One-electron orbit

It is possible to write down a many-body wavefiinction that will reflect the antisynmietric nature of the wavefiinction. In this discussion, the spin coordinate of each electron needs to be explicitly treated. The coordinates of an electron may be specified by rs. where s. represents the spin coordinate. Starting with one-electron orbitals, ( ). (r. s), the following fomi can be invoked ... 

In a number of classic papers Hohenberg, Kohn and Sham established a theoretical framework for justifying the replacement of die many-body wavefiinction by one-electron orbitals [15, 20, 21]. In particular, they proposed that die charge density plays a central role in describing the electronic stnicture of matter. A key aspect of their work was the local density approximation (LDA). Within this approximation, one can express the exchange energy as... 

VV e now wish to establish the general functional form of possible wavefunctions for the two electrons in this pseudo helium atom. We will do so by considering first the spatial part of the u a efunction. We will show how to derive functional forms for the wavefunction in which the i change of electrons is independent of the electron labels and does not affect the electron density. The simplest approach is to assume that each wavefunction for the helium atom is the product of the individual one-electron solutions. As we have just seen, this implies that the total energy is equal to the sum of the one-electron orbital energies, which is not correct as ii ignores electron-electron repulsion. Nevertheless, it is a useful illustrative model. The wavefunction of the lowest energy state then has each of the two electrons in a Is orbital ... 

Ihe one-electron orbitals are commonly called basis functions and often correspond to he atomic orbitals. We will label the basis functions with the Greek letters n, v, A and a. n the case of Equation (2.144) there are K basis functions and we should therefore xpect to derive a total of K molecular orbitals (although not all of these will necessarily 3e occupied by electrons). The smallest number of basis functions for a molecular system vill be that which can just accommodate all the electrons in the molecule. More sophisti- ated calculations use more basis functions than a minimal set. At the Hartree-Fock limit he energy of the system can be reduced no further by the addition of any more basis unctions however, it may be possible to lower the energy below the Hartree-Fock limit ay using a functional form of the wavefunction that is more extensive than the single Slater determinant. 

Although we are solving for one-electron orbitals, r /i and r /2, we do not want to fall into the trap of the last calculation. We shall include an extra potential energy term Vi to account for the repulsion between the negative charge on the first electron we consider, electron I, exerted by the other electron in helium, electron 2. We don t know where electron 2 is, so we must integrate over all possible locations of electron 2... 

There are m doubly occupied molecular orbitals, and the number of electrons is 2m because we have allocated an a and a spin electron to each. In the original Hartree model, the many-electron wavefunction was written as a straightforward product of one-electron orbitals i/p, i/ and so on... 

The aja, operator tests whether orbital i exists in the wave function, if that is the case, a one-electron orbital matrix element is generated, and similarly for the two-electron terms. Using the Hamiltonian in eq. (C.6) with the wave function in eq. (C.4) generates the first quantized operator in eq. (C.3). 

Following Mulliken, we shall use the word orbital to represent a one-electron orbital eigenfunction of an atom. 

The individual terms, e , in Eq. (2.27) are termed one-electron orbital energies and correspond to the ionization potential (-e ) of an electron in MO i r assuming that no reorganization of the core nuclei, or the other (2N — 1) electrons, takes place during ionization. The total energy of the system [Eq. (2.27)] is clearly not the sum of the one-electron energies because electron-electron interaction terms are included twice. 

Using an anti-symmetrized, rather than a simple product of one-electron orbitals, the expectation value of the energy becomes a sum of one-electron, Coulomb and exchange terms ... 

Most of the current implementations employ the original Car-Parrinello scheme based on DFT. The system is treated within periodic boundary conditions (PBC) and the Kohn-Sham (KS) one-electron orbitals are expanded in a basis set of plane waves (with wave vectors Gm) [48-50] ... 

Umt[p is the classical Coulomb energy and Exc[p] is the XC energy. It is the functional form of this XC functional, which is usually approximated in absence of an exact expression. The one-electron orbitals ipk(r) are obtained through self-... 

The Kohn-Sham equations can be obtained from the minimalization of the noninteracting kinetic energy after expressing it with one-electron orbitals. Because Tl 0 is generally a linear combination of several Slater determinants, the form of the... 

Using one-electron orbital picture, Fukui functions can be approximately defined as... 

Mulliken, Life, 90. On the "orbital," Mulliken wrote in 1932 "From here on, one-electron orbital wave functions will be referred to for brevity as orbitals. The method followed here will be to describe unshared electrons always in terms of atomic orbitals but to use molecular orbitals for shared electrons." In Robert Mulliken, "Electronic Structures of Polyatomic Molecules and Valence," Physical Review 41 (1932) 4971, on 50. 

The Aujbau (building up) Principle. One-electron orbitals are filled (occupied) by electrons sequentially, in the order of increasing energy. 

A second mechanism (the polarization mechanism) arises due to the polarization of the fully occupied (bonding) crystal orbitals formed by the eg. oxygen 2p. and Li 2s atomic orbitals in the presence of a magnetic field. A fully occupied crystal (or molecular) orbital in reality comprises one one-electron orbital occupied by a spin-up electron and a second one-... 

It could thus be stated that a pair description analogous to the Lewis picture is quite conveniently yielded by localized molecular orbitals. In other words, to the extent that the one-electron (orbital) method is in principle capable of faithful description of molecular electronic stmcture, the orbital analysis may provide a useful rationale of some molecular characteristics. Having generated LMOs, using any localizability criteria, the degree of spatial localization obtained may also be inquired. 

A. J. Coleman and I. Absar, One-electron orbitals intrinsic to the reduced Hamiltonian. Chem. Phys. Lett. 39, 609 (1976). 

These three types of clusters all involve one electron orbital. They provide a basis for the description of the d-d (i. e. ligand field) transitions and the ligand to metal charge transfer transitions which are responsible for most of the UV-visible spectra and opti-... 

SCF. Self Consistent Field. An iterative procedure whereby a one-electron orbital is determined under the influence of a potential made up of all the other electrons. Iteration continues until self consistency. Hartree-Fock, Density Functional and MP2 Models all employ SCF procedures. 

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