The fitting coefficients are collected in the vector x. The Coulomb matrix ¢ is defined as L he Coulomb matrix G 1s symmetric and positive aennite which guarantees the minimization in Eq. (11). In principle, the very efficient Cholesky decomposition algorithm can be used to solve the fitting equation system. ill However, if very large auxiliary function sets are used, G tends to become I-conditioned. In order to avoid numerical instabilities by solving the fi tting equations with the Cholesky decomposition approach, deMon2k performs a singular value decomposition (SVD) at the beginning of the SCF and uses a robust numerical solver’ that acts only on the nonredundant space of the SVD for subsequent solutions. Performing a full SVD is time consuming; however, the stability gained during the initial SCF and the application of the numerical solver even for different geometries (for exam- ple, in a geometry optimization or along a BOMD trajectory) overcompen- sate the computational load of the SVD. Bear in mind that the contribution from the Becke 88 exchange func- tional’* appearing in the B3LYP energy expression is only from the nonlocal part of the functional. The variation of any of these energy expressions with respect to the MO coefficients, under the MO ortho- normality constraints, leads to Roothaan—Hall equation systems. The hybrid ADFT Kohn—-Sham matrix has contributions from the variation of Egxx, the expansion of the auxiliary orbital products (Eq. 28). Note, however, that this is the result of a variational fitting as described above. If the approxi- mated exact exchange energy expression of Eq. (35) is added to the self- interaction including Hartree energy with Coulomb fitting, the variationally fitted self-interaction-free Hartree-Fock energy expression is obtained.”” On the other hand, if the fitted exact exchange energy is combined with exchange-correlation functionals, hybrid ADFT energy expressions without four-center ERIs are obtained. In particular, we find the following ADFT or DF-DFT energy expressions for the here discussed B3LYP (J (Eq. 36b), and M06-2X (Eq. 36c) functionals: Eq. 36a), PBI - EO E All energies are in kcal/mol. Vn order to validate our hybrid DFA implementations, we compared several properties that depend on energy differences. First, we performed calculations of standard enthalpies of formation AH?” using the G3/99X test set.””"’" The protocol for calculating AHf’* was the one established by Curtiss and collaborators.”” The values of mean absolute error (MAE) BI with respect to experiment’ are reported in Tables 1, 2, and 3 for B3LYP, PBEO, and MO “Heats of formation.” S 6-2X, respectively. They are listed under the entry tandard DFT results from NW Chem calculations for these three DFAs are also included in each table under the “DFT” column. As Tables 1-3 show, all the DFT reference. T | LDF-DFT MAI M06-2X functional. Al Fs are in excellent agreement with he largest deviation is only 0.05 kcal/mol for the though M06-2X uses a significant fraction of exact exchange (54%), part of the observed small deviations may also arise from LDF-EXX computation, for example, integration errors specific to the M06 suite.”” The use of LDF-ADFT increases the MAE for heats of formation with B3LYP, but decreases the same quantity for PBEO. This difference between LDF-DFT and LDF-ADFT is mainly sources other than the All energies are in kcal/mol. Figure 1 CPU timing breakdown for single-point calculations of linear alkane chains 2mploying the B3LYP hybrid functional and the auxiliary density functional theory ADFT) methodology. Timings are very different for the hybrid meta-GGA M06-2X. Figure 3 shows that the most demanding computational task here 1s the calculation of the exchange-correlation contribution. For the largest alkane, this task accounts for almost 50% of the total CPU time. The increased workload arises from the computation of the kinetic energy density t(r) and the t-dependent part of the Kohn—Sham matrix on the grid that prevents the use of ADFT. Nevertheless, the difference between LDF-DFT and LDF-ADFT timings is rather small for linear alkane chains (compare Figure 4 Comparison of amylose, helix (top) and linear alkane chain C49Hg2 (bottom) extension. End-to-end distance is about 50 A for the amylose helix and 49 A for the alkane chain. Figure 5 CPU timing breakdown for the first two SCF iterations for amylose helices described with the PBEO hybrid functional and the auxiliary density functional theory (ADFT) methodology. Figure 6 CPU timing breakdown for the first two SCF iterations for amylose helices described with the PBEO hybrid functional and the density-fitting density functional the- ory (DF-DFT) methodology.
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