Coupling constants averaging

By defining all these quantities as explicit functions of A, we can relate the density functional quantities to those more familiar from quantum chemistry. The exchange-correlation energy of density functional theory can be shown, via the Hellmann-Feynman theorem [38, 37], to be given by a coupling-constant average, i.e.. 

Note that, in the absence of explicit A-dependence, our notation implies the coupling-constant averaged quantity.) On the other hand, at full coupling strength, A = 1, we return to the fully-interacting system, so that, from Eqs. (6) and (7),... 

We do not distinguish here this density functional definition of exchange energy from that of Hartree-Fock (HF). This simplification is well-justified, if the HF electron density and the exact electron density differ only slightly [40]. Similarly, the coupling-constant averaged exchange-correlation hole is the usual... 

A special situation applies to compound 30. The exo-endo structure was identified from the two allyl vicinal proton coupling constants33. Only one stereoisomer could be detected. With increasing temperature above 210 K the latter two coupling constants average, the result of presumably sequential rotations around both allyl carbon-carbon... 

NMR spectroscopy in isotropic solution is characterised by the fact that free and random molecular reorientation averages many parameters that have a directional dependence, e.g. dipolar-coupling constants, chemical shifts, etc. Some parameters, such as dipolar-coupling constants, average to zero in isotropic solution so once the ability of a molecule to rotate and move freely and isotropically is removed, its NMR spectra immediately become more complex. 

Note that there is no coupling-constant average here. For large values of u, g r, r + u) -> 1, so that h(u) is typically a long-ranged function of u. We may separate out a reasonably short-ranged contribution, in the spirit of the LSD approximation. We write... 

INDO calculation of coupling constants. Average of N(l, exo) and N(l, endo). INDO calculation of hfs for bent and linear HNC structure. ... 

Because it satisfies all these constraints, the LSD model for the system-, spherically-, and coupling-constant-averaged hole of (1.101),... 

Coupling-Constant Average for the Spin-Polarized Electron Gas. [Erratum Phys. Rev. B, 56,... 

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