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

Dirichlet and von Neumann Basis Set for Quantum Mechanical Anharmonic Oscillators

1983 ◽  
Vol 38 (4) ◽  
pp. 473-476 ◽  
Author(s):  
Alejandro M. Mesón ◽  
Francisco M. Fernández ◽  
Eduardo A. Castro
Keyword(s):  
Variational Method ◽  
Harmonic Oscillator ◽  
Lower Bounds ◽  
Neumann Boundary ◽  
Basis Set ◽  
Basis Sets ◽  
Upper And Lower Bounds ◽  
Quantum Mechanical ◽  
Anharmonic Oscillators ◽  
Von Neumann

It is shown that accurate upper and lower bounds to the eigenvalues of anharmonic oscillators can be obtained by means of the Rayleigh-Ritz variational method and two trigonometric basis sets of functions which satisfy Dirichlet and Von Neumann boundary conditions. Numerical results show that the Dirichlet basis set is more appropriate than the harmonic oscillator one for calculating eigenvalues and the value of eigenfunctions at the origin.

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 Probing the Basis Set Limit for Thermochemical Contributions of Inner-Shell Correlation: Balance of Core-Core and Core-Valence Contributions

2018 ◽  
Author(s):  
Nitai Sylvetsky ◽  
Gershom Martin
Keyword(s):  
Lower Cost ◽  
Computational Cost ◽  
High Accuracy ◽  
Basis Set ◽  
Basis Sets ◽  
Convergence Behavior ◽  
The Core ◽  
Core Correlation ◽  
Cost Calculations

The inner-shell correlation contributions to the total atomization energies of the W4-17 computational thermochemistry benchmark have been determined at the CCSD(T) level near the basis set limit using several families of core correlation basis sets, such as aug-cc-pCVnZ (n=3-6), aug-cc-pwCVnZ (n=3-5), and nZaPa-CV (n=3-5). The three families of basis sets agree very well with each other (0.01 kcal/mol RMS) when extrapolating from the two largest available basis sets: however, there are considerable differences in convergence behavior for the smaller basis sets. nZaPa-CV is superior for the core-core term and awCVnZ for the core-valence term. While the aug-cc-pwCV(T+d)Z basis set of Yockel and Wilson is superior to aug-cc-pwCVTZ, further extension of this family proved unproductive. The best compromise between accuracy and computational cost, in the context of high-accuracy computational thermochemistry methods such as W4 theory, is CCSD(T)/awCV{T,Q}Z, where the {T,Q} notation stands for extrapolation from the awCVTZ and awCVQZ basis set pair. For lower-cost calculations, a previously proposed combination of CCSD-F12b/cc-pCVTZ-F12 and CCSD(T)/pwCVTZ(no f) appears to ‘give the best bang for the buck’. While core-valence correlation accounts for the lion’s share of the inner shell contribution in first-row molecules, for second-row molecules core-core contributions may become important, particularly in systems like P<sub>4</sub>and S<sub>4</sub>with multiple adjacent second-row atoms.<div>[In memory of Dieter Cremer, 1944-2017]</div>

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 &kappa;, 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 &kappa; come in pairs of opposite energy.

scholarly journals Probing the Basis Set Limit for Thermochemical Contributions of Inner-Shell Correlation: Balance of Core-Core and Core-Valence Contributions

2018 ◽  
Author(s):  
Nitai Sylvetsky ◽  
Gershom Martin
Keyword(s):  
Lower Cost ◽  
Computational Cost ◽  
High Accuracy ◽  
Basis Set ◽  
Basis Sets ◽  
Convergence Behavior ◽  
The Core ◽  
Core Correlation ◽  
Cost Calculations

The inner-shell correlation contributions to the total atomization energies of the W4-17 computational thermochemistry benchmark have been determined at the CCSD(T) level near the basis set limit using several families of core correlation basis sets, such as aug-cc-pCVnZ (n=3-6), aug-cc-pwCVnZ (n=3-5), and nZaPa-CV (n=3-5). The three families of basis sets agree very well with each other (0.01 kcal/mol RMS) when extrapolating from the two largest available basis sets: however, there are considerable differences in convergence behavior for the smaller basis sets. nZaPa-CV is superior for the core-core term and awCVnZ for the core-valence term. While the aug-cc-pwCV(T+d)Z basis set of Yockel and Wilson is superior to aug-cc-pwCVTZ, further extension of this family proved unproductive. The best compromise between accuracy and computational cost, in the context of high-accuracy computational thermochemistry methods such as W4 theory, is CCSD(T)/awCV{T,Q}Z, where the {T,Q} notation stands for extrapolation from the awCVTZ and awCVQZ basis set pair. For lower-cost calculations, a previously proposed combination of CCSD-F12b/cc-pCVTZ-F12 and CCSD(T)/pwCVTZ(no f) appears to ‘give the best bang for the buck’. While core-valence correlation accounts for the lion’s share of the inner shell contribution in first-row molecules, for second-row molecules core-core contributions may become important, particularly in systems like P<sub>4</sub>and S<sub>4</sub>with multiple adjacent second-row atoms.<div>[In memory of Dieter Cremer, 1944-2017]</div>

Continuum Contributions to Dipole Oscillator-Strength Sum Rules for Hydrogen in Finite Basis Sets

2017 ◽  
pp. 229-241 ◽  
Author(s):  
Jens Oddershede ◽  
John F. Ogilvie ◽  
Stephan P.A. Sauer ◽  
John R. Sabin
Keyword(s):  
Oscillator Strength ◽  
Sum Rules ◽  
Finite Basis ◽  
Basis Sets ◽  
Dipole Oscillator

scholarly journals An Overview of Self-Consistent Field Calculations Within Finite Basis Sets

Molecules ◽  
2020 ◽  
Vol 25 (5) ◽  
pp. 1218 ◽  
Author(s):  
Susi Lehtola ◽  
Frank Blockhuys ◽  
Christian Van Alsenoy
Keyword(s):  
Density Functional ◽  
Finite Basis ◽  
The Self ◽  
Basis Set ◽  
Alternative Methods ◽  
Basis Sets ◽  
Field Equations ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field

A uniform derivation of the self-consistent field equations in a finite basis set is presented. Both restricted and unrestricted Hartree–Fock (HF) theory as well as various density functional approximations are considered. The unitary invariance of the HF and density functional models is discussed, paving the way for the use of localized molecular orbitals. The self-consistent field equations are derived in a non-orthogonal basis set, and their solution is discussed also in the presence of linear dependencies in the basis. It is argued why iterative diagonalization of the Kohn–Sham–Fock matrix leads to the minimization of the total energy. Alternative methods for the solution of the self-consistent field equations via direct minimization as well as stability analysis are briefly discussed. Explicit expressions are given for the contributions to the Kohn–Sham–Fock matrix up to meta-GGA functionals. Range-separated hybrids and non-local correlation functionals are summarily reviewed.

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.

Sign in / Sign up

Export Citation Format

Share Document