Gradient Corrections and Hybrid Functionals

The initial implementation of DFT employed the so-called local density approximation, LDA (or, if we have separate a and [i spin, the local spin density approximation, LSDA). The basic assumption is that the density varies only slowly with distance -which it is locally constant. Another way of visualizing the concept of LDA is that we start with a homogeneous electron gas and subsequently localize the density around each external potential - each nucleus in a molecule or a solid. That the density is locally constant is indeed true for the intermediate densities, but not necessarily so in the high- and low-density regions. To correct for this, it was rec- 

In 1988, Becke proposed a gradient-corrected scheme for the exchange functional ( B88 ), ensuring that this should have the correct asymptotic limit (1/r) as oo [16]  

An alternative approach was offered by Lee, Yang, and Parr [19], who derived a gradient-corrected correlation functional ( LYP ) from the second-order density matrix in HF theory. Together with PW91, this functional is currently the most widely used correlation functional for molecular calculations. 

For the G2 set of compounds (a standardized test set of small molecules) the mean error to the atomization energy is approximately 2.5 kcal mol-1 at the B3LYP level, compared with 78 kcal -mol-1 for HF theory, and in the range of lkcalmor1 for the most accurate correlated ah initio methods. For most cases in which a moderately sized systems (10-50 atoms) is to be investigated, the B3LYP functional is currently the method of choice. 

That is, we now aim to describe in a more appropriate way the interaction part of the kinetic energy that is introduced to the ex-change-correlation functional in the Kohn-Sham scheme. Including the kinetic energy corrections increases the computational requirements substantially, but the accuracy is also much improved compared with conventional gradient-corrected functionals. 

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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. 

For a short exposition of the evolution from the local-density approximation to the local-spin-density approximation and gradient-corrected and hybrid functionals, see Levine IN (2000) Quantum chemistry, 5th edn. Prentice Flail, Upper Saddle River, NJ, pp 581-592... 

Gradient-Corrected and Hybrid Functionals. The LDA and LSDA are based on the uniform-electron-gas model, which is appropriate for a system where p varies slowly with position. The integrand in the expression (15.126) for E is a function of only p, and the integrand in is a function of only p and pP. Functionals that go... 

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. 

The HE, GVB, local MP2, and DFT methods are available, as well as local, gradient-corrected, and hybrid density functionals. The GVB-RCI (restricted configuration interaction) method is available to give correlation and correct bond dissociation with a minimum amount of CPU time. There is also a GVB-DFT calculation available, which is a GVB-SCF calculation with a post-SCF DFT calculation. In addition, GVB-MP2 calculations are possible. Geometry optimizations can be performed with constraints. Both quasi-Newton and QST transition structure finding algorithms are available, as well as the SCRF solvation method. 

Halls, M. D., Schlegel, H. B., 1998, Comparison of the Performance of Local, Gradient-Corrected, and Hybrid Density Functional Models in Predicting Infrared Intensities , J. Chem. Phys., 109, 10587. 

The application of density functional theory (DFT) to the study of the structure and reactivity of some molecules with unpaired electrons (radicals) performed by our group is presented. The results describe the application of LSD, gradient corrected and hybrid DFT methods to several small molecules. On average the results are as good as highly-correlated post-Hartree-Fock methods, but still some problems remain to be solved... 

We surveyed and illustrated in the previous sections some of the many present successful applications of TDDFT. In those applications, standard approximations (local, gradient-corrected, and hybrid) were used for both the ground-state calculation and the excitations, via the adiabatic approximation. In this section, we survey several important areas in which the standard functionals have been found to fail, and we explain what might be done about it. 

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