DFT-D method

One conceptually simple remedy for the shortcomings of DFT regarding dispersion forces is to simply add a dispersion-like contribution to the total energy between each pair of atoms in a material. This idea has been developed within localized basis set methods as the so-called DFT-D method. In DFT-D calculations, the total energy of a collection of atoms as calculated with DFT, dft> is augmented as follows ... 

Thanthiriwatte, K. S., Hohenstein, E. G., Burns, L. A., and Sherrill, C. D. [2011], Assessment of the performance of DFT and DFT-D methods for describing distance dependence of hydrogen-bonded interactions, ]. Chem. Theory Comput 7, pp. 88-96, doi 10.1021/ctl00469b. Torheyden, M., and Jansen, G. [2006]. A new potentiai energy surface for the water dimer obtained from separate fits of ab initio eiectrostatic, induction, dispersion and exchange energy contributions. Mol Phys. 104, pp. 2101-2138, doi 10.1080/00268970600679188. 

The energy differences calculated with the DFT-D method and QM/atom-atom hybrid approach with polarization are of a reasonable magnitude. However, the a form is known to undergo an exothermic solid-state transition to the /0 polymorph at 150 °C, with a transition enthalpy between 1.9 and 2.9kJmol . The T = 0K order of stability provided by aU of the computational methods is opposite to that seen experimentally at the transition temperature. A reversal in the sign of the enthalpy difference can only... 

The general emphasis in force field development is towards transferrable force fields, where the functional form and the values of associated parameters can be used in a wide variety of molecules and crystals. As the parameters are developed empirically, transferability implies a degree of reliability and confidence that the parameters will work for crystals for which they were not specifically parameterised. In a recent development of the so-called tailor-made force field, it was pointed out that for the specific case of crystal structure prediction, the force field does not need to be transferable and that in fact there are some important advantages to having a force field derived specifically for the molecule of interest. Given sufficiently accurate information from quantum mechanical calculations, the tailor-made force field can be obtained by fitting to the quantum mechanical potential energy surface. Neumann defined a number of quantum mechanical data sets which represented both the non-bonded and bonded interactions in the crystal. The parameters of the force field were then optimised to fit these data sets. The quantum mechanical method chosen for the calculations was the DFT(d) method which will be described below. 

Table 4.2 Summary of blind test results and results obtained with the DFT(d) method. 

No. Correet predietions/ Rank with No. Partieipants DFT(d) Method... 

For molecule II, whose crystal structure was predicted correctly by one of the 1999 blind test participants, the DFT(d) method finds the experimentally observed structure to be the second most stable crystal packing alternative with an energy difference of only 0.01 kcal mol compared to the lowest energy structure. Such a small energy difference falls within the numerical error of the calculations. The DFT(d) optimised structure shows a geometric deviation of nearly 0.5 A in comparison to the experimental structure, which is a much larger discrepancy than the values found for any of the other structures (around 0.1 A). So far, the reason for this large deviation remains unclear despite efforts to unravel the problem. 

A second polymorph of molecule IV was discovered after the second blind test. ° Since the new polymorph has two independent molecules in the asymmetric unit, it was not predicted by any of the participants because it was indicated at the start of the blind test that the structure would have just one molecule in the asymmetric unit. Two participants predicted the first polymorph correctly. The DFT(d) method ranks the two polymorphs as the first and second lowest energy structures among all the structures predicted by all the participants. ... 

Molecule IX was not considered in the re-evaluation study with the DFT(d) method because its van der Waals correction was originally not parameterised for iodine. This parameterisation has now been carried out and the crystal structures of molecule IX have been optimised using the DFT(d) method. The experimental structure is found to be the lowest energy crystal packing alternative among all the predicted structures submitted by the 2004 blind test participants. 

Bouteiller Y, Poully JC, Desfranfois C, Qregoire G (2009) Evaluation of MP2, DFT, and DFT-D methods for the prediction of infi ared spectra of peptides. J Phys Chem A 113 6301... 

Finally, we should also briefly discuss the performance of semiempirical methods. These are methods that neglect some of the more expensive integrals in Hartree-Fock molecular orbital theory and replace others with empirical parameters. Because semiempirical methods are based on Hartree-Fock theory, and because Hartree-Fock theory does not capture dispersion effects, semiempirical methods are not suitable for computing dispersion-dominated noncovalent interactions. Semiempirical methods yield repulsive potentials for the sandwich benzene dimer, just as Hartree-Fock does. However, given that semiempirical methods already contain empirical parameters, there is no reason not to fix this deficiency by adding terms proportional to r, as is done in force-field methods and the empirical DFT-D methods. Such an approach has been tested for some base pairs and sulfur-7t model systems. 

Performance of the DFT-D Method, Paired with the PCM Implicit Solvation Model, for the Computation of Interaction Energies of Solvated Complexes of Biological Interest. 

Can the DFT-D Method Describe the Full Range of Noncovalent Interactions Found in Large Biomolecules ... 

The MM method is used now not only in theoretical and computational works but also as a part of experimental studies (e.g., many X-ray and NMR-derived structures of proteins, nucleic acids, and their complexes with other molecules deposited in Protein Data Bank (Berman et al. 2003) and Nucleic Acid Data Bank (Berman et al. 1992) are the results of MM refinements). The MM semiempirical terms are used in some quantum mechanics computations, e.g., in the DFT-D method (Antony and Grimme 2006 Jurecka et al. 2007). We wUl follow the evolution of MM from the first precomputer and early computer-aided (i.e., before the era of personal computers) works to modern complex simulations impossible without supercomputers. 

The DFT-D method does not only work for molecular complexes and intramolecular dispersion effects but is rather general, and Sauer and coworkers extended this correction to periodic systems (Kerber et al. 2008). 

Somewhat removed is the density functional based tight-binding method (DFTB), which is based on a second-order expansion of the Kohn-Sham total energy, employing a self-consistent redistribution of Mulliken charges (SCC-DFTB) (Elstner et al. 1998). It also employs dispersion corrections similar to the DFT-D method and has successfully been applied, e.g., to nuclear-base stacking problems (Elstner et al. 2001). 

When the issue of London dispersion interactions is carefitUy considered with DPT, this yields a similar or often even better performance than MP2. Of the several approaches in this area, the DFT-D method has proven as an accurate and robust computational tool. 

The best accuracy is achieved by complete basis set (CBS) extrapolation methods, when two systematically improved basis sets are applied and the data is then extrapolated. The interaction energy computations, even with large basis sets, need to be corrected for basis set superposition error (BSSE). We oppose suggestions to ignore the BSSE correction or to attempt only its partial inclusion while assuming that the numbers can be correct due to error cancellation. This is a risky game. It is much better to provide BSSE-corrected numbers where a solid estimate of the underestimation of the interaction is typically possible. Fortunately, the CBS calculations are intrinsically BSSE-free. Similarly, computations with modern parameterized DFT-D methods (see below) do not require BSSE correction, since it is indirectly (effectively) included via parameterization. 

To correct for the overlap effects the dispersion energy is damped by distance-dependent damping function. The dispersion energy, represented by the Ce/R formula, is calculated separately from the DFT calculation and is simply added to the DFT energy. The disadvantage of DFT-D methods is obviously the need to combine electronic structure calculations with classical force field correction term, which also affects the transferability of these methods. Thus these methods are expected to be surpassed in the future by true DFT-based dispersion-including methods however for the moment it seems to us that DFT-D is more satisfactory for routine calculations of nudeobase interactions. One present difficulty is that we have so many new methods in the literature that is difficult to choose. This issue is beyond the scope of this chapter and we refer the reader to specialized literature (Bands et al. 2009 Sponer et al. 2008). We hope that standard (optimal) wide-spectrum dispersion-including DFT methods will soon be identified. 

The most recent method considered is DFT-D3 [35]. Previous DFT-D methods did not distinguish between different valence states of an atom in a molecule, that is the dispersion coefficients in Eq. (11.1) for sp and sp carbon atoms should differ, as dispersion coefficients decrease upon oxidation of an atom and increase upon reduction. To obtain accurate dispersion coefficients, the concept of atomic fractional coordination number was introduced in DFT-D3. The dispersion coefficients in Eq. (11.1) depend on the atomic fractional coordination number and the latter depends on an atom s geometrically closest neighbors. The D3-correction has continuous dispersion coefficients C even if chemical reaction occurs in a model system (i.e., dispersion coefficients change smoothly when an atom s valence or oxidation state changes), which is very efficient. Indeed, this allows smooth forces and therefore may be used in quantum molecular dynamics. For example, in the simple transition state of the Sj.j2 reaction [F CHj F ], the fractional coordination number of the carbon atom is 4.1 and that of fluorine atom is 0.57. DFT-D3 contains eighth-order terms with w = 8 and the eighth-order dispersion coefficients Cg in Eq. (11.1) are computed from for the same atom pairs. 

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