Generalized gradients spin functionals

Helgaker, T., Watson, M., Handy, N. C., 2000, Analytical Calculation of Nuclear Magnetic Resonance Indirect Spin-Spin Coupling Constants at the Generalized Gradient Aproximation and Hybrid Levels of Density Functional Theory , J. Chem. Phys., 113, 9402. 

The density functional calculations were performed using the Vienna Ab Initio Simulation Package (VASP). ° The spin-polarized generalized gradient approximation, Perdue—Wang exchange correlation function, and ultrasoft pseudopotentials were used. ... 

Three types of exchange/correlation functionals are presently in use (i) functionals based on the local spin density approximation, (ii) functionals based on the generalized gradient approximation, and (iii) functionals which employ the exact Hartree-Fock exchange as a component. The first of these are referred to as local density models, while the second two are collectively referred to as non-local models or alternatively as gradient-corrected models. 

The q-space imaging method, which deals with signals only after long diffusion times, discards all information relevant to dynamic aspects of water diffusion and transport, especially the restriction of water transport by membrane and cell wall permeability barriers in cellular tissues. This information is contained in the functional dependence of the pulsed gradient spin echo amplitude S(q,A,x) on the three independent variables q, A, and x (x is the 90-180 degree pulse spacing) [53]. As the tool to explore the q and A dependence of S(q,A,x), generalized diffusion times and their associated fractional populations are introduced and a multiple exponential time series expansion is used to analyze the dependence [53]. 

Also pure density-functional methods combined with plane-wave basis sets and ultrasoft pseudopotentials [58] were used in our studies of extended systems [59]. The computational efficiency of these methods enables larger systems and to some extent dynamical processes to be studied. Generalized-gradient approximation (GGA) or spin-polarized GGA DFT functionals [60, 61] were employed in the electronic structure calculations. 

New calculations using a more efficient CRYSTAL code [54] and optimization scheme [135] are presented here. In these, also the spin-polarized B3LYP hybrid functional was used and the orthorhombic structure of Li05TiO2 was studied. The same basis set was used as in [66]. Recently, the orthorhombic Li0.5TiO2 structure was studied by plane-wave generalized gradient-approximation (GGA) DFT calculations by Koudriachova et al. [136]. GGA calculations were performed also in the present study for comparison. 

In this section we define the local spin density (LSD) approximation, the workhorse of density functional theory. We then examine its extension to semilocal functionals, i.e., those which employ both the local density and its derivatives, also called generalized gradient approximations. We show how the PW91 functional obeys many exact conditions for the inhomogeneous system, as described in section 2, which earlier semilocal functionals do not. 

We will consider the general case of a gradient-corrected functional, using a spin-unrestricted formalism. The XC energy is formally given by... 

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