THE JOURNAL OF CHEMICAL PHYSICS 122,

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1 THE JOURNAL OF CHEMICAL PHYSICS 122, Benchmark calculations of the complete configuration-interaction limit of Born Oppenheimer diagonal corrections to the saddle points of isotopomers of the H+H 2 reaction Steven L. Mielke a Department of Chemistry, Northwestern University, Evanston, Illinois David W. Schwenke b National Aeronautics and Space Administration (NASA) Ames Research Center, Moffett Field, California Kirk A. Peterson c Department of Chemistry, Washington State University, Pullman, Washington Received 21 March 2005; accepted 28 March 2005; published online 15 June 2005 We present a detailed ab initio study of the effect that the Born Oppenheimer diagonal correction BODC has on the saddle-point properties of the H system and its isotopomers. Benchmark values are presented that are estimated to be within 0.1 cm 1 of the complete configuration-interaction limit. We consider the basis set and correlation treatment requirements for accurate BODC calculations, and both are observed to be more favorable than for the Born Oppenheimer energies. The BODC raises the H+H 2 barrier height by kcal/mol and slightly narrows the barrier with the imaginary frequency increasing by 2% American Institute of Physics. DOI: / I. INTRODUCTION a Electronic mail: slmielke@chem.northwestern.edu b Electronic mail: david.w.schwenke@nasa.gov c Electronic mail: kipeters@wsu.edu Advances in the treatment of electron correlation and the development of hierarchical basis sets, together with methods to accurately extrapolate the results to the complete basis-set CBS limit, now permit highly accurate estimates of the complete configuration-interaction CCI limit, i.e., full CI FCI with a complete basis. For many small systems, the remaining uncertainties are comparable to, or smaller than, the errors introduced by the Born Oppenheimer BO approximation. 1 This is especially true for the H+H 2 reaction, where uncertainties in the BO energies can be reduced 2, to within a few E h. Relativistic corrections for this system are very small, and have been estimated to change the barrier height by only 0.05 cm 1. The leadingorder correction to the BO approximation is the BO diagonal correction BODC, 4 which partially accounts for nuclear motion effects. The addition of the BODC to the BO energy is often referred to as the Born Huang BH energy, or the best adiabatic approximation, or simply the adiabatic approximation. This last usage is vague, since the BO approximation is also an adiabatic approximation and other adiabatic approximations exist ; on this basis a minority of authors refer to the BODC as a nonadiabatic effect although this term is usually reserved for cases that introduce coupling between individual adiabatic states. For reactions at low energies, where nonadiabatic effects can be expected to be small, the inclusion of the diagonal correction should be sufficient to allow highly accurate dynamics calculations. We have recently shown 5 that accounting for the BODC is necessary to accurately reproduce thermal rate measurements for the D+H 2 Refs and H +D 2 Refs. 5, 7, and reactions. In this article we will consider the level of ab initio treatment needed to accurately calculate the BODC for the H+H 2 saddle point, will explore the effect that the BODC has on the saddle-point properties of various isotopomers of this reaction, and will provide benchmark results of the BODC in the CCI limit. II. BACKGROUND A. BO diagonal correction The Born Oppenheimer approximation begins by separating the Hamiltonian, H, of the system into a nuclear kinetic-energy term T N = i 1 2m i i 2, where m i is a nuclear mass, and a remainder H BO. One then solves 1 H BO i = E i BO i, i = 1,2,,..., 2 where E i BO is the Born Oppenheimer potential energy surface PES for state i, which depends only on the nuclear coordinates. Fully accounting for the nuclear kinetic-energy term couples the BO eigenstates, so the exact system cannot be described by PESs. However, diagonal terms, /2005/ /2241/9/$ , American Institute of Physics

2 Mielke, Schwenke, and Peterson J. Chem. Phys. 122, G ii = i T N i, which introduce no coupling, can be included with the BO PESs, although the resulting Born Huang surfaces, E BH i, where E BH i = E BO i + E BODC i = E BO i + G ii, 4 now have explicit mass dependence. In the following, we will only be interested in the ground state i=1, and will drop the subscripts. We note that the BODC is a positive semidefinite quantity, and that isolated atoms have significant BODC values. Thus, the geometry dependence of the BODC is much smaller than its magnitude; for example, the BODC of H 2 at an internuclear separation of R=1.40a 0 is cm 1, but this is only 4.94 cm 1 less than the sum of the BODC for two isolated H atoms. For two-electron systems, such as H 2 Refs and H +, 16 highly accurate BODC values have been obtained by methods using explicitly correlated wave functions. Most calculations on larger systems have been limited to implementations at the Hartree Fock HF self-consistent-field SCF level of theory using either spin-restricted 17,18 RHF or unrestricted 19,20 UHF wave functions, although CI methods have been implemented, which have permitted more accurate calculations The first estimates of the magnitude of the BODC for the H+H 2 reaction were obtained by Garrett and Truhlar, 0 who employed a simple two-state diatomics-in-molecules DIMs approximation and a London potential. They predicted a barrier height correction of 0.21 kcal/mol 7 cm 1. Garashchuk et al. 20 calculated the BODC for the collinear H+H 2 exchange reaction using UHF and unrestricted Møller Plesset second-order UMP2 methods. The latter calculations were reported to yield a barrier height correction of 72 cm 1, while the UHF results were only presented graphically. A FCI calculation using a truncated 2-orbital CI natural orbital basis, obtained from the augmented, correlation consistent polarized valence quadruple-zeta aug-ccpvqz basis set, yielded a value of kcal/mol. In this article we report the details of multireference configurationinteraction MRCI calculations, already used elsewhere 5 to resolve the experimental-theoretical discrepancies for thermal reaction rates, which show that in the CCI limit the H barrier correction is kcal/ mol. The saddle-point configuration was studied in greater detail using many-body extrapolations with augmented modified correlation consistent basis sets, aug-mcc-pvnz, as large as n=7. 2, The MRCI code we use is not capable of treating k functions, but we have also optimized an aug-mcc-pv8z basis set the parameters for this basis set are listed in supporting information, and performed additional calculations with this set without k functions as well as with the aug-mccpv7z basis without i functions to form a hierarchical pairing; for convenience we will refer to these sets as n=7 and 8. The new variational result for the saddle-point energy with n=8 is E h, which is 22 E h below the n = 7 variational result. Many-body extrapolations with basisset pairs of 5,6, 6,7, and 7,8 yield saddle-point energies of , , and E h, respectively, and include three-body-extrapolation corrections of 61, 40, and E h, respectively. The many-bodyextrapolation scheme also yields plausible upper and lower bounds of the saddle-point energy. The upper bounds are above the variational bound of Komasa et al. 4 of E h obtained using a 600 function explicitly correlated Gaussian treatment, but the lower bounds of E h using data through n=7, or E h using data through n=8, remain useful. The marked consistency of the three many-body extrapolations mentioned above suggests that these bounds are extremely conservative and that the true energy is within a few E h of our best estimate of E h, which would correspond to a barrier height of kcal/mol. Quantum Monte Carlo QMC estimates are available from Diedrich and Anderson 5 of E h ±10 E h and from Riley and Anderson 6 of E h ±1.5 E h, where the uncertainties are one standard deviation statistical estimates. This latter value agrees perfectly with our two best many-body extrapolations, but biases and errors, as pointed out earlier, 2,,6 in prior QMC calculations 7,8 of more than 50 times the estimated uncertainties suggest that this level of agreement be viewed with caution. The CCI PES has a barrier height of kcal/ mol, which agrees with our best barrier estimate to within the stated uncertainties ±0.01 kcal/mol of the PES. It will be desirable to calculate the diagonal correction with uncertainties that are comparable to, or smaller than, those of the barrier height. B. The Born Oppenheimer barrier height The CCI potential-energy surface was obtained by extrapolating near-fci results MRCI calculations with CI errors of 1 E h using the aug-cc-pvtz and aug-cc-pvqz basis sets 1,2 via an extremely accurate many-body basis-set extrapolation method. 2 This method replaces the approximate one- and two-body contributions with accurate values, and estimates the remaining three-body contribution by assuming that it converges at the same rate as the sum of the two-body contributions. The resulting accuracy of the surface considering errors from the correlation treatment, fitting errors, and basis-set incompleteness is estimated to be 0.01 kcal/mol at most configurations. III. CALCULATIONS Most of the calculations reported here obtain the BODC via the method of Handy et al. 18 which is occasionally referred to 9 as the Born Handy method using a modification, as implemented by Schwenke and discussed in detail elsewhere, 24,40 of the internally contracted multireference configuration-interaction 41,42 icmrci code of the MOLPRO suite of programs. 4 The approach of Handy et al. 18 does not permit the use of symmetry, but it yields a mass-independent representation of the BODC that allows values for any isotopomer to be calculated via the expression

3 2241- Born Oppenheimer corrections to the H+H 2 reaction J. Chem. Phys. 122, atoms E BODC Ci =. 5 i=1 m i A number of calculations used an internal coordinate approach, which allows use of symmetry D 2h for H 2 and C s for triatomics, that has also been implemented 24,40 with the icmrci code. We also used the PSI package of Crawford et al. 44 to calculate UHF, single reference singles and doubles CI CISD, and FCI obtained via CISDT CI with single, double, and triple excitations calculations BODC values. For the present set of icmrci calculations, the internal contraction algorithm does not result in any reductions in the number of configurations; thus, the results are identical to those of standard MRCI calculations and we will drop the ic prefix. By choosing a single reference we can perform internally contracted CISD iccisd calculations; in this case the internal contraction scheme does reduce the number of configurations and the resulting BO energies are as much as 294 E h above the results of uncontracted CISD. The primary basis sets used in this study are the correlation consistent, cc-pvnz, 1 augmented correlation consistent, 2 aug-cc-pvnz, and doubly augmented correlation consistent, 45 d-aug-cc-pvnz, sets with n=d,t,q,5,or 6. In a few instances we also used the augmented modified correlation consistent sets, aug-mcc-pvnz, 2, with n=5 8, the f-limit basis set of Schwenke, 46 and extensions of these basis sets that will be discussed further below. Nuclear masses are used in the calculations, and are estimated by subtracting 1m e from the atomic masses of , , , and , for H, D, T, and Mu i.e., muonium, respectively. It is helpful to remember when making comparisons to other work that some authors 47 advocate the use of atomic rather than nuclear masses in the diagonal correction in an effort to obtain heuristic corrections beyond the Born Huang approximation. Several units of energy are used, and interconversion is achieved using 1E h = kcal/mol= cm 1 ; it may be convenient to keep in mind that 1 cm kcal/mol 4.6 E h. IV. RESULTS AND DISCUSSION Table I displays the convergence with respect to the basis-set size of the BODC for the H atom and for H 2 at the FCI level with an internuclear separation of R=1.4a 0. A subset of our results can be compared to those of Valeev and Sherrill, 25 who have implemented a CI treatment of the BODC, and these agree perfectly. We also note that Mitrushenkov et al. 2 have published BODC values for H 2 for five values of R at the FCI/cc-pVTZ and FCI/cc-pVQZ levels. If we adjust their data to correct for their use of atomic rather than nuclear masses, our calculations can reproduce their values to within cm 1. The remaining tiny differences are likely due to finite difference uncertainties present in both implementations; our calculations include an extrapolation to zero step size that greatly reduces these uncertainties typically to well below cm 1. In Table II we report mean unsigned deviations MUDs and maximum deviations between FCI calculations using TABLE I. Basis-set convergence of the BODC in cm 1 at the FCI level for H 2 at R=1.40a 0, and for the H atom. Basis H 2, present H 2, Ref. 25 H cc-pvdz cc-pvtz cc-pvqz cc-pv5z cc-pv6z aug-cc-pvdz aug-cc-pvtz aug-cc-pvqz aug-cc-pv5z aug-cc-pv6z d-aug-cc-pvdz d-aug-cc-pvtz d-aug-cc-pvqz mcc-pv5z mcc-pv6z mcc-pv7z mcc-pv8z-k s aug-mcc-pv5z aug-mcc-pv6z aug-mcc-pv7z aug-mcc-pv8z-k s Accurate a a b a Reference 15. b The exact value is 2m 1. various basis sets and the results of highly accurate studies for H 2 and H +. Specifically, we compare to the 51 BODC values of H 2 reported by Wolniewicz 14 having internuclear separations between 0.6 and 12a 0, and the 69 BODC values for H + reported by Cencek et al. 16 These calculations are tabulated in full in supporting information. The data in Tables I and II indicate that the use of double-zeta basis sets does not yield quantitative results, but that even calculations with triple-zeta sets can be very accurate. The basis-set convergence rate of the augmented sets is particularly rapid, with the H 2 MUDs dropping by a factor of TABLE II. Mean unsigned deviations MUDs and maximum deviations in cm 1 for various basis sets between FCI BODC calculations of H 2 and 51 accurate values of Wolniewicz Ref. 14 with R 0.6a 0, and between FCI BODC calculations of H + and the 69 accurate values of Cencek et al. Ref. 16. Basis H 2 MUD H 2 max. deviation H + MUD H + max. deviation cc-pvdz cc-pvtz cc-pvqz cc-pv5z cc-pv6z mcc-pv7z aug-cc-pvdz aug-cc-pvtz aug-cc-pvqz aug-cc-pv5z aug-cc-pv6z aug-mcc-pv7z

4 Mielke, Schwenke, and Peterson J. Chem. Phys. 122, TABLE III. The convergence of the FCI H 2 R=1.4a 0 BODC in cm 1 with respect to l max for various basis sets, and values extrapolated to the CCI limit. These can be compared to the accurate value of Cencek and Kutzelnigg Ref. 15 of cm 1. l max mcc-pv8z ete1-mcc-pv8z ete2-mcc-pv8z aug-mcc-pv8z f-limit ete1-f-limit a a a a Extrapolated: Two points b All points c a The relatively small number of i functions in the mcc-pv8z basis set suggests that these values are not yet near the radial-completeness limit. These data are not included in the extrapolation fits. b AfitofEq. 6 to the two converged points with the highest values of l max. c AfitofEq. 7 to all converged points. 4.6 in going from triple to quadruple zeta and dropping again by nearly a factor of 4 in going from quadruple to quintuple zeta. Curiously, the accuracy for the H atom is worse for the augmented sets than for the unaugmented sets; this behavior is not likely to be typical of larger systems. For H 2 at R=1.4a 0, results obtained with doubly augmented basis sets deviate from the singly augmented results by at most 0.01 cm 1. The unaugmented sets converge somewhat irregularly and from above, while the augmented sets converge monotonically from below. These trends are also observed with the mcc-pvnz and aug-mcc-pvnz sets, for which members exist as large as octuple zeta. Unfortunately, as indicated by the results in Table I, the BODC values obtained with the hierarchical basis sets at R=1.4a 0 display apparent convergence to a value between and cm 1 even though the accurate result 15 is cm 1. Furthermore, since the aug-mcc-pvnz results converge monotonically from below and are already higher than the accurate value, none of the conventional basis-set extrapolation schemes can improve the accuracy. Since neither the unaugmented nor the augmented results extrapolate precisely to the accurate value, and further augmentation according to the prescription of Woon and Dunning 45 does not alter this situation, we can conclude that the heuristic schemes 2,45 for optimizing diffuse functions, together with conventional extrapolation methods, are not sufficiently robust to yield arbitrarily high accuracy. Fortunately, the inaccuracy in the convergence limit of the augmented sets is quite small for the present purposes 0.01 cm 1, and as we will now demonstrate, it is possible to obtain much higher accuracy with alternative approaches. The correlation energy has been shown to converge asymptotically as O 1/l max, where l max is the maximum value of the angular momentum used in the basis set. Based on these findings, extrapolating results obtained with the correlation consistent basis sets according to a 1/n trend has been advocated, 52,5 but the convergence rate formally holds only for a single expansion center with basis sets that are complete with respect to the radial expansion for each l. The incremental energy lowerings from the correlation consistent sets are dominated by improvements in the radial quality of the basis set e.g., 76% of the incremental energy lowering in going from aug-mcc-pv6z to aug-mcc-pv7z for the total energy of H 2 at R=1.4a 0 is due to radial improvements so extrapolation of the correlation consistent sets in this manner is not formally well justified. Schwenke 46 has recently presented basis sets, denoted as f-limit sets, which are intended to be nearly radially complete for each value of l, and he has demonstrated that extrapolating energies obtained with these sets according to a 1/l max trend can yield very high accuracy. The radial quality of the mcc-pv8z basis sets is sufficiently close to completeness except for the i and k functions that results with angular truncations of these sets can also be extrapolated in this manner with high accuracy. Both the mcc-pv8z and f-limit basis sets are optimized to minimize the energy of H 2 at R=1.4a 0 using eventempered exponential parameters. Since the BODC is more sensitive to the diffuse region of the wave function than the energy is, extrapolating BODC results with these basis sets does not yield full accuracy. The aug-mcc-pv8z basis includes an additional diffuse function for each l, which is chosen to minimize the energy of H, and the addition of these functions significantly improves the quality of extrapolated results. The scheme of Woon and Dunning 45 for adding additional diffuse functions, i.e., even-tempered extensions of the augmented parameters, only adds very diffuse functions to which the BODC is not sensitive. A more systematic approach is simply to extend the unaugmented basis sets with additional functions in an even-tempered manner. We will indicate basis sets modified with such even-tempered extensions using a prefix of either ete1 or ete2 depending on whether one or two additional functions are added. Table III lists FCI BODC values for H 2 at R=1.4a 0 obtained from angular truncations of the mcc-pv8z, aug-mccpv8z, ete1-mcc-pv8z, ete2-mcc-pv8z, f-limit, and ete1- f-limit basis sets. In each case, the BODC monotonically decreases with increasing l max in a very regular fashion. We can also see that the results with the f-limit and mcc-pv8z basis sets are not fully converged with respect to the radial extension of the basis set, but that the addition of a single additional function is sufficient to yield high accuracy for both sets. Next, we consider the best way to extrapolate these results to the CBS limit. Since the mcc-pv8z basis set only contains two i functions, the results for any of the octuple-

5 Born Oppenheimer corrections to the H+H 2 reaction J. Chem. Phys. 122, TABLE IV. Optimized geometries a 0 and BODC energies cm 1 for various isotopomers of H 2. The Born Oppenheimer energies are obtained using the CCI PES Ref. rather than from direct ab initio calculations, and the diagonal correction is calculated at the FCI/aug-cc-pV5Z//FCI/aug-cc-pVTZ level. See the text. Species R e E BODC H HD D Mu zeta mcc sets with l max =6 are not expected to be fully converged, and will not be included in the extrapolations. If one has a functional form that is not optimal, or if some of the data is far from the asymptotic convergence regime, it would be best to use only the minimum number of results required for fitting the leading-order term. In this case, that would be a two-point fit to E BODC lmax = E BODC CBS +, 6 l max where E BODC CBS and are fitting parameters. For conventional ab initio data extrapolation schemes, using the minimum number of points is usually the best approach since the functional forms are only heuristic, but in the present case we can also consider fitting all of the data excluding l max =6 for the mcc basis sets to the form E BODC lmax = E BODC CBS + + 4, 7 l max l max where E BODC CBS,, and are fitting parameters, in an attempt to account for behavior beyond leading order. The results of both fits are included in Table III and agree with each other to within cm 1. Either of the extrapolated predictions, with any of the four basis sets that include additional diffuse functions, agree with the accurate result to within cm 1. Both the BO energy and the BODC both converge as O 1/l max, so their basis-set convergence rates for radially complete basis sets are comparable, but since the magnitude of the BODC is smaller than that of the BO energy, smaller basis sets are needed to achieve a given overall level of accuracy. Additionally, the convergence of the BODC seems significantly better behaved than that of the BO energy for basis sets that are substantially far from being radially complete. For example, the error in the H 2 R=1.4a 0 BODC obtained with the aug-cc-pvtz basis set is only 7% larger than the error obtained with the ete1-f-limit basis set with l max =2, whereas the error in the BO energy with the aug-cc-pvtz basis set is 72% larger than the error for the ete1-f-limit basis set with l max =2. Table IV gives the BODC and optimized geometries of various isotopomers of H 2. The optimizations are conducted using the CCI analytic PES for the Born Oppenheimer energies with the BODC calculated at the FCI/aug-cc-pVTZ level. The diagonal correction shown in the third column of Table IV was obtained at the FCI/aug-cc-pV5Z level of theory at these optimized geometries. Table V explores the basis set and correlation effects for BODC calculations of H at the saddle point, and Table VI does the same for the barrier height changes resulting from the inclusion of the diagonal correction. A number of different treatments are considered, including UHF, complete active space self-consistent field CASSCF, iccisd, CISD, MRCI, and FCI calculations. The CASSCF calculations are considered with both a valence orbital active space and an extended active space of 11 orbitals; these calculations will be denoted CAS and CAS 11, respectively. The MRCI calculations employed either a valence reference space, denoted MRCI, or an extended 11 orbital reference space, denoted MRCI 11, with orbitals obtained from the CASSCF calculations with active spaces of and 11 orbitals, respectively. The latter calculations are similar to those used in calculating the CCI PES, except that those employed CI natural orbitals rather than the CASSCF orbitals used here. Table VII shows MRCI and FCI results for BO energies calculated with the aug-cc-pvnz basis sets. The MRCI 11 calculations reproduce the FCI BO energies to within 0.78 E h, and are expected to reproduce the FCI limit of the BODC to within 0.01 cm 1 ; for the cc-pvdz, cc-pvtz, TABLE V. The BODC in cm 1 at the saddle point of collinear H with R 1 =R 2 =1.757a 0. Basis UHF CAS CAS 11 iccisd CISD MRCI MRCI 11 cc-pvdz a cc-pvtz a cc-pvqz a cc-pv5z cc-pv6z n/a aug-cc-pvdz a aug-cc-pvtz a aug-cc-pvqz aug-cc-pv5z n/a d-aug-cc-pvdz d-aug-cc-pvtz d-aug-cc-pvqz n/a a Full CI calculations were done with these five basis sets and agreed with the MRCI 11 values to within the quoted number of digits.

6 Mielke, Schwenke, and Peterson J. Chem. Phys. 122, TABLE VI. The barrier height correction due to the BODC in cm 1 for H. The saddle-point geometry is taken to be R 1 =R 2 =1.757a 0, and the reactant geometry is taken to be R=1.4015a 0. These values do not include the change in BO energy of the reactants due to the change in the optimized value of R, which would lower all values by cm 1. Basis UHF CAS CAS 11 iccisd CISD MRCI MRCI 11 cc-pvdz a cc-pvtz a cc-pvqz a cc-pv5z cc-pv6z n/a aug-cc-pvdz a aug-cc-pvtz a aug-cc-pvqz aug-cc-pv5z n/a d-aug-cc-pvdz d-aug-cc-pvtz d-aug-cc-pvqz n/a a Full CI calculations were done with these five basis sets and agreed with the MRCI 11 values to within the quoted number of digits. cc-pvqz, aug-cc-pvdz, and aug-cc-pvtz basis sets this was explicitly confirmed. The BO energies of the MRCI calculations deviate from the FCI limit by as much as 227 E h, but the diagonal corrections are observed to lie within 0.07 cm 1 of the FCI limit. Thus, the treatment of electron correlation is much easier for the BODC than for the BO energies. Calculations employing extended reference spaces can experience problems if the reference space is not consistent at each of the points used in the finite difference steps. The 11-orbital space is quite stable in this regard as is the valence reference space, but we experienced problems for intermediate reference spaces. It is also worth noting that when converging the larger CASSCF calculations we found it helpful to start with CI natural orbitals. RHF calculations yield extremely inaccurate results for the saddle-point diagonal correction; the RHF/aug-cc-pVTZ value is 46.8 cm 1, which is in error compared to the FCI result by over 200 cm 1. UHF values are too high at the saddle point by about 16 cm 1, but they also underestimate the reactant BODC by about 1 cm 1, resulting in a barrier height change that is too high by about 55%. Thus, correlation effects are very important for this system, which is in contrast with some observations with other systems. When Cencek et al. 16 compared their accurate BODC results for 69 configurations of H + to the earlier SCF calculations of Dinelli et al. 54 on the same set of configurations, they found that the SCF values were systematically too low by from TABLE VII. Born Oppenheimer energy calculations at the saddle point of collinear H with R 1 =R 2 =1.757a 0. Basis set FCI E h CI error E h MRCI CI error E h MRCI 11 CCI FCI E h a aug-cc-pvdz aug-cc-pvtz aug-cc-pvqz aug-cc-pv5z a The basis-set incompleteness error using an estimate of the CCI limit of E h to 14.9 cm 1. Due to the consistent nature of these errors, the geometry-dependent component i.e., the part that does not simply shift the overall zero of energy is small, with a mean unsigned deviation and maximum deviation of only 0.88 and.18 cm 1, respectively. Valeev and Sherrill 25 studied the diagonal correction of H 2,H 2 O, and BH using SCF and CI methods, and concluded that the effect of correlation corrections on relative energies was small. Tajti et al. 55 stated a belief that the use of correlated electronic wave functions has only a modest effect on the DBOC in support of their decision to include the BODC at only the SCF level in their recent ab initio thermochemistry model. The large correlation correction to the BODC observed for H suggests that SCF treatments are not always sufficient, and including correlation corrections for the BODC has been found necessary to achieve spectroscopic accuracy for the vibrationalrotational transitions in water. 24,27,40 CASSCF calculations with a valence active space predict the saddle-point BODC to be too high by about 9.2 cm 1 in the CBS limit, and get the reactant BODC too low by about 2. cm 1 for an overall error in the barrier height correction of about 11.6 cm 1 or nearly 22%. CASSCF calculations with extended active spaces are observed to yield high accuracy, but are not cost effective when compared to MRCI calculations. CISD calculations predict the BODC of the saddle point to within an accuracy of 2 4 cm 1, and the iccisd calculations have errors that are 55% 62% higher than the CISD results. As with the H 2 results, the aug-cc-pvtz basis set is sufficient to converge the BODC to within about 0.2 cm 1, and the augmented basis sets are again observed to yield more regular convergence trends. The inclusion of additional diffuse functions beyond those included in the aug-ccpvnz sets has only a very small effect for basis sets larger than double zeta in size. The MRCI BODC converges monotonically from above with increasing basis-set size for the cc-pvnz sets, and rapidly and monotonically from below for the aug-cc-pvnz sets. The basis-set requirements for accurate BODC calculations may be contrasted with those for the

7 Born Oppenheimer corrections to the H+H 2 reaction J. Chem. Phys. 122, TABLE VIII. Optimized geometries a 0, BODC cm 1, harmonic frequncies cm 1 at the saddle points, and the changes in the barrier heights due to the diagonal correction for various isotopomers of H. The BODC is calculated at the MRCI 11 /aug-cc-pv5z//mrci /aug-cc-pvtz level, while BO energies are obtained from the CCI PES. Ref.. See the text for additional details. Species R 1 R 2 sym bend asym /i BODC barrier cm 1 a H H H D D D T T T Mu Mu Mu D H H T H H Mu H H H D D Mu D D Born Oppenheimer b H H H D D D T T T Mu Mu Mu D H H T H H Mu H H H D D Mu D D a The difference between the barrier heights on the BH and BO surfaces. b As represented by the CCI analytical PES Ref.. BO energies see Table VII, where the aug-cc-pv5z result is still 256 E h below the CBS limit, and even the best extrapolated results have an uncertainty of several E h. The MRCI 11 BODC saddle-point calculation with the aug-cc-pv5z basis is expected to lie within 0.1 cm 1 of the CCI limit. This result can be used to obtain BODC values for isotopomers via Eq. 4, together with the values of C 1 =C = and C 2 = m e E h, where atom 2 is the center atom. We also conducted MRCI 11 calculations with the f-limit basis set truncated to either d or f functions and obtained BODC values of and cm 1, respectively. When these two values are extrapolated according to Eq. 6, we obtain a value of cm 1, which agrees with the aug-cc-pv5z value to within 0.01 cm 1. Table VIII details saddle-point properties of a number of isotopomers of H with and without inclusion of the diagonal correction. This table includes optimized geometries and frequencies Born Oppenheimer energies taken from the CCI potential-energy surface with diagonal corrections calculated at the MRCI /aug-cc-pvtz level, diagonal correction values calculated at the CCI+BODC optimized geometry at the MRCI 11 /aug-cc-pv5z level of theory, and the change in the barrier height due to the diagonal correction. For all isotopomers the BODC raises the barrier height. This can be contrasted with the heuristic finite nuclear mass correction FNMC approximation of Gonçalves and Mohallem, 56,57 where calculations 57 predict a lowering of the barrier height in qualitative disagreement with the present set of accurate results, as well as earlier approximate predictions, 20,0 The addition of the BODC can probably be expected to increase barrier heights for most chemical reactions. As pointed out by Garrett and Truhlar, 0 chemical reaction barriers typically arise from widely avoided intersections of two quasidiabatic potential surfaces one with reactant bonding character, and one with product bonding character. The magnitudes of the derivative couplings, and hence the diagonal correction which can be expressed in terms of the derivative couplings 0, will typically be higher in the region of such avoided intersections than at either the reactant or product configurations. As seen in Tables II and VIII, the optimized structures on the BH surface are very close to the BO optimized structures; the largest change for the reactions without Mu is a 0, and even the changes for Mu 2 and the MuD 2 saddle point are only and a 0, respectively. These configuration changes raise the BO energy at the reactants and lower it at the saddle point as compared to the BO energies at the geometries optimized on the BO PES ; thus, both changes slightly lower the observed barrier height correction these adjustments are included in the barrier corrections listed in Table VIII. For H this change is a trivial cm 1, but for Mu, MuH 2, and MuD 2 the changes are 0.44, 0.085, and cm 1, respectively. The differences between the harmonic frequencies observed for the BO and BH surfaces give some indication of the extent that the barrier shape changes on inclusion of the BODC. For the six isotopic variants considered in Table VIII that do not involve Mu, the frequencies of the symmetric stretch and the bend change by no more than 2 cm 1, and the imaginary frequencies increase by 6 cm 1, which suggests a slight narrowing of the barriers. In the limit that the barrier shape is unchanged we would expect the thermal reaction rates on the BH surface to be a factor of exp barrier /k B T smaller than those for the BO surface, where

8 Mielke, Schwenke, and Peterson J. Chem. Phys. 122, T is the temperature, barrier is the barrier height change due to the BODC, and k B is Boltzmann s constant. As reported previously, 5 this simple correction scheme is sufficient to bring thermal rate constants calculated on the CCI PES into nearly perfect agreement with the experimental measurements. The slight narrowing of the barrier due to the inclusion of the BODC that is suggested by the changes in the imaginary frequencies would result in small increases in the low-temperature rate constants due to increased tunneling effects, but these are expected to be much smaller than the uncertainties in the experimental measurements. For the Mu+H 2 and Mu+D 2 reactions, the BODC causes substantially larger barrier height increases, and the harmonic frequency changes are also substantially larger, which suggests that barrier shape changes might be sufficiently large that a simple rescaling of the BO thermal rate constants might not be accurate enough to reproduce the experimental results of Fleming and co-workers. 58,59 V. CONCLUDING REMARKS Born Oppenheimer diagonal correction BODC values are obtained for the saddle point of the H system, and various isotopomers, that are estimated to be accurate to within 0.1 cm 1 of the complete CI limit an uncertainty that is much smaller than that of the best estimates of the Born Oppenheimer saddle-point energy. Diffuse functions are important for obtaining precise results, but otherwise the basisset convergence is quite rapid, with even aug-cc-pvtz results being within about 0.2 cm 1 of the complete basis-set limit. Electron correlation effects are important for accurately estimating the barrier height change caused by the BODC; UHF and valence CASSCF calculations predict barrier height changes that are too high by 55% and 22%, respectively. MRCI calculations with only a valence reference space yield BODC values within less than 0.1 cm 1 of the FCI limit even for very large basis sets, which is two orders of magnitude better accuracy than this reference space provides for the Born Oppenheimer energies. The diagonal correction raises the barrier heights of the H+H 2, D+H 2, and H+D 2 reactions by 0.152, 0.195, and kcal/ mol, respectively, and very slightly narrows the barriers, with the imaginary frequencies for these three isotopomers being increased by less than 2.2%. While these changes might seem small, properly accounting for them is necessary to accurately predict 5 low-temperature thermal rate coefficients. Accounting for the BODC is expected to be even more important for reactions with muonium; the barrier height for the Mu+H 2 reaction increases by 0.67 kcal/mol and the imaginary frequency at the barrier increases by 6.7%. See Ref. for supporting information: Tables specifying the mcc-pv8z and aug-mcc-pv8z basis sets, and all the H 2 and H + BODC values reported in summary form in Table II, have been deposited with EPAPS. ACKNOWLEDGMENTS We are grateful to Dr. Bruce Garrett for helpful discussions. This work was supported by the National Science Foundation Grant No. CHE to KAP and the Chemical Sciences Division in the Office of Basic Energy Sciences of the U.S. Department of Energy. Part of this research was performed in the W. R. Wiley Environmental Molecular Sciences Laboratory, a national scientific user facility sponsored by the Department of Energy s Office of Biological and Environmental Research and located at Pacific Northwest National Laboratory. 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