When CSFs are used instead of determinants or HPHF functions, this has sub- stantial impacts on the spawning dynamics. Determinants are permitted to dif- fer by a maximum of two spin orbitals, but CSFs are permitted to have spatial structures which differ by two spatial orbitals. Generally the number of spatial excitations from a CSF is similar to the number of spin excitations from a deter- minant. As a consequence the potential number of CSFs connected to a starting spin eigenfunction is larger than the number of connected determinants by a factor of approximately ncsf(No, S), where No is the number of unpaired electrons of the
target site.
As explained in section 2.6, the maximum value for the time step, δτ , is inversely proportional to the maximum strength of each of the connections in equation 2.17, and as a consequence is proportional to the minimum generation probability pro- duced.
The restrictions on δτ tend to be due to sites with large numbers of unpaired electrons, which are extremely highly excited in comparison to the reference site (and also to the majority of occupied sites which contribute substantially to the calculated wavefunction). This is a consequence of spatial structures with large numbers of unpaired electrons having many available spin structures, and thus correspondingly lower generation probabilities. A perverse consequence of this is that a lot of unnecessary computational effort is spent treating the majority of occupied sites with a δτ value orders of magnitude below that which they require. For an alternative approach to dealing with restrictions on δτ , see chapter 7. If the restriction on one-to-one spawning is lifted, this problem may be mitigated. This corresponds to letting the set {k} considered in equation 2.15 contain all sites which share the same spatial structure. From an occupied site, a spatial excitation is made, and then spawning is attempted to all sites associated with the target
(a) Site-to-site spawning. A specific CSF asso-
ciated with the target of a spatial excitation is selected.
(b) Multiple-structure (‘site-to-blob’) spawn-
ing. The Hamiltonian matrix elements between the source particle and all CSFs sharing a target spatial structure are con- sidered.
(c) ‘Blob-to-blob’ spawning. The sum of the
Hamiltonian matrix elements between oc- cupied sites in a source spatial structure and all CSFs sharing a target spatial struc- ture are considered.
Figure 5.3: Spawning between different combinations of the CSFs within the source and
target spatial structures associated with a spatial excitation. This gener- ates spawning between multiple different sites during the same spawning step. The source spatial structure is shown to be sparsely occupied (only a few crosses). If a cross is black it is being considered as the source of a spawn, if it is grey it is not. All of the target sites being considered are indicated in grey.
spin structure∗ as demonstrated in figure 5.3b. The consequence of this is that the probability of generating site j given site i,
pgen(j|i) = pgen(J|i) = pgen(J|I),
is now given by the probability of making the spatial excitation from spatial struc- ture I to spatial structure J, increasing the generation probabilities by a factor of
ncsf(No).
This has the advantage that for the highly excited sites, with many unpaired electrons, which dominate the time step dependence, the generation probabilities are increased by the most — preventing these highly excited states from causing an increase in the cost of the low-lying states that do not require such a small time step. It is worth noting that this also improves the overall cost scaling of the
∗If the spawn being attempted is a same spatial structure spawn, spawning is attempted at
all sites that would give spawning-like behaviour, i.e. the source site is excluded from attempting to spawn to itself. This behaviour is already covered under the death step.
system — the total computational cost is inversely proportional to the time step, and as a consequence multiple-structure spawning removes an O(ncsf) ≈ O(eN)
term from the computational scaling. Figure 5.4 demonstrates clearly the benefit of using multiple-structure spawning on the time step values that may be used. The permitted time steps are still smaller than those available for determinental calculations, but they are substantially larger than if multiple-structure spawning were not available. As a consequence, Serber functions become comparable to HPHF functions in terms of computational cost.
When using Serber CSFs it is especially advantageous to make use of a multiple- structure spawning scheme. As described in section 4.5, once a Hamiltonian ma- trix element between two CSFs has been generated, generating the matrix element between different CSFs with the same spatial structures is the same, with only the indices into the permutation matrices used changing. The majority of the computational cost involved is spent generating the spatial excitation, and ma- nipulating the line-up permutations. It is trivial to return the matrix elements corresponding to one column of each of the permutation matrices used — giving substantially more convergence ‘bang’ for your computational ‘buck’ than other- wise. This modification of the algorithm is essentially computationally free, while permitting comparable values of δτ to determinental and HPHF calculations.
As an aside, it is worth noting that the total weight of particles spawned per unit imaginary time remains roughly the same if multiple-structure spawning is used. A substantial number of additional spawning attempts are made, but the generation probabilities are increased in proportion. As a consequence of this, the acceptance ratio (the ratio of spawned particles to attempted spawns) appears to plummet when multiple-structure spawning is in use.
5.5.1 ‘Blob-to-blob’ spawning
A logical extension of multiple-structure spawning is to consider all possible con- nections between source and target spatial structures in each spawning step. The sum over coefficients, cj, in equation 2.10a is considered in re-approximating equa-
N O F Ne N2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Serber Serber, multispawn
Figure 5.4: Variation in the ratio between the maximum δτ permitted without blooms
of three or more particles occurring between determinental calculations and those involving Serber functions, with or without multiple-structure spawning, as the number of electrons is increased. All of theses systems are modelled using a cc-pVDZ basis set. δτ is found by searching, as described in section 2.6. The lighter first row atoms are not included, as determining a bound on δτ for such a small system becomes meaningless.
tion 2.16, to give P j∈J|cj| γ × −δτγ P j∈J j6=k (Kkj− ESkj)cj pgen(K|J)Pj∈J|cj| −→ ∆c k ∀ k ∈ K ∈ {I ← J} .
As shown in figure 5.3c, all of the occupied sites, j, in a source spatial structure, J, are considered in generating the spawns to each of the available sites, k, associated with a target spatial structure, K. The total number of spawns attempted is determined from the cumulative weight on all of the occupied sites being spawned from, with the magnitude of each spawning attempt, γ, being determined as before.
By including as many terms, containing information from as many occupied sites and as many matrix elements as possible, blob-to-blob spawning acts to smooth out the stochastic changes in the wavefunction and thus to minimise the statistical noise.
−6.0 −5.5 −5.0 −4.5 −4.0 −3.5 −3.0 −2.5 −2.0 log10(pgen) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 (n or m alis ed ) fr eq u en cy Determinants HPHF Serber Serber, multispawn
Figure 5.5: Histogram of pgen values for stretched N2 (bond length 4.2 a0) in a cc-
pVDZ basis, using Slater determinants, HPHF functions and Serber CSFs (as a prototypical spin eigenfunction). Note the substantial improvement in generation probabilities when making use of multiple-structure spawning with Serber functions.
It is worth noting that such ‘blob-to-blob’ spawning has no substantial impact on the maximum time step that may be used compared to the normal multiple- structure spawning, as the generation probability depends only on the spatial excitation such that pgen(j|i) = pgen(J|I), and is therefore unchanged.