Hybrid functionals

B3LYP hybrid functional (Becke 1993), the PBEO hybrid functional (Adamo and Barone 1999), and the Heyd-Scuseria-Ernzerhof (HSE) hybrid functional (Heyd et al. 2003). 

The B3LYP hybrid functional (Becke 1993), the first hybrid functional, is the most frequently used functional (or method) in all functionals (or all theories) in quantum chemistry calculations. This functional uses three parameters as the mixing ratios to form the adiabatic connections between the Hartree-Fock exchange integral and the LDA exchange functional and between the LYP-GGA correlation functional and the LDA correlation functional, and to combine with the attenuated GGA term of the B88 exchange functional. 

Although it is often interpreted that this functional has no parameter ( 0 indicates no parameter), it contains, at least, fundamental constants in the PBE exchange-correlation functional. The advantages of this functional are its simple form, small number of parameters, and high reproducibility of chemical properties. However, this functional also has problems similar to those of the B3LYP functional. 

At the lower limit (A = 0) of the coupling constant integration of (1.87) is the exact exchange hole. This observation led Becke [94,95] to conclude that a fraction of exact exchange should be mixed with GGA exchange and correlation. The simplest such hybrid functional is 

The mixing coefficient a is not equal to or close to 1, because full exact exchange is incompatible with GGA correlation. The exact exchange hole in a molecule can have a highly nonlocal, multi-center character which is largely cancelled by an almost equal-but-opposite nonlocal, multicenter character in the exact correlation hole. The GGA exchange and correlation holes are more local, and more localized aroimd the reference electron. 

In this form, we can think of the correlation energy as the sum of two pieces The dynamic correlation energy modelled by E results from the tendency of electrons to avoid one another by swerving upon close approach, while the static correlation energy modelled by l—a) E — results from 

Hybrid functionals are perhaps the most accurate density functionals in use for quantum chemical calculations. Although based upon a valid physical insight, they do not satisfy any exact constraints that their underlying GGA s do not satisfy. 

We have repeatedly indicated that usually the exchange contributions are significantly larger in absolute numbers than the corresponding correlation effects. Therefore, an accurate expression for the exchange functional in particular is a prerequisite for obtaining meaningful results from density functional theory. However, we have seen in Chapters 1 and 5 that the 

Let us explore first the nature of the integrand E d for the limiting cases. At X = 0 we are dealing with an interaction free system, and the only component which is not included in the classical term is due to the antisymmetry of the fermion wave function. Thus, E d° is composed of exchange only, there is no correlation whatsoever.19 Hence, the X = 0 limit of the integral in equation (6-25) simply corresponds to the exchange contribution of a Slater determinant, as forexample, expressed through equation (5-18). Remember, that E d° can 

19 Keep in mind that dynamic electron correlation is always connected to the fact that electrons interact as charged species. 

Let us be specific. The simplest approximation to solve equation (6-25) is to assume that E cl is a linear function in X. This leads to 

For the B3LYP functional an unsigned error with respect to the G2 data base of only slightly above 2 kcal/mol was determined. 

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Density functional theory calculations have shown promise in recent studies. Gradient-corrected or hybrid functionals must be used. Usually, it is necessary to employ a moderately large basis set with polarization and diffuse functions along with these functionals. 

Both HF and DFT calculations can be performed. Supported DFT functionals include LDA, gradient-corrected, and hybrid functionals. Spin-restricted, unrestricted, and restricted open-shell calculations can be performed. The basis functions used by Crystal are Bloch functions formed from GTO atomic basis functions. Both all-electron and core potential basis sets can be used. 

All of the predicted structures are at least reasonably good. The two hybrid functionals produce the best structures, in excellent agreement with the experimental geometry. The SVWN and SVWNS functionals both produce good structures, while the BLYP geometry is the least accurate. 

Only the values computed by the hybrid functionals and MP2 are at all reasonable, and the B3PW91 value is in excellent agreement with experimental observations. Thi MP2 and B3LYP values are only modestly outside of the desired accuracy of 2 kcal-mol In Chapter 7, we will consider methods which were developed to consistently produce such very accurate thermochemical results. 

Seluflon Clearly, a hybrid functional is the best choice for this problem. We ran B3LYP calculations using the 6-31G(d), 6-31-i-G(d) and 6-311G(2d) basis sets. Here are the results ... 

Gaussian98 gives a choice of six exchange functionals and seven correlation functionals, together with a number of so-called hybrid functionals. These latter... 

Quantum-chemical calculations which utilize the density functional theory (DFT) are now perhaps amongst the most frequently performed because of their relatively low cost and high accuracy. Structural results obtained from DFT based methods are often as good as those derived from MP2 calculations. It is well documented that DFT methods, especially those involving hybrid functionals such as B3LYP, B3P86 and B3PW91, yield reliable... 

SCHEME 2.16 Additional reaction pathway for the generation of the quinone methide in the gas phase oxidation of 2-methylphenyl radical, investigated by the hybrid functional MPW1K (reproduced from Ref. [23] with permission from American Chemical Society). 

In addition, another computational study in the frame of DFT, using the hybrid functional MPW1K,53 had suggested that o-QM may be an intermediate in the reaction of the peroxy radical (HO2") with the benzyl radical at the ortho-position (Scheme 2.16),54 which should be significant in atmospheric processes and low-temperature combustion systems (T < 1500 K). 

Remarkably, the BLYP functional approaches or sometimes surpasses the accuracy of hybrid functionals if small basis sets are used (see Table 9-5). 

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