Functionals of quantities other than the electron density: Approximations to the exchange energy

ALGEBRAIC EQUATION AND ITERATIVE OPTIMIZATION FOR THE OPTIMIZED EFFECTIVE POTENTIAL IN DENSITY FUNCTIONAL THEORY

We further develop our recent direct method for the optimized effective potential (OEP) in density functional theory (DFT) [Yang and Wu, Phys. Rev. Lett.89, 143002 (2002)]. First, we show that the stationary condition in our optimization approach leads to a proper nonlinear algebraic equation for the OEP in a finite basis set, which differs from other finite basis set approaches. Then by constructing an approximate second derivative matrix of the energy functional in conjunction with the use of the Newton method, we significantly accelerate the convergence of the iterative optimization for OEP. Enhancement of the method is made in using the Tikhonov regularization method for the inversion of the second derivative matrix when it is singular or nearly singular and the direct inversion in the iterative space. It is shown that under a fixed stepsize condition, the optimization approach is equivalent to the self-consistent solution to the nonlinear algebraic equation for OEP. Because the approximate second derivatives are easy to compute and the iteration numbers are small now, the computation costs of OEP become comparable to that of regular DFT calculations as shown by calculations of some molecules, small and larger ones. We show how to find balanced results between energies and potentials when choosing a basis set for potentials.

Correlation energy functionals dependent on an effective number of electrons: Charged species and equilibrium geometries

Recently proposed spin-dependent and spin-independent correlation energy functionals [Perez-Jimenez et al., J. Chem. Phys. 116, 10571 (2002)] based on an effective number of electrons N are extended to deal with charged systems. By introducing the concept of an effective atomic number Z analogous to N, the spin-dependent functional in combination with Becke's exchange [Becke, Phys. Rev. A 38, 3098 (1988)] yields a mean absolute error (MAE) of 5.4 kcal/mol for the 88 ionization potentials and 58 electron affinities included in the extended G2 set, and a MAE of 4.1 kcal/mol for the 312 data comprising the above plus the 148 enthalpies of formation of the extended G2 set and the 18 total energies of the neutral atoms H through Ar. Geometry optimizations performed on the 53 molecules of the G2-1 test set with the above combination of exchange and correlation functionals yield MAEs of 0.017 A and 1.5 degrees for the 68 bond lengths and 29 angles analyzed as compared with the experimental estimates.

Analytic energy gradients of the optimized effective potential method

The analytic energy gradients of the optimized effective potential (OEP) method in density-functional theory are developed. Their implementation in the direct optimization approach of Yang and Wu [Phys. Rev. Lett. 89, 143002 (2002)] and Wu and Yang [J. Theor. Comput. Chem. 2, 627 (2003)] are carried out and the validity is confirmed by comparison with corresponding gradients calculated via numerical finite difference. These gradients are then used to perform geometry optimizations on a test set of molecules. It is found that exchange-only OEP (EXX) molecular geometries are very close to the Hartree-Fock results and that the difference between the B3LYP and OEP-B3LYP results is negligible. When the energy is expressed in terms of a functional of Kohn-Sham orbitals, or in terms of a Kohn-Sham potential, the OEP becomes the only way to perform density-functional calculations and the present development in the OEP method should play an important role in the applications of orbital or potential functionals.

Performance of the τ-dependent functionals in predicting the magnetic coupling of ionic antiferromagnetic insulators

The performance of some kinetic energy density (tau) dependent functionals in predicting the effective Heisenberg exchange has been explored using the KNiF3 and K2NiF4 insulators as case examples. Our results show that this new generation of functionals represents an important improvement with respect to the current local and gradient corrected functionals yielding a semi-quantitative description of the antiferromagnetic coupling without the need of hybrid approaches thus avoiding the calculation of exact, Hartree-Fock exchange. This feature opens a wide field of application especially in solid state.