"VECTAL" REPRESENTATION OF THE WAVE FUNCTION AND ITS PHYSICAL MEANING

1967 ◽  
Vol 45 (11) ◽  
pp. 3667-3676
Author(s):  
C. S. Lin
Keyword(s):  
Wave Function ◽  
Wave Functions ◽  
Electron Wave Function ◽  
Electron Wave ◽  
Expectation Values ◽  
Expectation Value ◽  
Hartree Fock ◽  
The One ◽  
The Relationship ◽  
Determinantal Form

A new form of one-electron wave function, "vectal," is introduced. It is shown that an arbitrary CI geminal and a certain class of many-electron wave functions can be represented in a single-determinantal form in terms of the vectal. Eigenvalue equations for the vectal, similar to that of the Hartree–Fock theory, are derived and the vectal representation is shown to enable a formal interpretation of the CI theory in the Hartree–Fock manner. The eigenvalue, vectal energy, is interpreted as the negative of an ionization potential, in Koop-man's sense, of the system described by the CI wave function. It is also shown that the expectation value of any one-electron operator, [Formula: see text], where Ψ is the CI wave function, is expressible in terms of the expectation values of the same operator with respect to the vectals. The vectals are interpreted as the one-electron wave function in the CI space.A possible application of the vectal representation is briefly described, and the relationship between the vectal representation and the "scalar representation" is discussed.

scholarly journals Evaluation of the one electron expectation values for different wave function of Be atom

2007 ◽  
Vol 4 (3) ◽  
pp. 393-396
Author(s):  
Baghdad Science Journal
Keyword(s):  
Wave Function ◽  
Slater Determinant ◽  
Open Shell ◽  
Expectation Values ◽  
Electronic Density ◽  
Expectation Value ◽  
Hartree Fock ◽  
Shell System ◽  
Electronic Wave Function ◽  
The One

The aim of this work is to evaluate the one- electron expectation value from the radial electronic density function D(r1) for different wave function for the 2S state of Be atom . The wave function used were published in 1960,1974and 1993, respectavily. Using Hartree-Fock wave function as a Slater determinant has used the partitioning technique for the analysis open shell system of Be (1s22s2) state, the analyze Be atom for six-pairs electronic wave function , tow of these are for intra-shells (K,L) and the rest for inter-shells(KL) . The results are obtained numerically by using computer programs (Mathcad).

Über die Berechnung der Korrelationsenergie der Atomelektronen

1960 ◽  
Vol 15 (10) ◽  
pp. 909-926 ◽  
Author(s):  
Levente Szász
Keyword(s):  
Wave Function ◽  
Numerical Calculation ◽  
Ground State ◽  
Correlation Functions ◽  
Correlation Energy ◽  
Wave Functions ◽  
Electron Wave ◽  
Hartree Fock ◽  
Experimental Values ◽  
The One

To calculate the correlation energy of an atom with N electrons we suggest the wave functionwhere à is the antisymmetrizer operator, φ1, φ2, ..., φN are one electron wave functions, and Wjk are correlation functions of the following form:where the constants c j km, n, l are variational parameters. The function (a) is a generalization of thewave function of Hylleraas for He. After a discussion of the properties of our function, an energy expression is derived. Numerical calculation is made for the ground state of the Be atom with the functionwhere φ1 and φ2 are ls wave functions, φ3 and φ4 are 2s wave functions, r1, r2, r3 and r4 are the radial coordinates of the four electrons, r12 and r34 are the distances between the corresponding electrons, and C1 and c2 are variational parameters. Using the one electron wave functions calculated by Roothaan and coll. with the Roothaan procedure, we got the energy value E= -14.624 a. u. while the Hartree-Fock and experimental values are EH,F= -14.570 a. u. and Eexp= -14.668 a. u. respectively. Thus the function (c) gives about one-half of the correlation energy of the Be atom.

Stationary properties of approximate wave functions

1969 ◽  
Vol 47 (21) ◽  
pp. 2355-2361 ◽  
Author(s):  
A. R. Ruffa
Keyword(s):  
Wave Function ◽  
Total Energy ◽  
Helium Atom ◽  
Wave Functions ◽  
Quantum Mechanical ◽  
Mechanical Wave ◽  
Expectation Value ◽  
Charge Densities ◽  
Hartree Fock ◽  
Approximate Wave

The accuracy of quantum mechanical wave functions is examined in terms of certain stationary properties. The most elementary of these, namely that displayed by the class of wave functions which yields a stationary value for the total energy of the system, is demonstrated to necessarily require few other stationary properties, and none of these appear to be particularly useful. However, the class of wave functions which yields both stationary energies and charge densities has very important stationary properties. A theorem is proven which states that any wave function in this class yields a stationary expectation value for any operator which can be expressed as a sum of one-particle operators. Since the Hartree–Fock wave function is known to possess these same stationary properties, this theorem demonstrates that the Hartree–Fock wave function is one of the infinitely many wave functions of the class. Methods for generating other wave functions in this class by modifying the Hartree–Fock wave function without changing its stationary properties are applied to the calculation of wave functions for the helium atom.

The dependence of the hyperfine interaction on the cellular potential in Li, Na, and K

1968 ◽  
Vol 46 (12) ◽  
pp. 1425-1434 ◽  
Author(s):  
R. A. Moore ◽  
S. H. Vosko
Keyword(s):  
Wave Function ◽  
Variational Method ◽  
Fermi Surface ◽  
Hyperfine Interaction ◽  
Electron Density ◽  
Electron Wave Function ◽  
Electron Wave ◽  
Surface Electron ◽  
Conduction Electrons ◽  
Hartree Fock

The effect of including the Hartree field due to the conduction electrons in the cellular potential on the Fermi surface electron wave function is investigated. It is found that the Fermi surface electron density at the nucleus is reduced by 10% to 20% by including this term. Also, an L dependent effective local potential constructed to simulate Hartree–Fock theory is calculated and applied to Li. All calculations are performed using the Wigner–Seitz spherical cellular approximation, and the Schrödinger equation is solved by the Kohn (1954) variational method.

Effect of Correlation on the One-Electron Wave Function in Atoms

1965 ◽  
Vol 139 (1A) ◽  
pp. A1-A3 ◽  
Author(s):  
S. Lundqvist ◽  
C. W. Ufford
Keyword(s):  
Wave Function ◽  
Electron Wave Function ◽  
Electron Wave ◽  
The One

scholarly journals Evaluation of the one electron expeetation values for different wave functions

2004 ◽  
Vol 1 (2) ◽  
pp. 336-339
Author(s):  
Baghdad Science Journal
Keyword(s):  
Wave Function ◽  
Slater Determinant ◽  
Open Shell ◽  
Wave Functions ◽  
Expectation Values ◽  
Electronic Density ◽  
Shell System ◽  
Positioning Technique ◽  
The One ◽  
Open Shell System

The aim of this work is to evaluate the onc-electron expectation values < r > from the radial electronic density funetion D(r) for different wave ?'unctions for the 2s state of Li atom. The wave functions used were published in 1963,174? and 1993 , respectavily. Using " " ' wave function as a Slater determinant has used the positioning technique for the analysis open shell system of Li (Is2 2s) State.

The dependence of the hyperfine interaction on the cellular potential in Na and K. II

1969 ◽  
Vol 47 (13) ◽  
pp. 1331-1336 ◽  
Author(s):  
R. A. Moore ◽  
S. H. Vosko
Keyword(s):  
Fermi Surface ◽  
Electron Density ◽  
Wave Functions ◽  
Electron Wave ◽  
Surface Electron ◽  
Conduction Electrons ◽  
Hartree Fock ◽  
A Value ◽  
Dependent Potential ◽  
Conduction Electron Density

The dependence of the Fermi surface electron wave functions in Na and K on (i) an L-dependent effective local cellular potential constructed to simulate Hartree-Fock theory and (ii) the inclusion of the Hartree field due to the conduction electrons in the cellular potential is investigated. All calculations are performed using the Wigner–Seitz spherical cellular approximation and the Schrödinger equation is solved by the Kohn variational method. It is found that to ensure a value of the Fermi surface electron density at the nucleus accurate to ~5%, it is necessary to use the L-dependent potential along with the Hartree field due to a realistic conduction electron density.

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