B3LYP/RI Performance Analysis Using Water Clusters

as Model Systems

Ahmad Huran

Univerity of Valencia

1. Model systems

The systems used in this study are different water clusters (H2O)n wheren = 4, 7, 10, 13, 16. All the optimized geometry where calculated at RHFlevel using 6-31G(d,p) basis set [1], in figure 1 a molecular drawing of themodel systems is shown.

Figure 1: (H2O)n, n = 4, 7, 10, 13, 16, all the structres are optimized at the RHF/6-31G(d)level.

2. Summary of computational technique

The main idea behind the RI approximation comes from the assumptionthat a four center two electron integral can be decomposed into two indepen-dent three indices sums by introducing an auxiliary index , in a way thatthe computational cost would be O (2M2M) where is an index that runsover an auxiliary basis set and M is the number of auxiliary basis functionand M is the number of the original basis functions. While its true that thiscan be done by different means, the RI approach achieves that by expressingan overlap distribution of two gaussians as a linear combination of a one elec-tron auxiliary basis, and the constants of the expansion are determined by aleast square-like fitting, hence the other name of the approximation Densityfitting, and after dealing with the mathematical details of the fitting onecan arrive at the following formula for the four center two electrons integrals

| =

| 1r12| | 1

r12| (1)

where {i} is the auxiliary basis set. Now if we consider a Hilbert spacewith a scalar product defined by the operator (1/r12), one can see where theResolution of Identity name comes from

I =

1

r12|| 1

r12(2)

3. Hardware specifications

All the calculations were performed serially.

CPU : Intel(R) Core(TM) i7 -3930K CPU @ 3.20 GHz

Tot. Mem. : 66051128 kB

Tot. Disk space: 4.5 TB

4. Results and discussion

In Table (1), we collect the execution time for both the ref. and theRI calculation along with number of basis function and the error in the RIenergy relative to the ref. energy. The premise of the RI approximation inthe reduction of the computational cost down to O (2M2M) depends on a

2

#Res. M tref tRI Erel.

4 192 9.893 11.642 9.63814e-077 336 41.151 47.440 9.29427e-0710 480 92.594 103.964 8.95224e-0713 624 170.845 184.137 9.08314e-0716 768 286.953 301.833 8.9329e-07

Table 1: B3LYP/RI performance results on the different water clusters. where #Res. isthe number of water molecules, M is the number of basis functions, tref., tRI are executiontime in seconds, for the ref. and the RI calculations respectively, and Erel is the errorin the RI energy relative to the ref. energy.

condition that is 2M < M2, other wise well have 2M2M > M

4, neverthe-less, while that condition is satisfied in all of the calculations of this study,one would find that the time needed in the RI calculations is bigger thanthe one needed for the ref. calculations(table 1), and that has nothing todo with the scaling things we mentioned above, this is a delay in the exe-cution time coming from the introduction of auxiliary bases sits as deployedin the RI approximation. However, in figure 2, a nonlinear regression fittingof the results to extrapolate the time needed for bigger basis set, shows thatat some point the effect of reducing the scaling will prevail over the addedexecution time, and the RI method would start to fruit. It also should benoted that the deployment of the RI approximation did not come at a bigcost of accuracy loss, where the relative error in the Energies was in all casessmaller than a millionth with small fluctuations from case to another(table 1).

In summary, the advantages of using the RI approximation might notbe so clear just by judgement based small system studies, but when evenbigger systems are considered this approximation is without a doubt a gamechanger. Even using todays available sheer computation power, the RI ap-proximation was the means to perform calculation that were otherwise un-feasible.

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Non-linear Regression results

Ref.RI

Figure 2: Non-linear regression fitting of the results to the function t = aM b wherearef = 3.17e 5 bref = 2.41 aRI = 7.69e 5 bRI = 2.28.

References

[1] S. Maheshwary, N. Patel, N. Sathyamurthy, A. D. Kulkarni, S. R. Gadre,Structure and stability of water clusters (H2O) n, n= 8-20: An ab ini-tio investigation, The Journal of Physical Chemistry A 105 (46) (2001)1052510537.

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