Adiabatic generalized gradient

The quantity ffffr.r. a ) is the frequency-dependent XC kernel for which common approximations are applied frequency-independent (adiabatic) local density approximations (LDA), adiabatic generalized gradient approximations (GGA), hybrid-DFT variants such as the popular functionals B3LYP and PBEO in which Kxc contains an admixture of Hartree-Fock ( exact ) exchange X,... 

As most of the electronic structure simulation methods, we start with the Born-Oppenheimer approximation to decouple the ionic and electronic degrees of freedom. The ions are treated classically, while the electrons are described by quantum mechanics. The electronic wavefunctions are solved in the instantaneous potential created by the ions, and are assumed to evolve adiabatically during the ionic dynamics, so as to remain on the Born-Oppenheimer surface. Beyond this, the most basic approximations of the method concern the treatment of exchange and correlation (XC) and the use of pseudopotentials. XC is treated within Kohn-Sham DFT [3]. Both the local (spin) density approximation (LDA/LSDA) [16] and the generalized gradients approximation (GGA) [17] are implemented. The pseudopotentials are standard norm-conserving [18, 19], treated in the fully non-local form proposed by Kleinman and Bylander [20]. 

Peclet number independent of Reynolds number also means that turbulent diffusion or dispersion is directly proportional to the fluid velocity. In general, reactors that are simple in construction, (tubular reactors and adiabatic reactors) approach their ideal condition much better in commercial size then on laboratory scale. On small scale and corresponding low flows, they are handicapped by significant temperature and concentration gradients that are not even well defined. In contrast, recycle reactors and CSTRs come much closer to their ideal state in laboratory sizes than in large equipment. The energy requirement for recycle reaci ors grows with the square of the volume. This limits increases in size or applicable recycle ratios. 

Clearly, in the absence of a radial temperature or velocity gradient, no radial mass transfer can exist unless, of course, a reaction occurs at the bed wall. When a system is adiabatic, a radial temperature and concentration gradient cannot exist unless a severe radial velocity variation is encountered (Carberry, 1976). Radial variations in fluid velocity can be due to the nature of flow, e.g. in laminar flow, and in the case of radial variations in void fraction. In general, an average radial velocity independent of radial position can be assumed, except from pathological cases such as in very low Reynolds numbers (laminar flow), where a parabolic profile might be anticipated. 

It is clear from (A.8) and (A.9) that the gradient difference and derivative coupling in the adiabatic representation can be related to Hamiltonian derivatives in a quasidiabatic representation. In the two-level approximation used in Section 2, the crude adiabatic states are trivial diabatic states. In practice (see (A.9)), the fully frozen states at Qo are not convenient because the CSF basis set l Q) is not complete and the states may not be expanded in a CSF basis set evaluated at another value of Q (this would require an infinite number of states). However, generalized crude adiabatic states are introduced for multiconfiguration methods by freezing the expansion coefficients but letting the CSFs relax as in the adiabatic states ... 

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