As we can see from these results, it looks like GVB wavefunctions are sufficient to study most of the molecules in the table. We can see that we get essentially the exact result for H2, the only wavefunction here with no nodes. Beyond this, we get decent results for almost all molecules except CN, CO, N2, NO, which fail catastrophically. This observation should be sufficient to dispel any remaining doubt that the fixed-node error can be quite large. For CO and N2, however, we see that an RCI wavefunction is sufficient to capture the remaining error in these nodes. On the other hand, even though RCI helps, apparently we are not yet using a wavefunction of significant quality to study CN or NO. The problem with these two ground state doublet molecules is that the unpaired electron has significant occupation in orbitals that would otherwise be GVB paired. For the molecule which we employed a CAS wavefunction, O2, we managed to measure a respectable atomization energy, even if the error is larger than we would like. A few of the other molecules expressed large errors, but did not take the time to isolate the problems. Our calculations produced a lower average error than those of Grossman, but since our calculations were run without pseudopotentials and his were, we were not able to run as many molecules as he.
We are glad to observe that most of our results, where we seem to have captured the essential chemistry, are within the error margins of chemical accuracy. Pointing out again that there is often as much error in the geometry as there is in the zero point energy, we should not necessarily expect better results than those we have presented here, given the survey nature of these results. However, we would have expected to do better for our ethylene atomization calculation because of the attention to detail from Chapter 4, which here is in error by about 2 kcal/mol. Our error for cyclobutane, for which our geometry is only mostly accurate, was in error by 3 kcal/mol. We assume this is because we are only adding perfect pairs to the CC bonds, and not for any of the CH bonds. With these considerations in mind, we are cautious about using QMC and our methodology to calculate atomization energies, even though we have seen several such calculations in the literature.
Equation 2.19)
p= T(r←r#) T(r→r#)
Ψ2T(r#)
Ψ2T(r), (5.11)
where T is some transition matrix. The acceptance probability
A= max[min[1, p],0] (5.12)
is compared to a uniformly distributed random number to determine whether a proposed trial configuration should be accepted for the walker in question. We present a distribution of the value p for all electron moves in Figure 5.2 using GVB/tz wavefunctions, where we can see that about half of the distribution is above 1. Looking to the left ofp = 0, which corresponds to crossing a node, we can see roughly how far walkers try to jump past the node. The fixed-node condition sets the probability of all such moves to zero, as shown in Equation 5.12. Figure 5.2 shows that the peak atp= 1 broadens as either the time step gets larger, or as the molecule gets larger, a feature which results in a lower average acceptance probability. Crudely assuming symmetry in the distribution, one might guess that the average acceptance probability would not drop below 0.5&p >1'+ 0.25&p <1'= 0.75, but this remains to be determined.
There is a very interesting feature visible when we bin the data after the fixed-node condition has been applied for the same data, as we have done for Figure 5.3. Here, we can see that all the p <0 tail has been mapped into thep >0 region, producing the large peak in the left-most bin. But what is more obvious now is that the peak does not appear to be discontinuous and that the abnormalities are seen out to about p = 0.05. This suggests a new strategy for the acceptance probability
A= max[min[1, p],0.05] (5.13)
which not only prohibits the p < 0 moves forbidden by the fixed-node condition, but also a few more moves. For our methylene 0.05 time step, the cumulative probability up to p = 0.05 is approximately 3.4%, of which 2.3% was in the p = 0 bin. This means that our new strategy would prohibit an additional 1.1% of the moves, possibly helping the calculation to avoid some of the instabilities that have made calculations difficult. If we apply this new rule, we get the results shown in Table 5.6, where both results have improved.
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Figure 5.2: The probability distribution function of the acceptance probability over the range -2 to 2. This data was collected by binning the acceptance probability before manip- ulation. Values outside of this range were added to the nearest bin for the histogram.
These very preliminary results are quite encouraging, and we believe that pursuit of this route will be fruitful.
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Figure 5.3: The probability distribution function of the acceptance probability over the range 0 to 2. This data was collected by binning the acceptance probability after the fixed- node condition has been applied. Values outside of this range were added to the nearest bin for the histogram.