Compton profiles from numerical Hartree–Fock wave functions

1976 ◽  
Vol 54 (21) ◽  
pp. 2155-2158 ◽  
Author(s):  
Robert Benesch
Keyword(s):  
Impulse Approximation ◽  
Wave Functions ◽  
Neutral Atoms ◽  
Hartree Fock ◽  
Hf Wave

Nonrelativistic Compton profiles, J(q)'s, calculated within the impulse approximation from numerical Hartree–Fock (HF) wave functions are reported for the neutral atoms As (Z = 33) through Yb (Z = 70). Comparison with results reported for relativistic HF wave functions indicates that the overall effect of using the relativistic functions is to produce total J(q)'s which are flatter at the centre than those computed from nonrelativistic HF wave functions.

Sternheimer Antishielding Functions ß(r) and γ(r) for Rare Earth Atoms

1986 ◽  
Vol 41 (1-2) ◽  
pp. 37-46 ◽  
Author(s):  
K. D. Sen ◽  
P. C. Schmidt ◽  
Alarich Weiss
Keyword(s):  
Rare Earth ◽  
Valence State ◽  
Wave Functions ◽  
Electric Field Gradients ◽  
Hartree Fock ◽  
Field Gradients ◽  
Lanthanide Atoms ◽  
Hf Wave ◽  
Self Consistency ◽  
Consistency Effects

The Sternheimer shielding-antishielding functions ß(r) and γ(r) are reported for all the fourteen lanthanide atoms at the uncoupled Hartree-Fock level of theory. Each atom is considered in two valence state configurations, 4fn 5d0 and 4 fn-1 5d1, and the nonrelativistic HF wave functions have been used. The 5d1 configuration leads to a smaller net antishielding than the 4fn configuration by ~ 6-12% in the series. The electron-electron self consistency effects are found to be less than 5% in the series. The importance of the calculated antishielding functions in the antishielding theory of electric field gradients in noncubic metals is discussed.

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.

scholarly journals PT-Symmetry in Hartree–Fock Theory

2019 ◽  
Author(s):  
Hugh G. A. Burton ◽  
Alex Thom ◽  
Pierre-Francois Loos
Keyword(s):  
Energy Landscape ◽  
Wave Functions ◽  
Pt Symmetry ◽  
Structure Theory ◽  
Hartree Fock ◽  
Self Consistent ◽  
Consistent Solution ◽  
Hf Wave ◽  
H2 Molecule ◽  
Occupied Orbital

<div> <div> <p>P T -symmetry — invariance with respect to combined space reflection P and time reversal T — provides a weaker condition than (Dirac) Hermiticity for ensuring a real energy spectrum of a general non-Hermitian Hamiltonian. PT -symmetric Hamiltonians therefore form an intermediate class between Hermitian and non-Hermitian Hamiltonians. In this work, we derive the conditions for PT-symmetry in the context of electronic structure theory, and specifically, within the Hartree–Fock (HF) approximation. We show that the HF orbitals are symmetric with respect to the P T operator if and only if the effective Fock Hamiltonian is PT -symmetric, and vice versa. By extension, if an optimal self-consistent solution is invariant under PT , then its eigenvalues and corresponding HF energy must be real. Moreover, we demonstrate how one can construct explicitly PT -symmetric Slater determinants by forming PT doublets (i.e. pairing each occupied orbital with its PT -transformed analogue), allowing PT -symmetry to be conserved throughout the self-consistent process. Finally, considering the H2 molecule as an illustrative example, we observe PT-symmetry in the HF energy landscape and find that the symmetry-broken unrestricted HF wave functions (i.e. diradical configurations) are P T -symmetric, while the symmetry-broken restricted HF wave functions (i.e. ionic configurations) break PT -symmetry.</p> </div> </div>

scholarly journals PT-Symmetry in Hartree–Fock Theory

2019 ◽  
Author(s):  
Hugh G. A. Burton ◽  
Alex Thom ◽  
Pierre-Francois Loos
Keyword(s):  
Energy Landscape ◽  
Wave Functions ◽  
Pt Symmetry ◽  
Structure Theory ◽  
Hartree Fock ◽  
Self Consistent ◽  
Consistent Solution ◽  
Hf Wave ◽  
H2 Molecule ◽  
Occupied Orbital

<div> <div> <p>P T -symmetry — invariance with respect to combined space reflection P and time reversal T — provides a weaker condition than (Dirac) Hermiticity for ensuring a real energy spectrum of a general non-Hermitian Hamiltonian. PT -symmetric Hamiltonians therefore form an intermediate class between Hermitian and non-Hermitian Hamiltonians. In this work, we derive the conditions for PT-symmetry in the context of electronic structure theory, and specifically, within the Hartree–Fock (HF) approximation. We show that the HF orbitals are symmetric with respect to the P T operator if and only if the effective Fock Hamiltonian is PT -symmetric, and vice versa. By extension, if an optimal self-consistent solution is invariant under PT , then its eigenvalues and corresponding HF energy must be real. Moreover, we demonstrate how one can construct explicitly PT -symmetric Slater determinants by forming PT doublets (i.e. pairing each occupied orbital with its PT -transformed analogue), allowing PT -symmetry to be conserved throughout the self-consistent process. Finally, considering the H2 molecule as an illustrative example, we observe PT-symmetry in the HF energy landscape and find that the symmetry-broken unrestricted HF wave functions (i.e. diradical configurations) are P T -symmetric, while the symmetry-broken restricted HF wave functions (i.e. ionic configurations) break PT -symmetry.</p> </div> </div>

The electrostatic calculation of molecular energies - IV. Optimum paired-electron orbitals and the electrostatic method

1956 ◽  
Vol 235 (1201) ◽  
pp. 224-234 ◽  
Keyword(s):  
Simple Procedure ◽  
Wave Functions ◽  
Optimum Form ◽  
Method Of Calculation ◽  
Hartree Fock ◽  
Electron Orbital ◽  
Electrostatic Method ◽  
Valence States ◽  
Paired Electron ◽  
Electrostatic Interpretation

Equations which determine the optimum form of paired-electron orbitals are derived. It is shown that for large nuclear separations these equations become the Hartree-Fock equa­tions for appropriate valence states of the separated atoms. An electrostatic interpretation of chemical bonding is developed using optimum paired-electron orbital functions. For these wave functions this simple procedure yields results identical with those obtained by the conventional method of calculation based on the Hamiltonian integral. Numerical computations by the electrostatic method are also discussed.

A comparison of theory and experiment for photo-ionization cross-sections I. Neon and the elements from boron to neon

1951 ◽  
Vol 208 (1094) ◽  
pp. 408-418 ◽  
Keyword(s):  
Cross Sections ◽  
General Trend ◽  
Wave Functions ◽  
Ionization Cross ◽  
Hartree Fock ◽  
Quantal Theory ◽  
Ionization Cross Sections ◽  
Photo Ionization ◽  
Near Future ◽  
Dipole Length

The quantal theory of the continuous photo-electric absorption of radiation is briefly summarized, pàrticular attention being given to the alternative formulae available and to the accuracy to be expected in practical calculations. Detailed calculations are described for the photo-ionization cross-section of neon, a system for which it is understood that experimental data should be available in the near future. The calculation is made using Hartree-Fock wave functions and the two formulae of the dipole length and the dipole velocity. The corresponding cross-sections are found to be 5.8 and 4.4 x 10- 18 cm 2 . at the spectral head and to rise slowly with increasing frequency until a broad maximum is reached for an energy of the ejected electron of about 11 eV. A comparison is made with previous calculations on the elements from boron to neon ; the general trend of the results is discussed and improved estimates for boron and fluorine are given (10 x 10 -18 cm 2 . for boron and 4.3 x 10- 18 cm 2 . for fluorine at the spectral head).

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