Double perturbation

The inclusion of the S W Hamiltonian leads to a system involving a double peituibation (137), namely, the electron-electron interaction and. Thus the reference state will be expressed as a double perturbed wavefunction... 

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 ... 

K. Szalewicz and B. Jeziorski. Symmetry adapted, double perturbation analysis of intramolecular correlation effects in weak intermolecular interactions. Molec. Phys., 38 191, 1979. 

More recently, Caves and Karplus71 have used diagrammatic techniques to investigate Hartree-Fock perturbation theory. They developed a double perturbation expansion in the perturbing field and the difference between the true electron repulsion potential and the Hartree-Fock potential, V. This is compared with a solution of the coupled Hartree-Fock equations. In their interesting analysis they show that the CPHF equations include all terms first order in V and some types of terms up to infinite order. They propose an alternative iteration procedure which sums an additional set of diagrams and thus should give results more accurate than the CPHF scheme. Calculations on Ha and Be confirmed these conclusions. 

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-... 

A general approach to the intramonomer correlation problem is known as the many-electron (or many-body) SAPT method88,141 213-215. In this method the zeroth-order Hamiltonian H0 is decomposed as H0 = F + W, where F = FA + FB is the sum of the Fock operators, FA and FB, of monomer A and B, respectively, and W is the intramonomer correlation operator. The correlation operator can be written as W = WA + WB, where Wx = Hx — Fx, X = A or B. The total Hamiltonian can be now be represented as H = F + V + W. This partitioning of H defines a double perturbation expansion of the wave function and interaction energy. In the SRS theory the wave function is obtained by expanding the parametrized Schrodinger equation as a power series in and A,... 

The expansion ofEqs. (1-166) and (1-167) leads to the so-called double perturbation expansion of the interaction energy,... 

CCSD(T) Coupled cluster with singles and doubles (+ perturbative triplets)... 

The details of SAPT are beyond the scope of the present work. For our purposes it is enough to say that the fundamental components of the interaction energy are ordinarily expanded in terms of two perturbations the intermonomer interaction operator and the intramonomer electron correlation operator. Such a treatment provides us with fundamental components in the form of a double perturbation series, which should be judiciously limited to some low order, which produces a compromise between efficiency and accuracy. The most important corrections for two- and three-body terms in the interaction energy are described in Table 1. The SAPT corrections are directly related to the interaction energy evaluated by the supermolecular approach, Eq.(2), provided that many body perturbation theory (MBPT) is used [19,28]. Assignment of different perturbation and supermolecular energies is shown in Table 1. The power of this approach is its open-ended character. One can thoroughly analyse the role of individual corrections and evaluate them with carefully controlled effort and desired... 

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 form [Eq. (3)] of the perturbation operator points out that formally we obtain a double perturbation expansion with the two-electron V2 and one-electron Vi perturbations. However, in the case of a Hartree-Fock potential the one-electron part of the perturbation is exactly canceled by some terms of the two-electron part. This becomes more transparent when we switch to the normal product form of the second-quantized operators2-21 indicated by the symbol. ... We define normal orders for second-quantized operators by moving all a ( particle annihilation) and P ( hole annihilation) operators to the right by virtue of the usual anticommutation relations [a b]+ = 8fl, [i j] = 8y since a 0) = f o) = 0. Then... 

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

Quadratic terms in die property expansions are considered to be first-order in electrical anharmonicity, cubic terms are taken to be second-order, etc. Similarly, cubic terms in the vibrational potential are considered to be first-order in mechanical anharmonicity, quartic terms are second-order, and so forth. The notation (n, m) is used hereafter for the order of electrical (n) and mechanical (m) anharmonicity whereas the total order (n -I- m) is denoted by I, II,. Although our definition of orders is reasonable other choices are possible. Two key questions are (1) How important are anharmonicity contributions to vibrational NLO properties and (2) What is the convergence behavior of the double perturbation series in electrical and mechanical anharmonicity Both questions will be addressed later. Here we note that compact expressions, complete through order II in electrical plus mechanical anharmonicity, have been presented [19]. The formulas of order I contain either cubic force constants or second derivatives of the electrical properties with respect to the normal coordinates. Depending upon the level of calculation at least one order of numerical differentiation is ordinarily required to determine these anharmonicity parameters. For electrical properties, the additional normal coordinate derivative may be replaced by an electric field derivative using relations such as d p./dQidQj = —d E/dldAj.ACd, = —dk,/rjF where F is the field and k j is... 

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. 

Finally, we would like to mention Pople s (1986) recent work this treats the derivatives of the (MP) correlation energy as a double perturbation problem, with respect to a physical perturbation (e.g. nuclear coordinate change) and a non-physical perturbation (electron correlation). This provides a unified theory for the treatment of geometry and property derivatives at the correlated level. 

In practical applications of the sapt approach to interactions of many-elect ron systems, one has to use the many-body version of sapt, which includes order-by-order the intramonomer correlation effects. The many-body SAPT is based on the partitioning of the total Hamiltonian as H = F+V+W, where the zeroth-order operator F = Fa + Fb is the sum of the Fock operators for the monomers A and B. The intermolecular interaction operator V = H — Ha — Hb is the difference between the Hamiltonians of interacting and noninteracting systems, and the intramonomer correlation operator W = Wa + Wb is the sum of the Moller-Plesset fluctuation potentials of the monomers Wx — Hx — Fx, X — A or B. The interaction operator V is taken in the non-expanded form, i.e., it is not approximated by the multipole expansion. The interaction energy components of Eq. (1) are now given in the form of a double perturbation series,... 

Double perturbation 198 Double zeta basis set 160 Dummy atom 176 Dynamic reactivity theories 280... 

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... 

---