Bloch and Wannier functions in momentum space

1988 ◽  
Vol 53 (9) ◽  
pp. 1890-1901 ◽  
Author(s):  
Jean-Louis Calais
Keyword(s):  
Momentum Space ◽  
Plane Waves ◽  
Wave Functions ◽  
Basis Set ◽  
Wannier Functions ◽  
Atomic Orbitals ◽  
Formal Properties ◽  
Direct Calculations

Direct calculations of wave functions in momentum space require a study of the formal properties of such functions. Here we discuss the momentum space counterparts of Bloch and Wannier functions, both in general and for two extreme kinds of basis set expansions: plane waves and atomic orbitals.

From Atomic Orbitals to Molecular Orbitals and Chemical Bonds

2020 ◽  
pp. 150-187
Author(s):  
Jochen Autschbach
Keyword(s):  
Hydrogen Molecule ◽  
Atomic Orbital ◽  
Plane Waves ◽  
Basis Set ◽  
Orbital Method ◽  
Chemical Bonds ◽  
Atomic Orbitals ◽  
Hartree Fock ◽  
Generalized Matrix

It is shown how an aufbau principle for atoms arises from the Hartree-Fock (HF) treatment with increasing numbers of electrons. The Slater screening rules are introduced. The HF equations for general molecules are not separable in the spatial variables. This requires another approximation, such as the linear combination of atomic orbitals (LCAO) molecular orbital method. The orbitals of molecules are represented in a basis set of known functions, for example atomic orbital (AO)-like functions or plane waves. The HF equation then becomes a generalized matrix pseudo-eigenvalue problem. Solutions are obtained for the hydrogen molecule ion and H2 with a minimal AO basis. The Slater rule for 1s shells is rationalized via the optimal exponent in a minimal 1s basis. The nature of the chemical bond, and specifically the role of the kinetic energy in covalent bonding, are discussed in details with the example of the hydrogen molecule ion.

Molecular orbitals in momentum space

1965 ◽  
Vol 286 (1406) ◽  
pp. 376-389 ◽  
Keyword(s):  
Momentum Space ◽  
Wave Functions ◽  
Basis Set ◽  
Equilibrium Distance ◽  
Numerical Work ◽  
Algebraic Expressions ◽  
Unitary Transformations ◽  
The Matrix ◽  
Internuclear Distances ◽  
Ground State Energies

The troublesome problem of developing cusps in ordinary molecular wave functions can be avoided by working with momentum-space wavefunctions for these have no cusps. The need for continuum wavefunctions can be eliminated if one works with a hydrogenic basis set in Fock’s projective momentmn space. This basis set is the set of R 4 spherical harmonics and as a consequence one may obtain, solely by the ordinary angular momentum calculus, algebraic expressions for all the integrals required in the solution of the momentum space Schrödinger equation. A number of these integrals and a number of R 4 transformation coefficients are tabulated. The method is then applied to several simple united-atom and l.c.a.o. wavefunctions for H + 2 and ground state energies and corrected wavefunctions are obtained. It is found in this numerical work that the method is most appropriate at internuclear distances somewhat less than the equilibrium distance. In Fock’s representation both l.c.a.o. and unitedatom approximations become exact as the internuclear distance approaches zero. The united-atom expansion can be viewed as an eigenvalue equation for the root-mean-square momentum, p 0 = √( — 2 E ). In the molecule, the matrix operator corresponding to p 0 is related to the operator for the united-atom by a sum of unitary transformations, one for each nucleus in the molecule.

SURFACE STATES OF NICKEL

1993 ◽  
Vol 07 (01n03) ◽  
pp. 546-549 ◽  
Author(s):  
V. CRISAN ◽  
A. VERNES ◽  
L. DULCA ◽  
V. POPESCU ◽  
D. KAPUSI
Keyword(s):  
Transition Metals ◽  
Brillouin Zone ◽  
Bound States ◽  
Plane Waves ◽  
Tight Binding ◽  
Surface States ◽  
Wave Functions ◽  
Basis Set ◽  
Scattering Method ◽  
Surface Brillouin Zone

In the present paper a calculation for the surface states of the 3d transition metals is reported. The scattering method is used in the combined LCAO-OPW representation for the wave functions. The basis set, in the presence of perturbation, is formed by tight-binding orbitals and plane waves. For the unreconstructed (100) surface of Ni, the surface Brillouin zone projected band1 structure and the bound states for the Γ point computed.

Ab initio band theory approach to electron energy loss near edge structures

1988 ◽  
Vol 46 ◽  
pp. 506-507
Author(s):  
Xudong Weng ◽  
O.F. Sankey ◽  
Peter Rez
Keyword(s):  
Ab Initio ◽  
Local Density ◽  
Interpolation Method ◽  
Atomic Orbital ◽  
Plane Waves ◽  
Large Set ◽  
Theory Approach ◽  
Band Theory ◽  
Atomic Orbitals ◽  
Pseudopotential Calculations

Single electron band structure techniques have been applied successfully to the interpretation of the near edge structures of metals and other materials. Among various band theories, the linear combination of atomic orbital (LCAO) method is especially simple and interpretable. The commonly used empirical LCAO method is mainly an interpolation method, where the energies and wave functions of atomic orbitals are adjusted in order to fit experimental or more accurately determined electron states. To achieve better accuracy, the size of calculation has to be expanded, for example, to include excited states and more-distant-neighboring atoms. This tends to sacrifice the simplicity and interpretability of the method.In this paper. we adopt an ab initio scheme which incorporates the conceptual advantage of the LCAO method with the accuracy of ab initio pseudopotential calculations. The so called pscudo-atomic-orbitals (PAO's), computed from a free atom within the local-density approximation and the pseudopotential approximation, are used as the basis of expansion, replacing the usually very large set of plane waves in the conventional pseudopotential method. These PAO's however, do not consist of a rigorously complete set of orthonormal states.

scholarly journals Reducing a Class of Two-Dimensional Integrals to One-Dimension with an Application to Gaussian Transforms

Atoms ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 53
Author(s):  
Jack C. Straton
Keyword(s):  
Quantum Theory ◽  
Arbitrary Function ◽  
Plane Waves ◽  
Wave Functions ◽  
One Dimension ◽  
Two Dimensional ◽  
Transition Amplitudes ◽  
Multidimensional Integrals ◽  
Coulomb Potentials

Quantum theory is awash in multidimensional integrals that contain exponentials in the integration variables, their inverses, and inverse polynomials of those variables. The present paper introduces a means to reduce pairs of such integrals to one dimension when the integrand contains powers multiplied by an arbitrary function of xy/(x+y) multiplying various combinations of exponentials. In some cases these exponentials arise directly from transition-amplitudes involving products of plane waves, hydrogenic wave functions, and Yukawa and/or Coulomb potentials. In other cases these exponentials arise from Gaussian transforms of such functions.

scholarly journals Linear scaling calculation of maximally localized Wannier functions with atomic basis set

2006 ◽  
Vol 124 (23) ◽  
pp. 234108 ◽  
Author(s):  
H. J. Xiang ◽  
Zhenyu Li ◽  
W. Z. Liang ◽  
Jinlong Yang ◽  
J. G. Hou ◽  
...  
Keyword(s):  
Basis Set ◽  
Linear Scaling ◽  
Wannier Functions

scholarly journals Origin Invariant Full Optical Rotation Tensor in the Length Dipole Gauge without London Atomic Orbitals

2021 ◽  
Author(s):  
Marco Caricato
Keyword(s):  
Electron Correlation ◽  
Density Functional ◽  
Optical Rotation ◽  
Density Functional Method ◽  
Functional Method ◽  
Basis Set ◽  
Rotation Tensor ◽  
Atomic Orbitals ◽  
Complete Basis Set ◽  
Approximate Wave

<div> <div> <div> <p>We present an origin-invariant approach to compute the full optical rotation tensor (Buckingham/Dunn tensor) in the length dipole gauge without recourse to London atomic orbitals, called LG(OI). The LG(OI) approach is simpler and less computationally demanding than the more common LG-London and modified velocity gauge (MVG) approaches and it can be used with any approximate wave function or density functional method. We report an implementation at coupled cluster with single and double excitations level (CCSD), for which we present the first simulations of the origin-invariant Buckingham/Dunn tensor in the length gauge. With this method, we attempt to decouple the effects of electron correlation and basis set incompleteness on the choice of gauge for optical rotation calculations on simple test systems. The simulations show a smooth convergence of the LG(OI) and MVG results with the basis set size towards the complete basis set limit. However, these preliminary results indicate that CCSD may not be close to a complete description of the electron correlation effects on this property even for small molecules, and that basis set incompleteness may be a less important cause of discrepancy between choices of gauge than electron correlation incompleteness. </p> </div> </div> </div>

Basis-Set Expansions of Relativistic Electronic Wave Functions

2007 ◽  
Author(s):  
Kenneth G. Dyall ◽  
Knut Faegri
Keyword(s):  
Relativistic Quantum ◽  
Finite Basis ◽  
Wave Functions ◽  
Basis Set ◽  
Electronic Wave ◽  
Nonrelativistic Case ◽  
Hartree Fock ◽  
Electronic Wave Functions ◽  
Basis Set Expansion ◽  
Molecular Calculations

There have been several successful applications of the Dirac–Hartree–Fock (DHF) equations to the calculation of numerical electronic wave functions for diatomic molecules (Laaksonen and Grant 1984a, 1984b, Sundholm 1988, 1994, Kullie et al. 1999). However, the use of numerical techniques in relativistic molecular calculations encounters the same difficulties as in the nonrelativistic case, and to proceed to general applications beyond simple diatomic and linear molecules it is necessary to resort to an analytic approximation using a basis set expansion of the wave function. The techniques for such calculations may to a large extent be based on the methods developed for nonrelativistic calculations, but it turns out that the transfer of these methods to the relativistic case requires special considerations. These considerations, as well as the development of the finite basis versions of both the Dirac and DHF equations, form the subject of the present chapter. In particular, in the early days of relativistic quantum chemistry, attempts to solve the DHF equations in a basis set expansion sometimes led to unexpected results. One of the problems was that some calculations did not tend to the correct nonrelativistic limit. Subsequent investigations revealed that this was caused by inconsistencies in the choice of basis set for the small-component space, and some basic principles of basisset selection for relativistic calculations were established. The variational stability of the DHF equations in a finite basis has also been a subject of debate. As we show in this chapter, it is possible to establish lower variational bounds, thus ensuring that the iterative solution of the DHF equations does not collapse. There are two basically different strategies that may be followed when developing a finite basis formulation for relativistic molecular calculations. One possibility is to expand the large and small components of the 4-spinor in a basis of 2-spinors. The alternative is to expand each of the scalar components of the 4-spinor in a scalar basis. Both approaches have their advantages and disadvantages, though the latter approach is obviously the easier one for adapting nonrelativistic methods, which work in real scalar arithmetic.

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