Variational principles 77-operator

The solvated Fock operator can be naturally derived from the variational principles [14] defining the Helmlioltz free energy of the system (fA) by... 

Note that this is also a functional of liaAr), Cas(r), and 4 ). Imposing constraints concerning the orthonormality of the configuration state function (C) and one-particle orbitals (pi) on the equation, one can derive the Eock operator from. A based on the variational principle ... 

If classical Coulombic interactions are assumed among point charges for electrostatic interactions between solute and solvent, and the term for the Cl coefficients (C) is omitted, the solvated Eock operator is reduced to Eq. (6). The significance of this definition of the Eock operator from a variational principle is that it enables us to express the analytical first derivative of the free energy with respect to the nuclear coordinate of the solute molecule R ,... 

Assume now that two different external potentials (which may be from nuclei), Vext and Vgjjj, result in the same electron density, p. Two different potentials imply that the two Hamilton operators are different, H and H, and the corresponding lowest energy wave functions are different, and Taking as an approximate wave function for H and using the variational principle yields... 

Nevertheless, the situation is not completely hopeless. There is a recipe for systematically approaching the wave function of the ground state P0> i- c., the state which delivers the lowest energy E0. This is the variational principle, which holds a very prominent place in all quantum-chemical applications. We recall from standard quantum mechanics that the expectation value of a particular observable represented by the appropriate operator O using any, possibly complex, wave function Etrial that is normalized according to equation (1-10) is given by... 

The variational principle now states that the energy computed via equation (1-11) as the expectation value of the Hamilton operator H from any guessed xTtrial will be an upper bound to the true energy of the ground state, i. e.,... 

Eq. (22) have been derived from the variation principle alone (given the structure of H) they contain only the single model approximation of Eq. (9) the typically chemical idea that the electronic structure of a complex many-electron system can be (quantitatively as well as qualitatively) understood in terms of the interactions among conceptually identifiable separate electron groups. In the discussion of the exact solutions of the Schrodinger equation for simple systems the operators which commute with the relevant H ( symmetries ) play a central role. We therefore devote the next section to an examination of the effect of symmetry constraints on the solutions of (22). 

It is found empirically and of course is predictable theoretically that, when using a model for molecular electronic structure, the set of eigenfunction equations associated with the operators commuting with H are constraints on the action of the variation principle if Et is computed from R subject to symmetry constraints and E2 is computed in the same model with no such constraints then (2)... 

The application of G-spinor basis sets can be illustrated most conveniently by constructing the matrix operators needed for DCB calculations. The DCB equations can be derived from a variational principle along familiar nonrelativistic lines [7], [8, Chapter 3]. It has usually been assumed that the absence of a global lower bound to the Dirac spectrum invalidates this procedure it has now been established [16] that the upper spectrum has a lower bound when the trial functions lie in an appropriate domain. This theorem covers the variational derivation of G-spinor matrix DCB equations. Sucher s repeated assertions [17] that the DCB Hamiltonian is fatally diseased and that the operators must be surrounded with energy projection operators can be safely forgotten. 

So, the question arises of how we might modify the HF wave function to obtain a lower electronic energy when we operate on that modified wave function with the Hamiltonian. By the variational principle, such a construction would be a more accurate wave function. We cannot do better than the HF wave function with a single determinant, so one obvious choice is to construct a wave function as a linear combination of multiple determinants, i.e.. 

Regardless of the nature of the problem, approximate upper bounds, follow from the variation principle, i. e. from outer projections. So, introducing the projection operator (41) we define ... 

The precise connection with finite dimensional matrix formulas obtains simply from Lowdin s inner and outer projections [21, 22], see more below, or equivalently from the corresponding Hylleraas-Lippmann-Schwinger-type variational principles [24, 25]. For instance, if we restrict our operator representations to an n-dimensional linear manifold (orthonormal for simplicity) defined by... 

The theory of the / -matrix can be understood most clearly in a variational formulation. The essential derivation for a single channel was given by Kohn [202], as a variational principle for the radial logarithmic derivative. If h is the radial Hamiltonian operator, the Schrodinger variational functional is... 

In the same way that a self-adjoint operator satisfies a variational principle, a pair of adjoint operators T and Tf satisfy a bivariational principle (5), which may also be applied to the transformed operators f and rT. Let us start by considering the variational expressions... 

The excitation operator St is symmetry-adapted, which discriminates between the SAC and ordinary CC methods. The Cj is the coefficient of the operator. Applying the variational principle, we obtain the variational SAC equations. 

In this paper, the general theory developed in Part I is applied to the Hartree-Fock Scheme for a transformed many-electron Hamiltonian. It is shown that, if the transformation is a product of one-electron transformations, then the Fock-Dirac operator as well as the effective Hamiltonian undergo similarity transformations of the one-electron type. The special properties of the Hartree-Fock scheme for a real self-adjoint Hamiltonian based on the bi-variational principle are discussed in greater detail. 

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