17.1 Diffusion Monte Carlo 539

The second term is called abranching term. When positive, it induces a growth of the number
of walkers and a decay if it is negative.
Before we expand on how to carry out DMC in practice, it may be helpful to consider an
analytically exact approach by focusing on the Green’s function corresponding to the time
evolution of the wave function. The approach is calledGreen’s function Monte Carlo(GFMC)
and shares its main ideas with the DMC method. Using Dirac’s bracket notation, we start
over by substituting the initial wave functionΨ(x)with its corresponding state ket|Ψ〉and
let the special evolution operator work on|Ψ〉

|Ψ〉t=e−(Ĥ−ET)t|Ψ〉,

Now we switch to position representation:

Ψ(x,t) =〈x|Ψ〉t=〈x|e−(Ĥ−ET)t|Ψ〉=

∫

〈x|e−(Ĥ−ET)t|x′〉〈x′|Ψ〉dx′

where we have just inserted a completeness relation, 1 =
∫
|x′〉〈x′|dx′. Defining the Green’s
function for this case

G(x,x′,t)≡〈x|e−(Ĥ−ET)t|x′〉=〈x|e−(K̂+V̂−ET)t|x′〉

we get:
Ψ(x,t) =

∫

G(x,x′,t)Ψ(x′)dx′ (17.5)

Calculating the Green’s functionGwould be greatly simplified if we could split it into separate
factors for each of the termsK̂and(V̂−ET)of the HamiltonianĤ, as with the following
factorization
eÂ+B̂=eÂeB̂

However, such a factorization requires that the operatorsÂandB̂commute,[A,B] = 0. Un-
fortunately, this is not the case for the termsK̂andV̂of the Hamiltonian. But, by expanding
the exponential of each part, using a descendent of the so called Baker-Campbell-Hausdorff
formula
eÂeB̂=eÂ+B̂e

(^12) [Â,̂B]
we get that:
e−(Ĥ−ET)t=e−(K̂+V̂−ET)t=e−Kt̂e−(V̂−ET)t+O(t^2 )
It is also possible to make approximations to higher orders oft, but we will here keep to the
simple first order case. Ast→ 0 , the error term disappears and the simple factorized form
becomes exact. Thus, an approximation of the form
e−(Ĥ−ET)t≈e−Kt̂e−(V̂−ET)t
is only good for smalltand is readily called ashort time approximation. The Green’s function
can by this be approximated as follows
G(x,x′,t) =〈x|e−(Ĥ−ET)t|x′〉
=〈x|e−Kt̂e−(V̂−ET)t|x′〉+O(t^2 )

∫
〈x|e−Kt̂|x′′〉〈x′′|e−(V̂−ET)t|x′〉dx′′+O(t^2 )

∫
GK(x,x′′,t)GV(x′′,x′,t)dx′′+O(t^2 )
For the kinetic energy part,GK, we get