Jan 24, 2002
Notes for Lecture
Notes for Lecture #4
Electrostatic energy
Section 1.11 in Jackson's text
The total electrostatic potential energy of interaction between
point charges {qi} at the positions {ri} is
given by
|
W = |
1
4pe0
|
|
å
i < j
|
|
qiqj
|ri-rj|
|
= |
1
8pe0
|
|
å
i ¹ j
|
|
qi qj
|ri-rj|
|
. |
| (1) |
In this expression, the first form explicitly counts all pairs,
while the second form counts all interactions and divides by 2 to
compensate for double counting.
For a finite system of charges, this expression can be evaluated
directly, however, for a large or infinite system, the expression
1 does not converge and numerical tricks must be used
to evaluate the energy. In fact, for the infinite system, one
has an infinite amount of charge and the energy of interaction is
undefined. If the system is neutral, it is possible to define a
meaningful interaction energy by use of an Ewald transformation.
The basic idea of the Ewald approach is as follows. The error
function erf(x) and its complement erfc(x) are
defined as:
|
erf(x) = |
2
|
|
ó
õ
|
x
0
|
e-t2 dt \and erfc(x) = 1 - erf(x) = |
2
|
|
ó
õ
|
¥
x
|
e-t2 dt. |
| (2) |
Ewald noted that
In this expression, the first term goes to a constant
(Ö{h/p}) as r ® 0, but has a long
range tail as r ® ¥. The second term has a
singular behavior as r ® 0, but vanishes
exponentially as r ® ¥. Thus, Ewald's idea is
to replace a single divergent summation with two convergent summations.
The first summation has a convergent summation in the form of its Fourier transform and the second
has a convergent direct summation. Thus the calculation of the
electrostatic energy would be evaluated using:
|
W = |
1
8pe0
|
|
å
i ¹ j
|
|
qiqj
|ri-rj|
|
= |
1
8pe0
|
|
æ
ç
ç
ç
ç
è
|
å
i ¹ j
|
|
|
qi qjerf( |
1
2
|
| Ö |
h
|
|ri-rj|) |
|ri-rj|
|
+ |
å
i ¹ j
|
|
|
qi qjerfc( |
1
2
|
| Ö |
h
|
|ri-rj|) |
|ri-rj|
|
ö
÷
÷
÷
÷
ø
|
. |
| (4) |
For an appropriate choice of the parameter h, the second
summation in Eq. 4 converges quickly and can be evaluated
directly. The first term in the summation of Eq. 4 must
be transformed into Fourier space.
In order to described these summations explicitly, we assume that
we have a periodic lattice so that every ion can be located by
ri = ta + T a location ta
within a unit cell and a periodic translation vector T,
In this way, the summation becomes:1
where N denotes the number of unit cells in the system. Since
we have a periodic system, N is infinite, but the energy per
unit cell W/N is well defined. The other identity that we must
use is that a sum over lattice translations T may be
transformed into an equivalent sum over ``reciprocal lattice
translations'' G according to the identity:
|
|
å
T
|
d3(r - T) = |
1
W
|
|
å
G
|
ei G ·r, |
| (6) |
where W denotes the unit cell volume. The proof of this
relation is given in the appendix.
The first term of Eq. 4 thus becomes
|
|
å
i ¹ j
|
|
|
qi qjerf( |
1
2
|
| Ö |
h
|
|ri-rj|)) |
|ri-rj|
|
= N |
æ
ç
ç
ç
ç
è
|
|
å
ab
|
qa qb |
å
T
|
|
|
erf( |
1
2
|
| Ö |
h
|
|ta-tb+ T |) |
|ta-tb + T|
|
- |
å
a
|
q2a |
æ
ú
Ö
|
|
ö
÷
÷
÷
÷
ø
|
, |
| (7) |
where the last term in Eq. 7 comes from subtracting
out the self-interaction (i = j) term from the complete lattice
sum. Using the short hand notation, tab º ta-tb, the lattice sum can be
evaluated:
|
|
å
T
|
|
|tab + T|
|
= |
ó
õ
|
d3r |
å
T
|
d3(r-T) |
|tab + r|
|
. |
| (8) |
This becomes,
|
|
1
W
|
|
å
G
|
|
ó
õ
|
d3r ei G ·r |
|tab + r|
|
= |
4 p
W
|
|
æ
ç
è
|
å
G ¹ 0
|
|
e- i G ·tab e- G2/h2
G2
|
+ |
1
2
|
|
ó
õ
|
1/2Ö{h}
0
|
|
du
u3
|
ö
÷
ø
|
, |
| (9) |
where the last term, which is infinite, comes from the G = 0
contribution.
If the last term of Eq. 9 cannot be eliminated, it
is clear that the electrostatic energy is infinite. The term can
be eliminated if and only if the system is neutral. Thus, it is
only meaningful to calculate the electrostatic energy of a
neutral periodic system. If the actual system has a net
charge of Q º åa qa, we could calculate
a meanful energy if we add to the actual charge density, a compensating
uniform density charge density of -Q/W. Taking all of the terms into
account, we find the final Ewald expression to be:
|
|
W
N
|
= |
å
ab
|
|
qa qb
8 pe0
|
|
æ
ç
ç
ç
ç
è
|
|
4 p
W
|
|
å
G ¹ 0
|
|
e- i G ·tab e-G2/h
G2
|
- |
æ
ú
Ö
|
|
dab + |
¢
å
T
|
|
|tab+T|
|
ö
÷
÷
÷
÷
ø
|
- |
4 pQ2
8 pe0 Wh
|
, |
| (10) |
where the ¢ in the summation over lattice translations
T indicates that all self-interaction terms should be
omitted.
Appendix I - comments on lattice vectors and
reciprocal lattice vectors
In this discussion, will assume we have a 3-dimensional periodic
system. It can be easily generalized to 1- or 2- dimensional
systems. In general, a translation vector can be described a
linear combination of the three primitive translation vectors
T1, T2, and T3:
where {n1, n2,n3} are integers. Note that the unit cell
volume W can be expressed in terms of the primitive
translation vectors according to:
The reciprocal lattice vectors G can generally be
written as a linear combination of the three primitive reciprocal
lattice vectors G1, G2, and
G3:
where {m1, m2,m3} are integers. The primitive reciprocal
lattice vectors are determined from the primitive translation
vectors according to the identities:
Note that the ``volume'' of the primitive reciprocal lattice is
given by
|
|G1 ·(G2 ×G3)| = |
(2 p)3
W
|
. |
| (15) |
Some examples of this are given below.
``Proof'' of Eq. 6
Consider the geometric series
|
|
+M
å
k = -M
|
ei k (G1 ·r) = |
|
sin |
æ
ç
è
|
(M+ |
1
2
|
)G1 ·r |
ö
÷
ø
|
|
sin( G1 ·r/2)
|
. |
| (16) |
The behavior of the right hand side of Eq. (16) is that it is
small in magnitude very except when the denominator vanishes. This occurs
whenever G1 ·r/2 = n1 p, where n1
represents any integer. If we take the limit M® ¥, we find that the function represents the behavior of
a sum of delta functions:
|
|
lim
M ® ¥
|
|
|
sin |
æ
ç
è
|
(M+ |
1
2
|
)G1 ·r |
ö
÷
ø
|
|
sin( G1 ·r/2)
|
= 2 p |
å
n1
|
d(G1 ·(r-n1T1). |
| (17) |
The summation over all lattice translations n1T1 is due
to the fact that sin( G1 ·r/2) = 0 whenever r = n1T1.
Carrying out the geometric summations in the right hand side of Eq. 6
for all three reciprocal lattice vectors and taking the limit as
in Eq. 17,
|
|
å
G
|
ei G ·r = (2 p)3 |
å
n1,n2,n3
|
d(G1 · (r-n1T1)) d(G2 ·(r-n2T2)) d(G3 · (r-n3T3)). |
| (18) |
Finally, the right hand side of Eq. 18 can be simplified
by eliminating the reciprocal lattice vectors from the d functions:
|
|
å
G
|
ei G ·r = |
(2 p)3
|G1 ·(G2 ×G3)|
|
|
å
T
|
d3(r-T) = Wd3(r-T), |
| (19) |
which is consistent with Eq. 6.
Appendix II - examples
In these examples, denote the length of the unit cell by a.
CsCl structure
There are two kinds of sites - tCs = 0 and
tCl = a/2 ([^(x)] + [^(y)] +[^(z)]).
|
T1 = a |
^
x
|
T2 = a |
^
y
|
T3 = a |
^
z
|
. |
| (20) |
|
G1 = |
2 p
a
|
|
^
x
|
G2 = |
2 p
a
|
|
^
y
|
G3 = |
2 p
a
|
|
^
z
|
. |
| (21) |
It is convenient to define h º m2/a2. Also we will
denote by q the unit charge. In these terms, Eq. 10
becomes for this case:
|
|
W
N
|
= |
q2
8 pe0 a
|
(T1+T2 ), |
| (22) |
where
|
T1 º |
1
p
|
|
å
m1,m2,m3 ¹ 0
|
2 |
(1-eip(m1+m2+m3)) e-[(4p2)/( m2)](m12+m22+m32)
m12+m22+m32
|
- |
2 m
|
|
| (23) |
and
|
T2 º |
å
n1,n2,n3 ¹ 0
|
|
|
2 erfc( |
m
2
|
|
| ___________
Ön12+n22+n32
|
) |
|
- |
å
n1,n2,n3
|
|
|
2erfc( |
m
2
|
|
æ
ú
Ö
|
|
( |
1
2
|
+n1)2+( |
1
2
|
+n2)2+( |
1
2
|
+n3)2 |
|
) |
|
|
æ
ú
Ö
|
|
( |
1
2
|
+n1)2+( |
1
2
|
+n2)2+( |
1
2
|
+n3)2 |
|
. |
| (24) |
In this expression, the sum over a and b has a total
of 4 contributions - 2 pairs of identical contributions. For
Cs-Cs or Cl-Cl interactions, tab = 0 and
qa = qb resulting in repulsive contributions. For
Cs-Cl or Cl-Cs interations,
tab = ±a/2([^(x)]+[^(y)]+[^(z)])
and qa = qb resulting in attractive contributions.
This expression can be evaluated using Maple, which gives the
result
|
|
W
N
|
= |
q2
8 pe0 a
|
(-4.071). |
| (25) |
NaCl structure
There are two kinds of sites - tNa = 0 and
tCl = a/2 [^(x)].
|
T1 = |
a
2
|
( |
^
x
|
+ |
^
y
|
) T2 = |
a
2
|
( |
^
y
|
+ |
^
z
|
) T3 = |
a
2
|
( |
^
x
|
+ |
^
z
|
). |
| (26) |
|
G1 = |
2 p
a
|
( |
^
x
|
+ |
^
y
|
- |
^
z
|
) G2 = |
2 p
a
|
(- |
^
x
|
+ |
^
y
|
+ |
^
z
|
) G3 = |
2 p
a
|
( |
^
x
|
- |
^
y
|
+ |
^
z
|
). |
| (27) |
Footnotes:
1 Note that for any two
lattice translations Ti and Tj,
Ti-Tj = Tk, where Tk is also a lattice
translation.
File translated from TEX by TTH, version 2.20.
On 24 Jan 2002, 10:31.