Wave function electron density from

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 technique used to extract the wave function in this work is conceptually simple the wave function obtained is a single determinant which reproduces the observed experimental data to the desired accuracy, while minimising the Hartree-Fock (HF) energy. The idea is closely related to some interesting recent work by Zhao et al. [1]. These authors have obtained the Kohn-Sham single determinant wave function of density functional theory (DFT) from a theoretical electron density. 

An expression for e(k) in the case of a Fermi gas of free electrons can be obtained by considering the effect of an introduced point charge potential, small enough so the arguments of perturbation theory are valid. In the absence of this potential, the electronic wave functions are plane waves V 1/2exp(ik r), where V is the volume of the system, and the electron density is uniform. The point charge potential is screened by the electrons, so that the potential felt by an electron, O, is due to the point charge and to the other electrons, whose wave functions are distorted from plane waves. The electron density and the potential are related by the Poisson equation,... 

As anticipated, the multipolar model is not the only technique available to refine electron density from a set of measured X-ray diffracted intensities. Alternative methods are possible, for example the direct refinement of reduced density matrix elements [73, 74] or even a wave function constrained to X-ray structure factor (XRCW) [75, 76]. Of course, in all these models an increasing amount of physical information is used from theoretical chemistry methods and of course one should carefully consider how experimental is the information obtained. 

The electronic SE focuses on the energy levels of the molecule. By obtaining the lowest energy, one assumes that the associated wave function will yield the electron distribution of the electronic ground state. An alternative theory has come into recent prominance, in which the SE is bypassed and attention focused on the electron density from which many desired properties including energy can derived directly [density functional theory (DFT)]. 

The electrostatic potential F(r,) at a given point i created in the neighboring space by the nuclear charges and the electronic distribution of a molecule can be calculated from the molecular wave function (strictly speaking from the corresponding first-order density function). As this quantity is directly obtainable from the wave function, it does not suffer from the drawbacks inherent in the classical population analysis. 

LCAO-MO Molecular Wave Functions. I. and II. Overlap Populations, Bond Orders, and Covalent Bond Energies. See also S. M. Bachrach, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1994, Vol. 5, pp. 171-227. Population Analysis and Electron Densities from Quantum Mechanics. 

Massa, L., Goldberg, M., Frishberg, C., Boehme, R. F. and LaPlaca, S. J. Wave functions derived by quantum modeling of the electron density from coherent X-ray diffraction beryllium metal. Phys. Rev. Lett. 55, 622-625 (1985). 

Let us conclude this section by returning to the electron density from the Coulson-Fischer wave function. Writing a normalization factor K on the RHS of Eq. (24), one finds... 

The sweeping theorem of Hohenberg and Kohn is that, like the wave function, the ground state s electron density determines all the properties of an electronic system [1]. The result is proved in three steps. First, one recalls that the number of electrons is determined from the electron density using Eq. (14). Next, one demonstrates that the external potential can be determined from the ground-state electron density. From N and v(r), we may determine the electronic Hamiltonian and solve Schro-dinger s equation for the wave function, subsequently determining all observable properties of the system. 

To get a first idea of what density-functional theory is about, it is useful to take a step back and recall some elementary quantum mechanics. In quantum mechanics we learn that all information we can possibly have about a given system is contained in the system s wave function, T. Here we will exclusively be concerned with the electronic structure of atoms, molecules and solids. The nuclear degrees of freedom (e.g., the crystal lattice in a solid) appear only in the form of a potential u(r) acting on the electrons, so that the wave function depends only on the electronic coordinates.2 Nonrelativistically, this wave function is calculated from Schrodinger s equation, which for a single electron moving in a potential v(r) reads... 

An alternative way of visualizing multi-variable functions is to condense or contract some of the variables. An electronic wave function, for example, is a multi-variable function, depending on 3N electron coordinates. For an independent-particle model, such as Hartree-Fock or density functional theory, the total (determinantal) wave function is built from N orbitals, each depending on three coordinates. 

Therefore, the square of the coefficient for in each wave function indicates the fraction of electron density to be found at the ith carbon atom when there is one electron in that i/ . If there are two electrons in a particular MO, then the electron density from that MO at the ith carbon atom is 2 c. To calculate the total electron density at a particular carbon atom, we must sum the electron density at that position in each of the MOs. The general expression for electron density at the ith position is then... 

According to quantum mechanics, we must describe the position of the electron in the hydrogen atom in terms of probabilities rather than exact locations. The information about the probability is contained in the wave functions, iff, obtained from Schrodinger s equation. The square of the wave function, called either the probability density or the electron density, as noted earUer, gives the probability that the electron is at my point in space. Because s orbitals are spherically symmetric, the value of iff for an s electron depends only on its distance from the nucleus, r. Thus, the probability density can be written as [ifi r)f, where i/ (r) is the value of iff at r. This function [iff r)f gives the probability density for any point located a distance r from the nucleus. 

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