CASINO manual - Theory of Condensed Matter

where {c n } are the determinant coefficients. Multi-determinant expansions are widely used in quantum
Chemistry methods and successfully describe correlations missing from the single-determinant wave
function for small systems. However, the length of the expansion required to achieve a given error in
the energy grows rapidly with system size, and for large systems multi-determinant expansions are
impractical.
In QMC one has the ability to use Jastrow factors (see Sec. 22), which do a good job at describing
electronic correlation. As a consequence, it is posible to use small multi-determinant expansions in
QMC and attain an excellent description of electronic correlations.
21.1 Compressed multi-determinant expansions
It is possible to apply certain algebraic tricks to a multi-determinant expansion in order to reduce
it to a much shorter expansion where the determinants are populated with modified orbitals. This
substantially reduces the cost of using large expansions in QMC. The compression method is described
in Ref. [21].
To use a multi-determinant wave function in casino, you will need to:
• Run casino with runtype set to gen mdet casl to produce an mdet.casl file describing the
multi-determinant expansion.
• Run the det compress utility to produce a cmdet.casl file describing the compressed multideterminant
expansion.
• Run casino as usual with the cmdet.casl file produced by det compress in the directory
where casino is being run.
22 The Jastrow factor
The Slater-Jastrow wave function is
Ψ(R) = exp [J] ∑ n
c n D ↑ nD ↓ n , (147)
where the D n are determinants of up and down spin orbitals and J is the Jastrow factor.
22.1 General form of CASINO’s Jastrow factor
casino uses the form of Jastrow factor proposed in Ref. [45]. casino’s Jastrow factor is the sum of
homogeneous, isotropic electron–electron terms u, a homogeneous, isotropic electron–electron–electron
term W , isotropic electron–nucleus terms χ centred on the nuclei, isotropic electron–electron–nucleus
terms f, also centred on the nuclei and, in periodic systems, plane-wave expansions of electron–electron
separation and electron position, p and q. The form is
J({r i }, {r I }) =
N−1
∑
N∑
i=1 j=i+1
N−1
∑
+
N∑
ions
u(r ij ) + W ({r ij }) +
N∑
i=1 j=i+1
p(r ij ) +
I=1
N∑
χ I (r iI ) +
i=1
N∑
ions
I=1
N−1
∑
N∑
i=1 j=i+1
f I (r iI , r jI , r ij )
N∑
q(r i ), (148)
i=1
where N is the number of electrons, N ions is the number of ions, r ij = r i − r j , r iI = r i − r I , r i is the
position of electron i and r I is the position of nucleus I. In periodic systems the electron–electron and
electron–nucleus separations, r ij and r iI , are evaluated under the minimum-image convention. Note
that u, χ, f, p and q may also depend on the spins of electrons i and j.
The plane-wave term, p, will describe similar sorts of correlation to the u term. In periodic systems
the u term must be cut off at a distance less than or equal to the radius of the sphere inscribed in
the WS cell of the simulation cell and therefore the u function includes electron pairs over less than
three quarters of the simulation cell. The p term adds variational freedom in the ‘corners’ of the
simulation cell, which could be important in small cells. The p term can also describe anisotropic
146