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

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.

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.

Introduction to Relativistic Quantum Chemistry

2007 ◽  
Author(s):  
Kenneth G. Dyall ◽  
Knut Faegri
Keyword(s):  
Dirac Equation ◽  
Quantum Chemistry ◽  
Relativistic Theory ◽  
Density Functional ◽  
Relativistic Quantum ◽  
Relativistic Effects ◽  
Basis Sets ◽  
Spin Orbit ◽  
Approximate Methods ◽  
Nonrelativistic Theory

This book provides an introduction to the essentials of relativistic effects in quantum chemistry, and a reference work that collects all the major developments in this field. It is designed for the graduate student and the computational chemist with a good background in nonrelativistic theory. In addition to explaining the necessary theory in detail, at a level that the non-expert and the student should readily be able to follow, the book discusses the implementation of the theory and practicalities of its use in calculations. After a brief introduction to classical relativity and electromagnetism, the Dirac equation is presented, and its symmetry, atomic solutions, and interpretation are explored. Four-component molecular methods are then developed: self-consistent field theory and the use of basis sets, double-group and time-reversal symmetry, correlation methods, molecular properties, and an overview of relativistic density functional theory. The emphases in this section are on the basics of relativistic theory and how relativistic theory differs from nonrelativistic theory. Approximate methods are treated next, starting with spin separation in the Dirac equation, and proceeding to the Foldy-Wouthuysen, Douglas-Kroll, and related transformations, Breit-Pauli and direct perturbation theory, regular approximations, matrix approximations, and pseudopotential and model potential methods. For each of these approximations, one-electron operators and many-electron methods are developed, spin-free and spin-orbit operators are presented, and the calculation of electric and magnetic properties is discussed. The treatment of spin-orbit effects with correlation rounds off the presentation of approximate methods. The book concludes with a discussion of the qualitative changes in the picture of structure and bonding that arise from the inclusion of relativity.

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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