RS perturbation theory

If A0, Bo are the unperturbed wavefunctions of molecules A (Na electrons) and B (NB electrons), and A,-, a pair of excited pseudostates describing single excitations on A and B, all fully antisymmetrized within the space of A and B, we have to second order of RS perturbation theory ... 

We turn now to the more recently proposed two-state model of long-range interactions (Magnasco, 2004b). It is of interest in so far as it avoids completely explicit calculation of the matrix elements (Equations 4.12-4.17) occurring in RS perturbation theory, being based only on the fundamental principles of variation theorem and on a classical electrostatic approach. 

As the ratio of the two perturbations is not known, the formalism of the double Rayleigh-Schrodinger (RS) perturbation theory can be used, which looks for the ground state of the total Hamiltonian... 

We saw earlier that a very simple form of the dispersion energy is obtained from frequency-dependent polarizabilities at the so-called uncoupled Hartree-Fock level. The sum over states appearing in second order RS perturbation theory is simply a sum over (occupied and virtual) orbitals. A first improvement of this simple model is obtained by including apparent correlation [140], i.e. by using frequency-dependent polarizabilities obtained from the TDCHF method [36,141]. This method was initially proposed in the context of the multipole expansion, but could be generalized [142-146] to charge density susceptibility functions (or polarization propagators), which avoids the use... 

The MP2 energy, the simplest correlation correction, is obtained from RS perturbation theory with the perturbation... 

The generalized Bloch equation (12) is the basis of the RS perturbation theory. This equation determines the wave operator and, together with Eq. (11), the energy corrections for all states of interest especially, it leads to perturbation expansions which are independent of the energy of the individual states, just referring the unperturbed basis states. Another form, better suitable for computations, is to cast this equation into a recursive form which connects the wave operators of two consecutive orders in the perturbation V. To obtain this form, let us start from the standard representation of the Bloch equation (16) in intermediate normalization and define... 

We shall analyze these matrix elements later in more detail. Here, let us note only that the characteristic energy denominators in the RS perturbation theory arises always from the energy difference of fwo unperfurbed eigenvalues, from which one of the corresponding eigenfunctions is inside and one outside of the model space. 

In this figure we have used both a harmonic-oscillator (HO) and a Brueckner-Hartree-Fock (BHF) basis for the single-particle wave functions in order to study the behavior of the RS perturbation theory at low orders. What can be seen from this figure is that the BHF basis yields a smaller overlap between states in the excluded space and the model space, reflected in the small change when going from second order to third order in the perturbation expansion. However, the BHF spectra are too compressed and in poor agreement with experiment. This is probably related to the fact that the radii obtained for the self-consistent single-particle wave functions are much smaller than the empirical ones [53]. 

Fig. 8. Theoretical and experimental low-lying spectrum for 0 obtained with the Bonn A potential defined in Table A.l of Ref. [7], using both a HO basis and a BHF basis. The terms H f, and denote the effective interaction through first, second and third order in Rayleigh-Schrodinger (RS) perturbation theory. All energies in MeV. Taken from Ref. [53].

In this subsection we first discuss the results obtained through second order in RS perturbation theory, which corresponds to setting the terms Fq and Rq equal to the second-order two-body diagrams displayed in Fig. 6. No folded diagrains are included at this level. Next, we discuss the differences between a second- and a thirdjarder Q-box. Furthermore, the third-order Q-box is used to evaluate the LS effective interaction R,-. 

In this section we will derive the standard expressions of Rayleigh-Schrodinger (RS) perturbation theory. Our formulas will be general and apply equally to one-particle or N-particle systems. Suppose we wish to solve the eigenvalue problem... 

We now generalize our previous development to obtain a diagrammatic representation of RS perturbation theory as applied to an N-state system. Consider the problem of finding the perturbation expansion for the lowest eigenvalue of such a system. Here we still have only one hole state, 1>, but there are now N - 1 particle states n>, n = 2, 3,..., AT. We draw the same set of diagrams as before. However, now we can label the particle lines with any index n. For example, the diagram... 

These expressions are identical to our previous results for the second- and third-order energies (Eqs. (6.12) and (6.15)) when i = 1. What if we want the perturbation expansion for some state i, which is not necessarily the lowest What do the diagrams look like One can easily verify that we get the same answers as before if we label our hole lines by the index i and the particle lines by the indices m, n, k,..., which can take on the values 1,2,..., i - 1, I + 1,..., N. Thus we now have a complete diagrammatic representation of RS perturbation theory, which is applicable to any perturbation and any zeroth-order state. 

In Section 62, we introduced a completely general diagrammatic repre sentation of RS perturbation theory. To adapt this to handle orbital perturbations we take the downward and upwaM lines to represent hole and particle spin orbitals, respectively, and the dots to correspond to the one-particle perturbation v. Ilien we draw the same set of diagrams as before, labeling the hole lines by indices a,b,...and the particle lines by indices r, s, Thus we have... 

The use of the above partitioning of the Hamiltonian, along with the general expressions of RS perturbation theory, is sometimes called M [Pg.350]

As with and E q this is just N times the third-order energy of a single H2 molecule (Eq. (6.78)). Recall that Eq. (6.78) was obtained by expanding the exact correlation energy within the basis in a Taylor series, so that the equivalence of the expressions derived in different ways provides a consistency check. Although this example is by no means a proof, we hope it will inspire some confidence in the statement that RS perturbation theory—in contrast to DCI—yields an approximation to the correlation energy which is size consistent (i.e., has the correct N-dependence). 

The Coulombic components alone give what is known as the polarization approximation [13], and can be obtained from ordinary RS perturbation theory for stationary states with the intermolecular potential V playing the role of a small perturbation. Not reflecting the full symmetry of the Hamiltonian if, the RS expansion converges very slowly, unlike expansions using antisymmetrized products, and an accurate energy cannot be obtained in finite order. There are two essential points in the RS expansion ... 

Elements of Rayleigh-Schrodinger (RS) Perturbation Theory Molecular Interactions... 

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