Gradient MCSCF

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. 

In this section we shall compute the energy gradient and the Hessian matrix corresponding to the energy expression (3 25). We introduce the variation of the Cl coefficients by operating on the MCSCF state I0> with the unitary... 

The Fock matrix (4 42) is in general not Hermitian for a non-converged MCSCF wave function. With optimized orbitals the gradient is zero. The MCSCF Fock matrix is thus Hermitian at this point on the energy surface. This condition has been used as a basis for optimization schemes in earlier developments of the MCSCF methodology. Convergence of such first order optimization schemes is, however, often poor, and they are not very much used today. 

However, what we want to achieve in an MCSCF calculation is to reach the point on the energy surface where the gradient of the energy is zero. How we get there is of less importance, as long as the process does not consume too much computer time. With this in mind one could try to simplify the Super-CI matrix d, with the hope that the simplified version still leads to solutions close to the exact solution, such that the iterations move the solution towards the... 

Lengsfield III, B.H., Saxe, P., and Yarkony, D.R. (1984). On the evaluation of nonadiabatic coupling matrix elements using SA-MCSCF/CI wavefunctions and analytic gradient methods. I, J. Chem. Phys. 81, 4549-4553. 

It is worth recalling here that the building blocks of a second-order MCSCF optimization scheme, the electronic gradient and Hessian, are also the key elements in the development of MCSCF response methods (see the contribution by Agren and... 

Following, we determine the effects of the interactions between the quantum and classical subsystems on the optimization procedures of the MCSCF electronic wavefunction by evaluating the contributions of the quantum-classical interactions to the gradient and Hessian terms in the above equation. 

This expression is very similar to those in the normal orthogonal case. We may therefore use any general gradient package. Our VB program generates the density matrices and the matrix L, which is used instead of the Lagrange multiplier matrix of for instance a MCSCF function. 

The ground state force field, vibrational normal modes and frequencies have been obtained with MCSCF analytic gradient and hessian calculations [176]. Frequencies computed with the DZ basis set are compared with experimental ones in Table 16. The T - So transition moments were obtained using distorted benzene geometries with atomic displacements along the normal modes, and with the derivatives in Eq. 97 obtained by numerical differentiation. The normal modes active for phosphorescence in benzene are depicted in Fig. 12. The final formula for the radiative lifetime of the k spin sublevel produced by radiation in all (i/f) bands is (ZFS representation x,y,z is used [49]) ... 

The MCSCF gradient expression was first given by Pulay (1977). The MCSCF Hessian and first anharmonicity expressions were derived by Pulay (1983) using a Fock-operator approach, and by Jprgensen and Simons (1983) and Simons and Jorgensen (1983) using a response function approach. 

The electronic gradient /(1) (which has no orbital part) has the same structure as the configuration part of the MCSCF electronic gradient [Eq. (84)] and may be constructed in the configuration basis, requiring /, SU), /<0>), and k(1 I(0) in the MO basis. (The k(1),/(0> integrals are needed since the orbital connection includes the MCSCF orbital reoptimization effects.)... 

The gradient Ww may be calculated using a procedure similar to that for MCSCF and MRCI wave functions, since H 1) is an operator of rank 2. The molecular gradient becomes... 

Calculations of property derivatives based on the expansion of the dipole moment follow an outline similar to the one used for the calculations based on the energy expansion. As an illustration we consider the MCSCF dipole gradient. 

The configuration part of the electronic gradient is for both MCSCF and MRCI wave functions... 

In order to optimize the MCSCF wave function we have to determine the Cl coefficients and the orbital parameters, that is, the AO expansion coefficients for the MOs. The condition for optimal parameters is that the gradient of the energy is zero with respect to variations of the parameters ... 

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