CASINO manual - Theory of Condensed Matter

27.1.2 ‘Standard’ QMC
Let the number of electrons in our system be N. In ‘standard’ QMC the orbitals are HF or DFT
eigenfunctions extending over the entire system. On the other hand, the basis functions used to
represent those orbitals are usually localized functions. Each time an electron is moved, all O(N)
occupied orbitals must be evaluated; however, only the O(1) basis functions in the vicinity of the
electron need to be computed to evaluate each orbital. So, the time taken to carry out a configuration
move of all N electrons in the system is O(N 2 ).
If the orbitals are represented by extended basis functions, such as plane waves, then the time taken
for a configuration move scales as O(N 3 ), because every basis function in every orbital has to be
computed for every electron.
27.1.3 Localized orbitals
It is easy to show that a nonsingular linear transformation of a set of orbitals can only change
the normalization of the Slater determinant of those orbitals. For insulating materials there exist
straightforward and efficient algorithms for determining linear transformations to highly localized sets
of orbitals [70, 71].
Highly localized orbitals can be truncated to zero at a certain distance from their centres without
introducing a substantial bias. Therefore, each time an electron is moved, only a few orbitals need to
be evaluated; the others must be zero because the electron lies outside their truncation radii. So the
number of orbitals to be computed after each electron move is O(1) [72].
The truncation of the orbitals results in small discontinuities in the Slater wave function. These are
potentially serious for QMC because they result in the presence of Dirac delta functions in the kineticenergy
integrand. The delta functions cannot be sampled, so their contribution to the total energy
is lost. However, in practice the resulting bias is extremely small, provided the truncation radii are
sufficiently large. In fact the bias can be made arbitrarily small by increasing the truncation radii.
The discontinuities can be avoided by bringing the localized orbitals smoothly to zero over a thin,
spherical skin. Surprisingly, it has been shown that the bias resulting from the use of smooth truncation
schemes is larger than the bias that occurs if the orbitals are truncated abruptly at their cutoff radii
[73]. The reason for this is that the smooth truncation schemes introduce a new, small length scale
(the skin thickness) into the problem, and the local kinetic energy in the truncation region is extremely
large. A time step that is physically reasonable for the system being simulated is much too large for
the length-scale introduced by the truncation region. This results in large time-step bias and frequent
population-explosion catastrophes. We therefore recommend truncating localized orbitals abruptly:
the bsmooth input parameter should be set to F.
27.1.4 Localized bases
Suppose the basis functions are zero outside fixed radii about their centres. Then the only functions
that have to be evaluated when an electron is moved are those with the electron inside their radii. So
the number of basis functions to be computed simply depends on the local environment of the electron
and is therefore independent of system size.
Gaussian basis functions can be regarded as localized if they are truncated to zero outside a certain
radius. This is done by default in casino: Gaussian functions exp(−ar 2 ) are assumed to be zero
when exp(−ar 2 ) < 10 −G T
, where G T is the gautol input parameter. Gaussian basis sets cannot yet
be used in conjunction with localized orbitals, however.
Alternatively, orbitals can be represented numerically using blip functions, which also constitute a
localized basis. If the orbitals are localized then the memory requirements are greatly reduced (by
a factor of O(N)), because we only have to store the blip coefficients needed to evaluate the orbital
within its truncation radius.
27.1.5 ‘Linear-scaling’ QMC
If the numbers of orbitals and basis functions to be evaluated are both O(1) then the CPU time for
a configuration move scales as O(N). This is what is meant by ‘linear-scaling QMC’. Note, however,
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