Electron localization function density construction

In quantum mechanics, the square of the wavefunction corresponds to the electron probability density, p, upon which we can construct ways to rationalize the concepts of chemical bonding. In this book, we shall discuss two related approaches to analyzing the topology of the electron density the electron localization function (ELF) and atoms in molecules (AIM). The latter is sometimes applied in the analysis of experimental data, as well as theoretical data, so we reserve discussion of this technique until Section 10.10. 

We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

The bond-pair wave functions in Equations 6.27, 6.28, and 6.29 were specially constructed to describe two electrons localized between two atoms as a single chemical bond between the atoms. These wave functions should not be called MOs, because they are not single-electron functions and they are not de-localized over the entire molecule. The corresponding single bonds (see Figs. 6.37, 6.38, and 6.39) are called cr bonds, because their electron density is cylindrically symmetric about the bond axis. There is no simple correlation between this symmetry... 

The above demonstrated possibility of obtaining numerical virtual orbitals indicate that the FD HF method can also be used as a solver of the Schrodinger equation for a one-electron diatomic system with an arbitrary potential. Thus, the scheme could be of interest to those who try to construct exchange-correlation potential functions or deal with local-scaling transformations within the functional density theory (32,33). 

The question of the locality of density-functional potentials of Kohn-Sham type, a central issue of the foundations of DFT, has been controversial for some time. Robert Nesbet has argued in several articles in the literature, in opposition to most of the DFT community, that the locality of DFT potentials has never been rigorously proven and he claims by means of a counter example that a local potential cannot exist for a system with more than two electrons. His conclusion is that a consistent density functional theory does not exist and that the only rigorous way to proceed is by constructing an orbital functional theory (OFT). This result has been challenged in the scientific literature by several authors criticisms that have been vigorously refuted by Nesbet. In the present volume four chapters appear on the subject. 

At present, the electronic structure of crystals, for the most part, has been calculated using the density-functional theory in a plane-wave (PW) basis set. The one-electron Bloch functions (crystal orbitals) calculated in the PW basis set are delocalized over the crystal and do not allow one to calculate the local characteristics of the electronic structure. As a consequence, the functions of the minimal valence basis set for atoms in the crystal should be constructed from the aforementioned Bloch functions. There exist several approaches to this problem. The most consistent approach was considered above and is associated with the variational method for constructing the Wannier-type atomic orbitals (WTAO) localized at atoms with the use of the calculated Bloch functions. Another two approaches use the so-called projection technique to connect the calculated in PW basis Bloch states with the atomic-like orbitals of the minimal basis set. 

Let us introduce another early example by Slater, 1951, where the electron density is exploited as the central quantity. This approach was originally constructed not with density functional theory in mind, but as an approximation to the non-local and complicated exchange contribution of the Hartree-Fock scheme. We have seen in the previous chapter that the exchange contribution stemming from the antisymmetry of the wave function can be expressed as the interaction between the charge density of spin o and the Fermi hole of the same spin... 

The Laplacian is constructed from second partial derivatives, so it is essentially a measure of the curvature of the function in three dimensions (Chapter 6). The Laplacian of any scalar field shows where the field is locally concentrated or depleted. The Laplacian has a negative value wherever the scalar field is locally concentrated and a positive value where it is locally depleted. The Laplacian of the electron density, p, shows where the electron density is locally concentrated or depleted. To understand this, we first look carefully at a onedimensional function and its first and second derivatives. 

Density functional approaches to molecular electronic structure rely on the existence theorem [10] of a universal functional of the electron density. Since this theorem does not provide any direction as to how such a functional should be constructed, the functionals in existence are obtained by relying on various physical models, such as the uniform electron gas and others. In particular, the construction of an exchange-correlation potential that depends on the electron density only locally seems impossible without some approximations. Such approximate exchange-correlation potentials have been derived and applied with some success for the description of molecular electronic ground states and their properties. However, there is no credible evidence that such simple constructions can lead to either systematic approximate treatments, or an exact description of molecular electronic properties. The exact functional that seems to... 

Here Ho is the kinetic energy operator of valence electrons Vps is the pseudopotential [40,41] which defines the atomic core. V = eUn(r) is the Hartree energy which satisfies the Poisson equation ArUn(r) = —4nep(r) with proper boundary conditions as discussed in the previous subsection. The last term is the exchange-correlation potential Vxc [p which is a functional of the density. Many forms of 14c exist and we use the simplest one which is the local density approximation [42] (LDA). One may also consider the generalized gradient approximation (GGA) [43,44] which can be implemented for transport calculations without too much difficulty [45]. Importantly a self-consistent solution of Eq. (2) is necessary because Hks is a functional of the charge density p. One constructs p from the KS states Ts, p(r) = (r p r) = ns Fs(r) 2, where p is the density matrix,... 

---