Theoretical Details To DBOC Corrections
Corrections for making the Born-Oppenheimer approximations in the quantum chemical treatment of atoms and molecules are easily computed using perturbation theory. The lowest-order correction is obtained in first-order and takes the form
This correction is known as the "diabatic Born-Oppenheimer correction (DBOC)" and its inclusion leads to the so-called adiabatic approximation with mass-dependent potentials that are different for the various isotopomers of a molecule.
As shown by Handy, Yamaguchi, and Schaefer (J. Chem. Phys. 84, 4481 (1986)), analytic derivative techniques can be used to compute E(DBOC). At the HF-SCF level, the DBOC correction can be thus calculated as
With the corresponding AO integrals and the CPHF coefficients , this can be recasted in the following form for RHF
with as the RHF-SCF density matrix
For UHF, this yields
with as the UHF-SCF density matrix
with the sum running over spatial orbitals instead of spin orbitals.
A slightly different, though equivalent expression for the DBOC at the HF-SCF level has been given earlier by Handy et al (J. Chem. Phys. 84, 4481 (1986)).
A corresponding scheme for correlated calculations of DBOC is currently implemented.