Electronic gradient

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... 

The second term in Eq. (69) is f 2)l(l where the electronic gradient /(2) is constructed from the Hamiltonian... 

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 left-hand side of Eq. (177) has a structure similar to the electronic gradient vector in variational wave function calculations. Unlike variational calculations, Eq. (177) cannot be used to determine the response parameters t(n) in a CC calculation. However, for the calculation of the nth geometrical derivative W n Eq. (177) eliminates r(n), which would otherwise appear in the calculation. In fact, we show below that the calculation of (3N-6)n response amplitudes t(n) is replaced by the solution of one set of linear equations of similar but simpler structure. By inserting Eq. (176) in Eq. (177) and rearranging terms, we see that X fulfills the equations... 

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

The Fock matrix appears naturally in the calculation of the orbital part of the electronic gradient ... 

Introducing the following notation for the electronic gradient and the electronic Hessian of the optimized wave function... 

From the expression for the electronic gradient Eq. 36, we note that the conditions Eq. 9 for a variational wave function... 

To evaluate the first- and second-order molecular properties, we choose the diagonal representation of the Hamiltonian. In this representation, the electronic energy, the electronic gradient, and the electronic Hessian of the electronic ground state 0) may be written in the following manner... 

Another difficulty with the metal oxide hypothesis is that the observed sum of the vertical electron equivalent gradients of Mn(II) and Fe(II) is much less than that of sulfide (Figure 7). In a simple vertical, steady-state system where the upward flux of Mn(II) and Fe(II) results in oxidized particulate metal oxides, which in turn settle to oxidize sulfide, the electron gradients of Mn(II) + Fe(II) would equal that for sulfide. The fact that they do not equal it suggests that the vertical flux of Mn(II) + Fe(II) would not produce sufficient particulate metal oxides. This problem would be solved if the particulate oxides were produced primarily at the boundaries and transported into the interior (40). 

For the optimization of Hartree-Fock wave functions, it is usually sufficient to apply the SCF scheme described in Sec. 3.1. By contrast, the optimization of MCSCF wave functions requires more advanced methods (e.g., the quasi-Newton method or some globally convergent modification of Newton s method, which involves, directly or indirectly, the calculation of the electronic Hessian as well as the electronic gradient at each iteration) [45]. 

The matrix-vector product (equation 67) may thus be evaluated in the same manner as an electronic gradient ... 

These equations have the same structure as the equations for the first-order orbital responses (equation 66), except that the right-hand side corresponds to the electronic gradient of the Cl energy ... 

Restricting ourselves to orthogonal transformations and taking the derivatives of the energy (equation 121) with respect to the elements of the antisymmetric matrix x, we obtain the following expressions for the electronic gradient and the electronic Hessian at = 0 ... 

Upon multiplication from the right by the negative electronic gradient (11.4.23), the Newton step (11.5.7) becomes... 

Sufficiently close to the solution, Newton s method exhibits quadratic convergence. To see how this convergence arises, we expand the electronic gradient and Hessian calculated at c = 0 about the exact solution at c = Cex (in a pseudo matrix notation) ... 

In the present section, we introduce the tools needed for optimization and manipulation of the MCSCF wave function. Having introduced the parametrization of the wave function in Section 12.2.1, we consider the expansion of the MCSCF energy in Section 12.2.2 and derive expressions for the electronic gradient and Hessian in Section 12.2.3. Some special properties of the MCSCF gradient and Hessian are then considered in Section 12.2.4 and 12.2.5. After a discussion of the important question of redundant orbital rotations in Section 12.2.6, we conclude this section with a discussion of the stmeture of MCSCF stationary points in Sections 12.2.7 and 12.2.8. 

Further simplifications occur since the gradient vanishes and since is the only nonzero block of see (12.2.10). Carrying out the differentiations in (12.2.23) and (12.2.24), we arrive at the following compact expres.sions for the MCSCF electronic gradient ... 

For these projectors, we note the following expressions involving the MCSCF electronic gradient and Hessian... 

The elements of the MCSCF Hessian matrix in (12.2.28)-( 12.2.30) contain contributions that depend on the electronic gradient and vanish at the stationary point. For future reference and manipulation, it is convenient to separate these gradient-containing terms from the remaining contributions to the electronic Hessian. Thus, introducing the notation... 

We may separate the MCSCF electronic gradient in the same manner. Thus, introducing the vector... 

We are now ready to consider the requirements for an optimized MCSCF state. In the present subsection, we examine the electronic gradient at stationary points in Section 12.2.8, we consider the electronic Hessian. According to (12.2.26) and (12.2.27), the stationary condition for an MCSCF wave function may be written as... 

In the remainder of the present section, we first consider the calculation of the electronic gradient in Section 12.5.1 and the evaluation of the matrix-vector products (12.5.4) in Section 12.5.2, building on the results for Hartree-Fock wave functions in Chapter 10 and for Cl wave functions in Chapter 11. Next, some aspects particular to the optimization of MCSCF wave functions are discussed in Section 12.5.3. After a discussion of the structure of the MCSCF Hessian (based on... 

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