Theory double perturbation

The ordering parameter perturbation theory can be extended to the case of more perturbations simultaneously present. The new assumption is the following form of the Hamiltonian 

This has a close relationship to the evaluation of magnetic parameters for example, X may by a component of the magnetic field and k may represent a component of the spin magnetic moment. 

After elementary manipulations similar to those followed above, the energy formulae result 

The last term is given by two equivalent expressions. The only need which survives is the expression for the first-order corrections to the state vector, which are 

Hildebrand, Introduction to Numerical Analysis, Dover Publications, Inc., New York, 1974. 

Helgaker, P. J0rgensen, Calculation of geometrical derivatives in molecular electronic structure theory, in S. Wilson, G. H. F. Diercksen (Eds.), Methods in Computational Molecular Physics, Plenum Press, New York, 1992. 

Sucher, in J. P. Briand (Ed.), Atoms in Unusual Simations, Plenum Press, New York, 1986. 

Sucher, InL J. Quant. Chem. Quant. Chem. Symp. 25 (1984) 3. 

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At the correlated level the many-body perturbation theory is applied, the localized version of which (LMBPT) has already proven to be useful in the study of molecular electronic structure. The LMBPT is a double perturbation theory, and the perturbational correction are calculated as ... 

The problem of the effect of an inaccurate zeroth-order wavefunction on dispersion calculations has been looked at again by Magnasco82 using double perturbation theory for the He---He case. Using hydrogenic, screened hydrogenic, and approxi-... 

It should be pointed out that Schwarz (20),using double perturbation theory,has demonstrated that it is possible to rationalize the relativistic bond length contraction in terms of the attractive Hellmann-Feynman force due to the relativistic change in electron density.In such an approach it would be necessary to analyze and get a physical picture of the relevant density changes... 

The remaining anharmonic square bracket terms are obtained by BK using double perturbation theory with the definition of orders given below. 

As in the BK treatment of non-resonant properties double perturbation theory may be used to evaluate the curly bracket quantities. The curly brackets are more difficult to evaluate than square brackets due to the presence of transition, as well as ground state, electrical properties. 

Thus the double perturbation theory enables to discuss the effects of electron correlation on molecular properties systematically, but there are not many numerical calculations proceeding along these lines instead of calculating the effect of /I2 by veuriation methods and subsequently dealing with d 1 by ordinary perturbation theory. The former procedure has some advantages, for, in principle, it should include ZI2 to infinite order (in other words, A1 and A 2 might enter on different levels of the perturbation treatment). 

If we want to study the nrl and the relativistic corrections of the first-order property E or the second-order property E we must apply double perturbation theory. To formulate this we first switch to the DE with modified metric... 

R 22 J. Autschbach and T. Ziegler, Double Perturbation Theory A Powerful Tool in Computational Coordination Chemistry , p. 83 R 23 I.P. Georgakaki, L.M. Thomson, EJ. Lyon, M.B. Hall and M.Y. Darens-bourg, Fundamental Properties of Small Molecule Models of Fe-Only Hydrogenase Computations Relative to the Definition of an Entatic State in the Active Site , p. 238 Vol. 243, 2003... 

J. Autschbach, T. Ziegler. Double perturbation theory a powerful tool in computational coordination chemistry. Coord. Chem. Rev., 238-239 (2003) 83-126. 

Double perturbation theory using Epstein-Nesbet partition. 

Symmetry-adapted perturbation theory is a well-motivated theoretical approach to compute the individual components of intermolecular interactions, namely, the electrostatic, induction, dispersion, and exchange-repulsion terms. The approach is a double-perturbation theory that uses a Hartree-Fock reference, with a Fock operator F written as the sum of Fock operators for the individual molecules. Both the intramolecular correlation potential W) and the intermolecular interactions (V) are treated as perturbations, so that the Hamiltonian is expressed as... 

We saw in the previous section that the perturbation theoretical expressions governing two molecules (or linear chains) at medium distances (where a multipole expansion for the electrostatic term alone is insufficient) are rather complicated even in second order. On the other hand, perturbation theory in this form cannot describe the simultaneous interactions of more than two molecules (only with the aid of the still more complicated double perturbation theory), and it is also not very accurate. Therefore one must develop a new method, which is nearly as accurate as the supermolecule approach (which, for larger interacting molecules, is not feasible because of the prohibitively large amount of computer time), can treat an arbitrary number of interacting molecules (or linear chains) at medium intermolecular (interchain) distances, and is much faster than perturbation theory (PT). This problem was solved at Erlangen in a series of papers for both molecules and linear chains. 

To overcome this problem, we can use double perturbation theory. The zeroth-order Hamiltonian is then normally some one-particle operator, such as the Fock operator, and the two perturbations are correlation and relativity. Formally, we write the Hamiltonian as... 

In each expression, the first two terms are equivalent, due to the interchange theorem of double perturbation theory (Dalgamo and Stewart 1956). This may be verified by making use of the equations for the two perturbations— but note that it is true only for exact wave functions. 

Krivdin and co-workers ° applied a double perturbation theory (DPT) at the second order level of approximation formalism to examine the dihedral angle dependence of the Fermi-contact (FC) contribution to nuclear spin-spin coupling constants. The authors have derived an analytical expression relating the FC term of /(H,H) across the aliphatic single carbon-carbon bond to the dihedral angle describing inner rotation around the C-C bond in the ten-electron ten-orbital moiety H-C-C-H. In particular, the authors have shown that extrema of H,H) are observed at q> = nn [n = 0, 1, 2,...), which provides a theore-... 

If one considers the magnetic fields caused by two nuclei, by means of double perturbation theory, one is led to the theory of (indirect) nuclear spin coupling. One deals with somewhat more complicated operators, but there is no gauge problem, since there is a natural gauge for either nucleus. 

The Judd-Ofelt model of the/ —s- / transitions extended by the third-order contributions to the transition amplitude is based on the double perturbation theory applied for the following Hamiltonian,... 

Following the standard procedure of double perturbation theory, the transition amplitude is now determined by the following contributions. 

If the R-ELHAV expansion is able to effectively reproduce the part of the interaction energy missing in Sr-srs in a low-order treatment (as will be seen in Sect. 4, this is the case), it is desirable to extend this theory to obtain a perturbation expansion that starts from 0o and takes both Vp and Vt into account. For this purpose, the most straightforward idea is to develop some double perturbation expansion in Vp and Vt which treats these two perturbations in an SRS-like and ELHAV-like way, respectively. The formulae defining the wave function corrections in this double perturbation theory can be obtained by expanding the equation... 

Double perturbation theory calculations are very time-consuming if one wants to go to high orders. Therefore, it would be highly advantageous to combine Vp and Vt in a single pertiubation theory, related to Eqs. 69-72 as closely as possible. We will present such a theory - the R-SRS+ELHAV method [72] - in the next subsection. 

To derive perturbation equations for a theory that uses < o as the zero-order function, takes into account both Vp and Vt, and avoids the comphcations of a double perturbation theory framework, we start from the following equation [72],... 

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