The pair function. Electron correlation

We now turn to 2-electron properties, which are completely determined by the function r(xi, X2)- It may be shown (McWeeny and Mizuno, 1961) that for a state with definite spin (quantum numbers 5, M) r is a six-component quantity. Let us write explicitly the off-diagonal components of r (p. 122), to avoid any ambiguity, putting 

This notation resembles that used in (5.3.12), but the subscripts after the commas now indicate the spin factors for the primed variables. We refer to the spinless factor in the first term as the aa, aa component, that in the second term as the nr/3, cr/5 component etc., and it follows at once, on integrating over spins, that i7 is the sum of four components only. With the usual brief notation for diagonal elements, namely 

Let us write down the pair-function components for the simple case of a system described by one determinant of spin-orbitals, with orbital factors t B, , pR,. The 1-electron density matrix has the form (5.3.12), namely (adding primes since off-diagonal elements will be needed) 

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. 

The proper description of correlation has been one of the main obstacles to progress in quantum chemistry, and it is therefore important to be familiar with its general features. If we suppose that W is an exact many-electron wavefunction then the following results hold (McWeeny, 1960). 

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