My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 3.
QUANTUM MONTE CARLO METHODS
3.4.1 The Jastrow factor
In section 2.3 it was shown that a single-determinant wave function takes account
of exchange but not of correlation; the Jastrow factor allows correlation effects to
be incorporated.
The most important correlations are those involving pairs of electrons. These
are included by having a term of the form
− ∑ i>j
u σi ,σ j
(|r i − r j |) (3.78)
in the Jastrow exponent J(X). Recall that the single-determinant wave function
does nothing to prevent electrons of opposite spin from coming together; this term
keeps these electrons apart, resulting in a significant lowering of energy. Electrons
of like spin are also kept apart more than before, although this affects the energy
less dramatically.
The two-body term of equation (3.78) does not simply keep electrons apart. Both
the long- and short-range behaviour of u are constrained by theoretical arguments.
When two electrons approach each other, the Coulomb energy diverges; for a
wave function to be an eigenstate of Ĥ, this divergence must be cancelled by a
corresponding divergence in the kinetic energy. Such a divergence is produced by
cusps in the wave function: discontinuities in the first derivative with respect to the
distance between the electrons. A full discussion of the cusp conditions is given in
appendix B.
The long-range behaviour of u may be determined by arguments based on the
random phase approximation of Bohm and Pines [9], and is the subject of chapter 7.
A connection is made between the long-range electron-electron correlations and the
long-wavelength density fluctuations known as plasmons; for a homogeneous system,
the resulting u function has the form 1/ω p |r i − r j | in the limit |r i − r j | → ∞, where
ω p = √ 4πn is the plasma frequency.
A function which combines the required short- and long-range behaviour is
u σi σ j
(|r i − r j |) =
1
)
(1 − e −|r i−r j |/F σi σ j
, (3.79)
ω p |r i − r j |
53