Gradient Corrected Methods

Perdew and Wang (PW86) proposed modifying the LSDA exchange expression to that shown in eq. (6.23), where x is a dimensionless gradient variable, and a, b and c being suitable constants (summation over equivalent expressions for the a and 0 densities is implicitly assumed).----------------------------------------------- 

Becke proposed a widely used correction (B or B88) to the LSDA exchange energy, ch has the correct —r asymptotic behaviour for the energy density (but not for the exchange potential).  

The / parameter is determined by fitting to known atomic data and x is defined in 

Another functional form (not a correction) proposed by Becke and Roussel (BR) has the form 

This functional contains derivatives of the orbitals, not just the gradient of the total density, and is computationally slightly more expensive. Despite the apparent difference in functional form, exchange expressions (6.24) and (6.25) have been found to provide 

One of the earliest and most popular GGA exchange functionals was proposed by A. D. Becke (B or B88) as a correction to the LSDA exchange energy.  

There have similarly been various GGA functionals proposed for the correlation energy. One popular functional is due to Lee, Yang and Parr (LYP), which has the rather intimidating form shown in eq. (6.40). 

c and d parameters are determined by fitting to data for the helium atom. Although not obvious from the form shown in eq. (6.40), the LYP functional does not include parallel spin correlation when all the spins are aligned (e.g. the LYP correlation energy for He is zero). The LYP correlation functional is often combined with the B88 or OPTX exchange functional to produce the BLYP and OLYP acronyms. 

The correlation part is similarly written as an enhancement factor added to the LSDA functional, where the t variable is related to the x variable by means of yet another spin-polarization function. 

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A more complex set of functionals utilizes the electron density and its gradient. These are called gradient-corrected methods. There are also hybrid methods that combine functionals from other methods with pieces of a Hartree-Fock calculation, usually the exchange integrals. 

Gradient corrected methods usually perform much better than LSDA. For the G2-1 data set (see Section 5.5), omitting electron affinities, the mean absolute deviations shown in Table 6.1 are obtained. The improvement achieved by adding gradient terms is impressive, and hybrid methods (like B3PW91) perform almost as well as the elaborate G2 model for these test cases. For a somewhat larger set of reference data, called the G2-2 set, the data shown in Table 6.2 are obtained. 

The atoms-in-molecules (AIM) analysis of electron density, using ab initio calculations, was considered in Section 5.5.4. A comparison of AIM analysis by DFT with that by ab initio calculations by Boyd et al. showed that results from DFT and ab initio methods were similar, but gradient-corrected methods were somewhat better than the SVWN method, using QCISD ab initio calculations as a standard. DFT shifts the CN, CO, and CF bond critical points of HCN, CO, and CH3F toward the carbon and increases the electron density in the bonding regions, compared to QCISD calculations [107]. 

Electron Correlation ivietnoas 98 6 2— Gradient Corrected Methods TS4... 

It is surprising to see that the gradient-corrected methods B-VWN and B-LYP, which exhibited the best performance in the previous study, underestimate the reaction barrier by up to 11 kcal/mol. On the other hand,... 

Although the total energy calculated by DPT methods should in principle converge to the experimental value (-76.438 au), there are no upper or lower bounds for the currently employed methods with approximate exchange-correlation functionals. Indeed, all the gradient-corrected methods used here (BLYP, PBE and HCTH) give total energies well below the experimental value with the pc-4 basis set. 

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