From:	Gustavo Mercier <mercie #*at*# med.cornell.edu>
Date:	Mon, 4 Oct 1993 19:37:18 -0400 (EDT)
Subject:	DFT

Hi, Netters!

Recently there have been a few questions about DFT. Although I don't
consider myself an expert, I have been listening to people talk about
DFT for a few years and recently started to do DFT computations on
metalloporphyrins. I hope the following will clarify some points. I certainly
welcome any corrections of my statements below!

DFT (Density Functional Theory) stands as a reformulation of the Schroedinger
Equation. Its origin really dates from the early days of quantum mechanics
(Thomas-Fermi-Dirac, Slater's work, etc), but its "modern" foundation is
based on the theorems of Hohenberg and Kohn developed in the '60's, and its
practical implementation in Quantum Chemistry rests on the Kohn - Sham
equations. For a beautiful and concise description of the historical
development of DFT I suggest you read the article by Hohenberg, Kohn,
and Sham in Advances of Quantum Chemistry v. 21 special ed. S. Trickey
pp 7 -26, 1990.

The HK theorems essentially state:

1) Given a density, rho, the external potential, Vext, is fixed.
For Quantum Chemistry using the electronic molecular hamiltonian
within the Born-Oppenheimer approximation, the Vext is the electrostatic
potential generated by the nuclei.

2) The energy is a UNIQUE Functional of the density.

As originally described, the above applies only to NON-DEGENERATE
GROUND STATE systems!

Following their work, issues described as the V-representability and
N-representability problems were identified and dealt with in a variety
of ways. For an explanation of these and the rigourous foundation of
DFT theory, try Kryachko and Ludena, Energy Density Functional Theory
of Many-Electron Systems by Kluwer Academic Publishers Understanding
Chemical Reactivity Series v. 4, 1990.

For computational chemists the Kohn - Sham equations are the origin of
most implementations of DFT theory. Essentially they apply a variation to

E[rho] = F[rho] + Eext[rho];
F = Ts + Eee + Exc

where Eee is the coulombic electron - electron repulsion and Exc is the
"exchange - correlation" term, a functional of rho. The key point to
understand is Ts. Ts is the kinetic energy for a collection of NON-INTERACTING
electrons!. As you can see the structure is similar to our familiar
Hartree - Fock, but there are subtle differences. In Hartree - Fock, the
Kinetic term is for an INTERACTING set of electrons. The difference is
THROWN into the undefined Exc term. KS showed that if Exc is known exactly,
the KS equations will yield the EXACT density! The KS equations are
generated using the above functional and the HK variational
principle that stems from Theorem #2. The key result is that due to the
choice of reference state, the KS equation is identical to the HARTREE
equation, the Schroedinger Eq. for a system of non-interacting electrons,
but with a new potential:

Veff = Vext + Vee + Vxc

Following the Hartree Eq., the density can be written EXACTLY as

rho = Sumi phi(i)^2

This form of rho is NOT an approximation as is the case when the
density is generated from the HARTREE - FOCK Equation. The rho
looks identical in form, but the phi's are very different!!

In solving the KS equations a Variational Method is used that includes
Lagrangians due to the constraint of the density to integrate to the total N
number of electrons. In Hartree - Fock, these correspond to
the orbital energies as shown through Koopman's Theorem. In DFT, the
Lagrangians do not have the same meaning. THEY DO NOT correspond to
orbital energies. Through some manipulations you can generate "orbital
energies" but they do not come explicitly from the KS equations.

The fact that orbital energies are not generated also means that the phi's
above don't have the same meaning they have within HF theory!
This is one of the most distressing things for us chemist who always
think in terms of orbitals!!! Only the density has "meaning", not its
"components". The physical interpretation associated with the orbitals
in HF theory which is useful in the generation of different configurations
for CI or GVB computations is lost!

Most of the work in DFT deals with defining Vxc. If it were known exactly,
we would have the exact equation and an exact solution within the
basis set expansion we chose for the phi's. In other words, within
Hartree - Fock we have an exact operator with its approximate solution
(single determinant), but in DFT we have an approximate functional with its
exact solution. The ADVANTAGE is that the approximate part is small, and
the density is a function of only three variables! In fact, using
Mathematica and the output of my DFT runs, I have written the analytic
expression for the the density of a 66 electron system using a double zeta
+ polarization basis set! All computed in an INDIGO R4000. It is satisfying
to actually see the analytic form of your result! Given the functional
associated with a molecular property, you can easily compute
the property. Particularly, standard numerical algorithms can fairly
easily be applied since the dimensionality of the problem has decreased
significantly! For example, in my case from 66*3 to 3.

When does DFT - KS fails? When your choice of Vxc or basis set is bad!
Also, from the practical point of view, many programs expand the
the electron density used to compute the Vee. If this basis is poor you
also get bad results. One thing that has been appreciated is that if
you want to adequately reproduce H-bonds, a critical point for the
many biochemically oriented people in this mailing list, you need
good descriptions of Vxc that include gradient expansions etc.

Sorry for the bandwith, but I hope the above was useful.

good luck
mercie[ AT ]cumc.cornell.edu




Similar Messages
03/11/1996:  Law of conservation of difficulty: violations.
05/15/1997:  Summary: Chemical Softness by DFT 
05/15/1997:  Re: CCL:G:Summary: Chemical Softness by DFT 
12/11/1995:  Summary: Koopmans' Theorem and Neglect of Bond in CAChe MOPAC Input
08/01/1996:  Re: CCL:M:Heat of formation calculation using MOPAC.
08/01/1995:  Spin contamination, effect on energy and structure.
02/04/1995:  Summary: DFT and H
12/16/1994:  Spin contamination & AM1 "ROHF" versus UHF
12/10/1995:  basis stes
05/03/1995:  Summary: DFT functionals


Raw Message Text