Wavefunction variation

Section V consists of a detailed discussion of redundant variables. The special case of MCSCF wavefunction optimization for two-electron systems is discussed in some detail. The relation between the configuration expansion space and the orbital variation space is quite straightforward for this case and this simplicity may be used to advantage in understanding the generalization to arbitrary numbers of electrons. There are two aspects of redundant variables that are important in the MCSCF method. First, if redundant variables are allowed to remain in the wavefunction variation space, then the optimization procedure becomes undefined or at least numerically ill-conditioned. Secondly, if the redundant variables are known for a given wavefunction then this flexibility may be used to transform the wavefunction to a form that is qualitatively easier to understand. The qualitative interpretation of MCSCF wavefunctions is one of the assets of the MCSCF method. 

The first energy expression of the above type results from truncating the expansion of the wavefunction variations to include only the first-order changes... 

The following restrictions are hereafter imposed on the two-electron wavefunction variations. The CSF expansion coefficients C are assumed to be real and any transformation applied to these coeffieients must be real. The orbital basis is allowed to be complex but any transformation applied to the orbitals must be real. These restrictions have no effect on the expectation values of real Hamiltonian operators. Finally, an orthonormal orbital basis is assumed and only orthogonal orbital transformations are allowed. This of course does place restrictions on some of the present discussions but is considered crucial for the extension of these results to the general N-electron case. 

Just as in the two-electron case of the full Cl expansion, the simultaneous effect of both the orbital variations and the CSF expansion coefficient variations is redundant. An arbitrary wavefunction variation may be expressed by considering only the variations of the expansion coefficients for some fixed set of orthonormal orbitals. The transformation of Eq. (235) is equivalent to the... 

In Eq. [5], A and B collect the Hessian components of the energy functional with respect to the wavefunction variational parameters. The eigenvalues (Oj give an approximation of the transition energies whereas the eigenvectors Yk) determine the corresponding transition matrix to the excited state K. 

Another approach for developing approximations to CC and CS reactive scattering calculations is to use distorted wave theory. In this approach, one considers that reaction is only a small perturbation on the nonreactive collision dynamics. As a result, the reactive scattering matrix can be approximated by the matrix element of a perturbative Hamiltonian operator using reagent and product nonreactive wavefunctions. Variations on this idea can be developed by using different approximations to the nonreactive wavefunctions. At the top of the hierachy of these methods is the coupled channel distorted wave (CCDW) method, followed by coupled states distorted wave (CSDW). 

Inserting the first- and second-order wavefunction variations into either of the expressions Eq. (6) or (12) yields the complete expressions for the gradient and Hessian. In doing this, it becomes clear that the computationally most expensive parts are related to the terms containing first-order variations in both bra and ket, e.g. 

The wavefunction variational principle implies the Hellmann-Feynman and virial theorems below and also implies the Hohenberg-Kohn [25] density functional variational principle to be presented later. 

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