Rayleigh-Schrodinger perturbation interaction

Note that the choice of non-orthogonal versus orthogonal basis functions has no consequence for the numerical variational solutions (cf. Coulson s treatment of He2, note 76), but it undermines the possibility of physical interpretation in perturbative terms. While a proper Rayleigh-Schrodinger perturbative treatment of the He- He interaction can be envisioned, it would not simply truncate at second order as assumed in the PMO analysis of Fig. 3.58. Note also that alternative perturbation-theory formulations that make no reference to an... 

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. 

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

If the distance between the interacting systems A and B is sufficiently large to enable the overlap of the respective electronic clouds to be disregarded, then the interaction energy may be calculated from the Rayleigh- Schrodinger perturbation theory, using as a basis of functions to describe the wave function of AB... 

The application of perturbation theory to many-body interactions leads to pairwise-additive and non-pairwise-additive contributions. For example, in the case of neutral, spherically symmetric systems which are separated by distances such that the orbital overlap can be neglected, the first non-pairwise-additive term appears at third order of the Rayleigh-Schrodinger perturbation treatment and corresponds to the dispersion energy which results from the induced-dipole-induced-dipole-induced-dipole78 interaction... 

Cwiok T, Jeziorski B, Kolos W, Moszynski R, Szalewicz K (1992) On the convergence of the symmetrized Rayleigh-Schrodinger perturbation theory formolecular interaction energies. J Chem Phys 97 7555-7559... 

This expression was deduced by the Rayleigh-Schrodinger perturbation theory. The Ramsey expression contains the electron exited states related to the energy E, and the operators A A and A which describe the involved electron interactions. The coupling of the external field to the orbital motion is expressed by a sum of orbital angular-momentum operators... 

The second order interaction energy, according to Rayleigh-Schrodinger perturbation theory is given by ... 

Table 1 Convergence of the Rayleigh-Schrodinger perturbation expansion for the interaction of two ground-state helium atoms at R = 1.0 and 5.6 bohr. The variational interaction energies of the fully symmetric state are equal to -1.01904 hartree and -166.5301 hartree, respectively. The Coulomb energy for R = 5.6 bohr is equal to -77.4764 /ihartree. The B66 and B71 basis sets were used for distances 1.0 and 5.6 bohr, respectively. E(n) denotes the sum of perturbation corrections up to and including the nth-order and S(n) the percent error of E(n) with respect to the variational interaction energy. Energies are in hartree for R = 1 bohr, and in /ihartree for R = 5.6 bohr.

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. 

Table 4 Convergence of the symmetrized Rayleigh-Schrodinger perturbation expansion for the interaction of a ground-state helium atom with a hydrogen molecule at R = 6.5 bohr. The B83 basis set was used. (n) and 6(n) are defined as in Table 1. Energies are in /ihartree.

Each electron in the system is assigned to either molecule A or B, and Hamiltonian operators and W for each molecule defined in terms of its assigned electrons. The unperturbed Hamiltonian for the system is then + W , and the perturbation XU consists of the Coulomb interactions between the nuclei and electrons of A and those of B. The unperturbed states, eigenfunctions of are simple product functions for closed-shell molecules, non-degenerate, Rayleigh-Schrodinger, perturbation theory gives the... 

The second-order effects comprise three contributions. The second-order Zee-man and hyperfine interactions involve obvious extensions of second-order Rayleigh-Schrodinger perturbation theory using the appropriate operators. If we introduce an arbitrary gauge origin Ra, and the variable tq — r — Rq, the second-order term involving both Zeeman and hyperfine operators is... 

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