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 H - Energy and stress densities
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 density is a field defined at each position r, for example the particle number density n(r), which is a well-defined, experimentally measurable function. It would be desirable to have expressions for other densities, in particular, energy and stress densities. However, energy and stress densities are not unique on a microscopic quantum scale, even though they are the basis of the theory of elasticity on a macroscopic scale. This appendix brings out three points: (1) certain integrals of energy and stress densities are unique and very useful; (2) there are important contributions to the energy or stress density that are completely unique – these include all terms that arise from the fact that electrons are a many-body system of fermions; (3) all other terms that are non-unique can be shown to involve only the single scalar number density – there are different possible choices for these terms, each involving only derivatives of the density n(r) or the classical Coulomb potential VCC(r) which is directly related to n(r). It follows that all the issues of non-uniqueness are exactly the same as in a one-particle problem.
Only one density is widely used in electronic structure – the particle density n(r). It is the fundamental measurable quantity in quantum mechanics and the fundamental density in density functional theory. Theoretical expressions for n(r) are well defined and lead to unique results. Here we emphasize that other densities have the potential to play a useful role in electronic structure theory.
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- Information
- Electronic StructureBasic Theory and Practical Methods, pp. 519 - 528Publisher: Cambridge University PressPrint publication year: 2004