Lowdin perturbation theory

All the calculations are performed in momentum space and (unless otherwise stated) plane waves with kinetic energy up to 9.15 Ry are included in the expansions of the wave functions. Only those with kinetic energy S 2.55 Ry are dealt with exact, the remaining ones are treated by Lowdin perturbation theory up to second order. This corresponds to approximately 21 + 125 waves when working with the two-atoms cells, 43 + 240 when working with the doubled (four atoms) ui it cells, 85 + 500 on quadrupled cells, etc. Two to five special k-points are used for Brillouin zone integration (corresponding to (222) in the notation of... 

In this chapter we will use extensively the so-called Lowdin perturbation theory, which can be formulated like this If a real symmetric matrix H can be written in form of two blocks... 

Some further remarks are in order on the subject of truncation of the pseudopotential. Most current semiempirical studies involve quite large secular determinants (say 50 x 50) but set f(g) equal to zero for g 2kp. However, a somewhat cruder procedure, that of truncating the basis set at g = 2kp, resulting in a smaller secular determinant, has also been widely used. This procedure may be put on a formal basis by the use of Lowdin perturbation theory, by which a larger secular determinant is reexpressed as a smaller one, with correction terms. For a local pseudopotential the correction terms are given by (Ref 51, pp. 78-83)... 

Lowdin, P-O., Studies in Perturbation Theory. X. Bounds to Energy Eigenvalues in Perturbation Theory Ground State, Physical Review, 1965 139A 357-364. 

Lowdin, who contributed in no small measure to the development of formal many-electron theory through his seminal work on electron correlation, reduced density matrices, perturbation theory, etc. many times expressed his concerns about the theoretical aspects of density functional approaches. This short review of the interconnected features of formal many-electron theory in terms of propagators, reduced pure state density matrices, and density functionals is dedicated to the memory of Per-Olov Lowdin. 

A review by P. O. Lowdin in "Perturbation Theory and its Applications to Quantum Mechanics" (Ed. C. H. Wilcox, John Wiley and Sons, New York, 1966)... 

P. O. Lowdin. Partitioning technique, perturbation theory, and rational approximations. Intern. J. Quantum Chem., 21 69, 1982. 

P.O. Lowdin, A Note on the Quantum-Mechanical Perturbation Theory, J. Chem. Phys. 19 (1951) 1396. 

P.-O. Lowdin, Studies in perturbation theory. IX. Connection between various approaches in the recent development-evaluation of upper bounds to energy eigenvalues in Schrodinger s perturbation theory. J. Math. Phys. 6, 1341-1353 (1965)... 

For a comprehensive account and many remarkable developments of general perturbation theory, see Lowdin (1962,1963,1964,1965,1966,1968,1982a) and Lowdin and Goscinski (1971). 

CIS(D) can be rigorously derived by applying the Lowdin-type (as opposed to Rayleigh-Schrodinger) perturbation theory [70] to CIS, according to Meissner [71]. Additional off-diagonal second-order corrections to CIS have been considered by Head-Gordon et al. [72],... 

Thus, for most non-closed shells many-electron theory in the form of Eqs. (77) and (85) may be safely assumed to apply with H.F. based on the average energy of the non-closed configuration with restrictions . With this H.F., the perturbation equation (64) can again be solved by operator techniques. Having the same for all the electrons simplifies matters also one notes that the projection operators that turn say into p in Eq. (93) commute with j/, Hq and H. Lowdin s projection operators should be useful in applying the many-electron theory and the perturbation theory to non-closed shells. 

P.-O. Lowdin. Studies in Perturbation Theory Part VIII. Separation of Dirac equation + Study of Spin-Orbit Coupling + Fermi Contact Terms. J. Mol. Spectrosc., 14(2) (1964) 131-144. 

The series of papers by Lowdin [109-121] entitled Studies in perturbation theory contributed much to our understanding of different perturbation theories and the relation between them. 

It may be of some interest to check whether the BSSE-free interaction operator of Eq. (15.27) could be used to solve this problem. This project was carried out in our laboratory (Surjan et al. 1985b, Surjan Poirier 1986). To utilize the full power of this interaction operator, we did not turn to a Lowdin basis, but applied a non-Hermitian perturbation theory similar to the former work by Kochanski and Gouyet (1975). 

This situation may be met when aiming to describe dynamic electron correlation starting from a single, multi-determinantal reference vector. Correction schemes based on perturbation theory (PT) have been applying successive Gram-Schmidt orthogonahzation in such circumstances [5-7], occasionally combined with Lowdin s symmetrical [8] or... 

Lowdin, P-O., 8cGoscinski, O. (1999). Studies in perturbation theory. XIV. Treatment of constants of motion, degeneracies and symmetry properties by means of multidimensional partitioning. International Journal of Quantum Chemistry, 5, 685. 

During the period 1963-1971, P.-O. Lowdin published a series of papers [42-53] with the general title Studies in Perturbation Theory , which afforded deep insight into perturbation theory expansions, the relation between different expansions and their application to quantum mechanical problems. We conclude this section with a brief overview of Ldwdin s work on perturbation theory. 

P.-O. Lowdin, in Perturbation Theory and its Application to Quantum Mechanics,... 

This leads to what Lowdin [6] has called generalized Brillouin-Wigner perturbation theory. 

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