Perturbation theory representation

These equations provide a convenient and accurate representation of the themrodynamic properties of hard spheres, especially as a reference system in perturbation theories for fluids. 

The early approaches to this model used perturbative expansion for weak coupling [Silbey and Harris 1983]. Generally speaking, perturbation theory allows one to consider a TLS coupled to an arbitrary bath via the term where / is an operator that acts on the bath variables. The equations of motion in the Heisenberg representation for the a operators, 8c/8t = ih [H, d], have the form... 

The perturbation theory presented in Chapter 2 implies that orientational relaxation is slower than rotational relaxation and considers the angular displacement during a free rotation to be a small parameter. Considering J(t) as a random time-dependent perturbation, it describes the orientational relaxation as a molecular response to it. Frequent and small chaotic turns constitute the rotational diffusion which is shown to be an equivalent representation of the process. The turns may proceed via free paths or via sudden jumps from one orientation to another. The phenomenological picture of rotational diffusion is compatible with both... 

Note that, in contrast to other forms of intermolecular perturbation theory to be considered below, the NBO-based decomposition (5.8) is based on a full matrix representation of the supermolecule Hamiltonian H. All terms in (5.8) are therefore fully consistent with the Pauli principle, and both /7units(0, and Vunits(mt) are properly Hermitian (and thus, physically interpretable) at all separations. 

The actual calculation were performed through fourth order, the perturbation corrections in both the canonical and the localized representation for the interval -2 > P > - 10. The results for C H(, and CioT/jo were compared to those obtained by full CL It was shown (Kapuy et al., 1984 Kapuy et al, 1988) that the perturbation theory recovers a fraction of the total correlation correction. The localization correction are relatively small (compared to the canonical ones) in the localized representation. 

The following important conclusion can be drown, that the diagrammatic many-body perturbation theory can be used in a localized representation. 

The main advantage suggested by the use of the localized many-body perturbation theory (LMBPT) is that the local effects can be separated from the non-local ones. The summations in the corrections at a given order can be truncated. As to the practical applicability of the localized representation, a localization (separation) method, satisfying a double requirement is highly desired. Well-localized (separated) orbitals with small off-diagonal Lagrangianmultipliers are required (Kapuy etal., 1983). 

There are also some unexpected problems, related to the fact that the stationarity conditions do not discriminate between ground and excited states, between pure states and ensemble states, and not even between fermions and bosons. The IBQ give only information about the nondiagonal elements of y and the Xk, whereas for the diagonal elements other sources of information must be used. These elements are essentially determined by the requirement of w-representability. This can be imposed exactly to the leading order of perturbation theory. Some information on the diagonal elements is obtained from the lCSE,t, though in a very expensive and hence not recommended way. The best way to take care of -representability is probably via a unitary Fock-space transformation of the reference function, because this transformation preserves the -representability. 

The question, what conditions are to be fulfilled by a density matrix to be the image of a wave function, that is, to describe a real physical system is opened till today. The contracted Schrodinger-equations derived for different order reduced density matrices by H. Nakatsui [1] give opportunity to determine density matrices by a non-variational way. The equations contain density matrices of different order, and the relationships needed for the exact solutions are not yet known in spite of the intensive research activity [2,3]. Recently perturbation theory corrections were published for correcting the error of the energy obtained by minimizing the density matrix directly applying the known conditions of N-representability [4], and... 

Second-order perturbation theory has allowed us to write the band-structure energy of a perfect crystal as a sum over reciprocal lattice vectors, since the structure factor vanishes unless q = G (cf eqn (6.59)). Thus, within the reciprocal lattice representation we have... 

In the language of perturbation theory, the two orbitals will constitute the unperturbed system, the perturbation is the interaction between them, and the result of the interaction is what we wish to determine. The situation is displayed in Figure 3.1a. The diagram shown in Figure 3.1 b conveys the same information in the standard representations of PMO or orbital interaction theory. The two interacting but unperturbed systems are shown on the left and the right, and the system after the interaction is turned on is displayed between them. Our task is to find out what the system looks like after the interaction. Fet us start with the two unperturbed orbitals and seek the best MOs that can be constructed from them. Thus,... 

Equations 2.85 and 2.86 may be considered the Schrodinger representation of the absorption of radiation by quantum systems in terms of spectroscopic transitions between states i) and /). In the Schrodinger picture, the time evolution of a system is described as a change of the state of the system, as implemented here in the form of the time-dependent perturbation theory. The results hardly resemble the classical relationships outlined above, compare Eqs. 2.68 and 2.86, even if we rewrite Eq. 2.86 in terms of an emission profile. Alternatively, one may choose to describe the time evolution in terms of time-dependent observables, the Heisenberg picture . In that case, expressions result that have great similarity with the classical expressions quoted above as we will see next. 

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