Wave function determination from electron density

To date, the only applications of these methods to the solution/metal interface have been reported by Price and Halley, who presented a simplified treatment of the water/metal interface. Briefly, their model involves the calculation of the metal s valence electrons wave function, assuming that the water molecules electronic density and the metal core electrons are fixed. The calculation is based on a one-electron effective potential, which is determined from the electronic density in the metal and the atomic distribution of the liquid. After solving the Schrddinger equation for the wave function and the electronic density for one configuration of the liquid atoms, the force on each atom is ciculated and the new positions are determined using standard molecular dynamics techniques. For more details about the specific implementation of these general ideas, the reader is referred to the original article. ... 

Zhao, Q. and Parr, R.G. (1993). Constrained-search method to determine wave functions from electronic densities, J. Chem. Phys. 98, 543-548. 

Theoretically, the radius of an ion extends from the nucleus to the outermost orbital occupied by electrons. The very nature of the angular wave function of an electron, which approaches zero asymptotically with increasing distance from the nucleus, indicates that an atom or ion has no definite size. Electron density maps compiled in X-ray determinations of crystal structures rarely show zero contours along a metal-anion bond. 

From the meaning of cp2 as probability or as charge density it follows that it will be desirable to provide the solution for the wave function of one electron always with a factor to be determined from the succeeding condition, such that integration over the whole space results in one J92dv = 1 in fact the probability is one of finding the electron in infinite space or... 

Each of these methods is based on the AFDF approach. Within the framework of the conventional Hartree-Fock-Roothaan-Hall self-consistent field linear combination of atomic orbitals (LCAO) ab initio representation of molecular wave functions built from molecular orbitals (MOs), the AFDF principle can be formulated using fragment density matrices. For a complete molecule M of some nuclear configuration K, using an atomic orbital (AO) basis of a set of n AOs density matrix P can be determined using the coefficients of AOs in the occupied MOs. The electronic density p(r) of the molecule M, a function of the three-dimensional position variable r, can be written as... 

Density functional theory (DFT). DFT is an alternative to the HF method, in which the fundamental role is played by the electron density rather than the wave function. The basis for this method is a proof by Pierre Ho-henberg and Kohn that all physical properties of a molecule are completely determined by its electron density. The computational savings that DFT offers come from the fact that the wave function of an -electron molecule depends on 3n spatial coordinates, whereas the electron density depends on just three spatial coordinates. Consequently, DFT calculations generally scale as the third power of the size of the basis set, rather than the fourth power of the HF methods. 

The premise behind DFT is that the energy of a molecule can be determined from the electron density instead of a wave function. This theory originated with a theorem by Hoenburg and Kohn that stated this was possible. The original theorem applied only to finding the ground-state electronic energy of a molecule. A practical application of this theory was developed by Kohn and Sham who formulated a method similar in structure to the Hartree-Fock method. 

In this formulation, the electron density is expressed as a linear combination of basis functions similar in mathematical form to HF orbitals. A determinant is then formed from these functions, called Kohn-Sham orbitals. It is the electron density from this determinant of orbitals that is used to compute the energy. This procedure is necessary because Fermion systems can only have electron densities that arise from an antisymmetric wave function. There has been some debate over the interpretation of Kohn-Sham orbitals. It is certain that they are not mathematically equivalent to either HF orbitals or natural orbitals from correlated calculations. However, Kohn-Sham orbitals do describe the behavior of electrons in a molecule, just as the other orbitals mentioned do. DFT orbital eigenvalues do not match the energies obtained from photoelectron spectroscopy experiments as well as HF orbital energies do. The questions still being debated are how to assign similarities and how to physically interpret the differences. 

The energy due to the external potential is determined simply by the density and is therefore independent of the wave function generating that density. Hence, it is the same for all wave functions integrating to a particular density and we can separate it from the kinetic and electron-electron repulsion contributions... 

---