Valence electron wave function

In the PP theory, the valence electron wave function is composed of two parts. The main part is the pseudo-wave function describing a relatively smooth-varying behavior of the electron. The second part describes a spatially rapid oscillation of the valence electron near the atomic core. This atomic-electron-like behavior is due to the fact that, passing the vicinity of an atom, the valence electron recalls its native outermost atomic orbitals under a relatively stronger atomic potential near the core. Quantum mechanically the situation corresponds to the fact that the valence electronic state should be orthogonal to the inner-core electronic states. The second part describes this CO. The CO terms explicitly contain the information of atomic position and atomic core orbitals. 

In the PP framework, the valence electron wave function % orthogonalized to the inner core electron wave function HA s is given by [12]... 

The microscope effect can be explained by using the fact that the r (-function is equivalently described in terms of the autocorrelation function of the valence electron wave functions as follows [7] ... 

The valence-electron wave functions of atoms, compressed beyond their ionization limits are Fourier sums of spherical Bessel functions corresponding to step functions (Compare 6.3.1) of the type... 

Schrodinger equation valence electrons wave function wavelength, X wave mechanical model wave-particle duality of nature... 

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. ... 

Problem 11-1. Consider three levels of approximation (a) Exact many-electron wave function, (b) Hartree-Fock wave function, (including all electrons), (c) Simple LCAO-MO valence electron wave function. For each of the following molecular properties, would you expect the Hartree-Fock approximation to give a correct prediction (to within 1% in the cases of quantitative predictions) Would you expect the LCAO-MO approximation to give a correct prediction ... 

The EH method (developed by Wolfsberg and Helmholz and by Hoffmann) is an extension of the Hiickel method in which the pi-electron approximation is not made, but all valence electrons are treated. The method is thus applicable to nonplanar, as well as planar, molecules. The valence-electron Hamiltonian is taken as the sum of one-electron Hamiltonians //va, = 2(/ eff(0, where Hcft(i) is not explicitly defined. The valence-electron wave function is the antisymmetrized product of spin-... 

The valence electron wave functions were taken to be zero outside of a near-metal region (see Fig. 6). The wave functions were expanded in a basis of plane waves. One-electron wave functions were computed through the following well-known [40] iterative process Compute the (/ + l)th iterate as... 

The valence electron wave function p can be determined from the one-electron self-consistent field eigenvalue equation... 

Ab initio density functional theory calculations were also carried out on the CH2=XH(A) and CH(A)=XH2 series of molecules. The basis set used was the CEP-TZDP+ described previously26 and is more extensive than the DZP basis set used in the CAS(4,4)-OVB calculations. In TZDP+ the valence electron wave function is expanded in a triple-zeta sp set of functions plus a double set of polarization d-type functions plus a set of diffuse sp-type functions. The B3LYP exchange-correlation functional20 as defined in the Gaussian 94 program set35 was used in all the DFT calculations. 

Physically Eq. (1) implies that the valence electron wave functions are strongly localized within the space between the bonded atoms, and that electrons belonging to spatially separated valence bonds do not interact with each other. In practice this condition is never fulfilled and the deformation of a given bond results in charge-density redistribution over the entire molecule, which is man-... 

In practical applications some groups can be held frozen in the iterative procedure. This is the case in core pseudopotential calculations where the core electron wave function represented by a suitable analytical form, remains fixed and the valence electron wave function is optimized in the effective field of the core. 

The characteristic properties of the metallic state of matter follow from two overwhelmingly important physical effects the overlap of valence electron wave functions on neighboring atoms and the Pauli exclusion principle. These effects are embodied in the nearly-free-electron (NFE) model of metals. 

The simplest many-electron wave function that satisfies the Exclusion Principle is a product of N different one-electron functions that have been antisymmetrized, or written as a determinant. Here, N is the number of electrons (or valence electrons) in the molecule. HyperChem uses this form of the wave function for most semi-empirical and ab initio calculations. Exceptions involve using the Configuration Interaction option (see page 119). HyperChem computes one-electron functions, termed molecular spin orbitals, by relatively simple integration and summation calculations. The many-electron wave function, which has N terms (the number of terms in the determinant), never needs to be evaluated. 

The basic idea of the pseudopotential theory is to replace the strong electron-ion potential by a much weaker potential - a pseudopotential that can describe the salient features of the valence electrons which determine most physical properties of molecules to a much greater extent than the core electrons do. Within the pseudopotential approximation, the core electrons are totally ignored and only the behaviour of the valence electrons outside the core region is considered as important and is described as accurately as possible [54]. Thus the core electrons and the strong ionic potential are replaced by a much weaker pseudopotential which acts on the associated valence pseudo wave functions rather than the real valence wave functions (p ). As... 

The main difficulty in the theoretical study of clusters of heavy atoms is that the number of electrons is large and grows rapidly with cluster size. Consequently, ab initio "brute force" calculations soon meet insuperable computational problems. To simplify the approach, conserving atomic concept as far as possible, it is useful to exploit the classical separation of the electrons into "core" and "valence" electrons and to treat explicitly only the wavefunction of the latter. A convenient way of doing so, without introducing empirical parameters, is provided by the use of generalyzed product function, in which the total electronic wave function is built up as antisymmetrized product of many group functions [2-6]. 

Most of the commonly used electronic-structure methods are based upon Hartree-Fock theory, with electron correlation sometimes included in various ways (Slater, 1974). Typically one begins with a many-electron wave function comprised of one or several Slater determinants and takes the one-electron wave functions to be molecular orbitals (MO s) in the form of linear combinations of atomic orbitals (LCAO s) (An alternative approach, the generalized valence-bond method (see, for example, Schultz and Messmer, 1986), has been used in a few cases but has not been widely applied to defect problems.)... 

Independently of each other, Pauling and Slater worked out a quantum mechanical explanation of the directional valences characteristic of chemical molecules. They did this by proposing directional properties for the p wave functions and for the sp3 wave functions resulting from "hybridization" of electron wave functions, or orbitals.73... 

Our model of positive atomic cores arranged in a periodic array with valence electrons is shown schematically in Fig. 14.1. The objective is to solve the Schrodinger equation to obtain the electronic wave function ( ) and the electronic energy band structure En( k ) where n labels the energy band and k the crystal wave vector which labels the electronic state. To explore the bonding properties discussed above, a calculation of the electronic charge density... 

It is known that the cohesion of a metal is ensured by the electrons partially filling a conduction (or valence) band. The wave functions of these conduction electrons are Bloch functions, i.e. amplitude modulated plane waves. Even though these wave functions are linear combinations of the electronic wave functions in the isolated atoms, reminiscence of the atomic orbitals is lost (or is eventually contained in the amplitude factor). The conduction electrons are, of course, originally, the outer or valence electrons of the atoms but in a metal, to describe them as s, p, d or f, i.e. with the quantum number proper to the atomic case, has little meaning. They may be considered to many purposes to be free electrons . 

The TMCs electronic wave function formalizing the CFT ionic model is one with a fixed number of electrons in the d-shell. In the EHCF method it is used as a zero approximation. The interactions responsible for electron transfers between the d-shell and the ligands are treated as perturbations. Following the standards semiempirical setting we restrict the AO basis for all atoms of the TMC by the valence orbitals. All the AOs of the TMC are... 

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