Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Notation
- Part I Overview and background topics
- Part II Density functional theory
- Part III Important preliminaries on atoms
- Part IV Determination of electronic structure: the three basic methods
- Part V Predicting properties of matter from electronic structure – recent developments
- Appendix A Functional equations
- Appendix B LSDA and GGA functionals
- Appendix C Adiabatic approximation
- Appendix D Response functions and Green's functions
- Appendix E Dielectric functions and optical properties
- Appendix F Coulomb interactions in extended systems
- Appendix G Stress from electronic structure
- Appendix H Energy and stress densities
- Appendix I Alternative force expressions
- Appendix J Scattering and phase shifts
- Appendix K Useful relations and formulas
- Appendix L Numerical methods
- Appendix M Iterative methods in electronic structure
- Appendix N Code for empirical pseudopotential and tight-binding
- Appendix O Units and conversion factors
- References
- Index
Appendix A - Functional equations
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Notation
- Part I Overview and background topics
- Part II Density functional theory
- Part III Important preliminaries on atoms
- Part IV Determination of electronic structure: the three basic methods
- Part V Predicting properties of matter from electronic structure – recent developments
- Appendix A Functional equations
- Appendix B LSDA and GGA functionals
- Appendix C Adiabatic approximation
- Appendix D Response functions and Green's functions
- Appendix E Dielectric functions and optical properties
- Appendix F Coulomb interactions in extended systems
- Appendix G Stress from electronic structure
- Appendix H Energy and stress densities
- Appendix I Alternative force expressions
- Appendix J Scattering and phase shifts
- Appendix K Useful relations and formulas
- Appendix L Numerical methods
- Appendix M Iterative methods in electronic structure
- Appendix N Code for empirical pseudopotential and tight-binding
- Appendix O Units and conversion factors
- References
- Index
Summary
Summary
A functional F[f] is a mapping of an entire function f onto a value. In electronic structure, functionals play a central role, not only in density functional theory, but also in the formulation of most of the theoretical methods as functionals of the underlying variables, in particular the wavefunctions. This appendix deals with the general formulation and derivation of variational equations from the functionals.
Basic definitions and variational equations
The difference between a function f(x) and a functional F[f] is that a function is defined to be a mapping of a variable x to a result (a number) f(x); whereas a functional is a mapping of an entire function f to a resulting number F[f]. The functional F[f], denoted by square brackets, depends upon the function f over its range of definition f(x) in terms of its argument x. Here we a describe some basic properties related to the functionals and their use in density functional theory; more complete description can be found in [93], App. A. A review of functional derivatives or the “calculus of variations” can be found in [861] and [862].
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- Information
- Electronic StructureBasic Theory and Practical Methods, pp. 476 - 478Publisher: Cambridge University PressPrint publication year: 2004