530 16 Improved Monte Carlo Approaches to Systems of Fermions
d
dt
(||A) = (||A)tr(
A−^1 dA
dt)
d
dt
ln||A(t) =tr(
A−^1
dA
dt)
=
N
∑
i= 1N
∑
j= 1A−i j^1 A ̇ji, (16.42)whereNis the number of entries in a row. What we have here is the expression for com-
puting the derivative of each of the determinants appearingin Eq. (16.41). Furthemore, note
that the specialization of this expression to the current problem implies that the termA−^1
appearing on the right hand side is the inverse of the Slater matrix, already available after
finishing each Monte Carlo cycle as deduced from the algorithms discussed in the previous
sections. It means that the only thing we have to do is to take the derivative of each single
wave function in the Slater matrix with respect to its variational parameter and taking the
trace ofΨSD(α)−^1 Ψ ̇SD(α). The implementation of this expression and its computationusing
analytical derivatives for the single state wave functionsis straighforward. The flow chart for
the Quantum Variational Monte Carlo method with optimization of the trial wave function is
shown in figure 16.1.16.12Exercises.
16.1.The aim of this project is to use the Variational Monte Carlo (VMC) method and evaluate
the ground state energy of the atoms helium, beryllium and neon.
We labelr 1 the distance from electron 1 to the nucleus and similarlyr 2 the distance be-
tween electron 2 and the nucleus. The contribution to the potential energy from the interac-
tions between the electrons and the nucleus is−2
r 1−
2
r 2, (16.43)
and if we add the electron-electron repulsion withr 12 =|r 1 −r 2 |, the total potential energy
V(r 1 ,r 2 )is
V(r 1 ,r 2 ) =−2
r 1−
2
r 2+
1
r 12, (16.44)
yielding the total HamiltonianĤ=−∇(^21)
2
−∇
(^22)
2
−^2
r 1−^2
r 2+^1
r 12, (16.45)
and Schrödinger’s equation reads
Ĥψ=Eψ. (16.46)
All equations are in so-called atomic units. The distancesriandr 12 are dimensionless. To have
energies in electronvolt you need to multiply all results with 2 ×E 0 , whereE 0 = 13. 6 eV. The
experimental binding energy for helium in atomic units a.u.isEHe=− 2. 9037 a.u..- Set up the Hartree-Fock equations for the ground state of the helium atom with two electrons
occupying the hydrogen-like orbitals with quantum numbersn= 1 ,s= 1 / 2 andl= 0. There is
no spin-orbit part in the two-body Hamiltonian.Make sure to write these equations using
atomic units. - Write a program which solves the Hartree-Fock equations for the helium atom. Use as input
for the first iteration the hydrogen-like single-particle wave function, with analytical shape
∼exp(−αri)whererirepresents the coordinates of electroni. The details of all equations