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.

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.

Matrix Approximations

2007 ◽  
Author(s):  
Kenneth G. Dyall ◽  
Knut Faegri
Keyword(s):  
Dirac Equation ◽  
Quantum Chemistry ◽  
Matrix Representation ◽  
Finite Basis ◽  
Basis Set ◽  
Basis Sets ◽  
Dirac Matrix ◽  
The Matrix ◽  
Proper Account ◽  
Matrix Approximations

In quantum chemistry, regardless of which operators we choose for the Hamiltonian, we almost invariably implement our chosen method in a finite basis set. The Douglas– Kroll and Barysz–Sadlej–Snijders methods in the end required a matrix representation of the momentum-dependent operators in the implementation, and the regular methods usually end up with a basis set, even if the potentials are tabulated on a grid. Why not start with a matrix representation of the Dirac equation and perform transformations on the Dirac matrix rather than doing operator transformations, for which the matrix elements are difficult to evaluate analytically? It is almost always much easier to do manipulations with matrices of operators than with the operators themselves. Provided proper account is taken in the basis sets of the correct relationships between the range and the domain of the operators (Dyall et al. 1984), matrix manipulations can be performed with little or no approximation beyond the matrix representation itself. In this chapter, we explore the use of matrix approximations.

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

Spurious states of the dirac equation in a finite basis set

2008 ◽  
Vol 105 (2) ◽  
pp. 183-188 ◽  
Author(s):  
I. I. Tupitsyn ◽  
V. M. Shabaev
Keyword(s):  
Dirac Equation ◽  
Finite Basis ◽  
Basis Set

scholarly journals Performance of polarization-consistent vs. correlation-consistent basis sets for CCSD(T) prediction of water dimer interaction energy

2019 ◽  
Vol 25 (10) ◽  
Author(s):  
Teobald Kupka ◽  
Aneta Buczek ◽  
Małgorzata A. Broda ◽  
Adrianna Mnich ◽  
Tapas Kar
Keyword(s):  
Interaction Energy ◽  
Cardinal Number ◽  
Reference Value ◽  
Basis Functions ◽  
Water Dimer ◽  
Basis Set ◽  
Basis Sets ◽  
Complete Basis Set ◽  
Correlation Consistent ◽  
Consistent Basis

Abstract Detailed study of Jensen’s polarization-consistent vs. Dunning’s correlation-consistent basis set families performance on the extrapolation of raw and counterpoise-corrected interaction energies of water dimer using coupled cluster with single, double, and perturbative correction for connected triple excitations (CCSD(T)) in the complete basis set (CBS) limit are reported. Both 3-parameter exponential and 2-parameter inverse-power fits vs. the cardinal number of basis set, as well as the number of basis functions were analyzed and compared with one of the most extensive CCSD(T) results reported recently. The obtained results for both Jensen- and Dunning-type basis sets underestimate raw interaction energy by less than 0.136 kcal/mol with respect to the reference value of − 4.98065 kcal/mol. The use of counterpoise correction further improves (closer to the reference value) interaction energy. Asymptotic convergence of 3-parameter fitted interaction energy with respect to both cardinal number of basis set and the number of basis functions are closer to the reference value at the CBS limit than other fitting approaches considered here. Separate fits of Hartree-Fock and correlation interaction energy with 3-parameter formula additionally improved the results, and the smallest CBS deviation from the reference value is about 0.001 kcal/mol (underestimated) for CCSD(T)/aug-cc-pVXZ calculations. However, Jensen’s basis set underestimates such value to 0.012 kcal/mol. No improvement was observed for using the number of basis functions instead of cardinal number for fitting.

Basis-set methods for the Dirac equation

2002 ◽  
Vol 80 (3) ◽  
pp. 181-265 ◽  
Author(s):  
C Krauthauser ◽  
R N Hill
Keyword(s):  
Sum Rules ◽  
Yukawa Potential ◽  
Bounded Operator ◽  
Finite Basis ◽  
Basis Set ◽  
Basis Sets ◽  
Upper And Lower Bounds ◽  
Projection Operators ◽  
Convergence Behavior ◽  
Inhomogeneous Problem

The pathologies associated with finite basis-set approximations to the Dirac Hamiltonian HDirac are avoided by applying the variational principle to the bounded operator 1 / (H Dirac – W) where W is a real number that is not in the spectrum of HDirac. Methods of calculating upper and lower bounds to eigenvalues, and bounds to the wave-function error as measured by the L2 norm, are described. Convergence is proven. The rate of convergence is analyzed. Boundary conditions are discussed. Benchmark energies and expectation values for the Yukawa potential, and for the Coulomb plus Yukawa potential, are tabulated. The convergence behavior of the energy-weighted dipole sum rules, which have traditionally been used to assess the quality of basis sets, and the convergence behavior of the solutions to the inhomogeneous problem, are analyzed analytically and explored numerically. It is shown that a basis set that exhibits rapid convergence when used to evaluate energy-weighted dipole sum rules can nevertheless exhibit slow convergence when used to solve the inhomogeneous problem and calculate a polarizability. A numerically stable method for constructing projection operators, and projections of the Hamiltonian, onto positive and negative energy states is given. PACS Nos.: 31.15Pf, 31.30Jv, 31.15-p

Relativistic AB Initio calculations of interaction energies: formulation and application to ionic solids

1986 ◽  
Vol 320 (1553) ◽  
pp. 71-105 ◽  
Keyword(s):  
Ab Initio ◽  
Inner Core ◽  
Basis Set ◽  
Basis Sets ◽  
Computational Techniques ◽  
Central Field ◽  
Atomic Orbitals ◽  
Molecule Formation ◽  
Ionic Solids ◽  
Free Atoms

The theory and computational techniques used in a computer program capable of performing fully relativistic ab initio electronic structure calculations for pairs of interacting atomic species are presented. If the species are ions in a crystal, a description of an ionic solid is obtained. If the two species are otherwise free, the program yields a wavefunction for a diatomic molecule. The molecular wavefunction is an antisymmetrized product of core and valence parts. The core is a Hartree product of the Dirac—Fock atomic orbitals of the free atoms. The largest contribution to the energy arises from the inner-core orbitals, each having negligible overlap with all other orbitals. The purely atomic inner-core energy does not contribute to the binding energy of the molecule, thus obviating the need to calculate the largest part of the molecular energy. The outer core consists of those remaining closed subshells of the isolated atoms that are not significantly affected on molecule formation. All the remaining orbitals, including at least the valence Dirac—Fock atomic orbitals of the free atoms plus further atomic functions needed to describe charge density changes upon molecule formation, are used to construct the valence wavefunction. This can be constructed to take account of correlation between the valence electrons. All atomic functions have central field form with the radial parts defined numerically. This method of constructing the molecular wavefunction avoids the need for large basis sets, ensures that the Dirac small components bear the correct relation to the large components and avoids basis set superposition errors. This program is used to initiate a non-empirical study of the properties of ionic solids. The results show that these properties cannot be reliably predicted by using free ion wavefunctions and that the Watson shell model for describing the non-negligible differences between free and in-crystal ion wavefunctions is not satisfactory. The results demonstrate the importance of inter-ionic dispersive attractions but show that it is not satisfactory to neglect the part quenching of the standard long-range form of these attractions arising from overlap of the ion wavefunctions.

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