Results and Discussion

  

Figure: number of significant ERIs as a function of $\omega$

IntegralsIntegrals Omega$\omega$ LegA C₆₀H₁₂₂ LegB C₅₄H₁₈ LegC C₃₅H₃₆ 10 x10

The first aspect to examine is just how many integrals Coulomb attenuation removes. Figure 7.2 shows the number of significant integrals (defined here as greater than 10^-9) as a function of $\omega$ for a linear alkane ( C₆₀H₁₂₂), a hydrogen terminated graphite sheet ( C₅₄H₁₈) and a hydrogen terminated diamond chunk ( C₃₅H₃₆). The 6-31+G* basis is used in all cases. The three molecules may be characterized as one-dimensional, two-dimensional and three-dimensional, respectively.

Examination of the alkane curve shows that extremely small attenuation has no effect on the number of integrals, but once attenuation starts to remove integrals, the removal of integrals is very efficient and, beyond mild attenuation ( $\omega = 0.2 \: a_{0}^{-1}$), most of the integrals are insignificant. The graphene is more compact than the the alkane and, as a result, much larger attenuation is required before the pruning begins to take effect. Once the pruning has begun, the curve shows a similar shape to the linear alkane, with a quite rapid removal of integrals. By $\omega = 1.0 \: a_{0}^{-1}$ only half of the original integrals remain. The effect of CASE on this diamond chunk, however, is less spectacular. C₃₅H₃₆ is very compact and a significant reduction in the number of integrals does not begin until large $\omega$ values. Once pruning of integrals begins, though, a similar rapid decrease in the number of integrals is observed. Clearly, the efficiency of CASE depends on the shape of the system, as well as the number of atoms present.

  

Figure: Log-Log plot of significant ERIs against $\omega$

Integrals $\ln(\mathrm{Integrals})$ Omega $\ln(\omega)$ LegA C₉₀H₁₈₂ LegB C₅₄H₁₈ LegC C₈₄H₆₄

The rate of decrease of the three systems is presented in Figure 7.3 through a log-log plot. An integral accuracy of 10^-6 has been used, with some slightly larger systems. The graph shows that, once integral removal has begun, the three-dimensional system ( C₈₄H₆₄) shows a faster rate of decrease than the two- and one-dimensional systems. This is as expected, as increasing $\omega$ decreases the interaction distance, which, in a one-dimensional system, produces a corresponding decrease in the number of integrals. However in a three-dimensional system, the same decrease in interaction distance will remove the the number of integrals removed for the one-dimensional system, raised to the third power, simply because there are now three dimensions within which particles interact.

  

Figure: HF/6-31G* timings of linear alkanes C_nH_2n+2 with $\omega = 0.25$

Time Time (s) Alkane_lengthn TraditionalPRISM BSBoxing and Screening SOScreening Only BOBoxing Only NBNSNo Boxing or Screening ONXQCTC/ONX

Figure 7.4 presents SGI Power-Challenge CPU times to calculate all the two-electron integrals required for a HF/6-31G* Q-CHEM calculation on a series of linear alkanes, using $\omega = 0.25 \: a_{0}^{-1}$ and an integral cutoff of 10^-9. The solid black curve shows the quadratic behaviour typical of the conventional O(N²) algorithm. Just above this lies the time for a CASE calculation using the traditional algorithm, clearly showing the cost of an extra interpolation. Included with these curves is the QCTC/ONX which shows an expensive O(N²) behaviour (QCTC is a linear Coulomb method, similar to the CFMM). A direct comparison with ONX can not really be made as the two algorithms use different integral code and different screening strategies, yet the overall nature of the curve is still useful. The poor performance of ONX is due to the high integral accuracy used. As shown later in this section, ONX becomes more competitive at lower integral accuracy. The two near-linear curves are for CASE using integral screening, and boxing with integral screening. The remaining curve is for boxing only. The fluctuation of this final curve is due to the additional computational overhead required when previously unsplit shell-pairs now cover two boxes, forcing the shell-pairs to be split for no computational gain. It is perhaps surprising that integral screening, although formally O(N²), performs better than boxing alone for these systems. Yet the quadratic nature of this curve can just be seen, and screening will become more expensive than boxing alone for extremely large systems. The fastest timings for large systems are, not surprisingly, when both boxing and integral screening are used. It should be noted though that there is a small window where screening alone is the fastest method.

  

Figure: HF/6-31G* CASE Boxing and Screening timings of linear alkanes C_nH_2n+2

Time Time (s) Alkane_lengthn Traditional $\omega = 0$ omega025 $\omega = 0.25$ omega05 $\omega = 0.50$ omega1 $\omega = 1.00$ omega2 $\omega = 2.00$

Figure 7.5 shows the effect on CPU times of altering $\omega$ for this same series of alkanes. $\omega = 0$ is the traditional 1/r code, exhibiting the expected quadratic trait, while higher $\omega$ values all show a linear behaviour. Notice that the time gained by doubling $\omega$ decreases as $\omega$ increases. For these systems there seems little to be gained by increasing $\omega$ past $1.0 \: a_{0}^{-1}$. By $\omega = 1.0 \: a_{0}^{-1}$ the integrals that remain involve diffuse orbitals, requiring very large $\omega$ values (much larger than $2\:a_{0}^{-1}$) to render them insignificant.

  

Figure: Ratio of PRISM time to Method time for C₅₀H₁₀₂ at various integral thresholds

Ratio
Ratio Method Method BS Box and Screen SO Screening Only BO Boxing Only ONX QCTC/ONX LegA10^-9 LegB10^-8 LegC10^-7 LegD10^-6 LegE10^-5

The effect of changing the integral accuracy (significance level) on the time to calculate the integrals for C₅₀H₁₀₂ is presented in Figure 7.6. The ratio of the conventional CPU time to each method's time reveals that all of these methods become relatively faster when the integral threshold is lowered. This is due to the extra chance (at the shell-quartet level) to screen out computational work based on the integral accuracy required. The greatest increase is seen for ONX, yet with these high accuracy computations it is still poor.

  

Figure: Ratio of PRISM time to method time for C₅₀H₁₀₂ with various basis sets

Ratio
Ratio Method Method BS Box and Screen SO Screening Only BO Boxing Only ONX QCTC/ONX LegA6-31G* LegB6-31G LegC3-21G LegDSTO-3G

Boxing and screening behave in similar ways with changes to the basis set. Figure 7.7 shows the the ratio of CPU times for the new CASE implementations with that of the traditional algorithm. The more uncontracted the basis set, the faster the method relative to the unattenuated calculation. For screening this is a reflection of the time taken to test each primitive shell-pair for its $\mathrm{T}_{\omega}$ value. For boxing, it is due to the splitting of shell-pairs. If a shell-pair has only a small degree of contraction, the amount of testing required is small. ONX shows a similar type of behaviour, which may just be a product of the different integral technology, as well as a significant slow-down with the addition of d functions, which is consistent with a delocalization of charge removing the benefits of density screening.

  

Figure: HF/6-31G* timings for graphite and diamond chunks with $\omega =0.5 \: a_{0}^{-1}$

Time Time (s) Carbon_numberNumber of carbon atoms TraditionalPRISM (graphite) BSBox and Screen (graphite) SOScreening only (graphite) Trad3DPRISM (diamond) SO3DScreening only (diamond)

The performance of CASE on two- and three-dimensional systems can be seen in Figure 7.8. The black curve is the timings for the traditional PRISM method on a series of graphite sheets. Below this curve lie the boxing with screening and screening only curves. These CASE curves show linear behaviour, but are not as efficient as those for the linear alkanes. The CASE timings for the diamond chunks however are disappointing. It is only for C₈₄H₆₄ that CASE is faster, and this is just the screening, without boxing. The answer for the poor performance lies in the physical size of the system. The largest three-dimensional structure included ( C₈₄H₆₄) is only $23 \: a_{0}$ long on its largest side, compared with $36 \: a_{0}$ for the largest graphite sheet used ( C₉₆H₂₄) and $195 \: a_{0}$ for the largest linear alkane ( C₉₀H₁₈₂).