522 16 Improved Monte Carlo Approaches to Systems of Fermions
which means that determiningRSDwhen only particleihas been moved, requires only the
evaluation of the dot product between a vector containing orbitals (evaluated at the new
position) and all the entries in theithcolumn of the inverse Slater matrix (evaluated at the
current position). This requires approximatelyO(N)operations.
Further optimizations can be done by noting that when only one particle is moved at the
time, one of the two determinants in the numerator and denominator of Eq. (16.3) is unaf-
fected, cancelling each other. This allows us to carry out calculations with only half of the total
number of particles every time a move occurs, requiring only(N/ 2 )doperations, wheredis
the number of spatial components of the problem, in systems with equal number of electrons
with spin up and down. The total number of operations for a problem in three dimensions
becomes(N/ 2 )^3 =N^3 / 8 , i.e., the total calculations are reduced up to by a factor ofeight.
16.4 Optimizing the∇ΨT/ΨTRatio
SettingΨD=|D|↑|D|↓in Eq. (16.2) we get,
∇Ψ
Ψ=∇(ΨDΨC)
ΨDΨC
=ΨC∇ΨD+ΨD∇ΨC
ΨDΨC
=∇ΨD
ΨD
+∇ΨC
ΨC
=
∇(|D|↑|D|↓)
|D|↑|D|↓ +
∇ΨC
ΨC,
or
∇Ψ
Ψ
=
∇(|D|↑)
|D|↑
+
∇(|D|↓)
|D|↓
+∇ΨC
ΨC
. (16.10)
16.4.1Evaluating the Gradient-determinant-to-determinant Ratio
The evaluation of Eq. (16.10) requires differentiating theNentries of the Slater matrix with
respect to all thedspatial components. Since the evaluation of the Slater determinant scales
asO(N^3 )this would involve of the order ofN·d·O(N^3 )≈O(N^4 )floating point operations. A
cheaper algorithm can be derived by noting that when only oneparticle is moved at the time,
only one row in the Slater matrix needs to be evaluated again.Thus, only the derivatives of
that row with respect to the coordinates of the particle moved need to be updated. Obtaining
the gradient-determinant ratio required in Eq. (16.10) becomes straigforward. It is analogous
to the procedure used in deriving Eq. (16.3). From Eq. (16.9)and Eq. (16.3) we see that
∇i|D(x)|
|D(x)|=
N
∑
j= 1∇iDi j(x)D−ji^1 (x) =N
∑
j= 1∇iφj(xi)D−ji^1 (x), (16.11)which means that when one particle is moved at the time, the gradient-determinant ratio is
given by the dot product between the gradient of the single-particle wave functions evaluated
for the particle at positionriand the inverse Slater matrix. A small modification has to be done
when computing the gradient to determinant ratio after a move has been accepted. Denoting
byythe vector containing the new spatial coordinates, by definition we get,