16.5 Optimizing the∇^2 ΨT/ΨTRatio 523
∇i|D(y)|
|D(y)|=
N
∑
j= 1∇iDi j(y)D−ji^1 (y) =N
∑
j= 1∇iφj(yi)D−ji^1 (y),which can be expressed in terms of the transpose and inverse of the Slater matrix evaluated
at the old positions [94] to get
∇i|D(y)|
|D(y)|=
1
R
N
∑
j= 1∇iφj(yi)D−ji^1 (x). (16.12)Computing a single derivative is anO(N)operation. Since there aredNderivatives, the total
time scaling becomesO(dN^2 ).
16.5 Optimizing the∇^2 ΨT/ΨTRatio
From the single-particle kinetic energy operator, the expectation value of the kinetic energy
expressed in atomic units for electroniis
〈K̂i〉=−1
2
〈Ψ|∇^2 i|Ψ〉
〈Ψ|Ψ〉, (16.13)
which is obtained by using Monte Carlo integration. The energy of each space configuration
is cummulated after each Monte Carlo cycle. For each electron we evaluate
Ki=−1
2
∇^2 iΨ
Ψ. (16.14)
Following a procedure similar to that of section 16.4, the term for the kinetic energy is ob-
tained by
∇^2 Ψ
Ψ=
∇^2 (ΨDΨC)
ΨDΨC
=
∇·[∇(ΨDΨC)]
ΨDΨC
=
∇·[ΨC∇ΨD+ΨD∇ΨC]
ΨDΨC
=
∇ΨC·∇ΨD+ΨC∇^2 ΨD+∇ΨD·∇ΨC+ΨD∇^2 ΨC
ΨDΨC
(16.15)∇^2 Ψ
Ψ
=
∇^2 ΨD
ΨD
+
∇^2 ΨC
ΨC
+ 2
∇ΨD
ΨD
·
∇ΨC
ΨC
=
∇^2 (|D|↑|D|↓)
(|D|↑|D|↓)
+
∇^2 ΨC
ΨC
+ 2
∇(|D|↑|D|↓)
(|D|↑|D|↓)
·
∇ΨC
ΨC
,
or
∇^2 Ψ
Ψ =
∇^2 |D|↑
|D|↑ +
∇^2 |D|↓
|D|↓ +
∇^2 ΨC
ΨC +^2
[
∇|D|↑
|D|↑ +
∇|D|↓
|D|↓
]
·
∇ΨC
ΨC, (16.16)
where thelaplace-determinant-to-determinant ratiois given by
∇^2 i|D(x)|
|D(x)|=
N
∑
j= 1∇^2 iDi j(x)D−ji^1 (x) =N
∑
j= 1∇^2 iφj(xi)D−ji^1 (x) (16.17)