The perturbation expansion

We start from the Dirac equation with modified metric (17), which contains as a natural expansion parameter [12]  

Expanding ip and E in powers of c (actually only even powers arise) = 00 + -f. . .  

We have explained in the Introduction why we prefer to count orders of rather than of c , although only even powers arise, such that one might as well regard as the expansion parameter. 

We take the scalar product from the left with tpo and use the hermiticity of Do — Dq5o as well as the normalization of ipo. 

These results are independent of the normalization condition (sec. 4.2), only the trivial normalization of ipQ has been used. 

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We can then identify each tenn in tlie expansion with one or more tenns in the perturbative expansion of... 

In the perturbation expansion [69] the first-order contribution to 4 consists of two terms which read... 

The MPn method treats the correlation part of the Hamiltonian as a perturbation on the Hartree-Fock part, and truncates the perturbation expansion at some order, typically n = 4. MP4 theory incorporates the effect of single, double, triple and quadruple substitutions. The method is size-consistent but not variational. It is commonly believed that the series MPl, MP2, MP3,. .. converges very slowly. 

The equilibrium structure is revised to the MP2/6-31G level of theory. For operational reasons, all electrons are used in the perturbation expansion. 

As shown in Table 4.2, the most important contribution to the energy in a Cl procedure comes from doubly excited determinants. This is also shown by the perturbation expansion, the second- and third-order energy corrections only involve doubles. At fourth order the singles, triples and quadruples enter the expansion for the first time. This is again consistent with Table 4.2, which shows that these types of excitation are of similar importance. 

We shall, however, proceed somewhat differently, and obtain directly the perturbation expansion for GA. Consider the case x0 > x 0 using the fret that... 

The second procedure, several aspects of which are reviewed in this paper, consists of directly computing the asymptotic value by employing newly-developed polymeric techniques which take advantage of the one-dimensional periodicity of these systems. Since the polarizability is either the linear response of the dipole moment to the field or the negative of the second-order term in the perturbation expansion of the energy as a power series in the field, several schemes can be proposed for its evaluation. Section 3 points out that several of these schemes are inconsistent with band theory summarized in Section 2. In Section 4, we present the main points of the polymeric polarization propagator approaches we have developed, and in Section 5, we describe some of their characteristics in applications to prototype systems. 

Why—and when—does the Fukui function work The first restriction—already noted in the original 1984 paper—is that the Fukui function predicts favorable interactions between molecules that are far apart. This can be understood because when one uses the perturbation expansion about the separated reagent limit to approximate the interaction energy between reagents, one of the terms that arises is the Coulomb interaction between the Fukui functions of the electron-donor and the electron-acceptor [59,60],... 

If we now turn back to the perturbation expansion (41) for P iP y 0> we may obviously express the r.h.s. as the sum of an arbitrary number of diagonal fragments preceded by a destruction fragment.. We have indeed identically ... 

Moreover, using the definition (10) of the streaming operator Sf and the perturbation expansion of its Laplace transform Rn(z) (see 28), it is easy to show that... 

Omitting all discussion of the mathematical properties and the subtleties with regard to the continuum of final state energies, we will hop to the perturbation expansion in three short equations. The initial state describing the unperturbed reactant (DA) system describes the solution to the zeroth order Schrodinger equation ... 

It is an important consequence that certain non-zero off-diagonal elements (Lagran-gian multipliers) Zij, Zab as extra terms enter the perturbation expansion at third and higher orders. 

Nakatsuji and Yasuda [56, 57] focused on the term appearing in A, which according to the perturbative expansion could be interpreted as a linked diagram of two A elements. In analogy to the Dyson equation, they proposed to estimate the A with a procedure whose main step, expressed in a spin-orbital basis, may be written... 

We expand the Hamiltonian and the IBCj in terms of a perturbation parameter p. in the spirit of M0ller-Plesset perturbation theory [34]. Details are found in Ref. [25]. We need not worry about the particle rank to which we have to go, since this is fully controlled by the perturbation expansion. We limit ourselves to a closed-shell state, such that the zeroth order is simply closed-shell Hartree-Fock. 

We do get information on the second-order corrections to y from the perturbation expansion of the CSE t [25]. However, in order to evaluate (72) we need to know 2)pr, which is not yet known, when one needs it. One can construct 2)%- in... 

If one could solve Eq. (203) exactly for exact energy— provided that the reference function is n-representable (e.g., is a normalized Slater determinant). The unitary transformation preserves the n-representability. Equation (203) is an infinite-order nonlinear set of equations and not easy to solve. However, the perturbation expansion terminates at any finite order. We have [6,12]... 

The correlation energy, Ecorrelation, is approximated by the second and higher order terms in the perturbation expansion -... 

Perturbation solutions for two special cases, namely, for n = 1 and n = 2, have been presented by Meadley and Rahman [47], Results based on Eq. (162) and the exact values computed from the perturbation expansion are shown in Tables VIII and IX (for n = 1 and 2, respectively). The agreement is again within 5 to 10%. The expressions of Meadley and Rahman are not accurate for large values of the ratio Da X/Gr /2. However, Eq. (162) can be used without any such restriction. 

In fact, as we discuss in section 4 below, pJT coupling gives surfaces resembling the type-II Renner-Teller surfaces shown in Fig. 1, but separated by a finite energy difference. Many theoretical formulations of the pseudo-Jahn-Teller effect have been proposed. The simplest is perhaps the perturbative expansion originally due to Pearson (51)... 

As for basis-set convergence, triple- calculations at the MP2 and CCSD levels are provided for comparison to die double- results. For this particular property, the results for p-benzyne are not terribly sensitive to improvements in the Ilexibility of the basis set. In the pyridynium ion case, die CCSD results are also not very sensitive, but a large effect is seen at the MP2 level. This has more to do with the instability of the perturbation expansion than any intrinsic difference between the isoelectronic aryiies. 

The other area in which projection of an unrestricted result has received attention is projected Mpller-Plesset perturbation theory, the PUMPn methods [31], where n is the order of the perturbation theory. In cases in which the UHF approximation is a poor starting point (considerable spin contamination, for example), the convergence of the MP perturbation expansion can be slow and/or erratic. The PUMP methods apply projection operators to the perturbation expansion, although usually not full projection but simply annihilation of the leading contaminants. This approach has met with mixed success again, it represents a rather expensive modification to a technique that was originally chosen partly for its economy — seldom a recipe for success. 

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