Two-electron density functional

In our discussion of the electron density in Chapter 5, I mentioned the density functions pi(xi) and p2(xi,X2). I have used the composite space-spin variable X to include both the spatial variables r and the spin variable s. These density functions have a probabilistic interpretation pi(xi)dridii gives the chance of finding an electron in the element dri d i of space and spin, whilst P2(X], X2) dt] d i dt2 di2 gives the chance of finding simultaneously electron 1 in dri dii and electron 2 in dr2di2- The two-electron density function gives information as to how the motion of any pair of electrons is correlated. For independent particles, these probabilities are independent and so we would expect... 

Averaging over the two-electron density function p leads to the global value t7 = 2 q. The constant part of the kinetic energy does not change with changes in p. Also, it has a negative value. 

Self-Consistent Hartree-Fock-Wigner Calculations A Two-Electron-Density Functional Theory... 

R.J. Boyd and J.M. Ugalde, Analysis of Wave Functions in Terms of One- and Two-Electron Density Functions, in Computational Chemistry. Structure, Interactions and Reactivity, edited by S. Fraga, Elsevier Science Publishers, Amsterdam, 1992. 

Figure 3 The one-electron and two-electron density functions of the Xg ground state of the H2 molecule. The upper plots contain the one-electron and two-electron densities of the uncorrelated Hartree-Fock description in a minimal basis the lower plots contain the corresponding densities of the two-configuration correlated FCI description in the same basis. In all cases, the electron density has been plotted on the molecular axis (one axis for the one-electron densities, two axes for the two-electron densities).

But, of course, the two-electron density function is unknown. However, the largest contribution to this two-particle density must, presumably, be the product of two copies of the single-particle density p x) so that it would seem sensible to write... 

As discussed in Section 2.7.3, the two-electron densities represent the probability of simultaneously locating two electrons at two given positions in the molecule. According to (5.2.17), the two-electron density functions of the bonding and antibonding configurations are given by... 

This factorization of the two-electron density functions is apparent in the two-electron plots in Figure 5.4, which represent the probability of simultaneously locating the two electrons at different positions on the molecular axis. Thus, the relative probabilities of locating one electron at different positions on the molecular axis are independent of the whereabouts of the other electron. For example, the probability of locating one electron at nucleus A is identical to the probability of locating the same electron at nucleus B even when the other electron is known to be located at B. 

Fig. 5.5. The one- and two-electron density functions of the two-configuration E+ ground (upper plots) and excited (lower plots) states of the hydrogen molecule on the molecular axis (atomic units). The two-electron densities are represented by surface and contour plots. In the contour plot for the excited state, the dashed lines represent two-electron nodes. The density functions have been calculated in a minimal basis of hydrogenic Is functions with unit exponents.

According to the aspherical-atom formalism proposed by Stewart [12], the one-electron density function is represented by an expansion in terms of rigid pseudoatoms, each formed by a core-invariant part and a deformable valence part. Spherical surface harmonics (multipoles) are employed to describe the directional properties of the deformable part. Our model consisted of two monopole (three for the sulfur atom), three dipole, five quadrupole, and seven octopole functions for each non-H atom. The generalised scattering factors (GSF) for the monopoles of these species were computed from the Hartree-Fockatomic functions tabulated by Clementi [14]. 

The wave function P contains all information of the joint probability distribution of the electrons. For example, the two-electron density is obtained from the wave function by integration over the spin and space coordinates of all but two electrons. It describes the joint probability of finding electron 1 at r, and electron 2 at r2. The two-electron density cannot be obtained from elastic Bragg scattering. 

In a more complex situation than that of two electrons occupying each its orbital one can expect much more sophisticated interconnections between the total spin and two-electron densities than those demonstrated above. The general statement follows from the theorem given in [72] which states that no one-electron density can depend on the permutation symmetry properties and thus on the total spin of the wave function. For that reason the difference between states of different total spin is concentrated in the cumulant. If there is no cumulant there is no chance to describe this difference. This explains to some extent the failure of almost 40 years of attempts to squeeze the TMCs into the semiempirical HFR theory by extending the variety of the two-electron integrals included in the parameterization. 

For the very small systems in Table 7.1, it is possible to approach the exact solution of the Schrodinger equation, but, as a rule, convergence of the correlation energy is depressingly slow. Mathematically, this derives from the poor ability of products of one-electron basis functions, which is what Slater determinants are, to describe the cusps in two-electron densities that characterize electronic structure. For the MP2 level of theory, Schwartz (1962)... 

First we discuss and construct monodisperse two-dimensional arrangements of impenetrable cylinders in terms of radial distance correlation functions, the lateral packing fraction and number density. In the second step, these hard cylinders are covered by the mean electronic density functions of the RISA chain segment ensemble. Last of all, the Fourier transformation and final averaging is... 

Carbo et al. introduced the concept of similarity index (SI) for measuring the similarity of two molecular electron density functions [99, 100]. The Carbo SI... 

Density functional calculations (DFT calculations, density functional theory) are, like ab initio and semiempirical calculations, based on the Schrodinger equation However, unlike the other two methods, DFT does not calculate a conventional wavefunction, but rather derives the electron distribution (electron density function) directly. Afunctional is a mathematical entity related to a function. 

The two delta terms which have been placed side by side encapsulate the main problem with DFT the sum of the kinetic energy deviation from the reference system and the electron-electron repulsion energy deviation from the classical system, called the exchange-correlation energy. In each term an unknown functional transforms electron density into an energy, kinetic and potential respectively. This exchange-correlation energy is a functional of the electron density function ... 

Density functional theory is based on the two Hohenberg-Kohn theorems, which state that the ground-state properties of an atom or molecule are determined by its electron density function, and that a trial electron density must give an energy greater than or equal to the true energy. Actually, the latter theorem is true only if the exact functional (see below) is used with the approximate functionals in use... 

The density functional theory of Hohenberg, Kohn and Sham [173,205] has become the standard formalism for first-principles calculations of the electronic structure of extended systems. Kohn and Sham postulate a model state described by a singledeterminant wave function whose electronic density function is identical to the ground-state density of an interacting /V-clcctron system. DFT theory is based on Hohenberg-Kohn theorems, which show that the external potential function v(r) of an //-electron system is determined by its ground-state electron density. The theory can be extended to nonzero temperatures by considering a statistical electron density defined by Fermi-Dirac occupation numbers [241], The theory is also easily extended to the spin-indexed density characteristic of UHF theory and of the two-fluid model of spin-polarized metals [414],... 

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