Large-order perturbation theory

Vol. 53 G.A. Arteca, F.M. Fernandez, E.A. Castro, Large Order Perturbation Theory and Summation Methods in Quantum Mechanics. XI, 644 pages. 1990. 

Jiri Cizek s research program centers on the quantum theory of molecular electronic structure and related developments in quantum chemical methodology, coupled-cluster approaches to many-electron correlation problems,105 large-order perturbation theory,106 dynamical groups and exactly solvable models, lower bounds, and the use of symbolic computation language in physics and in chemistry. 

We intend to publish in future volumes of the Advances in Quantum Chemistry two separate reviews on these problems, one on the large-order perturbation theory (LOPT) and one on continued fractions at large order (CFLO). 

Thus, perturbation theory based on the scaled hydrogenic basis can provide better accuracy and, when combined with the convenient calculation of matrix elements using so(4, 2) algebraic methods, results in an effective technique for large-order perturbation theory (Cizek and Vrscay, 1982 Clay, 1979 Vrscay, 1977 Bednar, 1973 Adams et al., 1980 Cizek et al 1980a,b Silverstone et al., 1979). 

Vrscay, E. R. (1983). Continued fractions, algebraic methods and quantum mechanical large order perturbation theory. Ph.D. Thesis, Univ. of Waterloo, Waterloo, Ontario. 

J.N. Silverman, C.A. Nicolaides, Complex Stark eigenvalues for excited states of hydrogenic ions from analytic continuation of real variationally based large-order perturbation theory, Chem. Phys. Lett. 164 (1991) 321. 

These concepts play an important role in the Hard and Soft Acid and Base (HSAB) principle, which states that hard acids prefer to react with hard bases, and vice versa. By means of Koopmann s theorem (Section 3.4) the hardness is related to the HOMO-LUMO energy difference, i.e. a small gap indicates a soft molecule. From second-order perturbation theory it also follows that a small gap between occupied and unoccupied orbitals will give a large contribution to the polarizability (Section 10.6), i.e. softness is a measure of how easily the electron density can be distorted by external fields, for example those generated by another molecule. In terms of the perturbation equation (15.1), a hard-hard interaction is primarily charge controlled, while a soft-soft interaction is orbital controlled. Both FMO and HSAB theories may be considered as being limiting cases of chemical reactivity described by the Fukui ftinction. 

The first-order perturbation theory in this case does not give good results, since the mutual interaction of two electrons on one nucleus is so large as to greatly deform the eigenfunctions it leads... 

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