ON THE SPHERICAL OSCILLATOR NUCLEUS

1959 ◽  
Vol 37 (8) ◽  
pp. 944-964 ◽  
Author(s):  
T. D. Newton
Keyword(s):  
Binding Energy ◽  
Model Potential ◽  
Wave Functions ◽  
Potential Well ◽  
Free Nucleon ◽  
Hartree Fock ◽  
Self Consistent ◽  
Low Energies ◽  
Spherical Oscillator ◽  
State Functions

The degree of consistency of an oscillator model of a nucleus is examined by means of a type of Hartree–Fock calculation based on a simple form of internucleon potential valid at low energies. An effective mass equal to 0.757 times the mass of a free nucléon is used. The oscillator wave functions are found to be not far from self-consistent and the oscillator frequency derived is physically reasonable, but the bound on the binding energy is not good. It is also shown that the oscillator wave functions are a good approximation for the state functions of particles bound in a finite potential well having the shape of a cutoff oscillator so that the Hartree–Fock calculation can be used to prescribe a shell model potential.

Accurate Numerical Hartree–Fock Self-Consistent-Field Wave Functions for La+, Tm+, and Yb+

1972 ◽  
Vol 50 (7) ◽  
pp. 708-709 ◽  
Author(s):  
K. M. S. Saxena
Keyword(s):  
Rare Earth ◽  
Virial Theorem ◽  
Wave Functions ◽  
Rare Earth Ions ◽  
Radial Wave ◽  
Hartree Fock ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Field Wave

Accurate numerical Hartree–Fock (HF) self-consistent-field (SCF) wave functions have been obtained for La+(4ƒ16S)3F and 1F, Tm+(4ƒ136S)3F and 1F, and Yb(4ƒ146S)2S rare-earth ions. In general, the total energy values have an accuracy of seven figures, the virial theorem is satisfied to seven significant digits, and the radial wave functions are self-consistent and without tail oscillations to three decimals. Several Hartree–Fock parameters are also evaluated with these functions.

HARTREE–FOCK WAVE FUNCTIONS FOR EXCITED STATES: II. SIMPLIFICATION OF THE ORBITAL EQUATIONS

1966 ◽  
Vol 44 (12) ◽  
pp. 3227-3240 ◽  
Author(s):  
Maurice Cohen ◽  
Paul S. Kelly
Keyword(s):  
Total Energy ◽  
Excited States ◽  
Wave Functions ◽  
Lower State ◽  
Hartree Fock ◽  
Orthogonality Constraint ◽  
State Functions

Hartree–Fock wave functions have been calculated for a number of excited states of the helium sequence, the wave functions being constrained to be orthogonal to all lower state functions. The effect of choosing the inner 1s orbital so that the orthogonality constraint is satisfied automatically has been examined, and it is shown that such a choice has a very small effect on the total energy. An extension to heavier systems is proposed.

APPROXIMATE WAVE FUNCTIONS OF Pb+++ BY THE METHOD OF SELF-CONSISTENT FIELD WITHOUT EXCHANGE

1959 ◽  
Vol 37 (9) ◽  
pp. 983-988 ◽  
Author(s):  
J. F. Hart ◽  
Beatrice H. Worsley
Keyword(s):  
Common Point ◽  
Wave Functions ◽  
The Self ◽  
Decimal Digit ◽  
Hartree Fock ◽  
The Core ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Approximate Wave

The FERUT program previously described for calculating Hartree–Fock wave functions by the method of the self-consistent field has been adapted to the configuration Pb+++. Although the exchange factors were omitted, the program was extended beyond its original scope in other respects, and an assessment of the difficulties so encountered is made. It might be noted, however, that, except in the case of the 4ƒ wave function, it was possible to begin all the integrations at a common point. Initial estimates were made from the Douglas, Hartree, and Runciman results for thallium. The estimates for the core functions were not assumed to be satisfactory. The errors in the final wave functions are considered to be no more than one or two units in the second decimal digit.

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>

Radial wave functions with exchange for Va 2+ , Kr and Ag +

1958 ◽  
Vol 247 (1250) ◽  
pp. 390-399 ◽  
Keyword(s):  
Wave Functions ◽  
Initial Slope ◽  
Radial Wave ◽  
University Of Toronto ◽  
Hartree Fock ◽  
Consistent Field ◽  
Self Consistent ◽  
Physical Approximation ◽  
Self Consistent Field ◽  
The University

A generalized program for calculating atomic radial wave functions with exchange has been prepared for the Ferranti computer (FERUT) at the University of Toronto, and is described in a separate paper. This program has now been applied to V 2+ , Kr and Ag + . The wave functions for these atoms, together with the energy and initial slope parameters, are presented to the accuracy justified by the physical approximation of the Hartree–Fock formulation. The configurations of Kr and Ag + are considerably larger than any which have previously been treated by the self-consistent field process with exchange.

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>

A qualitative investigation of the wave functions of metallic uranium

1958 ◽  
Vol 247 (1249) ◽  
pp. 199-213 ◽  
Keyword(s):  
Wave Function ◽  
Binding Energy ◽  
Binding Energies ◽  
Wave Functions ◽  
Atomic Sphere ◽  
Maximum Charge ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Negative Binding Energy ◽  
Metallic Uranium

The potential field and wave functions in metallic uranium have been calculated approximately by determining Hartree self-consistent fields for the four configurations (6 d ) 6 , (5 f ) 2 (6 d ) 2 (7 s ) 2 , (5 f ) 4 (6 d ) 2 , and (5 f ) 6 using the Wigner–Seitz boundary condition at the surface of the equivalent atomic sphere. In the self-consistent field obtained for each configuration, wave functions falling to zero at the surface of the sphere were evaluated for the outer electrons. The differences between the Hartree ∊-parameters for the two boundary conditions were used to give an indication of the relative band widths. The (5 f ) wave function has its principal maximum inside the (6 s ) (6 p ) shell, but is appreciable at the surface of the sphere and must participate in bonding. The binding energy of an electron in the (5 f ) wave function is 0.3934 rydbergs in the configuration (5 f ) 2 (6 d ) 2 (7 s ) 2 as given by the Hartree ∊-parameter. The (6 d ) function has its maximum charge density at the boundary and, with a binding energy of 0.1749 rydbergs in the same configuration, is likely to form a good metallic band. The (7 s ) function has a large negative binding energy and is not likely to occur. In the configuration (5 f ) 2 (6 d ) 2 (7 s ) 2 the band widths of the (5 f ), (6 d ) and (7 s ) functions are in the ratio 1:9∙8:22∙7. As the number of (5 f ) electrons is increased all the binding energies decrease and the band widths increase. At the same time the (6 s ) and (6 p ) functions are markedly perturbed, the functions (5 s ), (5 p ) and (5 d ) to a lesser extent.

HARTREE–FOCK WAVE FUNCTIONS FOR EXCITED STATES: THE 2 1S STATE OF HELIUM

1965 ◽  
Vol 43 (10) ◽  
pp. 1867-1881 ◽  
Author(s):  
Maurice Cohen ◽  
Paul S. Kelly
Keyword(s):  
Excited State ◽  
Ground State ◽  
Wave Functions ◽  
State Function ◽  
Hartree Fock ◽  
Singlet States ◽  
Low Energies ◽  
Orthogonality Constraint ◽  
Satisfactory Approximation ◽  
Excited Singlet States

Hartree–Fock wave functions for the first-excited singlet states of several members of the helium isoelectronic sequence have been calculated. It is shown that unphysically low energies result if the excited-state function is not constrained to be orthogonal to the ground-state eigenfunction, and the excited-state orbitals are chosen orthogonal.If the excited-state orbitals are not chosen orthogonal, the effect of the overall constraint is small, raising the total energy very slightly; however, the effect of imposing both constraints simultaneously is appreciable. It is concluded that the most satisfactory approximation involves nonorthogonal excited-state orbitals, together with an overall orthogonality constraint towards the ground state.

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