Variational perturbation theory

From the definition of molecular properties (29) and the above discussion it is clear that the proper theoretical framework for the description of molecular properties is variational perturbation theory. An excellent presentation of this approach is provided by Helgaker and Jprgensen [9]. Although they focus on the calculation of geometrical derivatives, the methods and proofs presented are straightforwardly extended to quasienergy derivatives, as shown by Christiansen eta/. [13]. 

In the remainder of this section we shall employ variational perturbation theory to derive general expressions for molecular properties to various orders. We will limit ourselves to variational wave functions 0) = 0(A)), in which A. is a vector containing the variational parameters. Using the form (20) of the perturbation part V t) of the total Hamiltonian, the time-averaged quasienergy can be written as 

The variational parameters X are determined from the variational condition 

We shall require that this variational condition is fulfilled at all perturbation strengths . This establishes the variational parameters as functions of the perturbation strengths A = A (fi), which is a key ingredient in the following derivations. For simplicity we choose A (0) = 0. 

By induction one easily establishes the following relation for total derivatives of the quasienergy to order n in the perturbation strengths 

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Wong K-Y, Gao J (2007) An automated integration-free path-integral method based on Kleinert s variational perturbation theory. J Chem Phys 127(21) 211103... 

Kleinert H (2004) Path integrals in quantum mechanics, statistics, polymer physics, and financial markets. 3rd edition. World Scientific Singapore River Edge, NJ, p xxvi, 1468 p. For the quantum mechanical integral equation, see Section 1.9 For the variational perturbation theory, see Chapters... 

E. Brandas, D.A. Micha, Variational Methods in the Wave Operator Formalism. Applications in Variation Perturbation Theory and the Theory of Energy Bounds, J. Math. Phys. 13 (1972) 155. 

For the evaluation of probabilities for spin-forbidden electric dipole transitions, the length form is appropriate. The velocity form can be made equivalent by adding spin-dependent terms to the momentum operator. A sum-over-states expansion is slowly convergent and ought to be avoided, if possible. Variational perturbation theory and the use of spin-orbit Cl expansions are conventional alternatives to elegant and more recent response theory approaches. 

Equation (57) can be solved for c. The result depends on P. If the basis for the Cl is complete, one gets c = 0/0, i.e. c is undetermined. If we consider an atomic state and the Cl basis is saturated up to some angular momentum quantum number / — 1, then the first and the last term in (57) — which involve identical matrix elements — go as (/ + ) 4, the second one as (l + ) 6 [11]. For a sufficiently large value of / the second term is negligible and from the variational treatment we get c = as required by the cusp condition. This exact result appears here as a consequence of variation-perturbation theory. 

Frequency-dependent higher-order properties can now be obtained as derivatives of the real part of the time-average of the quasi-energy W j- with respect to the field strengths of the external perturbations. To derive computational efficient expressions for the derivatives of the coupled cluster quasi-energy, which obey the 2n-(-1- and 2n-(-2-rules of variational perturbation theory [44, 45, 93], the (quasi-) energy is combined with the cluster equations to a Lagrangian ... 

Montgomery used Variational Perturbation Theory [46] to study the Is, 2p and 3d states of the CHA, obtaining energies that are very close to those reported by Goldman and Joslin [40]. The WKB method was first used by... 

The generalized perturbation theory expressions presented in this section for systems described by the homogeneous Boltzmann equation (excluding Section V,B,2) are in the form proposed by Stacey (40, 41). Had we assumed that the overall alteration in the reactor retains criticality, we would have achieved the Usachev-Gandini version of GPT. Stacey s version is often associated (41, 46, 48, 62) with the variational perturbation theory as distinguished from the GPT of Usachev-Gandini. Does the variational approach provide a different perturbation theory than the GPT derived (35,39) from physical considerations Is one of these versions of perturbation theory more general or more accurate than the other What does the term GPT stand for ... 

Kleinert, H., Pelster, A., Putz, M. V. (2002). Variational perturbation theory for markov processes. Physical Review E 65, 066128/1-7 (arXiv cond-mat/0202378). 

In actual practice, the perturbation expressions given above are better suited for displaying the structural features of the theory than for use in large-scale calculations. For nonrelativistic calculations, the most efficient approach to the ab initio calculation of NMR parameters has been variational perturbation theory (Helgaker et al. 1999). This, or equivalent approaches such as second-order propagator theory, will probably also turn out to be the optimal choice for the relativistic case. 

O. Christiansen, P. J0rgensen, and C. Hattig, Response Functions from Fourier Component Variational Perturbation Theory Applied to a Time-Averaged Quasienergy, Int. J. Quantum Chem. 68, 1 (1998). 

Christiansen, O, Jprgensen, P., and Hattig, C. (19986). Response functions from Fourier component variational perturbation theory applied to a time-averaged quasienergy. Int. J. Quantum Chem., 68, 1-52. 

For present purposes it is more useful to concentrate on other approaches, which start from the finite-basis form of the linear variation method. In many forms of variation-perturbation theory, exact unperturbed eigenfunctions are not required and the partitioning of the Hamiltonian into two terms is secondary to a partitioning of the basis. At the same time, as we shall see, it is possible to retrieve the equations of conventional perturbation theory by making an appropriate choice of basis. 

There are many ways of dealing with the small terms, most of them perturbational in character and presupposing knowledge (at least in principle) of exact eigenfunctions of Hq. We develop only one approach, based on the variation-perturbation theory of Section 2.5, which has the merits of simplicity, flexibility and generality and does not require knowledge of exact unperturbed wavefunctions. This approach also lends itself well to the physical interpretation of the mode of interaction of the small terms. 

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