Perturbation theory formal development

In this context equations (50) and (53) can be considered forming a completely general perturbation theory for nondegenerate systems, although a recent development permits to extend the formalism to degenerate states [lej. 

In this section we will discuss perturbation methods suitable for high-energy electron diffraction. For simplicity, in this section we will be concerned with only periodic structures and a transmission diffraction geometry. In the context of electron diffraction theory, the perturbation method has been extensively used and developed. Applications have been made to take into account the effects of weak beams [44, 45] inelastic scattering [46] higher-order Laue zone diffraction [47] crystal structure determination [48] and crystal structure factors refinement [38, 49]. A formal mathematical expression for the first order partial derivatives of the scattering matrix has been derived by Speer et al. [50], and a formal second order perturbation theory has been developed by Peng [22,34],... 

E. Schrodinger, Ann. Phys. 80 (1926), 437. The quantal formalism substantially follows the classical method developed by Lord Rayleigh (Theory of Sound [1894]) and is commonly referred to as Rayleigh-Schrodinger perturbation theory. ... 

Btiilding on atomic studies using even-tempered basis sets, universal basis sets and systematic sequences of even-tempered basis sets, recent work has shown that molecular basis sets can be systematically developed until the error associated with basis set truncation is less that some required tolerance. The approach has been applied first to diatomic molecules within the Hartree-Fock formalism[12] [13] [14] [15] [16] [17] where finite difference[18] [19] [20] [21] and finite element[22] [23] [24] [25] calculations provide benchmarks against which the results of finite basis set studies can be measured and then to polyatomic molecules and in calculations which take account of electron correlation effects by means of second order perturbation theory. The basis sets employed in these calculations are even-tempered and distributed, that is they contain functions centred not only on the atomic nuclei but also on the midpoints of the line segments between these nuclei and at other points. Functions centred on the bond centres were found to be very effective in approaching the Hartree-Fock limit but somewhat less effective in recovering correlation effects. 

S. A. Rice I agree with Prof. Kohler that the use of a density matrix formalism by Wilson and co-workers generalizes the optimal control treatment based on wave functions so that it can be applied to, for example, a thermal ensemble of initial states. All of the applications of that formalism I have seen are based on perturbation theory, which is less general than the optimal control scheme that has been developed by Kosloff, Rice, et al. and by Rabitz et al. Incidentally, the use of perturbation theory is not to be despised. Brumer and Shapiro have shown that the perturbation theory results can be used up to 20% product yield. Moreover, from the point of view of generating an optimal control held, the perturbation theory result can be used as a first guess, for which purpose it is very good. 

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. 

Basically the perturbative techniques can be grouped into two classes time-local (TL) and time-nonlocal (TNL) techniques, based on the Nakajima-Zwanzig or the Hashitsume-Shibata-Takahashi identity, respectively. Within the TL methods the QME of the relevant system depends only on the actual state of the system, whereas within the TNL methods the QME also depends on the past evolution of the system. This chapter concentrates on the TL formalism but also shows comparisons between TL and TNL QMEs. An important way how to go beyond second-order in perturbation theory is the so-called hierarchical approach by Tanimura, Kubo, Shao, Yan and others [18-26], The hierarchical method originally developed by Tanimura and Kubo [18] (see also the review in Ref. [26]) is based on the path integral technique for treating a reduced system coupled to a thermal bath of harmonic oscillators. Most interestingly, Ishizaki and Tanimura [27] recently showed that for a quadratic potential the second-order TL approximation coincides with the exact result. Numerically a hint in this direction was already visible in simulations for individual and coupled damped harmonic oscillators [28]. 

Using a numerical decomposition of the spectral density which describes the coupling of the system to the environment allows one to develop TL and TNL non-Markovian QMEs. Using the hierarchical approach the results can be extended from second-order perturbation theory to higher orders to be able to study the convergence properties of the different approaches. As shown in the example for bosonic baths, the TL formalism shows numerically almost converged results. Actually, this numerical finding has been analytically proven... 

Taking these introductory comments as a motivation, we shall turn to the formalism of response theory. Response theory is first of all a way of formulating time-dependent perturbation theory. In fact, time-dependent and time-independent perturbation theory are treated on equal footing, the latter being a special case of the former. As the name implies, response functions describe how a property of a system responds to an external perturbation. If initially, we have a system in the state 0) (the reference state), as a weak perturbation V(t) is turned on, the average value of an operator A will develop in time according to... 

Alternatively, analytical methods can be used such as coupled Hartree-Fock (CHF), where the perturbed HF equations are solved directly. Using standard perturbation theory one can also develop a sum-over-states (SOS) formalism and write... 

The incorporation of correlation effects in calculations for periodic solids requires the use of a many-body formalism. Second order many-body perturbation theory, in its MP2 form, should provide the basis of an efficient computational approach to this problem. In particular, the local MP2 methods originally developed for large molecules can be adapted for the treatment of periodic solids. 

In Volume 5 of this series, R. J. Bartlett and J. E Stanton authored a popular tutorial on applications of post-Hartree-Fock methods. Here in Chapter 2, Dr. T. Daniel Crawford and Professor Henry F. Schaefer III explore coupled cluster theory in great depth. Despite the depth, the treatment is brilliantly clear. Beginning with fundamental concepts of cluster expansion of the wavefunction, the authors provide the formal theory and the derivation of the coupled cluster equations. This is followed by thorough explanations of diagrammatic representations, the connection to many-bodied perturbation theory, and computer implementation of the method. Directions for future developments are laid out. 

In order to develop a diagrammatic approach to perturbation theory, it is convenient to work within a second-quantized formalism. The expressions for Ho and V then become... 

Nunes and Gonze [153] have recently extended DFPT to static responses of insulating ciystals for any order of perturbation theory by combining the variation perturbation approach with the modern theory of polarization [154]. There are evident similarities between this formalism and (a) the developments of Sipe and collaborators [117,121,123] within the independent particle approximation and (b) the recent work of Bishop, Gu and Kirtman [24, 155,156] at the time-dependent Hartree Fock level for one-dimensional periodic systems. 

The unique practical way to describe the theory of the interacting fields is by perturbation theory (PT). This is most justified for QED where the coupling parameter is a (see Introduction). The perturbation theory is most conveniently developed within the Hamilton formalism. Therefore we present the total Hamiltonian for the interacting electron-positron and electromagnetic fields as... 

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