The Pair Density. Orbital-dependent Exchange-correlation Functionals

5 The Pair Density. Orbital-dependent Exchange-correlation Functionals 

Density-functional theory, even with rather crude approximations such as LDA and GGA, is often better than Hartree-Fock LDA is remarkably accurate, for instance, for geometries and frequencies, and GGA has also made bond energies quite reliable. Therefore, the aura of mystery appeared around DFT (see discussion of this by Baerends and Gritsenko [367]). The simple truth is not that LDA/GGA is particularly good, but that Hartree-Fock is rather poor in the two-electron chemical-bond description. This becomes clear when one considers the statistical two-electron distribution, which is usually cast in terms of the exchange-correlation hole the decrease in probabihty to find other electrons in the neighborhood of a reference electron, compared to the (unconditional) one-electron probabihty distribution [337]. 

The concept of electron density (7.1), which provides an answer to the question how hkely is it to find one electron of arbitrary spin within a particular volume element while all other electrons may be anywhere can be extended to the probability of finding not one but a pair of two electrons with spins Ti and ct2 simultaneously within two volume elements dr and dr2, while the remaining N — 2 electrons have arbitrary positions and spins. The quantity that contains this information is the pair-electron density /02( i, 2) which is defined as 

The pair-electron density actually contains all the information about electron correlation. The pair density is a nonnegative quantity, symmetric in the coordinates and normalized to the total number of nondistinct electron pairs N N — 1). Obviously, if electrons would not interact, the probability of finding one electron at a particular point of coordinate-spin space would be completely independent of the position and spin of the second electron and it would be possible that both electrons are simultaneously found in the same volume element. In this case, pair density would reduce to a simple product of the individual probabilities, i.e. 

The N N — 1) factor enters because the particles are identical. From the antisymmetry of the many-elect ron wavefunction W xi.xn) it follows that P2 ( 1, 2) = —p2(x2,x ), i. e. the probability of finding two electrons with the same spin at the same point in space is zero. 

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