Stationary Direct Perturbation Theory

The problem raised at the end of the last section, that the exact solutions to the second-order perturbation equations are not known except for very simple cases, is not the only problem in the application of direct perturbation theory. The major problem is that the exact solutions of the zeroth-order problem, that is, of the Schrodinger equation, are not known except for the same simple cases. As a consequence, the zeroth-order wave function is not the eigenfunction of the zeroth-order Hamiltonian and the perturbation 

Another way of averting this problem is to confine the expansion space for the perturbed wave functions to a set of approximate functions, for which the matrix of the zeroth-order Hamiltonian may be made diagonal. The perturbation expansion must then be done on the matrix level, not on the operator level. In essence, we are performing a projection onto a finite space, which is what we are always doing when we solve in a finite basis set. 

In this representation both the Hamiltonian and the metric partition transparently into zeroth-order operators and a perturbation. 

What we wish to do is to make the energy stationary at various orders of perturbation theory by expanding the expectation of the Dirac operator. 

Developing the perturbation expansion gives the same basic expressions as before for the various orders of the energy, but in terms of the modified operators  

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The functionals Fik i>2k) play a central role in stationary direct perturbation theory. Fq iPq) has been called the Levy-Leblond functional [23], since its stationarity condition is the LLE. For 4( 2) th name Hylleraas-Rutkowski functional has been suggested [23], since this belongs to the class of Hylleraas functionals of second-order perturbation theory, and since it has first been proposed by Rutkowski [73, 74] in a slightly different form. 

Stationary Direct Perturbation Theory for Many-Electron Systems... 

The calculation of properties using direct perturbation theory follows exactly the same lines as we used for Breit-Pauli theory. As we noted above, stationary direct perturbation theory leads to precisely the same equations we would have obtained by simply expanding the perturbed wave functions in the set of eigenfunctions of the zeroth-order Hamiltonian, and on this basis we proceed with the development of multiple direct perturbation theory for properties. 

Importantly, the anti-Hermitian CSE may be evaluated through second order of a renormalized perturbation theory even when the cumulant 3-RDM is neglected in the reconstruction. The anti-Hermitian part of the CSE [27, 31, 63] is the stationary condition for two-body unitary transformations of the A-particle wave-function [31, 32], and hence the two-body unitary transformations may easily be evaluated with the anti-Hermitian CSE and RDM reconstruction without the many-electron Schrodinger equation. The contracted Schrodinger equation in conjunction with the concepts of reconstruction and purification provides a new, important approach to computing the 2-RDM directly without the many-electron wavefunction. 

The following stationary perturbation theory on the matrix A is also derived from the direct analogy of the present case to that of quantum mechanics. ... 

Time-dependent response theory concerns the response of a system initially in a stationary state, generally taken to be the ground state, to a perturbation turned on slowly, beginning some time in the distant past. The assumption that the perturbation is turned on slowly, i.e. the adiabatic approximation, enables us to consider the perturbation to be of first order. In TD-DFT the density response dp, i.e. the density change which results from the perturbation dveff, enables direct determination of the excitation energies as the poles of the response function dP (the linear response of the KS density matrix in the basis of the unperturbed molecular orbitals) without formally having to calculate a(co). 

The response functions theory the PCM method [1] is an extension of the response theory for molecules in the gas phase [2, 3], This latter is based on a variational-perturbation approach for the description of the variations of the electronic wave function and of the changes of the observables properties at the various orders of perturbation with respect to the perturbing fields, and no restrictions are posed on the nature of the observables and on the nature of the perturbing fields, and the theory gives also access to a direct determination of the transition properties (i.e. transition energies and transition probabilities) associated with transitions between the stationary states of the molecular systems. The PCM response theory adds to this framework several new elements. 

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