Potential Energy Surfaces for Ion-Molecule Reactions. Intersection of the 3A2 and 2B1 Surfaces of NH; BY CHARLES F. BENDER Laser Division, Lawrence Livermore Laboratory, University of California, Livermore, California 94550 AND JAMES H. MEADOWS AND HENRY F. SCHAEFER 111 Department of Chemistry, Lawrence Berkeley Laboratory University of California, Berkeley, California 94720 Received 10th May, 1976 The N+ + H2 system is one of the few ion-molecule reactions for which detailed molecular beam studies have been carried out. To complement this experimental research, we have performed a theoretical study of two of the low-lying NH; potential energy surfaces. The intersection and avoided intersection (for C, geometries) of the lowest 3Az and 3B1 surfaces allows a pathway by which the ground state of NH t may be accessed without a potential barrier.The electronic structure calculations employed a double zeta plus polarization basis set, and correlation effects were taken into account using the newly developed Vector Method (VM). To test the validity of this basis, additional self-consistent-field studies were performed using a very large contracted gaussian basis N( 13s 8p 3d/9s 6p 3d), H(6s 2p/4s 2p). The 3Az surface, on which N+ and Hz may approach, has a sur- prisingly deep potential minimum, -60 kcal mol-l, occurring at r,(NH) - 1.26 8, and B,(HNH) - 43'. Electron correlation is responsible for about 15 kcal of this well depth, which appears fairly insensitive to extension of the basis set beyond the double zeta plus polarization level. The line of intersection (or seam) of the 3Az and 3B1 surfaces is presented both numerically and pictorially.The minimum energy along this seam occurs at -51 kcal below separated N+ + H2. Thus for sufficiently low energies one expects N+ - H2 collisions to provide considerable " complex formation ". Fur- ther molecular beam experiments at such low energies (<0.5 eV) would be of particular interest. Simple ion-molecule reactions have provided some of the most fascinating examples to date of the interplay between different potential energy surfaces of a single chemical system.l Most noteworthy in this regard are the molecular beam studies of Mahan and co-~orkers,~-~ who have carefully investigated, among other systems, the Cf + H2, Nf + H2 and O+ + H2 reactions.These reactions are particularly appealing as prototypes, since they are sufficiently simple to be studied by both electronic structure theory5 and classical6 or semiclassical7 dynamics. In addition, the use of qualitative electronic correlation has also proved to be very helpful in understanding these simple reactions ; alternatively the experiments may serve as testing grounds for simple molecular orbital theory. A reasonable starting point for our discussion is the N+ + H2 electronic state correlation diagram of Fair and Mahan.3 This diagram is reproduced with their permission in fig. 1. As discussed by Fair and Mahan (and el~ewhere~*'~ in regard to the C+ + H2 reaction) the key feature in the interpretation of low energy (say less than -3 eV) molecular beam results is the intersection of two low-lying potential energy surfaces.For the Nf + H2 case (in C,, symmetry) these are the 3B1 and 3A2 surfaces. The 3B1 state is known11-14 to be the ground state ofNHz, the nitrenium60 POTENTIAL ENERGY SURFACES FOR ION-MOLECULE REACTIONS ion, while the 3A2 state is less understood. However, on the basis of orbital symmetry considerations l v 8 and earlier theoretical work9 on C+ + H2, the 3B1 surface is expected to be quite repulsive as the N+ initially approaches H2. The deep well of the 3B1 surface is “ protected ” from N+ - H2 collisions on the same surface by means of this large barrier. However, the 3A2 surface should be either much less repulsive9 or attractivelo as N+ approaches H2. And since the two surfaces are both of 3A” symmetry as soon as the N+ ion moves off the Hz perpendicular bisector, the C,, crossing of surfaces becomes an avoided intersection.If there are points along this crossing of 3Br and 3Az surfaces which lie at energies near or below the N+ + H2 asymptote, then there exists a barrier-free pathway 3A2 3 3A“ + 3B1 (1) for the formation of ground state NHT from separated N+ and Ha. Such a pathway for the analogous situation with respect to C+ + H2 has been recently demonstrated unequivocally in the important theoretical work of Pearson and Roueff.lo In their communication, Pearson and Roueff lo bring to light a critical ingredient in the proper theoretical treatment of this problem. That is, polarization functions5 (d functions on carbon andp functions on the hydrogen atoms in their case) critically affect the energy at which the seam or line of intersection occurs.Their finding is pertinent to the present discussion since Gittins and Hirst l5 have recently reported single configuration self-consistent-field (SCF) results for N+ + H2 using a basis set which is quite well-chosen and flexible 16*17 but lacks polarization functions. Gittins and Hirst conclude that access to the deep 3B1 potential well may be possible with only a small barrier, in the order of 4 kcal mol-? By comparison of the effects of polarization functions in the C+ + H2 sy~tem,~*~O it would appear likely that this barrier should disappear completely. The present paper, then, builds on the Gittins- Hirst work15 but goes well beyond it for the N+ + H2 system by the use of larger basis sets and the direct inclusion of correlation effects.These two theoretical exten- sions should allow for a meaningful comparison with the molecular beam experiments of Fair and Mahan.3 THEORETICAL APPROACH Two basis sets of contracted gaussian f ~ n c t i o n s ~ ~ ’ ~ were used here. The first was a standard Huzinaga-Dunning double zeta plus polarization (DZ + P) set, designated N(9s 5p ld/4s 2p Id), H(4s lp/2s lp). The polarization function exponents were 0.8 (nitrogen d functions) and 1.0 (hydrogen p functions), and a scale factor of < = 1.2 was used on the hydrogen s functions. This first basis is essentially the same (except for the obvious replacement of the C basis by one appropriate to N) as that used by Pearson and Roueff,lo and was used for both SCF and configuration interaction (CI) calculations.Since we were initially quite surprised by Pearson and Roueff’s demonstration lo of the critical importance of polarization functions, it was decided to test whether further extensions of their basis would be of qualitative importance to the shape of the N+ + H2 potential surfaces. Therefore, following the recent work of Meadows1* on CH2, a very large basis was adopted: N(13s 8p 349s 6p 3 4 and H(6s 2pj4s 2p). The polarization functions had gaussian orbital exponents a = 1.6, 0.8 and 0.3 for the nitrogen d functions and a = 1.4 and 0.25 for the hydrogen p functions based on past experien~e.~*~”~~ The nitrogen sp functions and hydrogen s functions were the appropriate primitive gaussian basis sets of van Duijneveldt,22 contracted to provide maximum flexibility in the valence region.That is, the five s functions with largestCHARLES F . BENDER, ET A L . 61 orbital exponents ai were grouped together according to the nitrogen atomic 1s orbital, and an analogous procedure followed for the three nitrogen p functions with largest exponents. Based in part on Clementi and Popkie's of the water molecule with many basis sets, we estimate that the present basis set for NHZ should yield total energies within 0.005 hartrees (-3 kcal) of the Hartree-Fock limits for the 3Az and 3Br potential surfaces. Relative errors, of course, should be much smaller. The electron configuration for the two states of primary interest are la! 2ai lb; 3a1 lbl 3B1 (2) la; 2a: 3a; lbz lbl 3A2 (3) and restricted SCF t h e ~ r y ~ ~ .~ ~ has been applied to both of these states. We also note that the first excited electronic state of NH; is of 'Al symmetry and several two-configuration C, la! 2ai lb; 3ai + C2 la! 2ai lbi lb: (4) SCF studies of this state were also made. Finally, it should be noted that the source of the large barrier in the N+ + H2 3B1 approach is the fact that for large N+ - H2 separations the la! 2a; 3 4 1 bl 4a, 3B1 (5) configuration, rather than (2), dominates the wave function. Here we report two such tests, the first with the N+ and H, species separated by a distance R = 100 bohr radii. R is the distance between the N+ ion and the H2 bond midpoint, while Y will designate the H-H internuclear separation.For R = 100, r = 1.4 (essentially the equilibrium internuclear separation of H2) the 3A2 SCF energies are -55.011 59 and -55.021 23 hartrees, the difference being 0.009 64 hartrees or 6.0 kcal mol-'. Secondly we report a point near the equilibrium 3A2 geometry, namely R = 2.0 and r = 1.8 bohr, where the two basis sets yield SCF energies -55.078 23 and -55.091 57 hartrees. The difference in the latter case is larger, 0.013 34 hartrees or 8.4 kcal mol-I. It is certianly not surprising that the near Hartree- Fock basis yields somewhat lower relative energies as N+ and H2 approach. And if SCF basis set errors are directly transmitted to CI results, one would expect our DZ + P basis to yield CI dissociation energies for N+ - H2 about 2.5 kcal less than the exact values.Of course, in the present case, the uncertainties in our treatment of the correlation problem are roughly of that same order of magnitude. In any case the potential surface differences arising from the two basis sets are small, about an order of magnitude less than those found by Pearson for CHZ in going from the DZ to the DZ + P basis set. Electron correlation was taken into account variationally using the newly developed vector method (VM) of Bender and co-workers.26 The CI calculations were carried out with the early version of the VM code. That is, all Slater determinants differing by one or two spin orbitals from (2) for the 3B1 calculations or (3) for the 3A2 calcula- tions were included. In this way 1810 and 1824 determinants were respectively employed in the 3B1 and 3A2 variational procedures.The above was carried out with the usual restriction that the lal orbital (essentially nitrogen 1s) be doubly occupied in all determinants. It is now well-established5 that such a CI procedure will provide at least 90% of the attainable valence shell correlation energy in cases A number of direct SCF comparisons of the two basis sets were made.62 POTENTIAL ENERGY SURFACES FOR ION-MOLECULE REACTIONS (such as the present) where the wave function is qualitatively described by a single determinant SCF wave function. Use of the near Hartree-Fock basis was restricted to the location of the equilibrium geometries of the 3A2, 3B1, and 'Al electronic states. With the DZ + P basis, a regular grid of points (available from the authors on request) for both the 3B1 and 3A2 states was mapped out.These were all combinations of R = 3.0, 2.5, 2.0, 1.75, 1.5 and 1.25 bohrs with Y = 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4 and 2.6 bohrs for a total of 6 x 8 = 48 points on the surface. To provide a reference point for the relative energies quote hereafter, we note that for the 3A2 state R = 100, Y = 1.4, the SCF and CI energies with the DZ + P basis were -55.011 58 and -55.123 29 hartrees. Thus for separated N+ plus H2 the calculated correlation energy is 0.11 1 708 hartrees. As we shall see, the correlation energy increases as the N+ and H2 are brought together. SINGLET-TRIPLET SEPARATION I N THE NITRENIUM ION Before going on to the primary purpose of this research, let us make a brief digression.Although the 3B1 - lAl separation in NH; is not known experimentally, there have been at least four theoretical predictions of this quantity. On the ab initio side the groups of Morokuma,l' Hayes12 and Harrison13 have predicted 45, 36 and 45 kcal mol-l, with the 3B1 state the lower lying in each case. More recently Haddon and Dewar14 have used their semi-empirical MIND0/3 method to predict 31 kcal mol-1 for this quantity. For comparison with these results, the near Hartree-Fock basis was used to predict the 3B1 and lAl equilibrium geometries. For the 3B1 state the predicted structure was r,(NH) = 1.018 A, B,(HNH) = 143.3', corresponding to a total SCF energy of -55.229 65 hartrees. The two-configuration SCF description (4) of the 'Al state yields re(NH) = 1.033 .$, B,(HNH) = 108.2' and a total energy of -55.183 29 hartree.Thus the singlet-triplet separation AE is predicted to be 0.046 36 hartrees = 29.1 kcal mol-l. The best means of evaluating the reliability of the above prediction is by compari- son of analogous theoretical procedures with experiment for CH2, for which an accurate - lAl) has recently become a~ailable.'~ The experimental value of 19.5 & 0.7 kcal may be compared with the 10.9 kcal obtainedl8 for CH2 by the method described in the previous paragraph. Thus it is evident that a two-configuration description of the lAl state overcompensates for the fact that the 3B1 state has less correlation energy. For CH2 the use of a single configuration SCF treatment of the 'Al state yields a separation of 24.8 kcal, too large as expected. More precisely the experimental result lies 61.9% of the way from the two-configuration lAl result to the one-configuration lAl result.With the above in mind, we carried out single configuration (la: 2a: lb: 3at) SCF calculations on NHZ, yielding r,(NH) = 1.032 A, O,(HNH) = 109.6", and E = -55.158 38. The singlet-triplet separation obtained in this way (44.7 kcal) is con- siderably greater than the two-configuration result, 29.1 kcal. It seems quite certain that the exact nitrenium separation lies between the two, and if the same 61.9% criterion is used, a semi-empirical prediction of 38.8 kcal is made. Partly because of the semi-empirical nature of our prediction and also because of the use of a Hartree- Fock limit basis, we suggest that the 38.8 kcal value is probably the most reliable prediction made to date.CHARLES F.BENDER, ET AL. 63 REGION OF INTERSECTION OF THE 3 A 2 AND 3B1 SURFACES Certainly the most interesting result found here is the rather deep potential well associated with the 3A2 state of NHf. Such a deep well is not anticipated from the C+ - H2 calculations of Liskow, Bender and Schaeferg or the correlation diagram (fig. 1) of Fair and Mahan. Such a well is implicit in the work of Pearson,l0 but he - 3 - 2 t- - -41- -t -6 4- i I: -6 FIG. 1,-Correlation diagram of Fair and Mahan3 for the N+ + H2 system. does not report the predicted 2B2 (analogous to the NH; 3A2 state) dissociation energy relative to separated C+ + H2. We do know that Pearson's CH; 2B2 state must be bound by at least 15 kcal, since that is the lowest energy at which the 2B2 and 2A1 electronic states are degenerate. Using the near Hartree-Fock basis, the 3A2 state of NHS is predicted by SCF theory to have an equilibrium geometry r,(NH) = 1.207 A, B,(HNH) = 46.4'.This small bond angle is characteristic of the early approach of N+ to H2; and the predicted equilibrium geometry corresponds to a near Hartree-Fock energy 44.6 kcal below separated N+ and H2. Using the DZ + P basis the 3A2 minimum is less precisely located since the grid (see previous section) is relatively sparse in this region (note that the density of grid points is greatest near the intersection of the 3A2 and 3B1 surfaces). With this disclaimer we note that the DZ + P SCF minimum is predicted by a 9-point fit to lie at r,(NH) = 1.25 A, 8, = 42', with energy 44.3 kcal below N+ + HZ.Realistically the true SCF minimum with this basis probably occurs about 2 kcal higher, when one considers the direct comparisons (between DZ + P and near Hartree-Fock basis sets) of the previous section. Similarly the CI equilibrium geometry is r,(NH) = 1.26 A, 0, = 43", and lies 60.4 kcal below N+ + H2. The lowest actual calculated point on the 3A2 surface occurs at R = 2.0 bohrs, Y = 1.8 bohrs (or r(NH) = 1.161 A, 8 = 48.5') for both SCF and CI methods. These points lie 41.8 and 56.8 kcal below the comparable asymptotic calculations and make it quite clear that electron correlation contributes -15 kcal to the well depth. If in turn this 15 kcal is added to the near Hartree-Fock well depth of 44.6 kcal, one obtains 59.6 kcal as the predicted dissociation energy relative to N+ + H,.In any case a value of -60 kcal for the dissociation energy consistently appears on the basis of the present theoretical research. A dissociation energy this large (nearly 3 eV) must be considered surprising as it certainly cannot be justified in terms of a classical electrostatic picture.64 POTENTIAL ENERGY SURFACES FOR ION-MOLECULE REACTIONS The 3A2 and 3B1 surfaces are illustrated in fig. 2 and 3. Note, of course, that since the region of interest here is that near the intersection, the actual position of the 3B1 NH; equilibrium geometry is not included. The fact that the 3B1 surface becomes very attractive in that direction is however quite clear.Also apparent is the large barrier ( -75 kcal) associated with the Woodward-Hoffmann forbidden least motion9 insertion of N+ into H2. To complement the two contour maps and the line of intersection indicated on each, table 1 gives some numerical values for the line of 10 R ( N - CM I/ bohrs FIG. 2.-3Az potential energy surface for NHf. R(N - CM) is the distance from the nitrogen nucleus to the H2 centre of mass. Contours are labelled in kcal mol-l relative to infinitely separated N+ plus HZ. Note that contours energetically below 25 kcal are labelled in 5 kcal intervals, while those above 25 kcal are spaced by 25 kcal. It seem quite clear from previous work9*'' on the related C+ - H2 system that the C,, approach along the 3A2 surface is by far the most likely to lead to the bound NH+, species.In this light one can make a rough picture of one important aspect of the dynamics. First, as fig. 2 implies, high energy C2, collisions will tend to be ~nreactive.~ That is, with r(H-H) fixed at 1.4 bohrs, the 3A2 surface becomes quite repulsive rather quickly. For example, at R = 1.5 bohrs the surface lies 35 kcal above separated N+ + H2. Therefore, a key feature leading to complex formation is the necessity that the collision occurs slowly enough such that the H-H separation can become sufficiently large to reach the area of the line of intersection. Inspection of fig. 1 and 2 or table 1 shows that the line of intersection reaches zero kcal relative energy at about R M 1.52 bohrs, r M 1.67. In other words the H-H separation must increase by nearly 0.3 bohrs - 0.15 A for the line of intersection to become dynamically meaningful in low energy collisions.A final noteworthy point is that the line of intersection for the N+ - H2 system passes through much lower relative energies (50 kcal as compared with 15 kcal) thanCHARLES F. BENDER, ET A L . 65 R ( N - C M ) bohrs / FIG. 3.-3B1 potential energy surface for NH;. R(N - CM) is the distance from the nitrogen nucleus to the H2 centre of mass. Contours are labelled in kcal mo1-I relative to infinitely separated N+ and H2. Note that contours energetically below 25 kcal are labelled in 5 kcal intervals, while those above 25 kcal are spaced by 25 kcal. TABLE LINE OF INTERSECTION OF THE LOWEST 3A2 AND 3B1 POTENTIAL ENERGY SURFACES OF NH; .THESE POINTS ARE GIVEN IN TWO COORDINATE SYSTEMS FOR EASE OF INTERPRETATION. AS NOTED IN THE TEXT, R IS THE DISTANCE BETWEEN THE N+ NUCLEUS AND THE Hz BOND MIDPOINT. ENERGIES ARE CONFIGURATION INTERACTION (CI) ENERGIES RELATIVE TO SEPAR- Rlbohrs 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 r / bohrs 1.240 1.330 1.425 1.523 1.623 1.726 1.834 1.946 2.061 2.179 2.301 2.426 2.556 ATED N+ + H2. r (N-H)/ii 0.762 0.796 0.83 1 0.867 0.902 0.939 0.976 1.014 1.052 1.091 1.130 1.171 1.212 B(HNH)/deg 51.0 52.4 53.9 55.4 56.8 58.2 59.6 61.1 62.4 63.8 65.2 66.5 67.9 mergy/kcal mol- 171.0 116.6 72.6 37.6 10.4 - 10.4 -26.1 - 37.3 - 44.7 -49.1 - 51 .O - 50.8 -48.8 the corresponding line of intersection for the C+ - H2 system." A naive interpreta- tion of this comparison would suggest that at low energies one should observe more complex formation for the N+ than the C+ reaction.At this point, however, we believe that detailed dynamical studies are called for. This work on N+ - H2 and Pearson's researchlo for C+ - H2 appear to provide rather accurate predictions of66 POTENTIAL ENERGY SURFACES FOR ION-MOLECULE REACTIONS some of the crucial potential surface features, and the greatest uncertainties are now of a dynamical nature. Of course, more information concerning these surfaces would be welcome, especially concerning the slopes of the two lowest 3Aff surfaces (arising from 3Az and 3B1 in C,, point group) in the region of their avoided inter- section. We thank Prof. Bruce H. Mahan for helpful discussions, encouragement, and patience.We also benefited from many illuminating conversations with Dr. Peter K. Pearson. The computational burden was shared by the Lawrence Livermore Laboratory CDC 7600 and the Harris Corporation Series 100 minicomputer, sup- ported by National Science Foundation Grants GP-39317 and GP-41509X. B. H. Mahan, Accounts Chem. Res., 1975, 8, 55. B. H. Mahan and T. M. Sloane, J. Chem. Phys., 1973, 59, 5661. J. A. 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