Rayleigh-Schrodinger formalism

Note that the wave operator arising in the Rayleigh-Schrodinger formalism, Q can be related to the wave operators in the Brillouin-Wigner method through the relation... 

The application of many-body perturbation theory to molecules involves the direct application of the Rayleigh-Schrodinger formalism with specific choices of reference Hamiltonian. The most familiar of these is that first presented by Mpller and Plesset... 

Let us now return to the Rayleigh-Schrodinger formalism which is used in developing the standard coupled cluster formalism. The single-reference coupled cluster expansion can be developed by first of all writing the Schrodinger equation for the non-degenerate case as... 

E. Schrodinger, Ann. Phys. 80 (1926), 437. The quantal formalism substantially follows the classical method developed by Lord Rayleigh (Theory of Sound [1894]) and is commonly referred to as Rayleigh-Schrodinger perturbation theory. ... 

The BW form of PT is formally very simple. However, the operators in it depend on the exact energy of the state studied. This requires a self-consistency procedure and limits its application to one energy level at a time. The Rayleigh-Schrodinger (RS) PT does not have these shortcomings, and is, therefore, a more suitable basis for many-body calculations of many-electron systems than the BW form of the theory, it is applicable to a group of levels simultaneously. 

In the above formula, Q is the nuclear coordinate, p, and I/r are the ground state and excited electronic terms. Here Kv is provided through the traditional Rayleigh-Schrodinger perturbation formula and K0 have an electrostatic meaning. This expression will be called traditional approach, which has, in principle, quantum correctness, but requires some amendments when different particular approaches of electronic structure calculation are employed (see the Bersuker s work in this volume). In the traditional formalism the vibronic constants P0 dH/dQ Pr) can be tackled with the electric field integrals at nuclei, while the K0 is ultimately related with electric field gradients. Computationally, these are easy to evaluate but the literally use of equations (1) and (2) definitions does not recover the total curvature computed by the ab initio method at hand. 

In the simplest symmetrized Rayleigh-Schrodinger (SRS) perturbation theory87-89 the symmetry forcing operator appears only in the energy expression. Hence, this formalism employs weak symmetry forcing and the operators v9, v9, and v9 are given by,... 

I. Hubac and P. Carsky, Phys. Rev., All, 2392 (1980). Correlation Energy of Open-Shell Systems. Application of the Many-Body Rayleigh-Schrodinger Perturbation Theory in the Restricted Roothaan-Hartree-Fock Formalism. 

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... 

It is worth noting that the convergence pattern of the polarization series for He2 is very similar to that found for Hj (9) and H2 (18). Thus, at the distances of the van der Waals minimum the Rayleigh-Schrodinger perturbation theory provides only a part of the interaction energy (15), and in practical applications symmetry-adapted perturbation formalisms must be used. 

Since both these models represent extended systems, the exploitation of the shell-model or Cl-type variational methods was a priori excluded. This placed emphasis on the development of perturbative approaches for this type of problem. It was soon realized that the most efficient methodological approach must be based on a mathematical formalism that was originally developed in quantum field theory. Moreover, in view of the extended character of the studied systems, it was absolutely essential that the method employed yields energies that are linearly proportional to the particle number N in the system or, in today s parlance, that it must be size extensive, so that the limiting procedure when N->-oo makes sense. In terms of MBPT, this imphes that only the connected or finked energy terms be present in the perturbation series, a requirement that automatically leads to the Rayleigh-Schrodinger PT. 

Formalism of the so-called response theory is another, quite universal language for the description of the more general approach to Rayleigh-Schrodinger perturbation theory suggested above, in which the summation over excited states is effectively replaced with solving a large system of linear equa-... 

Such second-order EPs have been used (Doll and Reinhardt, 1972 Purvis and Ohrn, 1974) to compute atomic and molecular ionization potentials, electron affinities, and even electron-atom shape resonance positions and lifetimes with some success. Based upon the experience gained to date, however, we cannot expect the accuracy of this approach to be better than 0.5 eV, even for systems that are described reasonably well by a singleconfiguration reference function. Often, this numerical accuracy is not satisfactory and hence the above formalism must be advanced to higher order (or replaced by another development that does not depend upon the Rayleigh-Schrodinger order concept). An example of such a second-order EP calculation is given in Problem 6.1. 

The renewal of interest in Brillouin-Wigner perturbation theory for many-body systems seen in recent years, is driven by the need to develop a robust multi-reference theory. Multi-reference formalisms are an important prerequisite for theoretical descriptions of dissociative phenomena and of many electronically excited states. Brillouin-Wigner perturbation theory is seen as a remedy to a problem which plagues multi-reference Rayleigh-Schrodinger perturbation theory the so-called intruder state problem. 

Brillouin-Wigner perturbation theory can be developed for both the single reference function case and the multireference function case using a common formalism. This contrasts with the situation for Rayleigh-Schrodinger perturbation theory. We shall, therefore, consider the single reference and multireference formalisms together. 

This use of the unperturbed population to calculate the change in revenue is a crade example of a certain level of estimation (called first order ) in Rayleigh-Schrodinger perturbation theory. We will now proceed to develop the theory more formally in the context of wavefunctions and energies. The above example has been presented to encourage the reader to anticipate that there is a lot of simple good sense in the results of perturbation theory even though the mathematical development is rather cumbersome and unintuitive. 

A key feature of the many-body perturbation theory is the use of the method of second-quantization. We therefore open this section by introducing the second quantization formalism. We then discuss the Rayleigh-Schrddinger perturbation theory in its many-body form, that is, many-body Rayleigh-Schrodinger perturbation theory. We close this section by presenting the many-body perturbation theory with an emphasis on its diagrammatic formulation. 

The state-selective approach to the multi-reference problem was further developed by Banerjee and Simons [118], by Laidig and Bartlett [119], by Hoffmann and Simons [120], by Li and Paldus [121, 122] and by Jeziorski, Paldus and Jankowski [123] who formulated extensive open-shell cc theory, based on the unitary group approach (uga) formalism. The Rayleigh-Schrodinger formulation of a state-selective approach to the multi-reference correlation problem has been developed more recently by Mukherjee and his collaborators [124-130] and also by Schaefer and his colleagues [131-133]. 

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