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Density Functional Methods

Density Functional theory[7, 8] is a formally exact theory based on the charge density of a system. Kohn-Sham Density Functional theory[9] is a formally exact one-electron theory. Working within the Born-Oppenheimer approximation, the many-body Schrödinger equation,


where tex2html_wrap_inline5833 is the many-body wavefunction, is replaced by a set of N one-electron equations of the form


where tex2html_wrap_inline5847 is a single-electron wavefunction. These one-electron equations contain a potential tex2html_wrap_inline5917 produced by all the ions and the electrons. Density Functional theory properly includes all parts of the electron-electron interaction, i.e. the Hartree potential


where tex2html_wrap_inline5919 is the charge density of all the electrons, a potential due to exchange and correlation effects, tex2html_wrap_inline5921 , and the external potential due to the ions, tex2html_wrap_inline5923 ,


Hohenberg and Kohn[10] originally developed Density Functional theory for application to the ground state of a system of spinless fermions. In such a system the particle density is given by


with tex2html_wrap_inline5925 being the many-body ground state wavefunction of the system. It can be shown that the total ground state energy of the system is a functional of the density, tex2html_wrap_inline5927 , and that if the energy due to the electron-ion interactions is excluded the remainder of the energy is a universal functional of the density, tex2html_wrap_inline5929 (i.e. tex2html_wrap_inline5929 does not depend on the potential from the ions). The most elegant proof of Density Functional theory is due to Levy[11] and is as follows:

For a particular N-representable densitygif (i.e. any density given by an antisymmetric N-electron wavefunction), a functional of the density corresponding to any operator tex2html_wrap_inline5939 can be defined via


The right hand side is evaluated by searching over wavefunctions, tex2html_wrap_inline5833 , which give rise to the density tex2html_wrap_inline5861 and looking for the one which makes the expectation value of the operator tex2html_wrap_inline5939 a minimum.

We can define tex2html_wrap_inline5929 in the same way, where




Now let tex2html_wrap_inline5925 be the ground state of an N-electron system and tex2html_wrap_inline5833 a state which yields a density tex2html_wrap_inline5861 and minimises tex2html_wrap_inline5957 . Then, from the definition of tex2html_wrap_inline5927 ,


Now tex2html_wrap_inline5961 is the electronic Hamiltonian, from Eq.(gif), and so tex2html_wrap_inline5927 must obey the variational principle (see section gif),


Also, from the definition of tex2html_wrap_inline5929 , in Eq.(gif), we have


since tex2html_wrap_inline5925 is just one of the trial wavefunctions that yield tex2html_wrap_inline5969 . Adding tex2html_wrap_inline5971 to the above equation gives


which in combination with Eq.(gif) yields the desired result


hence completing the proof.

next up previous contents
Next: Kohn-Sham Equations Up: The Many-Electron Problem Previous: Configuration Interaction

Andrew Williamson
Tue Nov 19 17:11:34 GMT 1996