Although density functional theory has its conceptual roots in the
Thomas-Fermi model, DFT was put on a firm theoretical footing by
the two Hohenberg-Kohn theorems. The original H-Ktheorems held only
for non-degenerate ground states in the absence of a magnetic field
, although they have since been generalized to encompass these.
The first H-K theorem demonstrates that the ground state properties
of a many-electron system are uniquely determined by an electron
density that depends on only 3 spatial coordinates. It lays the
groundwork for reducing the many-body problem of N electrons with
3N spatial coordinates to 3 spatial coordinates, through the use
of functionals of the electron density. This theorem can be extended
to the time-dependent domain to develop time-dependent density
functional theory, which can be used to describe excited states.
The second H-K theorem defines an energy functional for the system
and proves that the correct ground state electron density minimizes
this energy functional.
Within the framework of Kohn-Sham DFT, the intractable many-body
problem of interacting electrons in a static external potential is
reduced to a tractable problem of noninteracting electrons moving
in an effective potential. The effective potential includes the
external potential and the effects of the Coulomb interactions
between the electrons, e.g., the exchange and correlation interactions.
Modeling the latter two interactions becomes the difficulty within
KS DFT. The simplest approximation is the local-density approximation,
which is based upon exact exchange energy for a uniform electron
gas, which can be obtained from the Thomas-Fermi model, and from
fits to the correlation energy for a uniform electron gas. Non-
interacting systems are relatively easy to solve as the wavefunction
can be represented as a Slater determinant of orbitals. Further,
the kinetic energy functional of such a system is known exactly.
The exchange-correlation part of the total-energy functional
remains unknown and must be approximated.
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