Potential Energy Surfaces for Simple Chemical Reactions : Application of Valence-Bond Techniques to the Li +HF+LiF +H reaction BY GABRIEL G. BALINT-KURTI* AND ROBERT N. YARDLEY? School of Chemistry, Bristol University, Bristol BS8 1TS Received 3rd May, 1976 Ab initio multi-structure valence-bond calculations have been performed to determine the potential energy surface governing the reaction Li + HF --t LiF + H. Results for both linear and non-linear nuclear geometries are presented. The system is a prototype for many heavier alkali metal plus hydrogen halide reactions which have been studied using the crossed molecular beams technique. The ab initio valence-bond results are improved by applying corrections, within the framework of the orthogonalized Moffitt (OM) method, for the atomic errors present.The orbital basis set used was of double zeta quality, and was augmented by some extra orbitals. Preliminary calculations on the neutral and ionic diatomic species were performed to ensure the adequancy of the valence- bond structure basis sets used and care was taken to ensure that the basis sets provided an adequate description of F-, HF- and LiF-. The endoergicity of the reaction, ignoring the zero-point vibra- tional energies, was predicted by the ab initio and OM methods to be 5.8 and 2.5 kcal mol-' respectively, as compared with the experimental value of 2.6 kcal mol-I. Besides the ground state potential energy surface, several surfaces for excited electronic states have been calculated and are presented. The relationship of the ground state potential energy surface to the reactive cross section, and its variation with energy, is discussed.The ground and excited state potential energy surfaces are compared with previously proposed models and a mechanism for the production of alkali metal ions in hyperthermal alkali metal atom-hydrogen halide collisions is proposed. 1. INTRODUCTION The very first successful crossed molecular beam experiment was performed by Taylor and Datzl in 1955 on the reaction, K + HBr -+ KBr + H. In the period since these first experiments many groups of workers have performed progressively more refined experiments on the K + HBr system,2-6 on its isotopic variants K + DBr4 and K + TBr7, and also on the reverse reaction.8 The differential reactive cross sections for these reactions do not appear to fit into an easily rationalis- able pattern>** and there has been some speculation9 as to the type of potential energy surface which would be capable of simultaneously explaining the results of all of the experiments.Besides these experiments there have also been several experi- rnentsl0-l4 on related members of the family of reactions M + HX 3 MX + H. As an aid to the interpretation of these experiments we have calculated a potential energy surface for the Li + HF system. This is the lightest member of the M+ HX family of reactions and, besides being of interest in its own right, the surface might serve as a prototype for the heavier members of the family. The calculations were t Present Address: Department of Chemistry, University of Sheffield, Sheffield S3 7HF.78 POTENTIAL ENERGY SURFACES FOR SIMPLE CHEMICAL REACTIONS performed using two different methods.The first of these was an ab initio multi- structure valence-bond method. While the second method, which is called the orthogonalised Moffitt (OM) method, involved the application of corrections for the known atomic errors present in the ab initio calculation. Before examining the tri- atomic LiHF system, preliminary calculations were performed on all the constituent diatomics 15*16 so as to determine which atomic orbital and valence-bond structure basis sets should be used in the triatomic calculations. In section 2 of this paper we sketch very schematically the basic ideas behind the different methods used and outline the atomic orbital and valence-bond structure basis sets.We also summarise the results of the preliminary diatomic calculations. In section 3 the results for the col- linear ground state surfaces are presented and discussed, while in section 4 we present the ground state surfiaces for non-linear geometries. Section 5 is devoted to a dis- cussion of some of the low-lying electronically excited states of the system. In section 6 the relevance of the present results to reactive scattering experiments on the alkali metal atom-hydrogen halide family of systems is discussed, and comparison is made with previously published work. In this last section we also discuss collisions at hyperthermal energies and propose a mechanism for the production of alkali metal atom ions in such collisions.2. THEORETICAL BACKGROUND AND DESCRIPTION OF BASIS SETS The first step in the calculation is to construct approximate atomic eigenfunctions for all the atomic states which will play a role in the calculation. The approximate atomic eigenfunctions for different atoms are then multiplied together and anti- symmetrised to form composite functions (CF’s). These composite functions form the basis set for the expansion of the total electronic wavefunction of the system. The CF’s are constructed so as to be eigenfunctions of S, with the same eigenvalue. They are, however, not in general eigenfunctions of S2. The computer program has the facility to form linear combinations of CF’s which are eigenfunctions of S2 and to use these functions, which are generally just the traditional valence-bond structures, as the basis set for the expansion of the total electronic wavefunction for the system.The theory of the method has been described in detail elsewhere17 and the computer program which performs the ab initio multi-structure valence-bond calculations is available for general use.” The approximate atomic eigenfunctions are constructed from an atomic orbital basis set. In the present calculations we use atomic orbitals of double zeta quality, augmented by some additional orbitals to add flexibility to the basis set. These are, in fact, the same orbitals as those used in the preliminary diatomic calculations which have been published e1~ewhere.l~ The most notable aspect of the approximate atomic eigenfunction basis set is that both F-(lS) and H-(’S) were represented by functions which included a three-fold intra-atomic configuration interaction.This made it possible to obtain reasonably accurate electron affinities for fluorine and hydrogen. The calculated values for these quantities being respectively 0.1042 a.u. and 0.021 75 a.u., while the experimental values are 0.1273 a.u. and 0.027 75 a.u. Besides the extra s and p orbitals on fluorine and the extra s orbital on hydrogen which were required for the intra-atomic configuration interaction just mentioned, the following “ extra ” orbitals were also added to the atomic orbital basis set: 1. A set of p orbitals on hydrogen to allow for polarisation and a “ contracted ” 1s orbital to allow for distortion.GABRIEL G.BALINT-KURT1 AND ROBERT N. YARDLEY 79 2. 1s Gaussian orbitals at the centres of the LiF and HF bonds. 3. An extra very diffuse orbital on fluorine (exponent 0.0001). The last orbital was included to permit a correct description of HF- which auto-ionises at small H-F separations.16 The diffuse orbital effectively permits an electron to escape. In table 1 we tabulate the calculated and experimental dissociation energies for the diatomics HF, LiH and LiF [see ref. (15) for details]. As can be seen from the TABLE 1 .-DIATOMIC DISSOCIATION ENERGIES a large CF basis set small CF basis set ab initio OM ab initio OM experiment HF 0.202 1 0.208 2 0.190 1 0.198 3 0.224 7 LiH 0.080 97 0.081 02 0.079 93 0.079 96 0.092 46 LiF 0.185 8 0.199 1 0.180 9 0.194 3 0.220 6 (‘) All quantities in atomic units.1 a.u. of energy = 4.359 828 x 10-l8 J = 27.21 1 648 eV = 627.52 kcal mol-l. ( b ) The large CF basis set consisted of 113 functions and the small one of 28. (=) The large CF basis set consisted of 34 functions and the small one of 25. (‘) The large CF basis set con- sisted of 90 functions and the small one of 20. table, the valence-bond calculations yield between 82% and 95% of the experimental dissociation energies. These results compare quite favourably with other types of calculations in which much more extensive atomic orbital basis sets were used. Clearly for a triatomic system it is possible to construct a much greater number of CF’s, from the same set of approximate atomic eigenfunctions, than for a diatomic system. We, therefore, used the smaller diatomic CF basis sets as the starting point for the construction of our CF basis sets for the triatomic system.The CF basis set for LiHF was constructed in such a way that if any one of the atoms is removed to a large distance then the potential energy curves which result from a calculation are just those of the small basis set diatomic calculations for both the ground and excited states.15*16 Also included in the basis set were the CF’s needed to provide a reliable description of Li+ + HF- and H+ + LiF-.16 In the linear nuclear configuration, separate CF basis sets were constructed for I: and ll symmetries. These basis sets consisted of 278 and 175 CF’s respectively. For non-linear geometries two separate basis sets were again constructed, one of A’ symmetry which was even with respect to reflection in the molecular plane, and one of A” symmetry which was odd.The A’ basis set consisted of 447 CF’s and the A” of 267 CF’s. For non-linear geometries linear combinations of the CF’s were taken to form doublet spin states. The final basis sets consisted of 321 functions of 2A’ symmetry and 183 of 2A’’. One criterion for the reliability of a potential energy surface for a chemical reaction is the calculated value of the exo- or endo-ergicity of the reaction ( i e , the difference in energy between reactants and products ignoring the zero-point vibra- tional energy). In the present case the ab initio and OM methods yield 5.8 and 2.5 kcal mol-1 respectively for the endo-ergicity while the experimentalvalue is 2.6 kcal mol-’.The excellent agreement of the OM method with the experimental value is due to a fortuitous cancellation of errors. 3. THE COLLINEAR GROUND STATE SURFACE The surface which is perhaps of the greatest interest in discussing the Li + HF 3 LiF + H80 POTENTIAL ENERGY SURFACES FOR SIMPLE CHEMICAL REACTIONS reaction is that corresponding to the 2C+ ground state of the collinear Li - F - H geometry. Fig. 1 and 2 show contour maps of this surface as calculated by the ab initio multistructure valence-bond and OM methods respectively, while the corres- ponding numbers are presented in tables 2 and 3. The most significant point to 2 3 4 5 6 7 LI -F separation/a.u. FIG. 1 .--Contour map of the ground state, linear Li - F - H 'C+ potential energy surface calculated by the ab initio method.The zero of energy is taken to be the calculated energy of the separated ground state atoms. Energies are in atomic units. 5 ' 5 0 I i '. 1 I- I I I 1 1 2 3 4 5 6 7 Li-F separation / a.u. FIG. 2.-Contour map of the ground state, linear Li - F - H 'T,+ potential energy surface calculated by the OM method. The zero of energy is taken to be the experimental energy of the separated ground state atoms. Energies are in atomic units. notice when comparing the two calculations is that they agree on all qualitative aspects of the surface, and differ only slightly from a quantitative viewpoint. Fig. 3 shows a cut along the entrance valley of the reaction. As a lithium atom approaches an HF molecule from the fluorine end, with the HF internuclear distance fixed at its equilibrium value, the potential energy surface possesses a small but definite well.This well is predicted to have a depth of 2.2 kcal mol-' by the OM method and 1.6 kcal mol-1 by the ab initio method. Examination of the valence-bond wavefunctionGABRIEL G . BALINT-KURT1 AND ROBERT N . YARDLEY 81 1 .o 1.25 1.5 1.76 2.0 2.25 2.5 2.75 3 .O 3.5 4.0 4.5 5 .O Table 2" Ab initio GROUND STATE SURFACE FOR LINEAR Li - F - H 2.0 2.5 3.0 3.2 3.5 3.8 4.0 4.5 5.0 6.0 0.7178 0.4746 0.4169 0.4102 0.4066 0.4064 0.4069 0.4085 0.4096 0.4104 0.2942 0.0504 -0.0085 -0.0158 -0.0202 -0.0211 -0.0210 -0.0201 -0.0194 -0.0191 0.1541 -0.0893 -0.1487 -0.1561 -0.1608 -0.1620 -0.1619 -0.1611 -0.1605 -0.1602 0.1233 -0.1199 -0.1794 -0.1869 -0.1916 -0.1927 -0.1927 -0.1917 -0.1909 -0.1904 0.1336 -0.1105 -0.1697 -0.1769 -0.1813 -0.1821 -0.1818 -0.1805 -0.1795 -0.1785 0.1527 -0.0969 -0.1538 -0.1592 -0.1611 -0.1600 -0.1588 -0.1562 -0.1544 -0.1528 0.1560 -0.0989 -0.1541 -0.1574 -0.1546 -0.1479 -0.1430 -0.1334 -0.1284 -0.1248 0.1531 -0.1050 -0.1606 -0.1634 -0.1594 -0.1505 -0.1436 -0.1263 -0.1123 -0.0999 0.1510 -0.1100 -0.1662 -0.1691 -0.1647 -0.1552 -0.1477 -0.1284 -0.1107 -0.0848 0.1497 -0.1156 -0.1731 -0.1761 -0.1716 -0.1618 -0.1540 -0.1335 -0.1 143 -0.0828 0.1499 -0.1180 -0.1763 -0.1793 -0.1748 -0.1649 -0.1570 -0.1361 -0.1165 -0.0841 0.1503 -0.1190 -0.1776 -0.1806 -0.1761 -0.1661 -0.1581 -0.1371 - 0.1173 -0.0845 0.1506 -0.1195 -0.1781 -0.1811 -0.1765 -0.1664 -0.1584 -0.1373 -0.1 174 -0.0845 a All quantities are in atomic units.1 a.u. of length = 0.529 177 x 10-lo m. The zero of energy is the calculated energy of Li('S) + F('P0) + H('S). in the region of the well confirms our intuitive expectations that the well arises from the polarisation of a lithium atom by the permanent dipole moment of HF. A cut, through the same two surfaces along the exit valley of the reaction, with the LiF separation now fixed at its calculated equilibrium position, is shown in fig. 4. In order for the hydrogen atom to depart and for the reaction to be completed, the HF separation must increase from its equilibrium value, which is at the position of R H F YF 1 .o 1.25 1 .5 1.76 2.0 2.25 2.5 2.75 3 .O 3.5 4.0 4.5 5 .O TABLE 3" OM GROUND STATE SURFACE FOR LINEAR Li - F - H 2.0 2.5 3.0 3.2 3.5 3.8 4.0 4.5 5.0 6.0 0.6999 0.4569 0.3994 0.3927 0.3892 0.3891 0.3897 0.3914 0.3926 0.3934 0.2806 0.0369 -0.0219 -0.0291 -0.0334 -0.0341 -0.0339 -0.0328 -0.0320 -0.0316 0.1433 -0.1004 -0.1596 -0.1670 -0.1715 -0.1725 -0.1723 -0.1713 -0.1706 -0.1701 0.1142 -0.1294 -0.1888 -0.1962 -0.2008 -0.2018 -0.2016 -0.2004 -0.1995 -0.1987 0.1257 -0.1189 -0.1781 -0.1853 -0.1895 -0.1902 -0.1899 -0.1884 -0.1871 -0.1860 0.1455 -0.1047 -0.1616 -0.1670 -0.1687 -0.1675 -0.1662 -0.1634 -0.1615 -0.1596 0.1480 -0.1078 -0.1632 -0.1664 -0.1634 -0.1564 -0.1512 -0.1407 -0.1351 -0.1311 0.1444 -0.1149 -0.1708 -0.1737 -0.1696 -0.1606 -0.1536 -0.1360 -0.1211 -0.1062 0.1417 -0.1204 -0.1772 -0.1801 -0.1758 -0.1663 -0.1589 -0.1395 -0.1217 -0.0945 0.1398 -0.1268 -0.1850 -0.1881 -0.1837 -0.1740 -0.1663 -0.1460 -0.1269 -0.0960 0.1397 -0.1296 -0.1887 -0.1918 -0.1875 -0.1777 -0.1699 -0.1492 -0.1298 -0.0981 0.1399 -0.1308 -0.1903 -0.1935 -0.1891 -0.1793 -0.1714 -0.1505 -0.1310 -0.0989 0.1402 -0.1313 -0.1909 -0.1941 -0.1897 -0.1798 -0.1719 -0.1510 -0.1313 -0.0991 a All quantities are in atomic units.The zero of energy is the experimental energy of Li(2s> + F('P0) + H('S). the inner minimum in fig. 4, to infinity. There is clearly a sizeable barrier to the reaction which must be surmounted as the hydrogen atom departs and the system moves along the exit valley of the surface. The height of this barrier with respect to the reactants Li + HF is 21.3 & 0.5 kcal mo1-1 on both the ab initio and OM surfaces. The quoted uncertainty arises frem the fact that an insufficient number of calculations were performed to determine the barrier height more precisely.Fig. 5 shows the energy profile along the reaction coordinate. The reaction co- ordinate is taken to correspond to a lithium atom approaching to within its equili- brium distance of the fluorine atom while the HF distance remains fixed, and then82 POTENTIAL ENERGY SURFACES FOR SIMPLE CHEMICAL REACTIONS 3 4 5 6 Li -F separation 1 a.u. FIG. 3.Variation of potential energy along a cut in the ab initio and OM ground state, linear Li - F - H 'X+ potential energy surfaces with the H - F separation fixed at 1.76 a.u. The zero of energy is taken to be the calculated or experimental energy of the separated ground state atoms as appropriate. - 0.1 L -0 16 3 0 \ m i 2 -0.18 +- 0 W > 0 W x 0) W c ._ c - L L -0.20 1 I I 1 - 2 3 4 5 6 H - F separation 1a.u.FIG. 4.-Variation of potential energy along a cut in the ab initio and OM ground state, linear Li - F - H 'Z+ potential energy surfaces with the Li - F separation fixed at 3.2 a.u. The zero of energy is taken to be the calculated or experimental energy of the separated ground state atoms as appropriate.GABRIEL G. BALINT-KURT1 AND ROBERT N. YARDLEY 83 -o""rT--- - - 016/- I -2 -1 -_ . .. .. . i I 1 1 reaction coordinate l a u FIG. 5.-Energy profile along reaction coordinates of the ab initio and OM, linear, ground state Li - F - H potential energy surfaces. The zero of the reaction coordinate is taken to be at the coriier of the reaction (RLiF = 3.2 a.u., RHF = 1.76 a.u.). For negative values the reaction coordin- ate corresponds to (3.2 - RLIF) arid for positive values to (RHF - 1.76).the departure of the hydrogen atom while the LiF separation stays fixed at its equili- brium distance. That is, we treat the reaction as if there were no cutting-the-corner. As can be seen from the contour maps of the surface (fig. 1 and 2), this should be a reasonable approach. Fig. 5 demonstrates that by far the greater part of the barrier to reaction must be surmounted in the exit valley, after the system has turned the corner. 4. THE GROUND STATE SURFACE I N NONLINEAR The potential energy surfaces for the reaction were examined at selected nuclear configurations with the Li - F - H angle fixed at 135" and 90". In table 4 we compare the ground state potential energy surface as calculated by the OM method along the entrance channel of the reaction for the three different angles examined.In all three cases the H - F separation is held fixed at its equilibrium value of 1.76 a.u. while the Li - F separation is varied. We see from the table that the form of the surface is similar along the entrance valleys at all three angles. The depth of the shallow well in the entrance channel increases marginally (by about 0.25 kcal mol-') as the system is bent out of the linear configuration but then starts to decrease as the LiFH angle decreases below 120". This is demonstrated in curve A of fig. 6 where we show the variation of the ground state potential energy surface with angle for fixed H - F and Li - F separations. In curve A the internuclear separations correspond to the bottom of the shallow well in the entrance valley.We see that, as stated above, this curve starts to rise as the LiFH angle decreases below 120" and the potential NUCLEAR CONFIGURATIONS84 POTENTIAL ENERGY SURFACES FOR SIMPLE CHEMICAL REACTIONS TABLE 4.vARIATION OF THE GROUND STATE POTENTIAL ENERGY SURFACE ALONG ENTRANCE VALLEY OF THE Li + HF 3 LiF + H REACTION FOR DIFFERENT ANGLES OF APPROACH.- THE CALCULATIONS WERE PERFORMED USING THE ORTHOGONALTZED MOFFITT METHOD. THE H-F SEPARATION IS FIXED AT ITS CALCULATED EQUILIBRIUM VALUE (1.76 a.U.).' LiFH angle RLiF 180" 135" 90" 2.0 2.5 3 .O 3.2 3.5 3.8 4.0 4.5 5.0 7.0 9.0 11.0 co 0.1142 -0.1294 -0.1888 -0.1962 -0.2008 -0.2018 -0.201 6 -0.2004 -0.1995 -0.1986 -0.1985 -0.1984 -0.1983 - -0.1225 -0.1 88 1 -0.1963 -0.2013 -0.2023 - 0.2020 - 0.2006 -0.1994 - -0.1983 - -0.1098 - 0.1 829 -0.1925 - 0.1987 -0.2002 -0.2001 -0.1990 -0.198 1 -0.1982 -0.1984 -0.1983 - ' All quantities are in atomic units.The zero of energy is the experimental energy of the infinitely separated ground state atoms. -017 3 0 \ ul E z - 0 18 0 Q, > 0 c ._ c + -0 19 e !? x c al -0 20 \ - ', I I I t I I L L I I L i - F - H a n g l e / deg. FIG. 6.Variation of ground state Li - F - H potential energy surface with LiFH angle as calculated by OM method. The internuclear separations are &xed at: (A) RLiF = 4.0 a.u., RHF = 1.76 a.u. (B) RLiF = 3.2 a.u., R H ~ = 1.76 a.u.GABRIEL G. BALINT-KURT1 AND ROBERT N. YARDLEY 85 increases sharply for angles below 90".Curve B shows the variation of the potential with angle when the Li-F and H-F separations are fixed at values corresponding to the corner of the reaction path. In this case the potential is at first nearly constant as the angle is decreased, and then starts to rise at angles smaller than 120". In table 5 the ground state potential energy surface is compared along the exit TABLE s.-vARIATION OF THE GROUND STATE POTENTIAL ENERGY SURFACE ALONG EXIT VALLEY OF THE Li + HF -+ LiF + H REACTION FOR DIFFERENT LiFH ANGLES. THE CALCULATIONS WERE PERFORMED USING THE ORTHOGONALIZED MOFFITT METHOD. THE Li - F SEPARATION IS FIXED AT ITS CALCULATED EQUILIBRIUM VALUE (3.2 a.u.) (I LiFH angle RHF 180" 135" 90" 75" 1 .o 1.25 1.5 1.7 1.76 1.9 2.0 2.25 2.4 2.5 2.75 3 .O 3.5 4.0 4.5 5 .O 6.0 8 .o co 0.3927 - 0.0291 -0.1670 - 0.1 949 -0.1962 - 0.1921 -0.1853 - 0.1670 -0.1646 -0.1664 - 0.1737 - 0.1801 -0.1881 -0.191 8 -0.1935 - 0.1941 -0.1944 -0.1943 - 0.1942 - - 0.1659 -0.1948 -0.1963 - -0.1867 -0.1698 - 0.1665 -0.1675 - 0.1737 -0.1798 -0.1878 -0.191 6 - - - 0.1942 - -0.0157 -0.1 577 - 0.1897 -0.1925 -0.1894 -0.1808 -0.1788 -0.1789 -0.181 7 -0.1852 -0.1901 - 0.1926 - - - -0.1945 -0.1942 - 0.1820 0.1850 0.1858 0.1821 0.1820 0.1827 0.1858 - ~~ All quantities are in atomic units.The zero of energy is the experimental energy of the infinitely separated ground state atoms. valley of the reaction for different LiFH angles. At the first three angles considered the potential displays the same qualitative shape as shown in fig. 4. The height of the barrier to reaction, however, decreases quite dramatically with decreasing LiFH angle.Thus the barrier on the perpendicular surface (90") is about 12.2 kcalmol-'-a full 9 kcal mol-I lower than on the linear surface! 5. THE LOW-LYING ELECTRONICALLY EXCITED STATES OF LiHF Fig. 7 and 8 show some of the lower lying electronically excited states of the system, calculated using the OM method, along the entrance valley of the reaction Li + HF -+ LiF + H. Fig. 7 corresponds to a linear geometry with the lithium atom approaching the fluorine along the HF molecular axis, while fig. 8 corresponds to a perpendicular approach with the LiFH angle fixed at 90". The lowest curve in each corresponds to the ground state potential energy surface and correlates at large Li - F separations with a lithium atom and an HF molecule both in their ground states.The next group of states, two of them in fig. 7 and three in fig. 8, correlate86 POTENTIAL ENERGY SURFACES FOR S I M P L E CHEMICAL REACTIONS 3 5 7 9 Li -F separation / a u FIG. 7.-Cut through the five lowest linear Li - F - H potential energy surfaces as calculated by OM method with H - F separation fixed at 1.76 a.u. The zero of energy is taken to be the experi- mental energy of the separated ground state atoms. 1 1 - 1 7 1 1 7 6 L i --F - 1 i 2A’ / FIG. &--Cut through the six lowest perpendicular Li - F - H potential energy surfaces (LiFH angle = 90’) as calculated by OM method.GABRIEL G . BALINT-KURT1 AND ROBERT N. YARDLEY 87 at large distances with an excited lithium atom in its 2Po state plus a ground state HF molecule.At 9.0 a.u. the next state in both fig. 7 and 8 is found to correspond mainly to " Lif + HF(XIC+) + electron ". This comes about because HF- auto-ionises at small internuclear separations. As mentioned in section 2 our basis set included an extremely diffuse orbital which was intended effectively to permit the electron to escape and thus describe this auto-ionisation process. The highest curve in both figures corresponds, at an Li-F separation of 9.0 a.u., mainly to Li+ + HF- with the electron localised on the HF- molecule. Our preliminary calculations16 on the HF- ion confirm that the two states just discussed do in fact occupy their correct asymptotic positions relative to the other curves, although at an Li - F separation of 9 a.u.both curves are still rising and have not yet reached their corrected asymptotic values. In figs. 9 and 10 we show some of the low-lying electronically excited states of the system, calculated using the OM method along the exit valley of the reaction. In H-F separation h u . FIG. 9.-Cut through the nine lowest linear Li - F - H potential energy surfaces as calculated by OM method with Li - F separation fixed at 3.2 a.u. fig. 9 the system has a linear geometry while in fig. 10 the LiFH angle is fixed at 90". We have already commented on the fact that the barrier to reaction is much smaller for the perpendicular than for the linear nuclear arrangement. This barrier arises from an avoided curve crossing between states which can, to a first approximation, be described as Li(2S) + HF(X% +) for small H - F separations and LiF(XIX +) + H(2S) for larger separations. In the case of any avoided crossing of curves or surfaces the possibility of surface-hopping, or of a non-adiabatic transition, between the two surfaces arises. The probability of such a transition is increased by a close approach of the two surfaces.In the present case the two lowest surfaces approach closest to each other at an H - F separation of around 2.25 a.u. in the linear configuration.88 POTENTIAL ENERGY SURFACES FOR SIMPLE CHEMICAL REACTIONS 2 3 L 5 H -F separation / a u. I FIG. 10.-Cut through the seven lowest perpendicular Li - F - H potential energy surfaces, cor- responding to doublet spin states, as calculcated by OM method with Li - F separation fixed at 3.2 a.u.At this H - F separation the energy gap between the surface is 0.77 eV (17.8 kcal mol-') in the linear configuration and 1.73 eV (39.8 kcal mol-') in the perpendicular con- figuration. We see, therefore, that non-adiabatic transitions between these two surfaces will take place preferentially in the linear nuclear configuration. At large HF internuclear distances the nine curves shown in fig. 9 correspond in order of increasing energy to the states: 1. 'II -+ LiF(lII) + H(2S) 2. 2rI,411 -+ LiFCII) 3. H(2S) 3. 2Z+ -+ LiF(XIX+) + H(2,S') 6. 2X+ -+ LiF(2lC+) + H(2S) 7. 2C.+,4X+ + LiF(23X+) + H(2S) The curves shown in fig. 10 clearly have the same asymptotes, but only doublet states were examined for non-linear nuclear configurations.In comparing the curves along the entrance and exit valleys (figs. 7 and 9, or 8 and 10) it is very noticeable that there are no available low-energy excited states of the products. Indeed the threshold energy for producing excited products is 6.1 eV (calculated from experimental quanti- ties) and below this energy the only observable processes will be: 1. Reaction to produce products in their ground electronic state. 2. Non-reactive collisions to produce Li(2Po) + HF(XIX+). 3. Non-reactive collisions to produce " Li+ + HF(X'Z+) + electron ". 5.2c+:c+ -+ ~ i ~ ( 1 3 c + ) + H(~s)GABRIEL G . BALINT-KURT1 A N D ROBERT N. YARDLEY 89 6. DISCUSSION The main result of the present calculations is that they predict a ground state potential energy surface for the Li + HF LiF + H reaction which has a barrier to reaction in the exit valley.The barrier is predicted to decrease substantially as the LiFH angle decreases and the system becomes progressively more bent. Polanyi and co-workers 19s20 have performed trajectory calculations on systems with barriers to reaction in the exit valley. They find that the probability of reaction is increased to a much greater extent by the presence of vibrational energy in the reactant diatomic than by the presence of an equivalent amount of relative transla- tional energy. The fascinating experiments of Brooks and co-workers 10~11~13 have shown that, for the reaction K + HCl 4 KC1 + H, increasing the relative trans- lational energy available to the system has little effect, whereas excitation of the HCl with one quantum of vibrational energy results in an increase of the reactive cross section by two orders of magnitude.These experiments, together with Polanyi's calculations, lead to the conclusion that the potential energy surface governing the K + HCl reaction most probably has a barrier along its exit valley, and is, therefore, at least qualitatively similar to the surface reported here for its lighter analogue. In an attempt to reconcile the seemingly inconsistent, experimentally 4 9 determined, differential reactive cross sections for K + HBr and its isotopic analogues K + DBr and K + TBr, Roachg has proposed a model potential energy surface, together with an approximate discussion of the dynamics for the system. Roach's model potential for the K + HBr system is strikingly similar, on a qualitative level, to the one reported here for Li + HF.The main difference in the two potentials lies in their variation as the system is bent out of the linear geometry. Roach identifies two important nuclear geometries whose corresponding energies determine the ease with which reaction will occur. These geometries are for the LiHF system RHF = 1.76 a.u., RLiF = 3.2 a.u. which corresponds to the corner of the reaction, and the position of the maximum of the barrier along the exit valley. Using a very crude description of the dynamics of the system, it is the relative trans- lational motion which provides the energy to reach the corner of the reaction, while the vibrational energy in the reactant diatomic (i.e., HF) provides the energy difference between the corner and the top of the barrier. The variation of the energy at the corner of the reaction with LiFH angle is shown in fig.6(B). We see that this is essentially flat at first and starts rising steeply for angles smaller than 90". The variation of the energy at the top of the barrier can, to some extent, be deduced from table 5. From this table we see that the energy at the top of the barrier at first hardly varies as the system is bent out of a linear configuration and then decreases sharply between 135" and 90". Calculations of this quantity have only been performed at the four angles shown in the table. It seems reasonable to hypothesise that the energy at the top of the barrier will eventually rise as the angle is decreased below 75", and that the size of the hump (ie., the difference in energy between the corner and the top of the barrier) will continue to decrease and also that the hump itself might eventually vanish.The form of these changes of the potential with angle differ from those assumed by Roach. It seems likely, however, that the experimental results on K + HBr and its isotopic analogues could have been satisfactorily recon- ciled using a potential qualitatively similar to the LiHF potential presented here. There have been two previous calculations on the Li + HF system. The first of these was performed by Lester and Krauss2' and was an LCAO-SCF calculation which explored only the entrance valley of the reaction, with the H - F separation fixed at its equilibrium value. Our calculations are in good agreement with theirs90 POTENTIAL ENERGY SURFACES FOR SIMPLE CHEMICAL REACTIONS in that they predict a small well as the lithium approaches the fluorine end of the HF diatomic.Both calculations predict that the depth of the well increases marginally when the LiFH angle changes from 180" to 135" and then decreases sharply on going to 90". Our calculations predict a somewhat shallower well in the linear nuclear geometry but a deeper one for 90". The other previous set of calculations on the system were some performed by Baht-Kurti and Karp1~s.l~ These calculations used a smaller atomic orbital basis set, which was of a poorer quality than that used here. They also did not include any intra-atomic configuration interaction to improve the H- and F- wavefunctions as do the present ones.While several qualitative features of the potential surface reported in the previous work are similar to those reported here, there are important quantitative differences. These differences, which are evident in both the ab initio and OM calculations, are thought to arise mainly from an inadequate description of the F- wavefunction in the previous work. In the present case all features of the ab initio and OM potential surfaces are very similar. This is due to the fact that the approximate atomic eigenfunctions which are used to build up the molecular basis functions are all of a reasonably good and uniform q~a1ity.l~ Besides the experiments at thermal energies, experiments on alkali metal atom- hydrogen halide systems have been performed at hypthermal energies12*14 in the range 1 - 20 eV.In both these sets of experiments alkali metal ions were observed. Lacmann and Herschbach l2 in their experiments also looked for the production of electronically excited potassium atoms from K + HCl collisions, but failed to find them in significant quantities. The production of the alkali metal ions has been discussed in terms of avoided curve crossings which are represented as occurring as the alkali metal atom approaches the hydrogen halide molecule. Comparing Lac- mann and Herschbach's schematic picture of the low-lying potential energy curves of the system [fig. 3 of ref. (12)] with our calculated curves along the entrance valley of the reaction (fig.7 and S), we see that the figures show no qualitative similarity. As a consequence of this incompatibility we would like to suggest a possible alternative mechanism for the production of the alkali metal ions in alkali metal atom-hydrogen halide collisions. This mechanism involves the rapid departure of the hydrogen atom when the alkali metal atom has approached close to the halogen. The departing hydrogen atom takes away only a very small proportion of the initial relative kinetic energy, and therefore leaves the alkali halide molecule in a meta-stable state with enough " vibrational " energy to dissociate. This dissociation takes place to produce ionic fragments with a high p r ~ b a b i l i t y . ~ ~ , ~ ~ From our previous discussion we see that the departure of the hydrogen atom will be favoured (for the Li + HF system) for collisions having small LiFH angles.The viability of this mechanism depends on the continued decrease of the height of the hump in the exit valley with decreasing angle for angles below 90". At present we have had to hypothesise that this is the correct form of the potential, a more definitive discussion must await the outcome of further calculations. It should be noted that this proposed mechanism also rationalises the absence of excited alkali metal atoms in Lacmann and Hersch- bach's experiments, as such atoms are not produced by the dissociation of highly vibrationally excited alkali halide molecules. The authors are indebted to the Science Research Council for a grant of computer time and would like to express their gratitude to the staffs of the Atlas and Rutherford Computer Laboratories for their assistance.In particular, we are grateful for the provision of the ATMOL 2 integral evaluation package which was used in this work. R. N. Y. also wishes to thank the S.R.C. for financial support.G A B R I E L G . BALINT-KUR'I'I A N D ROBERT N. YAKDLEY 91 E. H. Taylor and S . Datz, J. Chem. Phys., 1955, 23, 1711. D. Beck, E. F. Green and J. ROSS, J. Chem. Phys., 1962,37,2895. J. R. Airey, E. F. Green, K. Kodera, G. P. Reck and J. Ross, J. Chem. Phys., 1967,46, 3287. K. T. Gillen, C. Riley and R. B. Bernstein, J . Chem. Phys., 1969, 50, 4019. C. Maltz, N. D. Weinstein and D. R. Herschbach, Mol. Phys., 1972, 24, 133. D. S. Y. Hsu and D. R. Herschbach, Furuduy Disc. Chern. Soc., 1973,55, 116. D. R. Herschbach in Proceedings of the Coilfereizce on Potential Energy Surfaces in Chemistry, ed. 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