MCSCF coupled perturbed equations

Importantly for direct dynamics calculations, analytic gradients for MCSCF methods [124-126] are available in many standard quantum chemistiy packages. This is a big advantage as numerical gradients require many evaluations of the wave function. The evaluation of the non-Hellmann-Feynman forces is the major effort, and requires the solution of what are termed the coupled-perturbed MCSCF (CP-MCSCF) equations. The large memory requirements of these equations can be bypassed if a direct method is used [233]. Modem computer architectures and codes then make the evaluation of first and second derivatives relatively straightforward in this theoretical framework. 

RPA, and CPHF. Time-dependent Hartree-Fock (TDFIF) is the Flartree-Fock approximation for the time-dependent Schrodinger equation. CPFIF stands for coupled perturbed Flartree-Fock. The random-phase approximation (RPA) is also an equivalent formulation. There have also been time-dependent MCSCF formulations using the time-dependent gauge invariant approach (TDGI) that is equivalent to multiconfiguration RPA. All of the time-dependent methods go to the static calculation results in the v = 0 limit. 

Analytical second derivatives for closed-shell (or unrestricted Hartree-Fock (UHF)) SCF wavefunctions are used routinely now. The extension to the MCSCF case is relatively new, however. In contrast to the first derivatives, the coupled perturbed SCF equations have to be solved in order to calculate the second and third energy derivatives. The closed-shell case is relatively straightforward, and will be discussed. The multiconfigurational formalism is... 

The terms Cr,(j) and Xr, are obtained by solving the standard coupled-perturbed SA-MCSCF equations. These are obtained by differentiating Newton-Raphson equations with respect to a nuclear distortion ... 

Coupled Perturbed State-Averaged MCSCF Equations Orbital Derivatives... 

The derivatives of the molecular orbitals are obtained from the coupled-perturbed SA-MCSCF equations. These equations come from the requirement that the SA-MCSCF conditions, Eqs. (36), should be satisfied at X - - (5X, given that they are satisfied at X, that is,... 

For the MEO optimizations using the CAS 1 active space, we found it necessary to use a quadratically convergent algorithm for the CASSCF wavefunction. The orbital rotation derivative contributions from the coupled perturbed multi-configurational self-consistent field (CP-MCSCF) equations (see Refs. [82-84] and the Chap. 3 of Ref. [85] for details) were neglected in the MECI optimizations using the CAS2 active space. 

The only aspect of the computational approach not explicitly given above is the determination of the U (R). The U (R) are obtained by differentiating the (state averaged) MCSCF equations. The resulting equations are referred to as the coupled perturbed (state averaged) MCSCF equations. The numerical effort required for the evaluation of equation (66b) can be further reduced by avoiding the explicit determination of U (R), Discussion of these aspects of the evaluation can again be found in Refs, 11 and 10,... 

The use of Cl methods has been declining in recent years, to the profit of MP and especially CC methods. It is now recognized that size extensivity is important for obtaining accurate results. Excited states, however, are somewhat difficult to treat by perturbation or coupled cluster methods, and Cl or MCSCF based methods have been the prefen ed methods here. More recently propagator or equation of motion (Section 10.9) methods have been developed for coupled cluster wave functions, which allows calculation of exited state properties. 

Contrary to response theory for exact states, in Section 3.11, or for coupled cluster wavefunctions, in Section 11.4, in MCSCF response theory the time dependence of the wavefunction is not determined directly from the time-dependent Schrodinger equation in the presence of the perturbation H t), Eq. (3.74). Instead, one applies the Ehrenfest theorem, Eq. (3.58), to the operators, which determine the time dependence of the MCSCF wavefunction, i.e. the operators hj ... 

Solutions to the eigenvalue equation (16) can be obtained by any of the standard quantum chemical methods, such as Hartree-Fock SCF, multiconfiguration SCF (MCSCF), Mpller-Plesset perturbation, coupled cluster, or density functional theories. The matrix elements of Hr, a one-electron operator, are readily computed, thus formally the inclusion of solvent effects in the quantum mechanical description of the solute molecule appears quite simple. Moreover, gradients of the eigenvalue E are readily computed. 

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