Relation between exchange-only optimized potential and Kohn–Sham methods with finite basis sets, and effect of linearly dependent products of orbital basis functions

2008 ◽  
Vol 128 (10) ◽  
pp. 104104 ◽  
Author(s):  
Andreas Görling ◽  
Andreas Heßelmann ◽  
Martin Jones ◽  
Mel Levy
Keyword(s):  
Finite Basis ◽  
Basis Functions ◽  
Basis Sets ◽  
Orbital Basis

scholarly journals Charge Conjugation Symmetry in the Finite Basis Approximation of the Dirac Equation

2020 ◽  
Author(s):  
Maen Salman ◽  
Trond Saue
Keyword(s):  
Angular Momentum ◽  
Dirac Equation ◽  
Free Particle ◽  
Finite Basis ◽  
Basis Functions ◽  
Charge Conjugation ◽  
Basis Set ◽  
Basis Sets ◽  
Pair Approximation ◽  
Central Field

4-component relativistic atomic and molecular calculations are typically performed within the no-pair approximation where negative-energy solutions are discarded, hence the symmetry between electronic and positronic solutions is not considered. These states are however needed in QED calculations, where furthermore charge conjugation symmetry becomes an issue. In this work we shall discuss the realization of charge conjugation symmetry of the Dirac equation in a central field within the finite basis approximation. Three schemes for basis set construction are considered: restricted, inverse and dual kinetic balance. We find that charge conjugation symmetry can be realized within the restricted and inverse kinetic balance prescriptions, but only with a special form of basis functions that does not obey the right boundary conditions of the radial wavefunctions. The dual kinetic balance prescription is on the other hand compatible with charge conjugation symmetry without restricting the form of the radial basis functions. However, since charge conjugation relates solutions of opposite value of the quantum number κ, this requires the use of basis sets chosen according to total angular momentum j rather than orbital angular momentum ` . As a special case, we consider the free-particle Dirac equation, where the solutions of opposite sign of energy are related by charge conjugation symmetry. We note that there is additional symmetry in those solutions of the same value of κ come in pairs of opposite energy.

scholarly journals Charge Conjugation Symmetry in the Finite Basis Approximation of the Dirac Equation

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1121
Author(s):  
Maen Salman ◽  
Trond Saue
Keyword(s):  
Angular Momentum ◽  
Dirac Equation ◽  
Free Particle ◽  
Finite Basis ◽  
Basis Functions ◽  
Charge Conjugation ◽  
Basis Set ◽  
Basis Sets ◽  
Pair Approximation ◽  
Central Field

Four-component relativistic atomic and molecular calculations are typically performed within the no-pair approximation where negative-energy solutions are discarded. These states are, however, needed in QED calculations, wherein, furthermore, charge conjugation symmetry, which connects electronic and positronic solutions, becomes an issue. In this work, we shall discuss the realization of charge conjugation symmetry of the Dirac equation in a central field within the finite basis approximation. Three schemes for basis set construction are considered: restricted, inverse, and dual kinetic balance. We find that charge conjugation symmetry can be realized within the restricted and inverse kinetic balance prescriptions, but only with a special form of basis functions that does not obey the right boundary conditions of the radial wavefunctions. The dual kinetic balance prescription is, on the other hand, compatible with charge conjugation symmetry without restricting the form of the radial basis functions. However, since charge conjugation relates solutions of opposite value of the quantum number κ , this requires the use of basis sets chosen according to total angular momentum j rather than orbital angular momentum ℓ. As a special case, we consider the free-particle Dirac equation, where opposite energy solutions are related by charge conjugation symmetry. We show that there is additional symmetry in that solutions of the same value of κ come in pairs of opposite energy.

Three-center molecular integrals and derivatives using solid harmonic Gaussian orbital and Kohn–Sham potential basis sets

2013 ◽  
Vol 91 (9) ◽  
pp. 907-915 ◽  
Author(s):  
Anguang Hu ◽  
Brett I. Dunlap
Keyword(s):  
Angular Momentum ◽  
Basis Function ◽  
Basis Sets ◽  
Factors Associated ◽  
Orbital Basis ◽  
Type Integral ◽  
Molecular Integrals ◽  
Potential Basis ◽  
Atom Position ◽  
Gaussian Orbital

Three-center integrals over Gaussian orbital and Kohn–Sham (KS) basis sets are reviewed. An orbital basis function carries angular momentum about its atomic center. That angular momentum is created by solid harmonic differentiation with respect to the center of an s-type basis function. That differentiation can be brought outside any purely s-type integral, even nonlocal pseudopotential integrals. Thus the angular factors associated with angular momentum and differentiation with respect to atom position can be pulled outside loops over orbital and KS Gaussian exponents.

scholarly journals Analysis of atomic orbital basis sets from the projection of plane-wave results

1996 ◽  
Vol 8 (21) ◽  
pp. 3859-3880 ◽  
Author(s):  
Daniel Sánchez-Portal ◽  
Emilio Artacho ◽  
José M Soler
Keyword(s):  
Plane Wave ◽  
Atomic Orbital ◽  
Basis Sets ◽  
Orbital Basis

Finite basis sets for the Dirac equation constructed fromBsplines

1988 ◽  
Vol 37 (2) ◽  
pp. 307-315 ◽  
Author(s):  
W. R. Johnson ◽  
S. A. Blundell ◽  
J. Sapirstein
Keyword(s):  
Dirac Equation ◽  
Finite Basis ◽  
Basis Sets

scholarly journals Perspectives on Basis Sets Beautiful: Seasonal Plantings of Diffuse Basis Functions

2011 ◽  
Vol 7 (10) ◽  
pp. 3027-3034 ◽  
Author(s):  
Ewa Papajak ◽  
Jingjing Zheng ◽  
Xuefei Xu ◽  
Hannah R. Leverentz ◽  
Donald G. Truhlar
Keyword(s):  
Basis Functions ◽  
Basis Sets

scholarly journals Gauge origin independence in finite basis sets and perturbation theory

2017 ◽  
Vol 683 ◽  
pp. 536-542 ◽  
Author(s):  
Lasse Kragh Sørensen ◽  
Roland Lindh ◽  
Marcus Lundberg
Keyword(s):  
Perturbation Theory ◽  
Finite Basis ◽  
Basis Sets
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