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.

An expansion method for calculating atomic properties X. 1 s 2 1 S -1 snp 1 P transitions of the helium sequence

1967 ◽  
Vol 301 (1466) ◽  
pp. 253-260 ◽  
Keyword(s):  
Cross Sections ◽  
Nuclear Charge ◽  
Expansion Method ◽  
Probable Error ◽  
Oscillator Strengths ◽  
Matrix Elements ◽  
Isoelectronic Sequence ◽  
First Order ◽  
Hartree Fock ◽  
Atomic Properties

The electric dipole matrix elements connecting the 1 s 2 1 S and 1 snp 1 P states of the helium isoelectronic sequence are calculated exactly to first order in inverse powers of the nuclear charge Z and the differences from the Hartree-Fock approximation are shown to correspond to virtual transitions of the 1 s electrons. Comparison of the oscillator strengths predicted by a screening approximation with more accurate values reveals a regular variation in the error contained in the screening approximation, the correction of which allows the prediction of oscillator strengths and probabilities of 1 s 2 1 S – 1 snp 1 P transitions for all values of n and all values of Z within a probable error of 2% (table 5). Values of the photoionization cross-sections at the spectral heads are also presented.

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).

An expansion method for calculating atomic properties, V. Transition probabilities of M shell electrons

1965 ◽  
Vol 286 (1407) ◽  
pp. 510-518 ◽  
Keyword(s):  
Transition Probabilities ◽  
Expansion Method ◽  
Atomic Properties

The expansion method is extended to M shell electrons and explicit calculations are carried out of the probabilities of all dipole transitions between configurations 1 8 2 2 8 2 2 p 6 3 l —1 8 2 2 8 2 2 p 6 3 l ´, and between configurations 1 8 2 2 8 2 2 p 6 3 l 1 3 l 2 —1 8 2 2 8 2 2 p 6 3 l 1 3 l 2 ´, account being taken of the mixing of degenerate configurations. Results are presented for all members of the sodium and magnesium isoelectronic sequences.

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.

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.

An expansion method for calculating atomic properties. VIII. Transitions in the Hartree-Fock approximation

1966 ◽  
Vol 293 (1434) ◽  
pp. 359-364 ◽  
Keyword(s):  
Quantum Number ◽  
Principal Quantum Number ◽  
Expansion Method ◽  
Matrix Elements ◽  
Order Error ◽  
First Order ◽  
Hartree Fock ◽  
Transition Matrix Elements ◽  
Active Electron

According to Brillouin’s theorem, diagonal matrix elements of one-electron operators are given correct to first order by the Hartree-Fock approximation. Transition matrix elements of one-electron operators are not given correct to first order because of the contributions from virtual excitations involving a change in the azimuthal quantum number of the active electron but no change in the principal quantum number. The electric dipole matrix elements connecting the 1 s 2 s 1,3 S states of helium to the 1 s 2 p 1,3 P states are calculated exactly to first order in inverse powers of the nuclear charge and the differences from the Hartree-Fock approximation are shown explicitly to correspond to virtual transitions of the inner shell 1 s electron of the type 1 s — np . A generalization of the Hartree-Fock approximation is discussed which eliminates most of the first order error.

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 Potential Energy Expectation Value for Lithium Excited State (1s2s3s)

2015 ◽  
Vol 12 (2) ◽  
pp. 394-399
Author(s):  
Baghdad Science Journal
Keyword(s):  
Wave Function ◽  
Excited State ◽  
Potential Energy ◽  
Atomic Number ◽  
Spin States ◽  
Expectation Value ◽  
Hartree Fock ◽  
Linear Behaviour ◽  
Energy Expectation ◽  
Energy Expectation Value

The purpose of the present work is to calculate the expectation value of potential energy for different spin states (??? ? ???,??? ? ???) and compared it with spin states (??? , ??? ) for lithium excited state (1s2s3s) and Li- like ions (Be+,B+2) using Hartree-Fock wave function by partitioning techanique .The result of inter particle expectation value shows linear behaviour with atomic number and for each atom and ion the shows the trend ??? < ??? < ??? < ???

"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 A study of some atomic properties for He-like selected ions

2007 ◽  
Vol 4 (2) ◽  
pp. 301-304
Author(s):  
Baghdad Science Journal
Keyword(s):  
Wave Function ◽  
Form Factor ◽  
Configuration Interaction ◽  
Diamagnetic Susceptibility ◽  
Wave Functions ◽  
Magnetic Shielding ◽  
Hartree Fock ◽  
Atomic Properties ◽  
Atomic Form Factor ◽  
He Atom

The atomic properties have been studied for He-like ions (He atom, Li+, Be2+ and B3+ions). These properties included, the atomic form factor f(S), electron density at the nucleus , nuclear magnetic shielding constant and diamagnetic susceptibility ,which are very important in the study of physical properties of the atoms and ions. For these purpose two types of the wave functions applied are used, the Hartree-Fock (HF) waves function (uncorrelated) and the Configuration interaction (CI) wave function (correlated). All the results and the behaviors obtained in this work have been discussed, interpreted and compared with those previously obtained.

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