524 16 Improved Monte Carlo Approaches to Systems of Fermions
for particle atxias deduced from Eq. (16.9) and Eq. (16.3). The comments givenin section
16.4 on performance yields applies also to this case. Moreover, Eq. (16.17) is computed with
the trial move only if it is accepted.
16.6 Updating the Inverse of the Slater Matrix
Computing the ratios in Eqs. (16.9), (16.11), (16.12) and (16.17) requires that we maintain
the inverse of the Slater matrix evaluated at the current position. Each time a trial position
is accepted, the row numberiof the Slater matrix changes and updating its inverse has to be
carried out. Getting the inverse of anN×Nmatrix by Gaussian elimination has a complexity
of order ofO(N^3 )operations, a luxury that we cannot afford for each time a particle move
is accepted. An alternative way of updating the inverse of a matrix when only a row/column
is changed was suggested by Sherman and Morris. It has a time scaling of the order of
O(N^2 )[94,128,129] and is given by
D−k j^1 (xnew) =
D−k j^1 (xcur)−D−ki (^1) (xcur)
R ∑
Nl= 1 Dil(xnew)D− 1
l j(x
cur)ifj 6 =i
D−ki^1 (xcur)
R ∑
N
l= 1 Dil(x
cur)D− 1
l j(x
cur) ifj=i
(16.18)
The evaluation of the determinant of anN×Nmatrix by standard Gaussian elimination
requiresO(N^3 )calculations. As there areNdindependent coordinates we need to evaluate
NdSlater determinants for the gradient (quantum force) andNdfor the Laplacian (kinetic
energy). With the updating algorithm we need only to invert the Slater determinant matrix
once. This can be done by the LU decomposition discussed in chapter 6.
Table 16.1 summarizes the computational cost associated with the Slater determinant part of
the trial wave function.
Operation No optimization With optimization
Evaluation ofR O(N^2 ) O
(N 2
2
)
Updating inverse O(N^3 ) O
(N 3
4
)
Transition of one particleO(N^2 )+O(N^3 ) O
(N 2
2
)
+O
(N 3
4
)
Table 16.1Comparison of the computational cost involved in the computation of the Slater determinant with
and without optimization.
16.7 Reducing the Computational Cost of the Correlation Form
The total number of different relative distancesri jisN(N− 1 )/ 2. In a matrix storage format,
the set forms a strictly upper triangular matrix^2
(^2) In the implementation, however, we do not store the entries lying on the diagonal.