The Pair Density

The concept of electron density, which provides an answer to the question how likely is it to find one electron of arbitrary spin within a particular volume element while all other electrons may be anywhere can now be extended to the probability of finding not one but a pair of two electrons with spins Oj and o2 simultaneously within two volume elements dFj and dr2, while the remaining N-2 electrons have arbitrary positions and spins. The quantity which contains this information is the pair density p2(xh x2), which is defined as 

This quantity is of great importance, since it actually contains all information about electron correlation, as we will see presently. Like the density, the pair density is also a non-negative quantity. It is symmetric in the coordinates and normalized to the total number of non-distinct pairs, i. e., N(N-l).8 Obviously, if electrons were identical, classical particles that do not interact at all, such as for example billiard balls of one color, 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. Since in our model we view electrons as idealized mass points with no volume, this would even include the possibility that both electrons are simultaneously found in the same volume element. In this case the pair density would reduce to a simple product of the individual probabilities, i.e., 

The (N-l)/N factor enters because the particles are identical and not distinguishable. Pictorially speaking, the probability that any of the N electrons is at x, is given by p(xj). The probability that another electron is simultaneously at x2 is only (N-l)/N p(x2) because the electron at xt cannot at the same time be at x2 and the probability must be reduced accordingly. 

However, billiard balls are a pretty bad model for electrons. First of all, as discussed above, electrons are fermions and therefore have an antisymmetric wave function. Second, they are charged particles and interact through the Coulomb repulsion they try to stay away from each other as much as possible. Both of these properties heavily influence the pair density and we will now enter an in-depth discussion of these effects. Let us begin with an exposition of the consequences of the antisymmetry of the wave function. This is most easily done if we introduce the concept of the reduced density matrix for two electrons, which we call y2. This is a simple generalization of p2(x, x2) given above according to 

When going from p2 to y2 we prime those variables in the second factor which are not included in the integration. The two sets of independent and continuous variables, i. e., x, x2 and Xj, x2, define the value of y2 (x, x2 iq, x2) which is the motivation for calling this quantity a matrix (for more information on reduced density matrices see in particular Davidson, 1976, or McWeeny, 1992). If we now interchange the variables x, and x2 (or xj and x2), y2 will change sign because of the antisymmetry of F  

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In the ease of intermediate topography, they assume that there is a eertain eharaeteristie (for a given surfaee) length, r or, whieh defines the size of small domains eharaeterized by the same value of adsorption energy, and write the pair density distribution as... 

In close relation to the fluctuations, one may introduce the correlation functions. The pair density distribution function for fluid particles (ri, r2) is defined as the average over all realizations of the matrix structure of the... 

In this chapter we make first contact with the electron density. We will discuss some of its properties and then extend our discussion to the closely related concept of the pair density. We will recognize that the latter contains all information needed to describe the exchange and correlation effects in atoms and molecules. An appealing avenue to visualize and understand these effects is provided by the concept of the exchange-correlation hole which emerges naturally from the pair density. This important concept, which will be of great use in later parts of this book, will finally be used to discuss from a different point of view why the restricted Hartree-Fock approach so badly fails to correctly describe the dissociation of the hydrogen molecule. 

Next, let us explore the consequences of the charge of the electrons on the pair density. Here it is the electrostatic repulsion, which manifests itself through the l/r12 term in the Hamiltonian, which prevents the electrons from coming too close to each other. This effect is of course independent of the spin. Usually it is this effect which is called simply electron correlation and in Section 1.4 we have made use of this convention. If we want to make the distinction from the Fermi correlation, the electrostatic effects are known under the label Coulomb correlation. 

It is now convenient to express the influence of the Fermi and Coulomb correlation on the pair density by separating the pair density into two parts, i. e. the simple product of independent densities and the remainder, brought about by Fermi and Coulomb effects and accounting for the (N-l)/N normalization... 

Under the conditions of maximum localization of the Fermi hole, one finds that the conditional pair density reduces to the electron density p. Under these conditions the Laplacian distribution of the conditional pair density reduces to the Laplacian of the electron density [48]. Thus the CCs of L(r) denote the number and preferred positions of the electron pairs for a fixed position of a reference pair, and the resulting patterns of localization recover the bonded and nonbonded pairs of the Lewis model. The topology of L(r) provides a mapping of the essential pairing information from six- to three-dimensional space and the mapping of the topology of L(r) on to the Lewis and VSEPR models is grounded in the physics of the pair density. 

Let us now look at the one particle density and compare it with the pair density x2(tx). We have to examine now the two possibilities, both electrons with the same spin or with different spin. In Figure 20.4, we have the pair density for the... 

We follow here the notation of Thakkar and Smith [15], and evaluated this and the other distribution functions mentioned below using the formulas given by them. The prefactor 2 in the definition of D(ri) causes it to describe the pair density contributions of the entire electron distribution (rather than that of one of the two electrons). 

In order to understand why approximate functionals yield accurate exchange-correlation energies, we decompose the exchange-correlation energy as follows [37]. We define the pair density of the inhomogeneous system as... 

This long-range correlation effect shows up in both the first-order density matrix and the exchange-correlation hole for finite systems [19]. We concentrate here on the exchange-correlation hole. The general asymptotic form of the pair density is then... 

There are only a few studies of 2-density functional theory for Q > 2 [2, 3, 6]. Most studies have concentrated on the pair density, or 2-density functional theory. Excepting the fundamental work of Ziesche, early work in 2-density functional theory focused on a differential equation for the pseudo-two-electron wavefunction [7-11] defined by... 

In retrospect, the importance of N-representability constraints on the pair density should have been clear from the very beginning. An A-representability... 

The importance of N-representability for pair-density functional theory was not fully appreciated probably because most research on pair-density theories has been performed by people from the density functional theory community, and there is no W-representability problem in conventional density functional theory. Perhaps this also explains why most work on the pair density has been performed in the first-quantized spatial representation (p2(xi,X2) = r2(xi,X2 xi,X2)) instead of the second-quantized orbital representation... 

The Slater hull constraints are not directly applicable to existing approaches to pair-density functional theory because they are formulated in the orbital representation. Toward the conclusion of this chapter, we will also address A-representability constraints that are applicable when the spatial representation of the pair density is used. 

Unfortunately, the A-representability constraints from the orbital representation are not readily generalized to the spatial representation. A first clue that the A-representability problem is more complicated for the spatial basis is that while every A-representable Q-density can be written as a weighted average of Slater determinantal Q-densities in the orbital resolution (cf. Eq. (54)), this is clearly not true in the spatially resolved formulation. For example, the pair density (Q = 2) of any real electronic system will have a cusp where electrons of opposite spin coincide but a weighted average of Slater determinantal pair densities,... 

There are useful two- and many-electron analogues of the functions discussed above, but when the Hamiltonian contains only one- and two-body operators it is sufficient to consider the pair functions thus the analogue of p(x x ) is the pair density matrix 7t(xi,X2 x i,x ) while that of which follows on identifying and integrating over spin variables as in (4), is H(ri,r2 r i,r2)- When the electron-electron interaction is purely coulombic, only the diagonal element H(ri,r2) is required and the expression for the total interaction energy becomes... 

Fig. 10. Plots of the pair density distribution versus distance for (a) rc = 0 (Smoluch-owski solution), (b) rc = 0.7 nm, (c) rc —0.7 nm. The distributions are shown for times of 10 12 to 10 6 s in decadic intervals. The calculations refer to a random initial condition and to the Smoluchowski boundary condition, i.e. p(ft,f) = 0, ft = 0.5 nm, D = 10 8 m2 s 1.

Equation (366) is a formal solution for the Laplace transform of the pair density [103]. This can be inverted to give a time-dependent form of P(r, t) t... 

J drfeSSp = j drfeSpet — J drA-(S>peq = whence the rate expression for the pair density is... 

For molecular pairs, the low-density limit P of the pair density operator is given by... 

Kinetic equations for the electron tunneling reactions in the case of non-pair distributions of reagents have been obtained [8-11]. Two methods have been used in the literature to obtain these equations. Both of them have been used earlier to describe the kinetics of energy transfer processes. These are the method of pair density and that of conditional concentrations. It has been shown [15] that these two different methods are, in fact, equivalent and lead to identical results. The detailed description of the pair density method can be found in refs. 3,5,7,13,16, and 33 and that of the method of conditional concentrations in refs. 5,8,15, and 17. 

When the concentrations of reagents have comparable values, it is necessary to pay attention to the correlation effect in the decay of different donors, i.e. to consider the fact that the spatial distribution of acceptors near the chosen donor can change as a result of the decay of the acceptors in the reactions with other donors neighbouring the chosen one. The rigorous derivation of kinetic equations with the consideration of such a correlation is, as far as we know, unavailable. The approximate description of the kinetics of a biomolecular electron tunneling reaction at n(t) = N t) can be given in terms of the pair density method with the help of eqn. (19) in which, however, N is not a constant quantity but depends on time in the same way as n(t), i.e. 

Following the ideas outlined in Ref. 28, let us consider the operator of the pair density... 

The operator (5.1) describes the pair density in the microscopic state. 

In statistical mechanics the pair density must be characterized by the average of the pair operator corresponding to the mixed state... 

Then the deviation of the microstate (5.1) and the mixed state (5.2), that is, the fluctuation of the pair density, is given by the operator... 

In Section IV, the correlation function of the fluctuations of the pair density was determined under the assumption of the weakening of the initial correlation between the pairs. This assumption is different from the total weakening of any initial correlation, because, in the bound-state case, we have correlations between the elementary particles of one bound pair. 

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