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’.

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.

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).

BOSE–EINSTEIN CONDENSATION IN BULK AND CONFINED SOLID HELIUM

2006 ◽  
Vol 20 (30n31) ◽  
pp. 5081-5092 ◽  
Author(s):  
L. REATTO ◽  
M. ROSSI ◽  
D. E. GALLI
Keyword(s):  
Wave Function ◽  
Ground State ◽  
Solid Helium ◽  
System Size ◽  
Einstein Condensation ◽  
Cylindrical Pore ◽  
Simulated System ◽  
The One ◽  
Bose Einstein ◽  
Energy Computation

We address the question if the ground state of solid 4 He has the number of lattice sites equal to the number of atoms (commensurate state) or if it is different (incommensurate state). We point out that energy computation from simulation as performed by now cannot be used to decide this question and that the presently best variational wave function, a shadow wave function, gives an incommensurate state. We have extended the calculation of the one–body density matrix ρ1 to the exact Shadow Path Integral Ground State method. Calculation of ρ1 at ρ = 0.031 Å-3 shows that Vacancy–Interstitial pair processes are present also in the exact computation but the simulated system size is too small to infer the presence of off–diagonal long range order. Variational simulations of 4 He confined in a narrow cylindrical pore are also discussed.

Inelastic Scattering of High Energy Electrons by Helium

1973 ◽  
Vol 51 (3) ◽  
pp. 311-315 ◽  
Author(s):  
S. P. Ojha ◽  
P. Tiwari ◽  
D. K. Rai
Keyword(s):  
Ground State ◽  
High Energy ◽  
Wave Functions ◽  
Oscillator Strengths ◽  
Final States ◽  
Interaction Type ◽  
Hartree Fock ◽  
High Energy Electrons ◽  
Approximate Wave ◽  
Accurate Wave

Generalized oscillator strengths and the cross section for excitation of helium by electron impact have been calculated in the Born approximation. Transitions from the ground state to the n1P (n = 2 and 3) states have been considered. Highly accurate wave functions of the Hartree–Fock and "configuration–interaction" type have been used to represent the ground state. Approximate wave functions due to Messmer have been employed for the final states. The results are compared with other calculations and with experiment.

scholarly journals Study of Deuteron System in a Two-Body Potential Model

2018 ◽  
Vol 46 ◽  
pp. 1860087
Author(s):  
Zahra Ghalenovi ◽  
Asadolah Tavakolinezhad
Keyword(s):  
Wave Function ◽  
Ground State ◽  
Potential Model ◽  
Energy Eigenvalue ◽  
Goldstone Boson ◽  
Body System ◽  
Relativistic Limit ◽  
Boson Exchange ◽  
The One ◽  
Body Potential

In this work, we study the Hamiltonian of Deuteron as a two-body system and solve the equation of the system in the non-relativistic limit. We obtain the ground state wave function as well as the corresponding energy eigenvalue of the deuteron system. The considered potentials are a combination of the confinement, Coulomb-like (as the one arising from one gluon exchange) and Goldstone boson exchange interaction.

An expansion method for calculating atomic properties. IX. The dipole polarizabilities of the lithium sequence*

1966 ◽  
Vol 293 (1434) ◽  
pp. 365-377 ◽  
Keyword(s):  
Nuclear Charge ◽  
Expansion Method ◽  
Zero Order ◽  
Isoelectronic Sequence ◽  
Hartree Fock ◽  
Atomic Properties ◽  
Dipole Polarizabilities

A theory is developed for expanding the dipole polarizabilities and shielding factors of an atom or ion in inverse powers of the nuclear charge Z in cases where the field links degenerate zero order configurations. Results for all members of the lithium isoelectronic sequence are presented both within the Hartree-Fock approximation and in a more accurate formulation, and are found to be in agreement with earlier work.

Correlated Atomic Pair Functions by the e-ϱ-Method. I. Ground State 11S and Lowest Excited States n1S (n > 1) and n3S of Helium

1999 ◽  
Vol 54 (12) ◽  
pp. 711-717
Author(s):  
F. F. Seelig ◽  
G. A. Becker
Keyword(s):  
Integral Equation ◽  
Wave Function ◽  
Helium Atom ◽  
Excited States ◽  
Ground State ◽  
Single Parent ◽  
Hartree Fock ◽  
Atomic Pair ◽  
Electronic Wave Function ◽  
S States

Abstract Some low n1S and n3S states of the helium atom are computed with the aid of the e-e method which formulates the electronic wave function of the 2 electrons ψ = e-e F, where ϱ=Z(r1+r2)–½r12 and here Z = 2. Both the differential and the integral equation for F contain a pseudopotential Ṽ instead of the true potential V that contrary to V is finite. For the ground state, F = 1 yields nearly the Hartree-Fock SCF accuracy, whereas a multinomial expansion in r1, r2 , r2 yields a relative error of about 10-7 . All integrals can be computed analytically and are derived from one single “parent” integral.

"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.

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