CASINO manual - Theory of Condensed Matter

every dmc decorr period iterations, which should therefore be given a suitably large value. The
total number of statistics-accumulation steps carried out should be sufficiently large that the required
number of configurations can be written out; however, the statistics-accumulation run will in fact
keep going until all the configurations have been written out, going beyond the specified number of
steps if necessary. Note that the statistics-accumulation phase of a preliminary DMC run cannot be
continued.
Whilst on the subject of reducing the expense of DMC equilibration, we point out that the length
of the equilibration period can be reduced if the initial configuration population is already partially
equilibrated. For example, if performing DMC calculations at a few different time steps, the initial
configuration population can be generated at one particular time step, then reused in the others. The
number of equilibration steps needed to re-equilibrate the population when the time step is changed
is small compared with the number of steps required to equilibrate the population from scratch.
14 Evaluation of Gaussian orbitals in the Slater wave function
casino can handle Gaussian basis sets up to and including angular momentum l = 4 (s, p, sp, d, f
and g functions). Suppose we have N Gaussians g w=1,...,N located at positions r w in the primitive
cell. Let R label the other primitive cells. There are therefore copies of the basic set of Gaussian
functions located at positions r w + R. We want to evaluate the Bloch orbitals at some point r. The
orbital is labelled by band ν and k point,
φ ν,k (r) = ∑ ∑
C ν,k g w (r − r w + R) exp[ik · R] . (56)
w
casino implicitly assumes the following about the k points when in Gaussian mode:
• The k points form a grid,
R
k lmn = l
q 1
b 1 + m q 2
b 2 + n q 3
b 3 + k s , (57)
where the q i are integers, the b i are the primitive reciprocal lattice vectors, 0 ≤ l ≤ q 1 − 1,
0 ≤ m ≤ q 2 − 1 and 0 ≤ n ≤ q 3 − 1. In crystal the offset k s is always zero, but casino does
not assume this.
• casino deals only with real orbitals, which can be formed by making linear combinations of
the states at k and −k. The list of k points in the file gwfn.data must contain only one of
each (k, −k) pair; the presence of the other is assumed. To make a many-body Bloch wave
function satisfying the condition that it is multiplied by a phase factor under the replacement
r i → r i + R s , where R s is a translation vector of the simulation cell, the offset must satisfy
k s = G s /2, where G s is a reciprocal lattice vector of the simulation cell [30]: see Sec. 15.
• Finite systems are treated as if they had a single k point.
15 Constructing real orbitals
The Bloch orbitals at an arbitrary point in k-space are complex. If the set of wave vectors consists of
±k pairs then one can always construct a set of real orbitals spanning the same space as the original
complex set. A necessary and sufficient condition for the mesh of Eq. (57) to consist of ±k pairs is
that k s = G s /2, where G s is a reciprocal lattice vector of the simulation cell lattice. It is four times
more efficient to use real orbitals than complex ones because it takes four multiplications to evaluate
the product of two complex numbers but only one multiplication for the product of two real numbers.
An orbital satisfying Bloch’s theorem can be written as
φ k (r) = u k (r)e ik·r , (58)
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