*ab initio*determination of the adiabatic potential energy function and the Born–Oppenheimer breakdown corrections for the electronic ground state of LiH isotopologues

^{1,2}, Péter G. Szalay

^{3,a)}, Julien Fremont

^{1}, Michael Rey

^{1}, Kirk A. Peterson

^{4}and Vladimir G. Tyuterev

^{1,b)}

### Abstract

High level *ab initio* potential energy functions have been constructed for LiH in order to predict vibrational levels up to dissociation. After careful tests of the parameters of the calculation, the final adiabatic potential energy function has been composed from: (a) an *ab initio* nonrelativistic potential obtained at the multireference configuration interaction with singles and doubles level including a size-extensivity correction and quintuple–sextuple ζ extrapolations of the basis, (b) a mass–velocity-Darwin relativistic correction, and (c) a diagonal Born–Oppenheimer (BO) correction. Finally, nonadiabaticeffects have also been considered by including a nonadiabatic correction to the kinetic energy operator of the nuclei. This correction is calculated from nonadiabatic matrix elements between the ground and excited electronic states. The calculated vibrational levels have been compared with those obtained from the experimental data [J. A. Coxon and C. S. Dickinson, J. Chem. Phys.134, 9378 (2004)]. It was found that the calculated BO potential results in vibrational levels which have root mean square (rms) deviations of about 6–7 cm^{−1} for LiH and ∼3 cm^{−1} for LiD. With all the above mentioned corrections accounted for, the rms deviation falls down to ∼1 cm^{−1}. These results represent a drastic improvement over previous theoretical predictions of vibrational levels for all isotopologues of LiH.

We acknowledge the support from ANR “IDEO” and LEFE-CHAT CNRS grants, from the Balaton French-Hungarian PHC exchange program and its Hungarian counterpart TéT, from the IDRIS computer centre of CNRS France and of the computer centre Reims-Champagne-Ardenne. F.H. thanks the CNRS for the PostDoc support at the University of Reims and for the opportunity to work on this project. P.S. acknowledges support by Orszagos Tudomanyos Kutatasi Alap (OTKA) (Grant No. F72423). The European Union and the European Social Fund have provided financial support to the project under the Grant agreement No. TÁMOP 4.2.1./B-09/1/KMR-2010-0003. J.F. acknowledges CNRS and Champagne-Ardenne Region for the Phd. thesis fund. The support of VEGA Grant No. 1/0648/10 is acknowledged. We thank David Schwenke for helpful discussions and Robert LeRoy for providing access to his LEVEL code.

I. INTRODUCTION

II. ELECTRONIC STRUCTURE CALCULATIONS

A. Nonrelativistic Born–Oppenheimer function

B. Adiabatic DBOC correction

C. Relativistic correction

III. CALCULATION OF NONADIABATIC COUPLINGS AND THE β(R) FUNCTION

IV. CALCULATION OF VIBRATIONAL ENERGY LEVELS

V. DISCUSSION AND CONCLUSIONS

### Key Topics

- Ab initio calculations
- 30.0
- Wave functions
- 23.0
- Non adiabatic reactions
- 19.0
- Basis sets
- 16.0
- Dissociation energies
- 15.0

## Figures

Convergence of the vibrational levels of ^{7}LiH calculated using different electron correlation methods with respect to the full-CI results. See Sec. II for details of the *ab initio* calculations.

Convergence of the vibrational levels of ^{7}LiH calculated using different electron correlation methods with respect to the full-CI results. See Sec. II for details of the *ab initio* calculations.

Relative differences of ^{7}LiH potential energy curves (hartree), scaled to the dissociation energy limit, calculated with the MR-CISD+Q_{P} method and different cc-pwCVXZ (*X* = Q,5,6) basis sets with respect to the highest level 6Z/(56) PEC. See Sec. II for details of the *ab initio* calculations.

Relative differences of ^{7}LiH potential energy curves (hartree), scaled to the dissociation energy limit, calculated with the MR-CISD+Q_{P} method and different cc-pwCVXZ (*X* = Q,5,6) basis sets with respect to the highest level 6Z/(56) PEC. See Sec. II for details of the *ab initio* calculations.

Convergence of the vibrational levels of ^{7}LiH calculated for potential energy curves obtained with the MR-CISD+Q_{P} method and different cc-pwCVXZ (*X* = Q,5,6) basis sets with respect to the highest level 6Z/(56) results. Upper panel: calculations with conventional basis sets. Lower panel: “noncorrelated” part of the energy obtained with the cc-pwCV6Z basis. See Sec. II for details of the *ab initio* calculations.

Convergence of the vibrational levels of ^{7}LiH calculated for potential energy curves obtained with the MR-CISD+Q_{P} method and different cc-pwCVXZ (*X* = Q,5,6) basis sets with respect to the highest level 6Z/(56) results. Upper panel: calculations with conventional basis sets. Lower panel: “noncorrelated” part of the energy obtained with the cc-pwCV6Z basis. See Sec. II for details of the *ab initio* calculations.

Comparison of the calculated vibrational levels of ^{7}LiH with the experimental determinations (Ref. 19). The figure also shows the importance of the different contributions [DBOC, relativistic effect, nonadiabatic contribution (β)] and the use of atomic versus nuclear masses. See text for more explanations.

Comparison of the calculated vibrational levels of ^{7}LiH with the experimental determinations (Ref. 19). The figure also shows the importance of the different contributions [DBOC, relativistic effect, nonadiabatic contribution (β)] and the use of atomic versus nuclear masses. See text for more explanations.

Comparison of the *ab initio* and empirical DBOC functions. The *ab initio* function has been obtained with the MR-CISD method using a 8-3-1 reference space and the cc-pwCVTZ basis. The empirical function is plotted using the parameters of Ref. 19. Curves are shifted to zero energy value near the equilibrium bond length (3 bohr).

Comparison of the *ab initio* and empirical DBOC functions. The *ab initio* function has been obtained with the MR-CISD method using a 8-3-1 reference space and the cc-pwCVTZ basis. The empirical function is plotted using the parameters of Ref. 19. Curves are shifted to zero energy value near the equilibrium bond length (3 bohr).

Comparison of the *ab initio* and empirical atomic DBOC functions. The *ab initio* function has been obtained with the MR-CISD method using a 8-3-1 reference space and the cc-pwCVTZ basis. The empirical functions are plotted using the parameters of Ref. 19. Curves are shifted to zero energy value near the equilibrium bond length (3 bohr).

Comparison of the *ab initio* and empirical atomic DBOC functions. The *ab initio* function has been obtained with the MR-CISD method using a 8-3-1 reference space and the cc-pwCVTZ basis. The empirical functions are plotted using the parameters of Ref. 19. Curves are shifted to zero energy value near the equilibrium bond length (3 bohr).

Perturbative mass–velocity-Darwin relativistic correction to the ground electronic state potential energy curve of LiH calculated with the internally contracted MR-CISD method using different cc-pwCVXZ (*X* = T,Q,5) basis sets.

Perturbative mass–velocity-Darwin relativistic correction to the ground electronic state potential energy curve of LiH calculated with the internally contracted MR-CISD method using different cc-pwCVXZ (*X* = T,Q,5) basis sets.

MR-CISD adiabatic potential energy curves of the eight lowest electronic states of LiH calculated using the cc-pwCV5Z basis set and a 4 electrons in 15 orbitals (8-3-1) CAS reference wave function.

MR-CISD adiabatic potential energy curves of the eight lowest electronic states of LiH calculated using the cc-pwCV5Z basis set and a 4 electrons in 15 orbitals (8-3-1) CAS reference wave function.

MR-CISD nonadiabatic couplings of the ground ^{1}Σ^{+} (ψ_{0}) state with three excited ^{1}Σ^{+} (ψ_{1}, ψ_{4}, ψ_{7}) electronic states of ^{7}LiH calculated using the cc-pwCV5Z basis set and a 4 electrons in 16 orbitals (8-3-1) CAS reference wave function.

MR-CISD nonadiabatic couplings of the ground ^{1}Σ^{+} (ψ_{0}) state with three excited ^{1}Σ^{+} (ψ_{1}, ψ_{4}, ψ_{7}) electronic states of ^{7}LiH calculated using the cc-pwCV5Z basis set and a 4 electrons in 16 orbitals (8-3-1) CAS reference wave function.

Dependence of the BO breakdown correction β(R) of ^{7}LiH on the number of nonadiabatic coupling terms *n* used in the summation of Eq. (3).

Dependence of the BO breakdown correction β(R) of ^{7}LiH on the number of nonadiabatic coupling terms *n* used in the summation of Eq. (3).

Mass dependence of BO breakdown correction β(R) for studied isotopologues ^{7}LiH, ^{6}LiH, ^{7}LiD, and ^{6}LiD.

Mass dependence of BO breakdown correction β(R) for studied isotopologues ^{7}LiH, ^{6}LiH, ^{7}LiD, and ^{6}LiD.

## Tables

Vibrational energies (with respect to ZPE), dissociation energy (cm^{−1}), and equilibrium bond length (bohr) of ^{7}LiH calculated from potential energy curves obtained by different correlated methods and the cc-pwCVQZ basis set. See Sec. II for details of the *ab initio* calculations.

Vibrational energies (with respect to ZPE), dissociation energy (cm^{−1}), and equilibrium bond length (bohr) of ^{7}LiH calculated from potential energy curves obtained by different correlated methods and the cc-pwCVQZ basis set. See Sec. II for details of the *ab initio* calculations.

Vibrational energies (with respect to ZPE), dissociation energy (cm^{−1}), and equilibrium bond length (bohr) of ^{7}LiH calculated for potential energy curves obtained with the MR-CISD+Q_{P} method and different cc-pwCVXZ (*X* = Q,5,6) basis sets [the abbreviation XZ (*X* = Q,5,6) is used for simplicity] with/without extrapolation to CBS included [the abbreviation (XY) (*X*,*Y* = Q,5,6) is used where *X* and *Y* denote cardinal numbers of basis sets used in the extrapolation]. See Sec. II for details of the *ab initio* calculations and extrapolations.

Vibrational energies (with respect to ZPE), dissociation energy (cm^{−1}), and equilibrium bond length (bohr) of ^{7}LiH calculated for potential energy curves obtained with the MR-CISD+Q_{P} method and different cc-pwCVXZ (*X* = Q,5,6) basis sets [the abbreviation XZ (*X* = Q,5,6) is used for simplicity] with/without extrapolation to CBS included [the abbreviation (XY) (*X*,*Y* = Q,5,6) is used where *X* and *Y* denote cardinal numbers of basis sets used in the extrapolation]. See Sec. II for details of the *ab initio* calculations and extrapolations.

Contribution of the DBOC to the vibrational levels and dissociation energy (cm^{−1}) of ^{7}LiH obtained with the MR-CISD+Q_{P} method using different basis sets and different reference CAS spaces. See Sec. II for details of the *ab initio* calculations.

Contribution of the DBOC to the vibrational levels and dissociation energy (cm^{−1}) of ^{7}LiH obtained with the MR-CISD+Q_{P} method using different basis sets and different reference CAS spaces. See Sec. II for details of the *ab initio* calculations.

Contributions (in cm^{−1}) to the vibrational energy levels using nuclear masses. All data refer to the most abundant isotopologue (^{7}LiH).

Contributions (in cm^{−1}) to the vibrational energy levels using nuclear masses. All data refer to the most abundant isotopologue (^{7}LiH).

Comparison of *ab initio* and empirical vibrational levels (in cm^{−1}) for different isotopologues of LiH. Nuclear masses and the BO breakdown correction β(R) have been used in the vibrational levels calculations.

Comparison of *ab initio* and empirical vibrational levels (in cm^{−1}) for different isotopologues of LiH. Nuclear masses and the BO breakdown correction β(R) have been used in the vibrational levels calculations.

Comparison of *ab initio* and empirical vibrational levels (in cm^{−1}) for different isotopologues of LiH. Atomic masses have been used in the vibrational level calculations.

Comparison of *ab initio* and empirical vibrational levels (in cm^{−1}) for different isotopologues of LiH. Atomic masses have been used in the vibrational level calculations.

Summary of rms deviations between *ab initio* and empirical values of vibrational energies.

Summary of rms deviations between *ab initio* and empirical values of vibrational energies.

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