15.2 Hartree-Fock theory 497
withsis the spin ( 1 / 2 for electrons),msis the spin projectionms=± 1 / 2 , and the spatial part
is
φnlml(r) =Rnl(r)Ylml(rˆ)
withYthe spherical harmonics discussed in chapter 14 andunl=rRnl. The other quantum
numbers are the orbital momentumland its projectionml=−l,−l+ 1 ,...,l− 1 ,land the
principal quantum numbern=nr+l+ 1 , withnrthe number of nodes of a given single-particle
wave function. All results are in atomic units, meaning thatthe energy is given byenl=
−Z^2 / 2 n^2 and the radius is dimensionless.
We obtain then a modified single-particle eigenfunction which in turn can be used as an
input in a variational Monte Carlo calculation of the groundstate of a specific atom. This
is the aim of the next chapter. Since Hartree-Fock theory does not treat correctly the role
of many-body correlations, the hope is that performing a Monte Carlo calculation we may
improve our results by obtaining a better agreement with experiment.
In the next chapter we focus on the variational Monte Carlo method as a way to improve
upon the Hartree-Fock results. Although the variational Monte Carlo approach will improve
our agreement with experiment compared with the Hartree-Fock results, there are still fur-
ther possibilities for improvement. This is provided by Green’s function Monte Carlo meth-
ods, which allow for an in principle exact calculation. The diffusion Monte Carlo method is
discussed in chapter 17, with an application to studies of Bose-Einstein condensation. Other
many-body methods such as large-scale diagonalization andcoupled-cluster theories are dis-
cussed in Ref. [117].
15.2 Hartree-Fock theory
Hartree-Fock theory [97, 118] is one of the simplest approximate theories for solving the
many-body Hamiltonian. It is based on a simple approximation to the true many-body wave-
function; that the wave-function is given by a single Slaterdeterminant ofNorthonormal
single-particle wave functions^1
ψnlmlsms=φnlml(r)ξms(s).
We use hereafter the shorthandψnlmlsms(r) =ψα(r), whereαnow contains all the quantum
numbers needed to specify a particular single-particle orbital.
The Slater determinant can then be written as
Φ(r 1 ,r 2 ,...,rN,α,β,...,ν) =√^1
N!
∣∣
∣∣
∣∣
∣∣
∣∣
∣
ψα(r 1 )ψα(r 2 )...ψα(rN)
ψβ(r 1 )ψβ(r 2 )...ψβ(rN)
..
...
.
... ..
.
ψν(r 1 )ψβ(r 2 )...ψβ(rN)∣∣
∣∣
∣∣
∣∣
∣∣
∣
. (15.1)
Here the variablesriinclude the coordinates of spin and space of particlei. The quantum
numbersα,β,...,νencompass all possible quantum numbers needed to specify a particular
system. As an example, consider the Neon atom, with ten electrons which can fill the 1 s, 2 sand
2 psingle-particle orbitals. Due to the spin projectionsmsand orbital momentum projections
ml, the 1 sand 2 sstates have a degeneracy of 2 ( 2 l+ 1 ) = 2 while the 2 porbital has a degeneracy
(^1) We limit ourselves to a restricted Hartree-Fock approach and assume that all the lowest-lying orbits are
filled. This constitutes an approach suitable for systems with filled shells. The theory we outline is therefore
applicable to systems which exhibit so-called magic numbers like the noble gases, closed-shell nuclei like^16 O
and^40 Ca and quantum dots with magic number fillings.