Computational Physics - Department of Physics

16.3 Computational Optimization of the Metropolis/Hasting Ratio 521

native algorithm not requiring the separated evaluation ofthe determinants will be derived
in the following. We start by defining a Slater matrixDwith its corresponding(i,j)−entries
given by
Di j≡φj(ri), (16.4)

whereφj(ri)is thejthsingle particle wave function evaluated for the particle atpositionri.

The inverse of a (Slater) matrix is related to its adjoint (transpose matrix of cofactors) and its
determinant by

D−^1 =ad jD

|D|

⇒|D|=ad jD

D−^1

, (16.5)

or

|D|=

N

∑

j= 1

Cji

D−i j^1

=

N

∑

j= 1

Di jCji, (16.6)

i.e., the determinant of a matrix equals the scalar product of any column(row) of the matrix
with the same column(row) of the matrix of cofactors.

In the particular case when only one particle is moved at the time (say particle at positionri),
this changes only one row (or column)^1 of the Slater matrix. An efficient way of evaluating
that ratio is as follows [94,129].

We define the ratio of the new to the old determinants in terms of Eq. (16.6) such that

RSD≡

|D(xnew)|

|D(xcur)|

=

∑Nj= 1 Di j(xnew)Cji(xnew)

∑Nj= 1 Di j(xcur)Cji(xcur)

.

When the particle at positionriis moved, theith−row of the matrix of cofactors remains
unchanged, i.e., the row numberiof the cofactor matrix are independent of the entries in the
rows of its corresponding matrixD. Therefore,

Ci j(xnew) =Ci j(xcur),

and

RSD=

∑Nj= 1 Di j(xnew)Cji(xcur)

∑Nj= 1 Di j(xcur)Cji(xcur)

=

∑Nj= 1 Di j(xnew)D−ji^1 (xcur)|D|(xcur)

∑Nj= 1 Di j(xcur)D−ji^1 (xcur)|D|(xcur)

. (16.7)

The invertibility ofDimplies that
N
∑
k

DikD−k j^1 =δi j. (16.8)

Hence, the denominator in Eq. (16.7) is equal to unity. Then,

RSD=

N

∑

j= 1

Di j(xnew)D−ji^1 (xcur).

Substituting Eq. (16.4) we arrive at

RSD=

N

∑

j= 1

φj(xnewi )D−ji^1 (xcur) (16.9)

(^1) Some authors prefer to express the Slater matrix by placing the orbitals in a row wise order and the position
of the particles in a column wise one.