CASINO manual - Theory of Condensed Matter

• It is possible to use blip orbitals to study systems that are finite, or periodic in just the x direction,
or in the (x, y) plane, or are periodic in all three dimensions by setting the blip periodicity
keyword to 0, 1, 2 or 3, respectively. The periodic keyword must be set to T if the system is
periodic in any direction. Note that if reduced-periodicity systems are studied then the atoms
should be placed in the centre of the unit cell spanned by the blip grid, i.e., the unit cell defined
by the lattice vectors.
• When excitations are specified, it should be noted that band indices start at the first nonlocalized
band. For example, if all bands apart from the highest occupied molecular orbital (HOMO) are
localized then the HOMO is band 1. Likewise, when casino reports the number of bands
occupied at a given k point, it does not include the number of localized orbitals.
• Multideterminant calculations involving localized orbitals are possible, but all localized orbitals
must be occupied in every determinant. (It only makes sense to carry out a localization transformation
between the set of occupied orbitals.)
• The bsmooth input parameter is used to specify whether localized orbitals with spherical
truncation surfaces should be abruptly or smoothly truncated. It is recommended that bsmooth
be set to F.
• For large systems it is advisable to set the sparse input parameter to T in order to exploit
the fact that most of the orbital values are zero when updating cofactor matrices used in the
evaluation of the Slater determinants.
28 Twist averaging in QMC
28.1 Periodic and twisted boundary conditions
Suppose we wish to calculate the energy per particle of a periodic solid. In one-electron theories
we can often reduce the problem to the primitive unit cell and integrate over the first Brillouin zone.
Reduction to the primitive cell is not possible in many-body calculations, however, because correlation
effects may be long-ranged; hence such calculations must be performed in periodic simulation cells
consisting of several primitive cells.
Suppose the simulation cell is of volume Ω and contains N electrons, and let {r 1 , . . . , r N } be the
electron coordinates. The Hamiltonian Ĥ must satisfy
Ĥ(r 1 , . . . , r i + R s , . . . , r N ) = Ĥ(r 1, . . . , r i , . . . , r N ) ∀i ∈ {1, . . . , N}, (236)
where R s is a simulation-cell lattice vector. This translational symmetry leads to the many-body
Bloch condition
(
Ψ ks (r 1 , . . . , r N ) = U ks (r 1 , . . . , r N ) exp ik s · ∑ )
r i , (237)
where U has the periodicity of the simulation cell for all electrons [74]. The use of a nonzero simulationcell
Bloch vector k s is sometimes described as the application of twisted boundary conditions [75].
In a finite simulation cell subject to periodic boundary conditions, each single-particle orbital is usually
taken to be of Bloch form ψ k (r) = exp[ik · r]u k (r), where u k has the periodicity of the primitive cell
and k lies on the grid of integer multiples of the simulation-cell reciprocal-lattice vectors within the
first Brillouin zone of the primitive cell, the grid being offset from the origin by the simulation-cell
Bloch vector k s . For metallic systems some of the ground-state orbitals are nonanalytic functions
of k, because the occupancy depends on k. Instead of integrating over single-particle orbitals inside
the Fermi surface to calculate the HF kinetic and exchange energies, one sums over a discrete set
of k vectors when a finite cell is used. As the system size is increased, the fineness of the grid of
single-particle Bloch k vectors increases, and the HF energy changes abruptly as shells of orbitals pass
through the Fermi surface.
One usually finds that the fluctuations in the QMC energy due to single-particle finite-size effects are
proportional to the corresponding fluctuations in the HF kinetic energy or the DFT energy. Hence
HF or DFT energy data can be used to extrapolate QMC energies to infinite system size. Note that a
judicious choice of k s (e.g., the Baldereschi point [76] for insulators) can greatly reduce single-particle
finite-size errors [74]. The common choice of k s = 0 generally maximizes shell-filling effects and is
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