Chemistry Reference
In-Depth Information
e
max
, the
limiting field, is an adjustable empirical parameter in the formulation. The total
polarization energy at a molecule is the sum of polarization energies at each of its
electron density pixels,
E
Pol
¼ S
The expression holds for
e < e
max
while
E
Pol,
i
¼
0 for
e > e
max
.
E
Pol,
i
.
Dispersion energies are calculated as a sum of pixel-pixel terms in a damped
London-type expression:
n
.
h
i
o
f
2
R
ij
6
pe
Þ
E
Disp
;
AB
¼
ð
=
Þ S
i;
A
S
j;
B
E
OS
a
i
a
j
ð
ð
R
Þ;
3
4
4
h
i
2
;
f
ð
R
Þ¼
exp
DR
ij
1
for
R
ij
<
D
;
damping function
where
D
is an adjustable empirical parameter.
E
OS
(the London “oscillator
strength”) is estimated from the ionization potential of the atom to whose basin
the pixel belongs.
For the repulsion energy, the total charge density overlap integral between
molecules A and B,
S
AB
, is calculated and is subdivided into contributions from
pairs of atomic species
m
and
n
,
S
mn
. Then
E
Rep
;mn
¼
ð
K
1
K
2
Dw
mn
Þ
S
mn
;
where
Dw
mn
is the corresponding difference in Pauling electronegativity.
K
1
and
K
2
are positive disposable parameters. The total repulsion energy is the sum over all
m
-
n
pairs.
The total intermolecular Pixel interaction energy is
E
Tot
¼
E
Coul
þ
E
Pol
þ
E
Disp
þ
E
Rep
:
The empirical parameters in the Pixel formulation were optimized by reprodu-
cing experimental heats of sublimation for organic crystals and interaction energies
between molecular dimers in comparison with ab initio calculations, and by semi-
quantitative agreement between Pixel partitioned energies and intermolecular per-
turbation theory (IMPT) partitioned energies. Universal values for the parameters
are available, but very minor adjustments can be made to fit any desired thermo-
chemical or structural property of the particular system under investigation. Since
each term of the PIXEL energy reproduces an actual physical effect, these adjust-
ments may be guided by an analysis of the electronic properties of the system under
consideration.
The PIXEL code and documentation (OPiX package) can be downloaded at the
author's site:
http://users.unimi.it/gavezzot
. Computing times on an ordinary PC are
of the order of a few seconds for a molecular dimer, to a few minutes for a complete
crystal structure. PIXEL has been repeatedly shown to give results of accuracy
comparable to ab initio MO methods [
51
]. For an example, Fig.
5
[
52
] shows a
comparison between MP2, PIXEL, uncorrected and corrected periodic-orbital
