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Density Functional Theory

 

Density Functional Theory (DFT) is based on the notion of the single particle electron density as a fundamental variable. This is a consequence of the Hohenberg - Kohn theorem [11] which states that the ground state electron density tex2html_wrap_inline2461 minimises the energy functional

  equation184

where tex2html_wrap_inline2463 is a universal functional. Furthermore, the minimum value of E is tex2html_wrap_inline2467 , the ground state electronic energy. This is an exact result and in principle, means that the ground state energy and electron density may be found using a variational minimisation over the electron density tex2html_wrap_inline2469 , a process with scales linearly with system size.

Unfortunately, the proof of the Hohenberg - Kohn theorem is not constructive, hence the form of the functional tex2html_wrap_inline2463 in Equation gif is not known. Kohn and Sham postulated [12] that this functional could be written

  equation190

Here the first term tex2html_wrap_inline2473 is the kinetic energy of a system of non-interacting electrons with density tex2html_wrap_inline2469 and the second is the electron - electron Hartree interaction. The final term, tex2html_wrap_inline2477 , is the exchange-correlation energy. By writing the electron density in terms of a set of single particle wavefunctions tex2html_wrap_inline2479 such that

equation199

the kinetic energy term may be written

equation201

Minimising tex2html_wrap_inline2481 with respect to tex2html_wrap_inline2469 , subject to the constraint that the number of electrons must be constant, leads us to a set of equations

  equation205

where tex2html_wrap_inline2487 is an `effective potential',

  equation213

and tex2html_wrap_inline2489 is the exchange-correlation potential tex2html_wrap_inline2491 . Equation gif can be seen to be a set of Schrödinger-like equations for the single particle Kohn-Sham orbitals

equation225

Thus, the problem of a system of interacting electrons has been mapped onto a system of non-interacting electrons moving in an effective potential given by Equation gif. However, the expression of the density in terms of a set of single particle orbitals has increased the complexity of the problem. The minimisation must now be performed over NM degrees of freedom where M is the number of basis functions used to represent the tex2html_wrap_inline2479 .


next up previous contents
Next: The Exchange-Correlation Term Up: The Total Energy Pseudopotential Previous: The Many-Body Schrödinger Equation

Matthew Segall
Wed Sep 24 12:24:18 BST 1997