Correlated pair functions

Perhaps the most constructive contribution to solve this problem is the study of pair-densities in hydrides of elements from lithium to fluorine by Bader and Stephens 116). One major advantage is that it is not a predetermined working hypothesis, but a quantitative test of validity of Lewis structures, which may be appropriate or not. Bader and Stephens conclude that spatially localized pairs are satisfactory in LiH, BeH2, BH3 and BHT that CH4 (among all molecules) is a border-line case, and that intra-correlated pair-functions would fail to recover a major fraction of the correlation energy in NH3, H20 and HF. It is also true for the neon atom and for N2 and F2 that most of the correlation energy comes from correlation between even the best optimized electron-pairs. There is no physical basis for the view that there are two separately localized pairs of non-bonded electrons in HzO. 

It should be not l that on the basis of an algebraic (geometrical) model there is no a priori justification for an ansatz like Equation 43. The most general form of the two-variable function J(AB) should contain a correlated pair-function 7(AB). 

From these results we can obtain a clear picture of the kind of electron correlation recognized in the 1-determinant approximation. Thus, for electrons of different spin, the form of Uap shows that the probability of two volume elements being occupied simultaneously by electrons, the first spin up the second down, is just the product of the probabilities of each of the two events occurring independently, i.e. without reference to the other. We say there is no correlation between the positions of electrons of opposite spin. This absence of correlation in the 1-determinant approximation is clearly a defect, since electrons repel each other and we should expect the probability of finding two of them close together to be reduced below the value for independent particles. Electrons of like spin (-l- ) are, however, described by a correlated pair function, namely Tlaa, and this clearly vanishes for rz— ri, since then the two terms become equal and cancel exactly. This special type of correlation prevents two electrons of like spin being found at the same point in space, and applies whenever the particles are fermions with antisymmetric wavefunctions it is described as Fermi correlation. 

We shall not consider in detail the many methods that have been devised for the actual computation of correlated pair functions there has been a great proliferation of techniques and approximations (see e.g. Hurley (1976) for a full survey of the earlier work, and Meyer (1977), Kutzelnigg (1977) and Ahlrichs (1979,1983) for reviews of further developments), but the underlying principles are quite simple. The most convenient starting point is (9.3.4), which, being formally exact, becomes... 

In all approaches to the calculation of correlated pair functions the computational machinery required is heavy, and applications in the forseeable future are likely to be limited to few-electron systems. A recent review has been given by Ahlrichs and Scharf (1987). 

In order to calculate the MP2 correlation energy from the explicitly correlated pair functions, Eq. (11), we start from a Hylleraas functional ... 

Explicitly correlated pair functions can thus be represented as excitations into the complete virtual orbital space, however, contracted with the integrals over the correlation factor ... 

Finally, the closure relations for the inhomogeneous pair functions must be chosen. The PY approximation for the fluid-fluid direct correlation function presumes that its blocking part vanishes. This implies that c, ii(/,y) = 0, and... 

Figure4.7 Relativistic bond contractions A re for Au2 calculated in the years from 1989 to 2001 using different quantum chemical methods. Electron correlation effects Acte = te(corn) — /"e(HF) at the relativistic level are shown on the right hand side of each bar if available. From the left to the right in chronological order Hartree-Fock-Slater results from Ziegler et al. [147] AIMP coupled pair functional results from Stbmberg and Wahlgren [148] EC-ARPP results from Schwerdtfeger [5] EDA results from Haberlen and Rdsch [149] Dirac-Fock-Slater...

Kofraneck and coworkers24 have used the geometries and harmonic force constants calculated for tram- and gauche-butadiene and for traws-hexatriene, using the ACPF (Average Coupled Pair Functional) method to include electron correlation, to compute scaled force fields and vibrational frequencies for trans-polyenes up to 18 carbon atoms and for the infinite chain. 

The symmetry requirements and the need to very effectively describe the correlation effects have been the main motivations that have turned our attention to explicitly correlated Gaussian functions as the choice for the basis set in the atomic and molecular non-BO calculations. These functions have been used previously in Born-Oppenheimer calculations to describe the electron correlation in molecular systems using the perturbation theory approach [35 2], While in those calculations, Gaussian pair functions (geminals), each dependent only on a single interelectron distance in the exponential factor, exp( pr ), were used, in the non-BO calculations each basis function needs to depend on distances between aU pairs of particles forming the system. 

In this form, the w-particle correlated Gaussian is a product of n orbital Gaussians centered at the origin of the coordinate system and n n — l)/2 Gaussian pair functions (geminals). 

To study the structure of the exchange-correlation energy functional, it is useful to relate this quantity to the pair-correlation function. The pair-correlation function of a system of interacting particles is defined in terms of the diagonal two-particle density matrix (for an extensive discussion of the properties of two-particle density matrices see [30]) as... 

Measurement of the cross-relaxation rate in either the laboratory or the rotating frame suffices. This assumes that the respective correlation time function, /(tc), is the same for all spin pairs. If the calibration of the crossrelaxation rate is not feasible (e.g., no suitable spin pair, spectral overlap, motion not isotropic, molecule has internal mobility), the value of /(tc) for each spin pair (or for a group of selected pairs), i.e., the correlation time, must be explicitly determined. 

Pj, is a projection operator ensuring the proper spatial symmetry of the function. The above method is general and can be applied to any molecule. In practical application this method requires an optimisation of a huge number of nonlinear parameters. For two-electron molecule, for example, there are 5 parameters per basis function which means as many as 5000 nonlinear parameters to be optimised for 1000 term wave function. In the case of three and four-electron molecules each basis function contains 9 and 14 nonlinear parameters respectively (all possible correlation pairs considered). The process of optimisation of nonlinear parameters is very time consuming and it is a bottle neck of the method. 

As the temperature stimulation is switched off, the static kinetics is governed by equation (4.1.40) with the initial distribution function y(r) from equation (4.2.11). However, all attempts [102] to describe in such a way the experimental tunnelling luminescence decay for F and in KBr (Fig. 4.18) were unsuccessful. Both this observation and the absence of the plateau of 7(f) during the temperature stimulation, characteristic for the quasi-steady states, argue that the tunnelling recombination takes place in correlated pairs. This is in line with the conclusion [107] that for ordinary defect concentrations 1016 cm-3 (X-ray sample excitation for minutes) and the time 105 s the slope is dose-independent but 7(f) oc dose [95]. 

Gdanitz and Ahlrichs devised a simpler variant of CPF, the averaged coupled-pair functional (ACPF) approach [30]. This produces results very similar to CPF for well-behaved closed-shell cases and is completely invariant to a unitary transformation on the occupied MOs. Its big advantage is that it can be cast in a multireference form. Multireference ACPF is probably the most sophisticated treatment of the correlation problem currently available that can be applied fairly widely, although it can encounter difficulties with the selection of reference spaces, as discussed elsewhere. 

Given the formal similarity between the Hamiltonians defined in Eqs.(2) and (5), it follows that the ground-state energy, E, is given in terms of a universal functional of the pair (or n-particle) density, n(x), which attains its minimum value for the exact pair density. Furthermore, within a Kohn-Sham scheme, the form of this functional is identical to the functional of ordinary DFT but is given in terms of the correlated pair density. The details of this derivation... 

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