Short introduction of the Fourier Transform Coulomb (FTC) method
The Coulomb part of the DFT calculations using ‘ordinary’ Gaussian representations can be speeded up dramatically using plane waves as secondary basis set by replacing the most costly analytical electron repulsion integrals with numerical integration techniques. The main advantages to keep the Gaussians as the primary basis set is that the diagonalization step is much faster then using primarily plane wave basis set and allelectron calculations can be performed.
The FTC technique is precise and tunable; all results are practically identical with the traditional analytical integral calculations. As table 1 shows the FTC technique is at least 23 orders of magnitude more accurate then the most popular plane wave based GAPW method (developed by Krack and Parinello) using the same energy cutoff. It is also at least 23 orders of magnitude more accurate then the so called density fitting or resolution of identity technique.
Performance wise the FTC method replaces analytical integral evaluations whose computational cost scales as O(N 4) (where N is the number of basis functions) with O(N 2) procedures. The asymptotical scaling of computational costs with system size is linear while in the analytical integral evaluation case it is quadratic. A first implementation of the FTC technique already resulted in more then an order of magnitude speed up in the evaluation of Coulomb matrixes and the speed up is increasing with the basis set size. QCHEM offers a general and very efficient implementation of the FTC method which is working together with the Jengine and the CFMM (Continous Fast Multipole Method) technique. The following figure illustrates the computational performance of our current implementation for 3D diamond clusters with two different and relatively big basis sets. Further development along this line is on the way in QCHEM.
Short scientific history of the FTC method
The first version of the FTC technique was introduced in the paper of [ L. Fusti Molnar and P. Pulay, J. Chem. Phys, 117, 7827 (2002) ] This work was based on two techniques described in the article of [ L. Fusti Molnar and P. Pulay, J. Chem. Phys., 116, 7795 (2002) ]. A more detailed paper is [ L. Fusti Molnar, J. Chem. Phys, 119, 11080 (2003) ] which describes a new technique to localize the so called filtered core functions and an efficient way to implement the Coulomb energy forces. Systematic studies of the accuracy, the scaling of the computational cost with system size and in basis set size are also included in this article. The current implementation of the FTC technique is published in the following article: [ L. Fusti Molnar and J. Kong, J. Chem. Phys, 122, 074108 (2005) ]
Table 1. Accuracy comparison between the current implementation of the FTC method and the GAPW method.
Errors ^{b} using GAPW / µE h 
Errors ^{c} using FTC / µE h 
Errors ^{d} using FTC / µE h (with tighter thresholds) 

SF 6 
2670 
6 
0.15 
CO 2 
730 
1 
0.00 
H 2O 
220 
0 
0.01 
^{a} All errors are the total energy differences between the given method and an analytical integral evaluation scheme using standard Gaussian basis sets. 631G* basis set were used.
^{b} 200 Rydberg energy cutoff was used [M. Kranck and M. Parinello, Phys. Chem. Chem. Phys, 2, 2105 (2000)].
^{c} 4.5 grid density (about 200 Rydberg) was used for fair comparison with GAPW results. The integral threshold was 10 ^{12} in both the analytical and FTC calculations and the SCF convergence criteria was 10 ^{7} in Brillouin condition.
^{d} 5.0 grid density (about 247 Rydberg) was used with tighter thresholds in the localization of the core functions and in the screening processes. The integral threshold was 10 ^{12} in both the analytical and FTC calculations and the SCF convergence criteria was 10 ^{7} in Brillouin condition.
Table 2. Accuracy comparison of the current implementation of the FTC method with a recent version of the density fitting scheme

Errors ^{b} using Density Fitting (Resolution of Identity) / µE h 
Errors ^{c} using FTC / µE h (with default thresholds) 
Errors ^{e} using FTC / µE h (with tighter thresholds) 
C_{2}H_{2}H 
103 
0 
0.02 
C_{6}H_{6} 
534 
3 
0.26 
glycine (neutral form) 
550 
4 
0.13 
^{a} All errors are total energy differences between the given method and an analytical integral evaluation scheme using ccpVDZ Gaussian basis set.
^{b} Results from [F. R. Manby, P. J. Knowles and A. W. Llold, J. Chem. Phys, 115, 9144 (2001)]
^{c} 3.75 grid density (about 140 Rydberg) was used. The integral threshold was 10 ^{12} in both the analytical and FTC calculations and the SCF convergence criteria was 10 ^{7} in Brillouin condition.
^{e} See footnote d in table1.
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