Chemistry Reference
In-Depth Information
with
4
2
π
π
sin
kr
kr
δδ
+
βα
0
β
N
β
2
U
=
⋅
∆
k k
⋅
⋅
1
-
(6.38)
βα
α
3
()
2
βα
with α and β as in Equation 6.36. As stated above, the middle step of converting
ĥ
ij
(
t
)
to
ĉ
ij
(
t
)
is carried out by solving the linear system of Equation 6.31 for each value of β.
If
n
ij
denotes indexing such that
r
n
ij
≤
R
ij
≤
r
n
ij
+ 1
, and
h
ij
(
t
)
and
c
ij
(
t
)
denote vectors con-
taining the elements of
h
ij
(
t
)
and
c
ij
(
t
)
, respectively, with
n
ij
+1 ≤ α ≤
N
,
h
ij
(
t
)
is updated
at each iteration step according to
(
t
+
1
)
()
t
()
t
h
=+
h
∆
h
(6.39)
ij
ij
ij
Here, Δ
h
ij
(t)
in Newton's method is found by solution of the linear system,
11
11
11
()
t
()
t
J
J
J
∆
∆
∆
h
h
h
∆
∆
∆
c
c
c
11
12
22
11
11
1
12
1
12
12
()
t
()
t
J
J
J
=
(6.40)
22
12
12
J
22
J
22
J
22
()
t
()
t
11
12
22
22
22
where the right-hand side represents the difference between the approximation of the
long-range DCF to be enforced and the currently computed DCF,
()
t
()
t
∆
c
=
tr
()
−
c
,
α
=+…
n
1
,
,
N
(6.41)
ij
,
α
ij
α
ij
,
α
ij
The Jacobian has the elements,
()
t
()
t
∂
∂
c
h
∂
c
ij n
,
+
1
ij n
,
+
1
ij
...
ij
()
t
()
t
∂
h
αβ
,
n
+
1
αβ
,
N
αβ
ij
J
αβ
≡
...
...
...
(6.42)
∂
c
()
t
∂
∂
c
h
()
t
ij N
,
ij N
,
...
(()
t
()
t
∂
h
αβ
,
n
+
1
αβ
,
N
αβ
These are partial derivatives that can be expanded by the chain rule to,
N
N
∂
∂
c
h
∂
∂
c
c
∂
∂
c
h
∂
h
h
∂
∂
c
h
∑
∑
0
ij
,
α
ij
,
α
ij
,
β
ab
,
β
′
ij
,
β
=
=
U
T
(6.43)
αβ
βα
′
∂
ab
,
α
′
ij
,
β
ab
,
β
′
ab
,
α
′
ab
,
β
ββ
=
00
,
′
=
β
=
The last equality is due to the result,
∂
∂
c
h
∂
∂
c
h
ij
,
β
ij
,
β
=
δ
(6.44)
ββ
′
ab
,
β
′
ab
,
β
