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.

"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.

Resonating UHF Study on Electron Correlation in a Ground State of Two Electrons Confined in 2D Quantum Dot

2012 ◽  
Vol 1423 ◽  
Author(s):  
Takuma Okunishi ◽  
Kyozaburo Takeda
Keyword(s):  
Quantum Dot ◽  
Electron Correlation ◽  
Ground State ◽  
Electron State ◽  
Wave Functions ◽  
Electron Wave ◽  
Temporal Fluctuation ◽  
Semiconductor Quantum Dot ◽  
Hartree Fock ◽  
Reference Description

ABSTRACTWe theoretically study the spatial and temporal fluctuation of two electrons confined in a semiconductor quantum dot (QD). Eigenstates are determined by the resonating unrestricted Hartree-Fock (res-UHF) approach in order to take into account the electron correlation via the configuration interaction (CI). The time-dependent (TD) wave function is, then, expanded by the UHF solutions, and the CI treatment is combined with the TD Schrödinger equation (TD-CI). The present TD-CI approach has an advantage to study how the electron correlation fluctuates the multi-electron state spatially and/or temporally through the multi-reference description of many-electron wave functions.

PARITY-PROJECTED RESONATING HARTREE-FOCK APPROXIMATION TO THE LIPKIN MODEL

1999 ◽  
Vol 08 (05) ◽  
pp. 443-460 ◽  
Author(s):  
SEIYA NISHIYAMA ◽  
MORIYOSHI IDO ◽  
KAZUHIRO ISHIDA
Keyword(s):  
Wave Function ◽  
Ground State ◽  
Correlation Energy ◽  
Energy Functional ◽  
Hartree Fock ◽  
Parity Symmetry ◽  
Lipkin Model ◽  
Model Hamiltonian ◽  
Ground State Correlation ◽  
Hf Wave

An exact eigenstate of the Lipkin-model Hamiltonian has a parity-symmetry. The resonating Hartree-Fock (Res-HF) ground-state wave function however breaks the parity-symmetry. To restore the parity-symmetry, we develop a parity-projected (P-P) Res-HF approximation to the Lipkin model. The P-P Res-HF wave function is superposed by two P-P Slater determinants. Self-consistent calculation is carried out so as to optimize the P-P Res-HF energy functional. A Res-HF ground state with definite parity brings us the ground-state energy much nearer to the exact one than that the Res-HF and explains most of the ground-state correlation energy. We also show behaviour of P-P Res-HF orbital energies and mixing coefficients calculated by the P-P Res-HF approximation and compare them with those calculated by the parity-unprojected Res-HF approximation. From this comparison, we find very interesting relations between them which have never been seen in the parity-unprojected Res-HF calculation.

scholarly journals Hartree–Fock implementation using a Laguerre-based wave function for the ground state and correlation energies of two-electron atoms

2018 ◽  
Vol 376 (2115) ◽  
pp. 20170153 ◽  
Author(s):  
Andrew W. King ◽  
Adam L. Baskerville ◽  
Hazel Cox
Keyword(s):  
Wave Function ◽  
Ground State ◽  
Nuclear Charge ◽  
Analytical Form ◽  
Theoretical Chemistry ◽  
Variational Parameter ◽  
Isoelectronic Sequence ◽  
Hartree Fock ◽  
The One ◽  
Accurate Wave

An implementation of the Hartree–Fock (HF) method using a Laguerre-based wave function is described and used to accurately study the ground state of two-electron atoms in the fixed nucleus approximation, and by comparison with fully correlated (FC) energies, used to determine accurate electron correlation energies. A variational parameter A is included in the wave function and is shown to rapidly increase the convergence of the energy. The one-electron integrals are solved by series solution and an analytical form is found for the two-electron integrals. This methodology is used to produce accurate wave functions, energies and expectation values for the helium isoelectronic sequence, including at low nuclear charge just prior to electron detachment. Additionally, the critical nuclear charge for binding two electrons within the HF approach is calculated and determined to be Z HF C =1.031 177 528. This article is part of the theme issue ‘Modern theoretical chemistry’.

scholarly journals History of the development of the Half-Projected Hartree-Fock method. Application to the calculation of excited states of the same symmetry as the ground state

2021 ◽  
Author(s):  
Maria Belen Ruiz
Keyword(s):  
Wave Function ◽  
Excited State ◽  
Formic Acid ◽  
Excited States ◽  
Ground State ◽  
Correlation Energy ◽  
Spin Correlation ◽  
Theoretical Spectrum ◽  
Hartree Fock ◽  
History Of

Spin projected wave functions are known as generalizations of the Hartree-Fock wave function. Among them, the Half-Projected Hartree-Fock (HPHF) model represents a good compromise between the restricted (RHF) and unrestricted (UHF) Hartree-Fock methods. The HPHF wave function is a nearly pure wave function of spin and recovers a small part of the spin correlation energy. This paper reviews the history of the HPHF theory, not only from the conceptual point of view but also providing a compilation of the publications of this method over the years until now. In addition, the extension of the HPHF method to the calculation of non-orthogonal excited states to the ground state will be treated. The variational collapse during the calculation of singlet excited states with the same symmetry as the ground state is avoided by orthogonalizing the excited orbital to the corresponding occupied orbital. As an example, the potential energy surface of the S0 ground and 1S1(n, π∗) first excited state of the formic acid HCOOH are calculated. Formic acid exhibits complex energy surfaces with respect two large amplitude motions, the torsional rotation of the O-H group and the waving out-of-plane angle of the H atom. In the excited state, the molecule adopts a pyramidal structure. The obtained energy results are fitted to curves that can be used for the calculation of the theoretical spectrum.

scholarly journals THE METHOD OF INCREMENTS — A WAVEFUNCTION-BASED AB-INITIO CORRELATION METHOD FOR SOLIDS

2007 ◽  
Vol 21 (13n14) ◽  
pp. 2204-2214 ◽  
Author(s):  
BEATE PAULUS
Keyword(s):  
Experimental Data ◽  
Ab Initio ◽  
Ground State ◽  
Infinite System ◽  
Correlation Energy ◽  
Correlation Method ◽  
Many Body ◽  
Hartree Fock ◽  
The Many ◽  
Good Agreement

The method of increments is a wavefunction-based ab initio correlation method for solids, which explicitly calculates the many-body wavefunction of the system. After a Hartree-Fock treatment of the infinite system the correlation energy of the solid is expanded in terms of localised orbitals or of a group of localised orbitals. The method of increments has been applied to a great variety of materials with a band gap, but in this paper the extension to metals is described. The application to solid mercury is presented, where we achieve very good agreement of the calculated ground-state properties with the experimental data.

LONG RANGE FORCES BETWEEN HYDROGEN MOLECULES

1955 ◽  
Vol 33 (11) ◽  
pp. 668-678 ◽  
Author(s):  
F. R. Britton ◽  
D. T. W. Bean
Keyword(s):  
Wave Function ◽  
Long Range ◽  
Ground State ◽  
Close Agreement ◽  
Van Der Waals ◽  
Wave Functions ◽  
Numerical Factor ◽  
Hydrogen Molecules ◽  
Static Quadrupole

Long range forces between two hydrogen molecules are calculated by using methods developed by Massey and Buckingham. Several terms omitted by them and a corrected numerical factor greatly change results for the van der Waals energy but do not affect their results for the static quadrupole–quadrupole energy. By using seven approximate ground state H2 wave functions information is obtained regarding the dependence of the van der Waals energy on the choice of wave function. The value of this energy averaged over all orientations of the molecular axes is found to be approximately −11.0 R−6 atomic units, a result in close agreement with semiempirical values.

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).

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.

Determination of hyperon properties through the variational method considering the hyperfine interaction

2019 ◽  
Vol 28 (10) ◽  
pp. 1950087 ◽  
Author(s):  
S. M. Moosavi Nejad ◽  
A. Armat
Keyword(s):  
Experimental Data ◽  
Wave Function ◽  
Ground State ◽  
Radiative Decay ◽  
Magnetic Moments ◽  
Wave Functions ◽  
Decay Widths ◽  
Free Parameters ◽  
Theoretical Results

Performing a fit procedure on the hyperon masses, we first determine the free parameters in the Cornell-like hypercentral potential between the constituent quarks of hyperons in their ground state. To this end, using the variational principle, we apply the hyperspherical Hamiltonian including the Cornell-like hypercentral potential and the perturbation potentials due to the spin–spin, spin–isospin and isospin–isospin interactions between constituent quarks. In the following, we compute the hyperon magnetic moments as well as radiative decay widths of spin-3/2 hyperons using the spin-flavor wave function of hyperons. Our analysis shows acceptable consistencies between theoretical results and available experimental data. This leads to reliable wave functions for hyperons at their ground state.

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