MCSCF higher derivatives

Relatively little has been done for the calculation of higher derivatives of correlated wavefunctions which are not of the MCSCF type. In an impressive theoretical paper, Simons etal. (1984) have worked out general formulas for higher derivatives of the Cl wavefunction up to fourth order. Using second quantization notation, and an orthonormal basis set, Simons et al. arrive at fairly compact formulas which must be, however, expanded considerably to make them directly programmable. It appears, however, that these methods are not practicable at the present time. Indeed, it is perhaps useful to remember that derivative methods are not a goal in themselves but a means to study potential surfaces. Unless a derivative method avoids some redundancy in a competing numerical scheme, it cannot be expected to be more efficient than a numerical method it replaces. 

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. 

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 ... 

Olsen and Jorgensen (1985, 1995) have derived and discussed response functions for exact, HF, and MCSCF wave functions in great detail, while Koch and Jorgensen (1990) presented a derivation for CC wave functions. The latter was modified by Pedersen and Koch (1997) to ensure proper symmetry of the response functions. Christiansen et al. (1998) have presented a derivation of dynamic response functions for variational as well as non-variational wave functions that resembles the way in which static response functions are deduced from energy derivatives. Linear and higher order response functions based on DFT have been presented by Salek et al. (2002). Damped response theory has been discussed by Norman et al. (2001) in the context of HF and MCSCF response theory. Nonpertur-bative calculations of static magnetic properties at the HF level have been presented by Tellgren et al. (2008, 2009). 

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