MCSCF first derivatives

MCSCF first derivatives, the integral derivatives do not need to be stored or transformed explicitly. 

We shall in this chapter discuss the methods employed for the optimization of the variational parameters of the MCSCF wave function. Many different methods have been used for this optimization. They are usually divided into two different classes, depending on the rate of convergence first or second order methods. First order methods are based solely on the calculation of the energy and its first derivative (in one form or another) with respect to the variational parameters. Second order methods are based upon an expansion of the energy to second order (first and second derivatives). Third or even higher order methods can be obtained by including more terms in the expansion, but they have been of rather small practical importance. 

The first-order MCSCF response equations were first derived by Dalgaard and Jdrgensen (1978). For geometrical perturbations these equations were derived by Osamura et al. (1982a) using a Fock-operator approach. 

To summarize, the first anharmonicity may be evaluated for an MCSCF wave function with second and third derivative integrals in the AO basis, first derivative integrals in the MO basis with two general and two active indices, and undifferentiated integrals with three general and one active indices. 

Let us summarize. The calculation of Cl first anharmonicities requires no storage or transformation of second and third derivative two-electron integrals, but the full set of first derivative MO integrals is needed. One must construct and transform one set of effective density elements for third derivative integrals and 3M — 6 sets of effective densities for second derivative integrals. In addition to the 3N — 6 MCSCF orbital responses k(1) and the Handy-Schaefer vector Cm needed for the Hessian, the first anharmonicity requires the solution of 3JV — 6 response equations to obtain (1). 

In contrast with first derivatives, second derivatives involve couplings with all states (sums over M in (A.15)) that correspond to a second-order Jahn-Teller effect. Such contributions from higher-lying states (M > 2) do not exist in a pure two-level model (see Sec. 2), but they are part of the actual MCSCF calculation, where the number of eigenstates is equal to the number of CSFs. Limiting the values of M to 0 and 1 leads to ... 

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 calculation of third and fourth derivatives is attractive for two reasons these are the quantities needed in the lowest-order treatment of vibrational anharmonicities, and a quartic surface is the simplest to exhibit a double minimum, i.e. the simplest model of a reaction surface. Moccia (1970) did apparently first consider the SCF third derivative problem. A detailed derivation of SCF and MCSCF third derivatives was given by Pulay (1983a) independently, Simons and J )rgensen (1983) also considered the calculation of MCSCF third, and even fourth, derivatives in a short note. As pointed out in Section II, third derivatives of the energy require only the first derivatives of the coefficients, and are thus computationally attractive. By contrast, fourth derivatives require the solution of the second-order CP MCSCF equations. The only computer implementation so far is that of Gaw etal.( 984) for closed shells, although the detailed theory has been worked out for the MCSCF case... 

Third derivatives have so been formulated for the SCF and MCSCF energies by a number of groups and implemented by at least one group. In addition to the first, second and third derivatives of the integrals, only the first derivatives of the coefficients are needed. This is similar to perturbation theory, where the third-order energy can be computed with the first-order wavefunction. 

The solution of the SCF equations involves a number of technical tricks decisive for actual calculations. In the end, they represent an optimization problem that can be tackled in many different ways. An interesting aspect to mention, though, is that usually only the first derivative (the linear variation) is considered. The set of spinors obtained produces a stationary energy but it is not guaranteed that this is a true minimum rather than a saddle point. Nevertheless, this possibility is usually not tested unless a special type of MCSCF calculations that we introduce in section 10.6 is carried out. 

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. 

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. 

There has been some recent concern (8,9) however, that this bent bond description of multiple bonds derived from the GVB-PP model may be an artifact of the model. The concern takes two forms first, that the SOPP restrictions on the GVB wave function are the source of the bent bonds and the full GVB model will produce the usual <7,7r-bond description and second, if an MCSCF or Cl wave function which is more general than GVB is used, this will give back the c,7r description. 

Calculations of analytic excited state properties for correlated methods have been reported by several groups [107-118]. Excited state dynamic properties from cubic response theory were first obtained by Norman et al. at the SCF level [55] and by Jonsson et al. at the MCSCF [56] level, and in a subsequent study a polarizable continuum model was applied to account for solvation effects [119]. Hattlg et al. presented a general theory for excited state response functions at the CC level using a quasi-energy formulation [120] which was subsequently implemented and applied at the CCSD level [121, 122]. The first ID DFT calculation of dynamic excited state polarizabilities, which we will shortly review here, was presented in [58] for pyrimidine and -tetrazine utilizing the double residue of the cubic response function derived in Section 2.7.3. 

For long MCSCF expansions, construction of the AO density matrix becomes a major computational task, and an alternative method, similar to Cl derivative calculation, must be used. This method, outlined by Meyer (1976), and first used by Brooks et al. (1980), consists of the explicit transformation of the MO two-particle density matrix to AO basis, prior to gradient evaluation. Transformation of the density matrix is obviously superior to the transformation of the 3N gradient components of each integral. 

In this equation, X stands for the elements of the matrix X, A denotes the Cl coefficients, and the elements of g on the right-hand side are the derivatives of the energy function W with respect to the MCSCF parameters X or A, and the nuclear coordinate a, W in our general notation (Eq. (6)). The gradients g have two components the first is the gradient evaluated with the derivative integrals, and the second arises from the change of the metric, i.e. from the effect of the matrix T. These terms, or course, correspond to the derivative constraint terms in the constrained formulation (Eq. (14)). The solution of the above equation can be carried out by a code similar to the one used to determine the MCSCF wavefunction itself. 

For SCF methods, p represents elements of the one- and two-particle density matrices. In correlated methods, the one-particle part of p is the sum of the aaual reduced density and a contribution that is proportional to the derivative of the energy with respect to orbital rotations. For the MCSCF method, the orbitals are variationally optimum, and this latter term vanishes. Similarly, for FCI there is no orbital contribution. However, it is required for other Cl, CC, and MBPT correlated methods, because the energy in these approaches is not stationary with respect to first-order changes in the molecular orbitals. This... 

Hence, if we are to make bE = 0, we can avoid these terms. But to do so we have to have E optimum with respect to the location of the atomic basis functions, t (R) the MO coefficients, c(R) and the Cl coefficients, C(R). The first cannot be satisfied unless the atomic orbital basis set is floated off the atomic centers to an optimum location [105], while the second requires optimum MO coefficients, and the third optimum Cl coefficients. In practice, we will introduce atomic orbital derivatives explicitly, so the AOs can follow their atoms. Now focusing only on the MO and Cl coefficients, in SCF we have optimum MOs and no Cl term. In MCSCF, both terms would vanish, whUe in Cl, the MO derivatives would remain, but the Cl coefficients contribution would vanish. In the non-variational coupled-cluster theory, neither will vanish and this means that CC theory forces us into some new considerations for analytical forces. 

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