Computational Physics - Department of Physics

520 16 Improved Monte Carlo Approaches to Systems of Fermions

If for each spin configurationσ= (σ 1 ,...,σN)we replace the total antisymmetric wave func-
tion by a version with permuted arguments arranged such thatthe firstN↑arguments are
spin up and the restN↓=N−N↑are spin down we get

Ψ(x 1 ,...,xN)→Ψ(xi1,...,xiN)

=Ψ({ri1,↑},...,{riN↑,↑},{riN↑+ 1 ,↓},...,{riN,↓})

=Ψ({r 1 ,↑},...,{rN↑,↑},{r1N↑+ 1 ,↓},...,{rN,↓}).

Because the operatorÔis symmetric with respect to the exchange of labels in a pair of
particles, each spin configuration gives an identical contribution to the expectation value.
Hence,

〈Ô〉=

〈Ψ(r)|Ô(r)|Ψ(r)〉

〈Ψ(r)|Ψ(r)〉

The new state is antisymmetric with respect to exchange of spatial coordinates of pairs of
spin-up or spin-down electrons. Therefore, for spin-independent Hamiltonians, the Slater de-
terminant can be splitted in a product of Slater determinants obtained from single particle
orbitals with different spins. For electronic systems we get then

ΨD=D↑D↓,

where

D↑=|D(r 1 ,r 2 ,...,rN/ 2 )|↑=

∣∣

∣∣

∣∣

∣∣

∣

φ 1 (r 1 ) φ 2 (r 1 ) ··· φN/ 2 (r 1 )

φ 1 (r 2 ) φ 2 (r 2 ) ··· φN/ 2 (r 2 )

φ 1 (rN/ 2 )φ 2 (rN/ 2 )···φN/ 2 (rN/ 2 )

∣∣

∣∣

∣∣

∣∣

∣

↑

. (16.1)

In a similar way,D↓=|D(rN/ 2 + 1 ,rN/ 2 + 2 ,...,rN)|↓. The normalization factor has been removed,
since it cancels in the ratios needed by the variational Monte Carlo algorithm, as shown
later. The new stateΨD(r)gives in this case the same expectation value asΨ(x), but is more
convenient in terms of computational cost.The Slater determinant can now be factorized as

ΨT(x) =D↑D↓ΨC. (16.2)

16.3 Computational Optimization of the Metropolis/Hasting Ratio

In the Metropolis/Hasting algorithm, theacceptance ratiodetermines the probability for a
particle to be accepted at a new position. The ratio of the trial wave functions evaluated at
the new and current positions is given by

R≡Ψ

Tnew

ΨTcur

=

|D|new↑

|D|cur↑

|D|new↓

|D|cur↓

︸ ︷︷ ︸

RSD

ΨCnew

ΨCcur

︸︷︷︸

RC

. (16.3)

16.3.1Evaluating the Determinant-determinant Ratio

Evaluating the determinant of anN×Nmatrix by Gaussian elimination takes of the order of
O(N^3 )operations, which is rather expensive for a many-particle quantum system. An alter-