of functions Oi and the selection of the subspace of 0; is of fundamental importance in ob- taining the best possible representation of 0. Since the expansion in (16) is frequently truncated even further for a particular subspace of 0;, the search for the subset which provides the most rapidly convergent expansion be- comes even more important. Two common forms for the functions or orbitals 0; for polyatomic systems have been atomic orbitals and molecular orbitals (MO) which are often represented by linear combinations of atomic orbitals (LCAO). The fa— miliar valence bond method results if atomic orbitals are used. The use of molecular orbitals, which we have em- ployed, results in the LCAO-MO method. If the full ex- pansion in (16) is used, the two methods will, of course, yield the same results. The larger the number of atomic orbitals (basis set or basis functions) used, the larger the expansion will be. The use of atomic or molecular orbitals is deter- mined by the severity of the truncation of the eXpansion 78 (16). As the number of terms retained in the expansion decreases, the more important the form of the orbitals used becomes from the point of view of generating the most rapidly convergent expansion. Historically, this has usually been accomplished by the use of self- consistent field (SCF) molecular orbitals. If only one term, iLEL, a single determinant, is kept in equa- tion (16), the best single determinant wave function may be generated by the Hartree-Fock-Roothaan (HFR) SCF method. We will make use of the prOperties of SCF molecular orbitals in our discussion in this section, although the HFR—SCF method will be outlined in the next section. In Hartree-Fock (HF) theory the potential ex— perienced by an electron due to the other electrons is constructed to be an average potential of the other electrons. This potential felt by an electron should be governed by Coulomb's-law which provides for instantaneous electron-electron interactions. As a result, the electrons in HF theory are not allowed to adequately correlate their motions in response to the actual potential they feel. This effect causes the HF energy, EHF' to be higher than the exact non-relativistic energy, E by an amount exact' called the "correlation energy," E - E = E E c‘ c exact ’ HF' Although Ec is only a small fraction of the total energy (less than 1% for the systems of concern here), it is of 79 the same order of magnitude as bond energies, for example. HF wave functions frequently provide good results for properties which are relatively insensitive to electron correlation-prOperties such as geometries, ionization potentials and some one-electron prOperties. Attempts to use HF theory for the description of reactions have provided some of the most glaring examples of the failure of HF theory. In chemical reactions bonds are broken and new bonds are formed. In this process electrons may become paired and unpaired, and they must redistribute in response to a completely new environment. It is in just such a situation as this that we need a description which allows the electrons to correlate their motions, if we hope to provide a physically reasonable picture of a reaction. For each atom in a molecule we must provide at least one atomic orbital or basis function for every or- bital occupied in the electron configuration of that atom. The number of basis functions, M, is invariably larger than this minimum number required. The result is that in the solution of the Schrodinger equation we ob- tain No occupied SCF orbitals and M-No unoccupied or virtual SCF orbitals. Since the complete set of SCF or- bitals is by definition an orthonormal set, the space of the virtual orbitals must be othogonal to the space of the occupied orbitals. The increased flexibility of allowing 80 electrons to be in virtual orbitals will let the electrons correlate their motions better. Virtual orbitals provide a convenient way of allowing the electrons the flexibility of getting further from each other. There is, however, a serious drawback in.the use of SCF orbitals for our CI eXpansion. The original motiva- tion for the use of SCF orbitals was to facilitate the convergence of equation (16) when the expansion is se- verely truncated. Since the virtual orbitals are part of an orthogonal set of SCF orbitals, they are frequently used because of this property rather than as a result of their convergence prOperties when used in a CI expansion. There is nothing in HF theory which ascribes to them this desired prOperty. Rather than try to overcome this flaw by a multi-configurational SCF or natural orbital method, we chose a very modest size set of basis functions and retained the majority of the resultant terms which rise in equation (16), in describing the C + H2 and N+ + H2 reactions. Slater determinants provide a convenient basis in which to expand our wave function, since they are orthonormal. Since a full CI is one characterized by including all possible determinants in (16), the number of terms in a full CI frequently becomes totally un- manageable. If, for example, we were considering a 6 81 electron system using 10 basis functions, we would have 6 electrons to be placed in 20 spin—orbitals. A full CI would have 1% = 38,760 terms in it. Besides arbitrarily truncating the CI expansion (16), we may use spin and spatial symmetry to reduce the size of the expansion. Our orbitals for non-linear geometries were constructed to be symmetry orbitals which transform according to one of the irreducible representa- tions of the symmetry group of the molecule. The CI ex- pansion for a B1 state of a system with C symmetry can 2V be restricted to include only those determinants with B1 symmetry. Our CI expansion will be a linear combination of configurations rather than a simple sum of Slater deter- minants. A configuration is a linear combination of determinants which is a spin eigenfunction; i.e., a C , (17) where C = 2 b D . The details of determining the bIJ values such that each CI is an eigenfunction of the §2 operator are given in Appendix II. The CI expansion for a 381 state may be re- stricted to only those configurations with B1 symmetry 82 and S2 eigenvalue = 2. We may further reduce the size of a CI expansion by considering, for example, only the MS = 0, ML = 0 component of the 9-fold degerate atomic carbon 3P state.~ The problem of determining the wave function (17) and energy is solved by applying the variational principle to E = . The energy is minimized by varying the expansion coefficients. The result is the usual matrix eigenvalue problem or secular equation H a = a E , (18) where HIJ = . A "double scripting" of the symbols or "parallax" notation will be used to denote matrices. The H matrix is con- structed and then diagonalized to yield P eigenvalues, EI' and P eigenvectors, a P is the number of configurations I; in (17). Open Shell Self-Consistent Field Theory There are a number of open shell self-consistent field (OS-SCF) theories currently being used. The OS-SCF theory outlined here is that given by Roothaan [3]. Rather than giving all the details of his theory, an overview will be presented. 83 The 2N electron wave function is an antisymmetrized product of No spin-orbitals or No one-electron orbitals w = :3 l¢1(1)$1<2)...¢N0_1<2N—1)¢N0<2Nn . (19) The spatial orbitals which are doubly occupied are referred to as the closed shell and are placed at the front of the orbital product. The spatial orbitals which are singly occupied (with a spins) are referred to as the Open shell and are placed at the end of the orbital product. A matrix notation is appropriate for simplicity and clarity. The subscripts i,j will denote Open or closed shell orbitals, k,l will denote closed shell or- bitals, and m,n will denote open shell orbitals. The orbitals are constrained such that (1) ==l, (2) <¢iI¢j> = 6ij' and (3) the ”C or closed shell space is orthogonal to the 60 or open shell space. The combined set of N orthonormal orbitals is collected into a row +0 + + Our Hamiltonian for the 2N electron system is A 2N A 2N ’= 2 H + 2 (l/r ) , }\ =1 a “<8 as where Ha is a one electron operator which includes the kinetic energy of the ath electron and the ineteraction of the ath electron with the nuclei 84 2. all nuclei a - 2 (zy/roy) . Y Roothaan's OS-SCF theory is valid when the energy may be written as E= <¢Ifi|¢>=EC+EO+ECO , (20) Where EC = 2 i Hk +- £1 (ZJkl-Kkl) , E0 = flzriHm + f mzn (2aJmn - menH . Eco é 2f Q; (2Jkm - Kkm) . Unless otherwise indicated, all summations are over the No orbitals, and the following definitions have.been used “1 a “1'3”? _ 1 Jij - <°1‘1’°j‘2’ I;1—2-|¢i(1)¢j(2)> 1 Kij = <¢i(l)¢i(2) I 2:1“; I ¢j(l)¢j (2)) . EC, E0, and ECO are energies associated with the closed, open, and closed and open shell orbitals, respectively. The calculations performed on the 3A2 and 382 states of CH; made use of this theory. Both states have wave functions whose energies are in the form dictated by equation (20) with a = l, b = 2 and f a %. 85 The variational principle is now applied to equa- tion (2) subject to the contraints Sij E <¢iI¢j> = 5ij' We want to minimize E with respect to the set of orbitals *, and we use Lagrangian multipliers, Aij to impose the constraints. Therefore, Eli.(6S..) = 0 , (21) ISE) _ ij 3 13 where 6E E(¢l+6¢l,¢2+6¢2,...) - E(¢l,¢2,...) , and 5515 <¢i+5¢il¢j+6¢j> — <¢iI¢j> 0 When equation (21) is written out, a lengthy integro- differential equation results. Since this equation must vanish for arbitrary variations of the set of orbitals 8, the coefficients of all 6¢i must be zero. If we set the coefficients of 6¢i = 0 and note that A is a Hermitian matrix, we obtain two coupled integro-differential equa- tions, (H + 2JC — KC + 2J0 - K0) ¢k = i ¢lelk + i ¢nenk , (22) and f(H + 2J - K + 2aJo - bKo) ¢m = 2 ¢191 + z ¢ 9 (23) I C C 1 m n n nm where 01k = -21kl . 86 In equations (22) and (23) the Coulomb Operator associated with orbital Oi, the closed shell Coulomb Operator, the open shell Coulomb Operator, and the total Coulomb Operator are respectively,- ’ 1 J1‘1’°3‘1’ <¢i(2)|§I;|¢i(2)>¢j(l) J = 2 J C k k J = f 2 J 0 m m Similarly, the analogous exchange operators are ' l K1‘1’°j‘1’ <¢i(2)|;I;I¢j(2>> ¢i(1) _ 2 KC ‘ k Kk K = f 2 K 0 m m The Operators in equations (22) and (23) are invariant under the unitary transformations + + + -.> I_' O- ‘c ’ ¢c ”c and *0 ‘ 80 ‘10 ° (24) These transformations can be chosen to eliminate the Off- diagonal multipliers 81k and enm within the closed and 87 Open shells. They will not in general eliminate the Off- diagonal multipliers enk and elm coupling the closed and open shells. In order to uncouple equations (22) and (23), Roothaan defines a new set of Hermitian operators. The Coulomb, closed shell Coulomb, Open shell Coulomb, and total Coulomb coupling Operators are L1°j = <¢ilJol¢j> ¢i + <¢il¢j> Jo°i L = 2 CkLk L0 = f i Lm Similarly, the analogous exchange operators are 2 9- ll 1 j <¢1IKo|°j> °1 + <¢il¢j> Ko°1 3 ll With these coupling Operators, equations (22) and (23) are easily uncoupled to yield two pseudo-eigenvalue equa- tions. However, the closed shell Fock Hamiltonian is not very 88 different from the Open shell Fock Hamiltonian. The re- sult is that the bc space will be quite similar to the o0 space, and by definition the two spaces should be orthog- onal. With the aid of the coupling Operators and by using some straight-forward manipulations the problem may be avoided by rewriting the closed and open shell equations with a common Fock Hamiltonian, 9' figc = fc“c ' 1‘30 40.10 , where § = n + ZJT - KT + 2a(LT - J0) - 8(MT - K0) . («Jame-ek1 + <¢kl2aJo - BKOI¢1> . emn "‘0’ = T + f <¢mlzaJo " BKo|¢n> ' mn a= (l-a) z]-f5 ' e- (l-b) _. Tmf , The p matrices may now be diagonalized by the transforma- tions of equation (24) PO. = e.¢. (25) i 1 1 ' where e = U+uU . 89 In terms of the Si’ the energy becomes E = 2 (H + e ) + f 2 (H + e ) k k k m m m - f fi;(2oka — BKkm) - f3 g; (2aJmn - men) . The solution of the Hartree-Fock-Roothaan open shell equation, equation (25), for molecules is based on the linear combination of atomic orbitals (LCAO) method. This procedure was first introduced by Roothaan for the closed shell Hartree-Fock equations [4]. The orbitals, which are now molecular orbitals, are expanded in a linear combination of atomic orbitals or basis functions, xj, which are centered on the nuclei co ¢i = jil ijji , (26) where the C's should not be confused with configurations. In a matrix representation, equation (26) becomes oi = §Ei and the total transformation is 5 = QC where 3 and § are row vectors and Ei is a column vector. Practical con- siderations require a truncation of the set of basis functions. For a system with N0 occupied orbitals and M basis functions, c is an M x N0 matrix of expansion coef- ficients and M > NO' By substituting equation (26) into equation (25) and writing the result in a matrix notation, 90 we obtain the Hartree-Fock-Roothaan Open shell self- consistent field (HFR—OS-SCF) equations méi =-ACi€i . (27) The following definitions are used Aij F = H + P - Q + R = M X M matrix 5 ll (XiIHIXj> '3 ll ZJT - KT 0 = ZoJo - BK The matrices JT or KT, Jo or K0, are distinguished by the density matrix, 0T or 90, that is used in their defini- tion. In the following definitions, 0 a T or O: M (J ) = z °§1 G'ij k,l M ° I’1‘I < (K ) a z p T closed 0 .. t and pkl" i Crkcrl + pkl 0 open _ * °k1 ‘ f 2 Crkcrl ° r 91 The energy may now be written as E = trace {(H + F) 0T - leT + (f - 1)00} . The constraint, <¢I¢> = M = unit matrix results in the constraint, CIAC = I. This constraint allows us to re- write FC = AC8, which is another form of equation (27), as ¢+FC = e . (28) Equation (28) is solved by an iterative process: 1) 2) 3) 4) 5) 6) an initial estimate is made for the C matrix 9'1' and DO are constructed T is assembled F is diagonalized to yield a new C and a matrix (C is a.M x M matrix whose first No columns define the occupied orbitals and whose remaining M - No columns define the virtual orbitals; e is a di- agonal matrix whose diagonal elements are the 81) O a new 9T and 0 are constructed from the first No columns of C. 0 are compared with the old ones: the new OT and 0 if they have not changed, self-consistency has been solved; if they have changed, it is necessary to go back to step 3 and repeat steps 3 to 6. APPENDIX II COMPUTATIONAL TECHNIQUES FOR THE POTENTIAL ENERGY SURFACES The Basis Set In this study we employed a minimal basis set, 112;! ls, 23, 2px, 2py, and sz orbitals for C and N+ and a ls orbital for H. Each of these orbitals was rep- resented by a linear combination of 3 nuclear centered gaussian functions djigij ' where gij a XZszne.aijr (unnormalized). The angular dependence of these nuclear centered gaussian functions is determined by XZYmZ". An s type function implies Z é m = n = 0, while a px type function implies Z = l and m a n = 0. The expansion coefficients, dji' and the exponents, aij' which are those given by Ditchfield, Hehre and Pople [5], were determined by minimizing the 3 2 energy of the P, 4S and S states of C, N, and H, re- spectively. These are collected in Table 5. 92 93 Table 5.--The coefficients (dj) and exponents (cj) of N: the gaussian functions which represent the C, and H atomic orbitals. ‘j = 1 j = 2 j = 3 C18 .djw- .06960382 .3936907 .6658730 ajv- ' 155.2622 23.28926 4.948442 C28 dj'B .08215337 .6034712 .4736710 aj 8 5.793223 .4472592 .1440200 C2p dj = .1124423 .4657363 .6227623 oj 3 4.152398 .8464664 .1981786 N18 dj = .06913578 .3934835 .6657830 oj a 214.1064 32.15723 6.866035 N28 dj = .08165721 .5980757 .5802935 aj = 8.394697 .6465681 .2050868 N2P d3 = .1164253 .4705657 .6176643 “j = 6.085492 1.252070 .2896766 “18 dj 8 .07047866 .4078893 .6476689 - 4.500225 .6812745 .1513748 94 V and C°°V geometries are given in Figure 31. We will, for conveni- The CH2 and NH; orientations in C2 ence, define AH2 as representing CH2 and NH; when no dis- tinction is to be made between them. We will also let AH and A represent CH and NH+, and C and N+, respectively. 2v wv Y Y /\ /\ H A > z A II————Ii-—~9» z Figure 31.—-The CH2 and NH; orientations in C and C0° 2V V geometries. Our basis set for AH2 in C symmetry was composed of four 2V al functions x18, x23. X2pz' and le + tz ; two b2 functions, X2p and le - tz; one b1 function, Y xsz; and no a2 functions. Our basis set for AH2 in va symmetry was composed of five 0 functions 95 xls’ XZS’ szz' XHl' XH2 and two w functions x2px and XZpY' An orthonormal set of 7 orbitals was constructed by applying the Gram-Schmidt technique [7] to our 7 basis functions and then normalizing the resultant orbitals. The forms of the unnormalized orbitals for sz and C°°V geometries are given in Tables 6 and 7, respectively. The coefficients al through all should not be confused with configuration expansion coefficients. For sz geometries, 81 through ¢4 were al orbitals, ¢5 and ¢6 were b2 orbitals, and ¢7 was of b1 symmetry. For va geometries, we obtained five a or— bitals, o1 through ¢5, and two 8 orbitals, ¢6 and o7. At this point, it is apprOpriate to discuss the rationale for our use of orthogonalized symmetry orbitals as the basis. The orbitals (and therefore the electrons in them) in AH2 systems may be divided into two subsets, core and valence orbitals. The core orbitals are those which are essentially unaffected in the reactions, i;g;, the ls orbital of N+ and C. Since these core electrons are not involved in the electronic redistribution, their correlation energy should be relatively constant. And, since our main concern is with energy differences and not absolute ener- gies, we froze the core orbitals in the CI expansion. This means that all determinants which were used contained a 96 Table 6.--Orthogona1 Gram-Schmidt Molecular Orbitals in sz Geometries ¢l a xls °2 ' X29 ‘ a1I°1> G U I (XHI + XHZ) ’ a2N1> ‘ a3I¢2> szz XZPY ¢6 ' (x31 ’ an) ' a4M5> XZPX Table 7.--Orthogonal Gram-Schmidt Molecular Orbitals in Coov Geometries W ¢8=xls ¢9"“‘28 - a1|¢8> 4’10‘x2pz ¢11“xm " "Isms> " °6I°9> ‘ °7I°1o’ ¢12""82 ' °8I°8> ' °9I‘I’9> " aloI°1o> '°11I¢11’ °13' xzpY ¢l4B x2px 97 doubly occupied ls orbital on the heavy atom. By selecting the first orbital in the Gram-Schmidt process to be the ls orbital of C or N, we were assured that there were no com- ponents of the valence orbitals in the core orbital. This insured that the full CI (1:21! all possible arrangements of 6 valence electrons using the remaining 6 Spatial or- bitals or 12 spin-Orbitals were allowed for) within the valence orbitals completely spanned that subspace. The Determinants and Configurations In this section we will discuss the number and type of configurations which were used in our CI wave functions. The specific details of constructing con- figurations from determinants will be discussed in the next section. As pointed out earlier, the AH2 determinants were generated by keeping the lowest energy or first orbital doubly occupied. The determinants generated were all possible electron occupations within this constraint. The determinants for the C, N+, CH, NH+, and NH systems were generated by using the same constraint. All possible occupations of the 12 valence spin- 12: - 924 de- orbitals with 6 electrons results in terminants for AH2‘ If we consider only the MS a 0 com- ponents of the higher spin multiplets, we have three a spin 98 electrons to be placed in six 6: spin-orbitals and three 8 spin electrons to be placed in sixfi33pin-orbita1s. The result is (5%éT) (3IIT) = 400 MS a 0 AH2 determinants. By taking the appropriate linear combinations of these 400 deter- minants, we may construct 175 singlet, 189 triplet, 35 quitet and l septet MS = 0 AH2 configurations. The 189 triplet configurations may be further broken down by con- sidering their symmetry with respect to the reflection plane, OV.(YZ). There are 99 triplet configurations which are antisymmetric with respect to 0v (YZ). These 99 triplets are characterized by a single electron in the b1 (sz) or n ) orbital. x (cw In sz geometries we generated 51 331 configura- tions from the 99 triplets by selecting only those which were symmetric with respect to 0v (x2). The remaining 48 triplets, which were 3A2 configurations, were not con- sidered. The 3B1 surfaces for the insertion reactions were represented by a 51 configuration wave function. The abstraction reactions, which are characterized by C symmetry, were represented by configurations whose form :as not determined as easily. While the orbitals in sz geometries transform according to the symmetry Opera- tions of sz, the orbitals in va geometries do not trans- form according to the symmetry Operations of va' The n orbitals, Ox and Oy, do not have definite symmetry with respect to the infinite number of reflections and 99 and rotations containing the molecular axis. The n or- bitals are not eigenfunctions of the E Operator. The Z angular dependence of gaussian pX and pY functions is a real representation of the Yil spherical harmonics. Aside from normalization constants, we may write the angular dependence of Ox and ¢Y as cpx = ~21- (¢+ + ¢_) i where ¢i = Y: . For convenience, let ¢i¢§ represent the Slater determinant 99 [¢l$1...¢x¢Y] , where all the orbitals before 45x are doubly occupied. The ¢§, Oi, ¢i¢%$%, and ¢i$§¢§ deter- minants are not eigenfunctions of £2 and thus are not strictly H determinants. Since the potential is cylin- drically symmetric and independent of ML' our H deter- minants are just unitary transformations of the eigen- functions of £2 and therefore are equivalent representa- tions. A p2 configuration gives rise to, among other possibilities, the following determinants and terms, 100 ¢+$; 1A(ML = + 2) ¢_$; 1A(ML = - 2) ¢+$_ - $+¢_ 12+ ¢+¢_ 3;- 8+1 ¢+76_ + $+ “8 bination of (l) the sum of all permutations of unpaired a spin electrons with unpaired 8 spin electrons and (2) the sum of all permutations of a with 8 spin electrons within a doubly occupied spatial orbital, for all of the doubly occupied orbitals. A determinant with 2 unpaired elec- trons, for example, is not an eigenfunction of S2: 62810137102763) 8 1310102331 +11 [0176137203] + I (o + 2(4)) 91' (0161027531 The general form for a Spin projection operator is O = H (X_:X') k ifik k' i A where 11 is the eigenvalue of 82 corresponding to the spin state we Wish to project out. The 1k are the eigen- values of?)2 corresponding to the spin states we wish to annihilate. A determinant with 4 unpaired electonrs, is in general, a mixture of singlet, triplet, and quintet spin. A pure triplet spin state may be obtained by annihilating the singlet and quintet components by using A §Z_)O A BTH—fla <§§§>=—§s (S-6). The result of operating on a determinant With a projection operator is a linear combination of determinants which is a pure spin function. In general, if there are N determinants in the subspace of determinants with the same spatial orbitals singly and doubly occupied, N linearly dependent configurations will result. The M linearly independent configurations were determined by trial-and-error, when M was less than N. The application of this method to 3 unpaired electrons will be detailed. There are 3 possible MS = + %~determinants with 3 unpaired electrons: 01 = «1 10131420374) 02 =14 101711273041 D3 =A [1111121542] If we use the doublet projection operator, OD = - § (1932 T-S). we obtain 124 ODD1=ID1+D2 +1’3 ,. 7 G1302=DI+YT92+D3 A 7 GDD3=D1+D2+ID3° Two linearly independent configurations may be generated from these linearly dependent configurations: C1 = ODDZ +GDD3 + 2131 - I)2 - D3 2 _ _f2 _ _ and C2 - @DDZ SD03 - D2 D3 . If we use the quartet projection Operator, 60-3156 we Obtain AD = SD = ($13 =1(D +13 +11) CS(11 02 03 3' 1 2 3° The linearly independent quartet configuration is The determinants and linearly independent configura- tions which‘were used are listed in Tables 15, 16, and 17 for 4, 5, and 6 unpaired electrons, respectively. The configurations given in Tables 15, 16, and 17 are for 125 Table 15.--Determinants and Configurations Arising From Four Unpaired Electrons (with MS - 0) 01 =- 71 1010237371741 04 = )4 1616203041 02 =- 34 (0176203341 05 = )4. 1310203041 03 - 91 1010233041 06 = & [$10203T4] 3c1 = D1 - 04 3c2 = 02 - 05 3 0 11 U I U Table 16.—-Determinants and Configurations Arising From 126 Five Unpaired Electrons (with MS a + 1/2) D1 = 71 _— I [7172030405] 06 = :1 [01720337405] 161020304051 07 = :1 101720304751 171020364051 a. = A 101026374051 171020304751 09 = 5. 1010273043751 1017276304051 010 = :7; (01020321747651 (D6+D9) - (D7+D8) 2(D5+D10) - (D6+D7+D8+Dg) 2(D3-D4) + (D7+D9) - (D6+D8) 4D +2D 2 10 - 2(D3+D4+D5) + (D6+D7) - (08+Dg) 3D1 + (08+D9+D10) - (D2+D3+D4+D5+DG+D7) (D2+D3+D4) - (D5+D6+D7) (D2+D5+D8) - (D4+D7+Dlo) 2(D3+D6-D9) + (08+Dlo) - (D2+D4+D5+D7) 6D - 4(D8+D9+Dlo) + (02+D3+D4+D5+D6+D7) l 127 Table l7.--Determinants and Configurations Arising From Six Unpaired Electrons (with M = 0) s 017 91 (0112137421576) “11 = :1 17517273441516) D2 =71 [0102730475766] 012 = .21 17617203740506) 03 = A 1010273740576] 0135 11, (7162030475061 0,, =71) [41027371747506] 0“ =51 [7172030405761 05 = ~11 10172430475766) 015 = .23 (7102730405061 06 =31 (01720374057661 016 =13. [7102763047506] 07 = 1) (01720376475061 017 =2; 17142730405761 D8 = 93 1017273740506) “13 = 17102030475761 09 = 7‘1 101‘2730475061 019 = :1, (7102037405761 1’10 3 43“ [¢1—2$3¢4¢5$6] D20 = ":1 [“1¢2¢3$4$5¢6] 3C1 - 9(D1+Dll) - (DZ+D3+D4+D5+D6+D7+D8+D9+Dlo) ‘ ID12+Dl3+914+015+D16+D17+018+D19+D20I 3c2 — 8(D2+D12) - (D3+D4+D5+D6+D7+D8+D9+Dlo) ‘ (D13+D14+D15+D16+Dl7+Dl8+D19+DZO) 3c3 = 7(D3+D13) - (D4+D5+D6+D7+D8+D9+Dlo) ‘D14+D15+D16+Dl7+918+D19+Dz0I Table 128 l7.--Continued 0 1| 0 ll 0 ll 0 II “(D4+D14) (D5+D6+D7+D8+D9+Dlo) _ (D15+016+Dl7+D18+D19+D20) 5(DS+D15) (D6+D7+D8+D9+Dlo) ‘ (D16+D17+Dl8+Dl9+D20) 4(DGTDls) ‘ (D7+D8+D9+D10+D17+D18+D19+D20) 3(“7‘“317) (D8+D9+D10+Dl8+D19+DZO) 2(D8+D18) (D9+D10+D19+D20) (“9+DI9’ (“10+020’ 129 subspaces of determinants with 4, 5, and 6 total electrons, respectively. The results were directly applicable to the same subspaces which arose in our A, AH, and AH2 systems which had 6, 7, and 8 total electrons, respectively. The Quality of OurfRepresentations The failures of an SCF description, especially with respect to dissociation products, required that we use a CI description. The major stumbling block to a CI approach is the large number of configurations which can arise, even when Spatial and spin symmetry have been fully exploited. There are two methods which can alle- viate this problem. A reduction in the basis set size can significantly reduce the size of the CI expansion. If a larger and better basis set is employed, then a truncation of the CI expansion may be required. The quality of our representation is dictated mainly by our choice of a minimal basis set and not the relatively less significant truncation of the CI expansion. Since this was our first eXperience with potential energy surfaces, the errors which were bound to arise were less costly than if a more extensive basis were used. Since our CI was complete, our description of the reactants, inter- mediates and products involved in the reactions is at a consistent level of accuracy. 130 Previous experience has shown that minimal basis set calculations provide only qualitatively significant absolute energies, but provide a more reliable descrip- tion of geometries and relative properties such as energy differences. Characterization of the Reactants and Products Our basis set (Table 5) is the 3 gaussian represen- tation of the atomic orbitals given by Ditchfield, Hehre, and Pople [5]. As a check or our integral [6], Gram- Schmidt [7], and CI [8] programs and possible input errors, we duplicated their results by computing the energy of the apprOpriate single determinants. We Obtained ener— gies for C(3p), u(451, and H(2S) of -37.45306, -54.06288, and -.49698 a.u., respectively, in complete agreement with the calculated energies of Ditchfield, Hehre, and Pople. The energies of the 3P, 1D, and 1S states of C, calculated using a 36 configuration wave function, are -37.47011, -37.41099, and -37.36966 a.u., respectively. The corresponding N+ energies are, with the same quality wave function, -53.5347, -53.46275, and -53.40599 a.u. The energies of the N atom, using the 51 configuration 12 NH: wave function at an N—H; separation of 1 x 10+ bohrs, are -54.06288, -53.9550, and -53.91882 a.u. for 131 the 4S, 2D, and 2P states, respectively. These results are compared with experimental results in Table 18. The comparisons are quite reasonable with respect to the quality of representation. There is a certain consistency which seems reasonable. Our results were obtained by using a basis set whose parameters were Optimized for only the lowest term. + 2 + 21 29). cn(zn), CH142’). NH(32‘). NH+(2H), and NH+(4Z-) diatomics were characterized by . 1 + The 112(29): H using the CI wave functions described earlier. The mini- mum energies and the internuclear separation at the minimum, Re' were determined by interpolation. Except l + 3 for the H2 29 and + 2 + 4, the 112(29). H 2: curves, which are shown in Figure 3.22;), CH(2H), CH142‘), NH+(2H), NH+(4Z-) and NH(32') energy curves as a function of the internuclear separation are shown in Figures 32, 32, 33, 34 35, 36, and 37, respectively. A comparison of our re- sults with experiment, when possible, is given in Tables 19 and 20. The 42' state of CH has not been observed eXperimentally. We obtained values for Re which are predictably and consistently larger (~ 20%). The De values, which we calculated with the assumption that as the diatmoics dissociated, the only maximum in the energy was at infinite internuclear separation, are consistently smaller than the experimental results. These results 132 Table 18.--C, N+ and N Energy Differences in Electron Volts Species AE(1D-3P) AE(ls-3P) C - This work 1.6088 2.7334 C - Moore [9] 1.26387 2.6841 C - % difference 27.3 1.8 N+- This work 1.9572 3.5017 N+- Moore [9] 1.89892 4.05272 N+- % difference 3.1 -13.6 AE(2D-4S) 63(22-48) N - This work 2.936 3.920 N - Moore [9] 2.3834 3.5757 N - 8 difference 23.2 ' 9.6 133 _.30 — 22+ 11 :5 S 6‘ g + z -.40 m _ H(ZS)+H+ -.50 - ' 22+ 0. g 1 1 2 I3 4 6 8 l w 2- “3 H-H (BOHRS) Figure 32.--The H w-t 2 0+ and 2 + zu energy curves. -37.96 -37.98~ -38.00 ENERGY (A. U.) -38.02 134 C(BP)+H(ZS) b---- 1 11 ’1 .. 2.552 3 4 C‘H (BOHRS) Figure 33.--The CH (2n) energy curve. -37.98 :5 6 M lid 2 It: -38.02 _ 2.335 135 C(3P)+H(ZS) C—H (BOHRS) Figure 34.-~The CH (42-) energy curve. ,5— 136 N+(3P)+H(28) -54.04 *54.06 -54.08 5 5 E E -54. 10 m '54.12 -54.14 Figure 35. 2 2.44 3 4 5 w N-H (BOHRS) --The NH+ (2H) energy curve. 137 -54.06 '- N(4S)+H+ m -54.08 - -54.10 b =1 5 S ‘1‘ 1. g -54.12 [:1 -54.14 - -54.16 I- I I 1 I 1 2 2.481 3 4 5 9° N-H (BOHRS) Figure 36.--The NH+ (427) energy curve. 138 -54.54I- N(4S)+H(ZS) -54.56 )- ..54.58 - ENERGY (A. U .) -s4.60 - ,- —54.62 1. l l 2 2.37 3 4 v 9° N-H (BOHRS) Figure 37.--The NH (32-) energy curve. 139 Table l9.--Diatomic Re Values in Bohrs State Re (here) Re (eXpt.) % difference 1 + - H2(X 29) 1.667 1.4008 [10] 19.0 H:(X22;) 2.508 2.003 [11] 25.2 cn(xzn) 2.552 2.116 [121 20.6 4— CH(a 2 ) 2.335 NH(x3z‘) 2.37 1.9614 [11] 20.8 + 2 NH (x H) 2.477 2.0428 [13] 21.3 + 4 - NH (a 2 1 ~ 2.481 2.088 (131 18.8 Table 20.--Diatomic De Values in Electron Volts .— L Dissociation % dif- State Products.' De (here) De (expt.) ference H2(X123) 8125), 8(25) 3.243 4.747 [14] 31.7 H;(xzz;) H(2S), H+ 1.814 2.788 (11] 34.9 2 3 2 . cn(x n). C( p), H( 5) 1.838 3.63 112] 49.4 08(642‘) c13p), H(28) 1.477 3 - 4 2 NH(X 2 1 N( S), H( 5) 1.474 3.41 (151 56.8 NH+(X2n) N+(3p), H125) 2.903 3.90 [16] 25.6 4. NH+(a4z-) N145), H 2.559 .140 provide a useful yardstick by which our results for the reactions may be measured. Our bond distances can be estimated to be about 10 to 30% too large. Our AH and energy barrier for a reaction may be estimated to be about 25 to 50% too small and too large, respectively. We will compare our 331 geometries for CH2 and NH: with the “best" ab_initio results available. Our criterion for "best" is the common although not always justified criterion of lowest energy. For the lowest 3B1 surface of CH2, we obtained a global minimum at R = 2.32 bohrs and 8 = 126°, CH HCH where E = -38.64563 a.u. A high quality calculation by Langhoff and Davidson [17] on CH2 (3B1) predicted RCH = 2.07 bohrs and eHCH = 132°. Our value of RCH is 12.1% larger than the value given by Langhoff and Davidson. For the lowest 3B1 surface of NH3, we obtained a global minimum at RNH = 2.27 bohrs and 6 = 160°, where HNH E = -54.81995 a.u. The NH; ion is characterized by a lack of pertinent eXperimental results andab initio calculations of the same quality as may be found for CH2. There appears to be no previously reported calculation 3 which Optimized the value of RNH for the El state of NH3. Lee and Morokuma [18] have done the best calcula- l tions to date on NH; (331). They obtained 0 a 180° HNH and a very flat potential energy curve (energy vs. eHNH) 141 around eHNH = 180°. They did not indicate their bond length, however. Chu, Sin, and Hayes [19] obtained eHNH = 140° with RNH fixed at 2.0 bohrs. Harrison and Eakers (201 used RNH = 1.9055 bohrs for their 3B1 state calculations. Their value was obtained by minimizing the SCF energy of the 1A1 state.’ They obtained eHNH = 150° and a very low barrier to linearity. A very high quality calculation is needed to give a good prediction of 0 due to the extreme flatness of the surface HNH' _ 0 around eHNH - 180 . 10. 11. 12. 13. 14. 15. REFERENCES G. Herzberg, Proc. Roy Soc. (London), 262A, 291 (1961). G. Herzberg and J. W. C. Johns, J. Chem. Phys.'§4, 2276 (1971). C. C. J. Roothaan, Revs. Modern Phys. 32, 179 (1960). C. C. J. Roothaan, Revs. Modern Phys. 33, 69 (1951). R. Ditchfield, W. J. Hehre and J. A. Pople, J. Chem. Phys. 52, 5001 (1970). A modified IBMOL (VERSION 2), written by E. Clementi and A. Veillard, IBM Research Laboratory was used; modifications were made by D. A. Wernette. D. A. Wernette, unpublished work. V. Nicely, unpublished work; modifications were made by D. A. Wernette. C. E. Moore, "Atom Energy Levels" volume 1, National Bureau of Standards Circular 467 (1949). B. P. Stoicheff, Can. J. Phys. 35, 730 (1957); Advan. Spectry. l, 91 (1959). G. Herzberg, "Spectra of Diatomic Molecules," (Van Nostrand, Princeton, NJ, 1950). G. Herzberg and J. W. C. Johns, Astrophys J. 158, 399 (1969). R. Colin and A. E. Douglas, Can. J. Phys. 46, 61 (1968). G. Herzberg and A. Monfils, J. Mol. Spectry 5, 482 (1960). W. J. Stevens, J. Chem. Phys. 58, 1264 (1973); W. E. Kaskan and M. P. Nader, J. Chem. Phys. 56, 2220 (1972). 142 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 143 H. P. D. Liu and G. Verhaegen, J. Chem. Phys. 53, 735 (1969), calculated D (NH+) from experimental results 0 +9 2 0 3 - as follows: Do (NH , x H) = Do (NH, X 2 ) - I. P. (NH, x3z’) + I. P. (H, 2S). S. R. Langhoff and E. R. Davidson, International J. of Quantum Chem. VII, 759 (1973). S. T. Lee and K. Morokuma, J. Am. Chem. Soc. _3, 6863 (1971). S. Y. Chu, A. K. Sin and E. F. Hayes, J. Am. Chem. Soc. 24, 2969 (1972). J. F. Harrison and C. W. Eakers, J. Am. Chem. Soc. 95, 3467.(l973). C. F. Bender, P. K. Pearson, S. V. O'Neil and H. F. Schaeffer, J. Chem. Phys. 56, 4626 (1972). D. R. Stull and H. Prophet, "JANAF Thermochemical Tables," Second Edition, N.B.S. 47 (1871). The cal- culated AH of reaction is from the AHf (0°K) for the species involved. J. L. Franklin, J. G. Dillard, H. M. Rosenstock, J. T. Herron, K. Draxl and F. H. Field, "Ionization Poten- tials, Appearance Potentials and Heats of Formation of Gaseous Positive Ions," N.B.S. 26 (1969). The calculated AH of reaction is from A (298°K) for the species involved. If H (28) is nvolved, see reference [22] for AH? (298°K). W. Braun, A. M. Bass, D. D. Davis and J. D. Simmons, Proc. Roy Soc. A312, 417 (1969). J. A. Fair and B. H. Mahan, J. Chem. Phys. 62, 515 (1975). J. F. Harrison and L. C. Allen, J. Am. Chem. Soc. 21, 807 (1969). W. J. Hehre, R. F. Stewart and J. A. Pople, J. Chem. Phys. 55, 162 (1971). R. F. Stewart, J. Chem. Phys. 52, 431 (1970). 29. 30. 31. 144 POLYATOM (Version 2), D. B. Neumann, H. Basch, R. L. Kornegoy, L. C. Snyder, J. W. Moskowitz, C. Hornback and S. P. Liebmann; Program 199, QCPE, Indiana Uni- versity. Modifications were made by G. Shalhoub. S. V. O'Neil, H. F. Schaefer and C. F. Bender, J. Chem. Phys. 25, 162 (1971). C. A. Coulson and J. G. Stamper, Mol. Phys. 6, 609 (1963).