Analytic non-adiabatic derivative coupling terms for spin-orbit MRCI wavefunctions. I. Formalism

Abstract

Analytic gradients of electronic eigenvalues require one calculation per nuclear geometry, compared to at least 3n + 1 calculations for finite difference methods, where n is the number of nuclei. Analytic nonadiabatic derivative coupling terms (DCTs), which are calculated in a similar fashion, are used to remove nondiagonal contributions to the kinetic energy operator, leading to more accurate nuclear dynamics calculations than those that employ the Born-Oppenheimer approximation, i.e., that assume off-diagonal contributions are zero. The current methods and underpinnings for calculating both of these quantities, gradients and DCTs, for the State-Averaged MultiReference Configuration Interaction with Singles and Doubles (MRCI-SD) wavefunctions in COLUMBUS are reviewed. Before this work, these methods were not available for wavefunctions of a relativistic MRCI-SD Hamiltonian. Calculation of these terms is critical in successfully modeling the dynamics of systems that depend on transitions between potential energy surfaces split by the spin-orbit operator, such as diode-pumped alkali lasers. A formalism for calculating the transition density matrices and analytic derivative coupling terms for such systems is presented.

Electronic Structure Methods for the Description of Nonadiabatic Effects and Conical Intersections

“…127,128,132,385 The NAC in COLUMBUS has also been expanded recently to the relativistic MRCI-SD method. 386,387 MRCISD NAC has also been developed by Hoffmann and coworkers extending to state-averaged MCSCF where the states do not need to have the same weights. 388 Implementations of CASSCF NAC are the most widespread and exist in most packages with MCSCF capabilities, such as COLUMBUS, 132 GAMESS, 187,188 GAUSSIAN, 389 MOLPRO, 176 MOL-CAS, 175 and OPENMOLCAS.…”

“…Implementations of analytic gradients and derivative couplings at the MRCI level are not as widespread as CASSCF ones. The Columbus software is a main publicly distributed software that has analytic gradients readily available and extended them to NAC. ,,, The NAC in Columbus has also been expanded recently to the relativistic MRCI-SD method. , MRCISD NAC has also been developed by Hoffmann and co-workers extending to state-averaged MCSCF where the states do not need to have the same weights …”

TD-DFT spin-adiabats with analytic nonadiabatic derivative couplings

“…25 to time-dependent density functional theory (TD-DFT) within the Tamm-Dancoff approximation (TDA), which corrects the orbital energies relative to Hartree-Fock and CIS and yields better excitation energies while retaining the same structure as the CIS. 26 Moreover, we will also calculate CIS and TDA NADCs, extending the previous work with semiempirical 27 or MR-CISD wavefunctions 28 into the realm of modern TD-DFT calculations (where calculating NADCs has been a hot topic in the recent years [29][30][31][32][33][34] given the formal absence of a TD-DFT wavefunction). In principle, our algorithm should also be extendable to spin-flip methods.…”

Theoretical Cross Sections of the Inelastic Fine Structure Transition M(2P1/2) + Ng ↔ M(2P3/2) + Ng for M = K, Rb, and Cs and Ng = He, Ne, and Ar

“…The representation of Ĥ R in the Hund’s case (c) basis takes the following form: where The radial derivative coupling terms are expressed in integral form: where Φ j ,ω ( r⃗ ; R ) are electronic states and r⃗ are electronic coordinates. For these calculations F and G are assumed to be negligible in the Hund’s case (c) basis . ,− …”