The molecular Hamiltonian for M atoms with N electrons, can be written in Atomic Units. (See the appendix: Units) as,
where
is the distance between the atomic centers,
is the distance between the electrons,
is the distance between an electron and an atomic center.
The total energy, which is the expectation value of the Hamiltonian, is written as
In matrix form, the Hartree-Fock-Roothan equations are,
where C is the expansion coefficient matrix,
and
is the energy.
Using a cubic division of space, the kinetic energy of the electrons can be expressed as
where
This is a relation between the total kinetic energy and the electron density. This is the Thomas-Fermi kinetic energy functional.
This is based on statistical mechanics and Thomas and Fermi applied this to electrons in atoms and molecules!
Electronic properties are determined as a functional of the electron density by applying locally relations appropriate for a homogeneous electronic system.
The usefulness of this approach is that long range computations are removed. The accuracy of this approach is remarkable. (No more surprising than the accuracy of Hartree-Fock schemes)
The fundamental quantity is the electron density, N, given by,
TO begin, consider extending the Thomas-Fermi kinetic energy functional to cover the total energy of the system. Recall the object is to obtain an expression for the energy of the system. Initially neglecting exchange-correlation terms, using terms first order in the density matrix and classical electrostatic interactions,
with the constraint of conservation of the number of electrons;
For the ground state, then
which gives;
where
is the electrostatic potential at point
due to the nuclei and the entire electron density.
Hohenberg and Kohn (1964) and Kohn and Sham (1965) Theorems.
Theorem 1: The electron density
,
through normalization, determines the electron number N and the external
potential V(|vecr). And with Schrodinger's equation,
all the molecular properties.
Theorem 2: The energy functional
is a minimum for the true electron density. This requires electron number
conservation.
Thus, minimize the energy functional
with respect to variations in
(not variations in the wave functions as in Hartree-Fock) with the
constraint of electron number conservation
where
is the basis function and
are the expansion coefficients.
These are the Kohn-Sham orbital equations.
This looks a lot like the equations arising from the variational approach for Hartree-Fock theory. However there are crucial differences;