Perturbation theory for atomic systems

1959 ◽  
Vol 251 (1265) ◽  
pp. 282-290 ◽  
Keyword(s):  
Wave Function ◽  
Perturbation Theory ◽  
Quantitative Assessment ◽  
Atomic System ◽  
The Other ◽  
Atomic Systems ◽  
First Order ◽  
Hartree Fock ◽  
Alternative Procedures ◽  
Variation Iteration Method

The equations of the Hartree and Hartree-Fock formulations for a perturbed atomic system are discussed. It is pointed out that there are two alternative procedures, one of which is correct to first order in the error of the unperturbed wave function, but not the other, and explicit expressions are written down for the error in the derived perturbed energies in the two cases. A quantitative assessment of the accuracy of the two procedures is provided by the calculation of the dipole and quadrupole polarizabilities of helium, a variation-iteration method being used to solve the relevant equations.

scholarly journals Theory of electron correlation in atoms and molecules

1961 ◽  
Vol 260 (1302) ◽  
pp. 379-392 ◽  
Keyword(s):  
Wave Function ◽  
Perturbation Theory ◽  
Electron Correlation ◽  
Electron System ◽  
Exclusion Principle ◽  
First Order ◽  
Hartree Fock ◽  
Order Energy ◽  
F Wave ◽  
Pair Function

Corrections to the Hartree-Fock (H. F. ) wave function, ϕ 0 , and energy of a many-electron system are given. By the use of operator techniques in perturbation theory, the first-order w. f., X ' 1 , is obtained in terms of pair functions. These satisfy equations just like those of an actual two-electron system, except that now each electron moves in the H. F. field of the entire N -electron 'medium’ added to the field of nuclei. Every pair function must be orthogonalized to each of the two H. F. orbitals associated with it to get the complete X ' 1 . This X ' 1 determines both E 2 and E 3 . The second-order energy, E 2 , comes out as the sum of pair interactions and three- and four-body correlations due to the exclusion principle. The latter may be incorporated into the pair energies if each pair function is orthogonalized also to the remaining H. F. orbitals of ϕ 0 . The approach allows the valence and inner shells, etc., to be discussed separately and extends some of the concepts of quantum chemistry based on orbital approximations so as to include correlation.

Perturbation theory of short-range atomic interactions

1970 ◽  
Vol 317 (1531) ◽  
pp. 545-574 ◽  
Keyword(s):  
Perturbation Theory ◽  
Short Range ◽  
Similarity Transformation ◽  
Order Theory ◽  
First Order ◽  
Hartree Fock ◽  
First Order Theory ◽  
Atomic Interactions ◽  
Cusp Conditions ◽  
Molecular Calculations

A new united atom perturbation theory of the interaction of two atoms at small separations is described. The key feature is a similarity transformation of the Schrödinger equation which enables the cusp conditions to be satisfied at both nuclei and preserves the correct molecular symmetry. The first-order theory is examined in detail and compared with other united atom theories. Numerical calculations are presented for the ground states of the systems H + 2 , HeH 2+ HeH, He 2 and Li + He, based mainly on Hartree-Fock wavefunctions for the united atoms, and are compared with accurate molecular calculations. The agreement is remarkably good for separations up to 1 bohr.

CALCULATION OF THEORETICAL ISOTROPIC COMPTON PROFILE FOR MANY PARTICLE SYSTEMS

2010 ◽  
Vol 24 (14) ◽  
pp. 1601-1614
Author(s):  
ALI A. ALZUBADI ◽  
KHALIL H. ALBAYATI
Keyword(s):  
Wave Function ◽  
Impulse Approximation ◽  
Particle Systems ◽  
Electron Interaction ◽  
Compton Profile ◽  
High Momentum ◽  
Atomic Systems ◽  
Hartree Fock ◽  
Hf Wave ◽  
Momentum Region

Theoretical isotropic (spherically symmetric) Compton profiles (ICP) have been calculated for many particle systems' He , Li , Be and B atoms in their ground states. Our calculations were performed using Roothan–Hartree–Fock (RHF) wave function, HF wave function of Thakkar and re-optimized HF wave function of Clementi–Roetti, taking into account the impulse approximation. The theoretical analysis included a decomposition of the various intra and inter shells and their contributions in the total ICP. A high momentum region of up to 4 a.u. was investigated and a non-negligible tail was observed in all ICP curves. The existence of a high momentum tail was mainly due to the electron–electron interaction. The ICP for the He atom has been compared with the available experimental data and it is found that the ICP values agree very well with them. A few low order radial momentum expectation values 〈pn〉 and the total energy for these atomic systems have also been calculated and compared with their counterparts' wave functions.

Perturbation treatment of the Hartree–Fock equations for the ground states of the boron-like ions

1969 ◽  
Vol 47 (7) ◽  
pp. 699-705 ◽  
Author(s):  
C. S. Sharma ◽  
R. G. Wilson
Keyword(s):  
Ground State ◽  
Atomic System ◽  
Ground States ◽  
Great Accuracy ◽  
Second Order ◽  
Expectation Values ◽  
Third Order ◽  
First Order ◽  
Hartree Fock ◽  
The Third

The first-order Hartree–Fock and unrestricted Hartree–Fock equations for the ground state of a five electron atomic system are solved exactly. The solutions are used to evaluate the corresponding second-order energies exactly and the third-order energies with great accuracy. The first-order terms in the expectation values of 1/r, r, r2, and δ(r) are also calculated.

The Polarizability of Molecular Hydrogen H2

1936 ◽  
Vol 32 (2) ◽  
pp. 260-264 ◽  
Author(s):  
C. E. Easthope
Keyword(s):  
Wave Function ◽  
Perturbation Theory ◽  
Molecular Hydrogen ◽  
Applied Field ◽  
Minimum Energy ◽  
Wave Functions ◽  
The Other ◽  
System A ◽  
The One ◽  
Two Parameter

1. The problem of calculating the polarizability of molecular hydrogen has recently been considered by a number of investigators. Steensholt and Hirschfelder use the variational method developed by Hylleras and Hassé. For ψ0, the wave function of the unperturbed molecule when no external field is present, they take either the Rosent or the Wang wave function, while the wave functions of the perturbed molecule were considered in both the one-parameter form, ψ0 [1+A(q1 + q2)] and the two-parameter form, ψ0 [1+A(q1 + q2) + B(r1q1 + r2q2)], where A and B are parameters to be varied so as to give the system a minimum energy, q1 and q2 are the coordinates of the electrons 1 and 2 in the direction of the applied field as measured from the centre of the molecule, and r1 and r2 are their respective distances from the same point. Mrowka, on the other hand, employs a method based on the usual perturbation theory. Their numerical results are given in the following table.

An expansion method for calculating atomic properties IV. Transition probabilities

1964 ◽  
Vol 280 (1381) ◽  
pp. 258-270 ◽  
Keyword(s):  
Wave Function ◽  
Matrix Element ◽  
Transition Probabilities ◽  
Expansion Method ◽  
The State ◽  
Diagonal Matrix Element ◽  
Expectation Value ◽  
First Order ◽  
Hartree Fock ◽  
Atomic Properties

An earlier expression for the expectation value of a single-electron operator which isstationary with respect to first-order variations of the state wave function has been generalized to the case of an off-diagonal matrix element connecting two different states. Explicit calculations are carried out of the probabilities of dipole transitions between configurations 1 s a 2 s b 2 p c and 1 s a 2 s b–1 2 p c+1 for all members of the isoelectronic sequences from helium to neon and the importance of taking into account the mixing of degenerate configurations is demonstrated. The accuracy is at least comparable to that of the Hartree-Fock approximation and in cases where degeneracy is important it is much superior.

scholarly journals OPTIMIZED QUASIPARTICLE DENSITY FUNCTIONAL APPROACH FOR MULTIELECTRON ATOMIC SYSTEMS

2021 ◽  
pp. 38-44
Author(s):  
A. Glushkov ◽  
V. Kovalchuk ◽  
A. Sofronkov ◽  
A. Svinarenko
Keyword(s):  
Density Functional ◽  
Atomic System ◽  
Mass Operator ◽  
Fermi Liquids ◽  
Dft Theory ◽  
Functional Theory ◽  
Atomic Systems ◽  
Hartree Fock ◽  
The Third ◽  
Quasiparticle Density

We present the optimized version of the quasiparticle density functional theory (DFT), constructed on the principles of the Landau-Migdal Fermi-liquids theory and principles of the optimized one-quasiparticle representation in theory of multielectron systems. The master equations can be naturally obtained on the basis of variational principle, starting  from a Lagrangian of an atomic system as a functional of  three quasiparticle densities. These densities  are similar to the Hartree-Fock (HF)  electron density and kinetical energy density correspondingly, however the third density  has no an analog in the Hartree-Fock or the standard  DFT theory and appears as result of account for the energy dependence of the mass operator S. The elaborated  approach to construction of the eigen-functions basis can be characterized as an improved one in comparison with similar basises of other one-particle representations, namely, in the HF,  the standard Kohn-Sham approximations etc.

Hypervirial theorems and perturbation theory in quantum mechanics

1965 ◽  
Vol 283 (1393) ◽  
pp. 229-237 ◽  
Keyword(s):  
Wave Function ◽  
Perturbation Theory ◽  
Wave Functions ◽  
First Order ◽  
Optimum Energy ◽  
Arbitrary Operator ◽  
Approximate Wave Function ◽  
Hypervirial Theorem ◽  
Variational Wave Functions ◽  
Approximate Wave

Previous ideas about the way in which hypervirial theorems might be used to improve approximate wave functions are discussed. To provide a firmer foundation for these ideas, a link is established between hypervirial theorems and perturbation theory. It is proved that if the first-order perturbation correction to the expectation value of an arbitrary operator vanishes, then the approximate wave function used satisfies a certain hypervirial theorem. Conversely, if an arbitrary hypervirial theorem is satisfied by the wave function, then it is proved that the expectation values of certain operators have vanishing first-order corrections. Some consequences of the theory as applied to variational wave functions with optimum energy are developed. The results are illustrated by the use of a simple approximate wave function for the ground state of the helium atom.

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