Biomedical Engineering Reference
In-Depth Information
20.4 Numerical Integration (Quadrature)
Given the functionals and the electron density, we have to integrate over the space of the
molecule to obtain the energy. These extra integrals cannot usually be done analytically and
so a numerical integration has to be done each KS-LCAO cycle. What we do is to replace
the integral by a sum over quadrature points, e.g.
w
A
i
f
P
1
,
P
1
, grad
P
1
, grad
P
1
;
r
A
i
ε
X/C
=
(20.19)
A
i
where the first summation is over the atoms and the second is over the numerical quadrature
grid points. The
w
A
i
are the quadrature weights and the grid points are given by the sum of
the position of nucleus A and a suitable one-centre grid:
r
A
i
=
R
A
+
r
i
The keynote paper for numerical integration is that due to Becke (1988a), and once again
the paper is so important that you should see the abstract.
We propose a simple scheme for decomposition of molecular functions into single-center
components. The problem of three-dimensional integration in molecular systems thus reduces
to a sum of one-center, atomic-like integrations, which are treated using standard numerical
techniques in spherical polar coordinates. The resulting method is tested on representative
diatomic and polyatomic systems for which we obtain five- or six-figure accuracy using a few
thousand integration points per atom.
Becke's scheme rigorously separates themolecular integral into atomic contributions that
may be treated by standard single-centre techniques. Numerical integration grids are usually
specified as a specific number of radial shells around each atom, each of which contains a
set number of integration points. For example, Gaussian 03 uses a grid designated (75,302)
which has 75 radial shells per atom, each containing 302 points giving a total of 22 650
integration points.
Over the years, a large number of exchange and correlation functionals have been
proposed.
20.5 Practical Details
There are two versions of DFT, one for closed shell systems analogous to the Roothaan
closed shell treatment, and one for the open shell case analogous to the UHF technique
and we implement the Kohn-ShamLCAO (KS-LCAO) equations by including the relevant
exchange-correlation term(s) instead of the traditional HF exchange term in the HF-LCAO
equations. That is, for the UHF case we write
+
U
C
(
r
)
ψ
α
(
r
)
h
(1)
(
r
)
+
J
(
r
)
+
U
X
(
r
)
=
ε
α
ψ
α
(
r
)
