Rayleigh-Schrodinger-perturbation theory

In perturbation theory, one often starts with an a priori partitioning of the many-body atomic or molecular Hamiltonian, 

For fhe model operator Ho, the Schrodinger equation has a complete (and orthonormal) set of eigenfunctions 

These eigenfunctions define a basis in the N-electron Hilbert space. To account for the effects of the rest interaction V perturbatively, the full Hilbert space H of Hg is separated into two parts the model space M and its complement, TL M, which includes all the remaining dimensions. To these subspaces, we assign the projection operators 

In the following, we suppose that each of the desired solution of Eq. (8) has a significant part inside of this model space. We shall denote the projections of these physical states IV a) onto M as the model functions and assume, that they are linearly independent. Below, however, we shall find that these model functions 

The distinction of the model functions = l.d enables us to (formally) define the wave operator for the reversed projection. This wave operator maps the model functions back onto the exact states [26,27], 

Rayleigh-Schrodinger Perturbation Theory.—In Rayleigh-Schrodinger perturbation theory the unknown energy in the denominators of the Lennard-Jones Brillouin Wigner expansion is avoided. This enables a size-consistent theory to be derived. 

The wave operator, may be written in an alternative form by replacing by 0+ Qo E0Q0 and by 2 x- QakE0Q0 giving 

Rearranging this expression in terms of powers of the perturbation, we obtain 

Explicitly, the first few terms in the Rayleigh-Schrodinger perturbation expansion may be written in the form  

Clearly, the terms other than the first in the expressions for Ell E( E etc. depend on the number of electrons in a non-linear fashion. These terms exactly cancel components of the first terms in each of the expressions which also have a non-linear dependence on the number of electrons. 

In Rayleigh-Schrodinger perturbation theory, the expansion for the exact ground state energy of the perturbed system can also be written in the form 

The Rayleigh-Schrddinger perturbation theory is developed by substituting the power series for the wave function (1.22) and that for the energy (1.21) into the Schrodinger equation (1.3) in the form 

Equating powers of A leads to the basic equations of Rayleigh-Schrodinger perturbation theory. The zero-order equation is simply the eigenvalue problem for the reference or model system with respect to which the perturbation expansion is developed  

Without loss of generality, we can require that the p order perturbed wave functions Xo be orthogonal to the reference function, Po - 

Multiplying the zero-order equation (1.27) from the left by and integrating gives 

Our definition of molecular properties in the previous section is restricted to static and periodic perturbations V t) =V t + T) where T is the period. The perturbed wave function then has the form [17] 

In this section we will investigate the nature of the exact perturbed wave function through Rayleigh-Schrodinger perturbation theory. We will employ intermediate normalization 

We have not explicitly written out the corrections jo t) to the phase-isolated wave function as they follow irmnediately from the relation 

Consider now the time average of the nth order correction to the time-dependent quasienergy. In shorthand-notation we write this as 

As an alternative to Brillouin-Wigner perturbation theory, we may consider Rayleigh-Schrddinger perturbation theory, which is size consistent. In this method the total energy is computed in a stepwise manner 

If we insert into the right-hand side of (10.79) and (10.80) the expansion Eq = 

YlT=o 0 then, by using the usual perturbation theory argument, we equalize the terms of the same order, we get for n = 0  

Let us consider the case of a closed shell. In the Mdller-Plesset perturbation theory we assume as the sum of the Hartree-Fock operators (from the RHF 

We may carry out calculations through a given order for such a perturbation. A very popular method relies on the inclusion of the perturbational corrections to the energy through the second order (known as MP2 method) and through the fourth order (MP4). 

Excited-state energies and wave functions are automatically obtained from Cl calculations. However, the quality of the wave functions is more difficult to achieve. The equivalent of the HF description for the ground state requires an all-singles Cl (SCI). Singly excited configurations do not mix with the HF determinant, that is, 

If the solutions (energies and wave functions P ) of the Schrodinger equation for the unperturbed system Tf(°) P = F,1,01 4/jl°l are known, and the operator form of the perturbation, Hp, can be specified, the Rayleigh-Schrodinger perturbation theory will provide a description of the perturbed system in terms of the unperturbed system. Thus, for the perturbed system, the SE is 

The parameter is introduced to keep track of the order of the perturbation series, as will become clear. Indeed, one can perform a Taylor series expansion of the perturbed wave functions and perturbed energies using X to keep track of the order of the expansions. Since the set of eigenfunctions of the unperturbed SE form a complete and orthonormal set, the perturbed wave functions can be expanded in terms of them. Thus, 

The superscripts in parentheses indicate successive levels of correction. If the perturbation is small, this series will converge. Substitution of equations (A.92) and (A.93) into equation (A.91) and collecting powers of yields 

Equation ( .95) is a power series in which can only be true if the coefficients in front of each term are individually zero. Thus, 

If the solutions (energies E and wave functions XP ) of the Schrodinger equation for the unperturbed system //(°)xp(°) = are known, and the operator form of the 

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An essential thing to stress concerning the above development of so-called Rayleigh-Schrodinger perturbation theory (RSPT) is that each of the energy corrections... 

Wigner, E. P., Phys. Rev. 94, 77, "Application of the Rayleigh-Schrodinger perturbation theory to the hydrogen atom." The whole electrostatic potential is considered as a perturbation. 

The Rayleigh-Schrodinger Perturbation Theory (see [2]) leads then to the following system of linear equations for the determination of cj (j=l,. ..M) ... 

The mathematical procedure that we present here for solving equation (9.15) is known as Rayleigh-Schrodinger perturbation theory. There are other procedures, but they are seldom used. In the Rayleigh-Schrodinger method, the eigenfunctions tpn and the eigenvalues E are expanded as power series in A... 

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. ... 

From this starting point, London employed standard techniques of Rayleigh-Schrodinger perturbation theory to evaluate the leading effects of the intermolecular... 

Angyan, J. G. Rayleigh-Schrodinger perturbation theory of non-linear Schrodinger equations with linear perturbation, IntJ.Quantum Chem., 47 (1993), 469-483... 

It should be apparent that the expressions for the wave functions after interaction [equations (3.38) and (3.39)] are equivalent to the Rayleigh-Schrodinger perturbation theory (RSPT) result for the perturbed wave function correct to first order [equation (A.109)]. Similarly, the parallel between the MO energies [equations (3.33) and (3.34)] and the RSPT energy correct to second order [equation (A. 110)] is obvious. The missing first-order correction emphasizes the correspondence of the first-order corrected wave function and the second-order corrected energy. Note that equations (3.33), (3.34), (3.38), and (3.39) are valid under the same conditions required for the application of perturbation theory, namely that the perturbation be weak compared to energy differences. 

Let us define the operator of the effective Hamiltonian in Rayleigh-Schrodinger perturbation theory in the following way ... 

The first- and second-order electronic wavefunctions necessary for further development are obtained from Rayleigh-SchrOdinger perturbation theory (see ref. (28) for details). 

The method is based on the following procedure.33 All possible doubly excited configurations are generated from the Hartree-Fock function and their contributions to the second-order Rayleigh-Schrodinger perturbation theory energy computed. Approximately 100 of the most important are used for a Cl calculation, all singly... 

IV. Many-Body Rayleigh-Schrodinger Perturbation Theory.108... 

VIII. Use of the Many-Body Rayleigh-Schrodinger Perturbation Theory for the Interaction Between Two Closed-Shell Systems... 

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