Pair, density

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... 

Figure 3-10. Relative electron crowding for different orbital pair densities.

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. 

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 o, and o2 simultaneously within two volume elements dr, and dr2, while the remaining N-2 electrons have arbitrary positions and spins. The quantity which contains this information is the pair density p2(xl5 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.,... 

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(x1 x2) given above according to... 

It should be obvious that the diagonal elements of this matrix (i. e., for x, = x and x2 = x2) bring us back to our pair density p2(x1 x2) defined above. If we now look at the special situation that x, = x2, that is the probability that two electrons with the same spin are found within the same volume element, we find that... 

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 can easily be shown that the HF approximation discussed in Chapter 1 does include the Fermi-correlation, but completely neglects the Coulomb part. To demonstrate this, we analyze the Hartree-Fock pair density for a two-electron system with the two spatial orbitals ()> and < )2 and spin functions a and o2... 

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... 

Perdew, J. P., Ernzerhof, M., Burke, K., Savin, A., 1997, On-Top Pair-Density Interpretation of Spin Density Functional Theory, With Applications to Magnetism , Int. J. Quant. Chem., 61, 197. 

The theory as presented so far is clearly incomplete. The topology of the density, while recovering the concepts of atoms, bonds and structure, gives no indication of the localized bonded and non-bonded pairs of electrons of the Lewis model of structure and reactivity, a model secondary in importance only to the atomic model. The Lewis model is concerned with the pairing of electrons, information contained in the electron pair density and not in the density itself. Remarkably enough however, the essential information about the spatial pairing of electrons is contained in the Laplacian of the electron density, the sum of the three second derivatives of the density at each point in space, the quantity V2p(r) [44]. 

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... 

Note that the bond order index defined by Mayer accounts for the covalent contribution to the bond (this is why of late it is often mentioned as shared electron pair density index, SEDI). As such, the index cannot be expected to produce the integer values corresponding to the Lewis picture if a bond has a significant ionic contribution. The bond order index defined in this way measures the degree of correlation of the fluctuation of electron densities on the two atoms in question [7]. 

An approximation of the lifetime in PS at RT using an electron-hole pair density equal to one pair per crystallite and the radiative recombination parameter of bulk silicon give values in the order of 10 ms [Ho3]. The estimated radiative lifetime of excitons is strongly size dependent [Sa4, Hi4, Hi8] and increases from fractions of microseconds to milliseconds, corresponding to an increase in diameter from 1 to 3 nm [Hy2, Ta3], as shown in Fig. 7.18. For larger crystallites a recombination via non-radiative channels is expected to dominate. The experimentally observed stretched exponential decay characteristic of the PL is interpreted as a consequence of the randomness of the porous skeleton structure [Sa5]. 

Fig. 6. A) Orbital representation B) Electron densities q ) C) Electron pair densities Q ri, ) of the singlet and triplet functions of two electrons in a one-dimensional box. The two axes are n and j-2...

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... 

Ernzerhof M, Burke K, Perdew JP, Long-range asymptotic behavior of ground-state wavefunc-tions, one-matrices, and pair densities, submitted to J Chem Phys... 

Perdew JP, Ernzerhof M, Burke K, Savin A. On-top pair-density interpretation of spin-density functional theory, with applications to magnetism to appear in Int. J. Quantum Chem. 

The diagonal elements D(ri ri) and D(rir2 rir2) are the electron density and pair density, respectively. Then Eq. (30) can be rewritten... 

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... 

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