Chemistry Reference
In-Depth Information
system..The.long-range.electrostatic.energy.is.computed.via.integration.of.the.inter-
action.energies.between.the.charges.of.the.unit.cell.and.all.the.charges.of.the.lattice,.
with.an.ininitely.large.cell.that.is.a.collection.of.the.unit.cells.of.the.initial.periodic.
boundary.conditions.cubic.box.(Equation.2.11):
direct
reciprocal
self
correction
E
=
E
+
E
+
E
+
E
.
(2.11)
.
elec
where
N
N
erfc
(
α
r
n
)
∑
=
i j
,
,
direct
E
=
q q
.
(2.12)
i
j
r
i j
,
,
n
i
j
i
+
1
.
⎡
⎤
−
⎛
2
2
⎞
⎟
+
π
m
exp
i
2
π
m r
⋅
(
−
r
)
⎢
⎥
⎜
j
i
N
N
α
2
1
∑
∑
∑
⎣
⎦
reciprocal
E
=
q q
.
(2.13)
i
j
2
2
π
V
m
i
=
1
j
=
1
m
≠
0
.
N
α
π
∑
self
2
E
= −
q
.
(2.14)
i
.
i
=
1
2
⎛
N
⎞
2
1 2
π
ε
∑
correction
.
(2.15)
E
=
q
r
⎜
⎟
i
i
(
+
)
V
⎝
⎠
i
=
1
.
E
direct
.is.the.energy.in.real.space,.
N
.is.the.number.of.atoms.and.
erfc
(
x
).is.the.compli-
mentary.error.function.deined.by
x
2
∫
2
−
u
erfc x
( ) = −
1
e
du
.
(2.16)
π
.
0
E
reciprocal
. is. the. energy. in. reciprocal. space,.
V = a
1
.×.
a
2
.×.
a
3
. is. the. volume. of. the. box.
with.unit.vector.of.
a
1
,.
a
2
,.and.
a
3
,
.
m
=
m
1 1
*
*
*
+ +
.is.the.vector.of.the.recip-
rocal.space,.where.
a
1
*
,.
a
2
*
,.and.
a
3
*
.are.the.unit.vectors.of.the.simulation.box.in.recip-
rocal.space,.and.
m
1
,.
m
2
,.and.
m
3
.are.integers.deining.the.length.of.the.unit.vector.
The.value.of.α.(α.>.0).is.used.to.control.the.convergence.of.Ewald.sum..If.α.is.
large,.the.value.of.
E
direct
.decreases,.which.means.the.value.of.
E
direct
.can.be.neglected.
when. α. is. large. enough.. Thus,. the. minimum. image. convention. can. be. employed.
to.calculate.the.value.of.
E
direct
..
E
self
.is.the.interaction.energy.of.an.atom.with.itself,.
E
correction
.is.the.dipole.moment.correction,.and.ε.is.the.dielectric.constant..Another.
commonly.used.method.is.the.Particle-Mesh.Ewald.(PME).method,
54
.which.calcu-
lates.the.
E
reciprocal
.term.using.point.charges.distributed.on.surrounding.grid.points,.
a
m m
m m
2
2
3
3
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