Real-Space Mesh Techniques in Density Functional Theory

Thomas L. Beck
Department of Chemistry, University of Cincinnati, Cincinnati, OH 45221-0172

Abstract

This review discusses progress in efflcient solvers which have as their
foundation a representation in real space, either through finite-difference or
finite-element formulations.  The relationship of real-space approaches to
linear-scaling electrostatics and electronic structure methods is first
discussed.  Then the basic aspects of real-space representations are
presented.  Multigrid techniques for solving the discretized problems are
covered; these numerical schemes allow for highly efflcient solution of the
grid-based equations.  Applications to problems in electrostatics are
discussed, in particular numerical solutions of Poisson and Poisson-Boltzmann
equations.  Next, methods for solving self-consistent eigenvalue problems in
real space are presented; these techniques have been extensively applied to
solutions of the Hartree-Fock and Kohn-Sham equations of electronic structure,
and to eigenvalue problems arising in semiconductor and polymer physics.
Finally, real-space methods have found recent application in computations of
optical response and excited states in time-dependent density functional
theory, and these computational developments are summarized.  Multiscale
solvers are competitive with the most efflcient available plane-wave
techniques in terms of the number of self-consistency steps required to reach
the ground state, and they require less work in each self-consistency update
on a uniform grid.  Besides excellent efficiencies,the decided advantages of
the real-space multiscale approach are 1) the near-locality of each function
update, 2) the ability to handle global eigenfunction constraints and
potential updates on coarse levels, and 3) the ability to incorporate adaptive
local mesh refinements without loss of optimal multigrid efflciencies.

CONTENTS
I. INTRODUCTION 2
II. DENSITY FUNCTIONAL THEORY 5
A. Kohn-Sham equations 5
B. Classical DFT 6
III. LINEAR-SCALING CALCULATIONS 7
A. Classical electrostatics 8
B. Electronic structure 9
IV. REAL-SPACE REPRESENT A TIONS 10
A. Finite differences 11
1. Basic finite-difference representation 11
2. Solution by iterative techniques 12
3. Generation of high-order finite-difference formulas 13
B. Finite elements 14
1. Variational formulation 14
2. Finite-element bases 15
V. MULTIGRID TECHNIQUES 16
A. Essential features of multigrid 16
B. Full approximation scheme multigrid V-cycle 17
C. Full multigrid 18
VI. ELECTROST A TICS CALCULA TIONS 18
A. Poisson solvers 19
1. Illustration of multigrid efficiency 19
2. Mesh-refinement techniques 20
B. Poisson-Boltzmann solvers 21
C. Computations of free energies 23
D. Biophysical applications 25
VII. SOLUTION OF SELF-CONSISTENT EIGENVALUE PROBLEMS 26
A. Fixed-potential eigenvalue problems in real-space 26
1. Algorithms 26
2. Applications 28
B. Finite-difference methods for self-consistent problems 29
C. Multigrid methods 31
D. Finite-difference mesh-refinement techniques 34
E. Finite-element solutions 35
F. Orbital-minimization methods 37
VIII. TIME-DEPENDENT DFT CALCULATIONS IN REAL SPACE 37
A. TDDFT in real time and optical response 38
B. TDDFT calculation of excited states 39
IX. SUMMARY 40
ACKNOWLEDGMENTS 41
Appendix A 41
References 42