Hybrid HF/DFT methods

Popular forms for the correlation functional have been developed by Perdew and Wang and by Lee, Yang, and Parr. In recent years a method known as Becke3LYP (or equivalently, B3LYP) is emerging as a favorite of DFT practitioners. This is actually a hybrid HF/DFT method, in which E is composed of both HF and DFT exchange terms (recall HF does include exchange interactions) and DFT electron correlation functionals. These various terms... 

The CO-LCAO calculations based on both the HF method and DFT allow one not only to make a comparison between the results obtained within these two approximations but also to employ a combination of these approximations used in hybrid HF-DFT methods. The HF self-consistent electron density of the crystal can be used to calculate correlation corrections to the total HF energy a posteriori [390]. In some cases it is useful to use the HF self-consistent density matrix to make the convergence of the DFT LCAO self-consistent procedure faster [23]. 

It was already mentioned in Chap. 9 that the HF method typically overestimates the optical gap, whereas DFT underestimates it. This can affect the electronic-density distribution (chemical-bond covalency) and defect-level positions within the gap (even determined with respect to the valence-band top). In this respect, the hybrid HF-DFT methods widely used in the molecular chemistry, e.g., B3LYP, seem to be more promising tools. 

Most of the remaining error in DFT calculations comes from the assumption that the so-called exchange holes are localized, which causes them to fail to connect adiabatically between the KS independent-electron reference system and the real molecule. As a cure, the exchange functional is modified so it incorporates a degree of the HF exchange density, which leads to hybrid HF/DFT methods, also known as adiabatic connection models. These functionals constitute the current stare of the art in the field of DFT calculations. 

The SM1-SM3 methods model solvation in water with various degrees of sophistication. The SM4 method models solvation in alkane solvents. The SM5 method is generalized to model any solvent. The SM5.42R method is designed to work with HF, DFT or hybrid HF/DFT calculations, as well as with AMI or PM3. SM5.42R is implemented using a SCRF algorithm as described below. A description of the differences between these methods can be found in the manual accompanying the AMSOL program and in the reviews listed at the end of this chapter. Available Hamiltonians and solvents are summarized in Table 24.1. 

The calculations were performed employing either pure ab initio Hartree-Fock (HF) methods, or hybrid HF-DFT functionals, in particular B3LYP [22]. The hybrid functionals have several advantages. One is that they are commonly applied with great success in computational studies of molecules and clusters, thus making it possible to benefit from the gathered experience from molecular studies. Another is their recently noted ability to accurately model band gaps in semiconductor compounds [57]. 

First, a series of calculations was performed to determine the interaction of halide ions with a Cu atom. The simultaneous tests of the basis sets and the functional were performed by use of the Gaussian92 program. Several different DFT variants were tested, for example SVWN, BP86 and B3LYP that are representative of the pure local DFT, pure non-local DFT and the hybrid HF/DFT non-local functional. Other DFT alternatives were also tested, but the trend in results seems to be close to that obtained with the methods mentioned above. 

The second DFT LCAO linear-scaling method by Scuseria and Kudin (SK method) [379] uses Gaussian atomic orbitals and a fast multipole method, which achieves not only linear-scaling with system size, but also very high accuracy in aU infinite summations [397]. This approach allows both all-electron and pseudopotential calculations and can be applied also with hybrid HF-DFT exchange-correlation functionals. 

The important advantage of the Scuseria-Kudin implementation of the DFT LCAO 0 N) method for solids is the possibility to use hybrid HF-DFT exchange-correlation functionals, including the recently developed screened Coulomb hybrid functional, discussed in the next subsection. 

Fig. 9.8. Band structure of cubic crystals (a),(b),(c)-SrXi03, (d),(e),(f)-SrZr03. HF LCAO method-(a),(d) hybrid HF-DFT(PBEO) LCAO method-(b),(e) DFT(PBE) LCAO method-...

All DFT methods perform better than HF and MP2 the hybrid techniques are even better than the costly QCISD. Both GGA functionals show scaling factors close to unity which means that they can be used without scaling, but they do not perform quite as well as... 

The hybrid DFT methods used here are B3LYP (35,36), PBEIPBE 39,40,47), mPWlPW91 37), and MPWIK 38). The ab initio methods discussed in this article include HF, MP4SDQ 44), and QCISD(T) (27). We consider only one pure DFT method, namely BLYP 48,49). 

A comparison of HF, MP2 and density functional methods in a system with Hartree-Fock wave function instabilities, ONO—OM (for M = Li, Na and K), shows that DFT methods are able to avoid the problems that ab initio methods have for this difficult class of molecules. The computed MP2 frequencies and IR intensities were more affected by instabilities than HF. The hybrid B3LYP functional reproduced the experimental frequencies most reliably. The cis,cis conformation of ONO—OM was highly preferred because of electrostatic attraction and was strongest in the case where M = Li. The small Li cation can fit in best in the planar five-membered ring. This is completely different from the nonionic... 

In addition to the moments of the charge distribution, molecular polarizabilities have also seen a fair degree of study comparing DFT to conventional MO methods. While data on molecular polarizabilities are less widely available, the consensus appears to be that for this property DFT methods, pure or hybrid, fail to do as well as the MP2 level of theory, with conventional functionals typically showing errors only slightly smaller than those predicted by HF (usually about 1 a.u.), while the MP2 level has errors only 25 percent as large. In certain instances, ACM functionals have been more competitive with MP2, but still not quite as good. 

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