Kinetic energy, non interacting

E c, it is the part which remains after subtraction of the non-interacting kinetic energy, and the ifne and J potential energy terms. 

Of course, the non-interacting kinetic energy is not equal to the true kinetic energy of the interacting system, even if the systems share the same density, i. e., Ts T.13 Kohn and Sham accounted for that by introducing the following separation of the functional F[p]... 

Since the uniform electron density is perfectly neutralized by a rigid uniform positive background, the total energy per electron of the uniform gas is just + xe, where the non-interacting kinetic energy is... 

In the above expressions, p(r) is to be interpreted as the exact ground-state density of Hx. The form (51) was anticipated by Percus [71] who gave an expression for the non-interacting kinetic energy functional. 

Here the subscript s on the potential denotes that we are now solving single- electron equations, since we have a non-interacting system. But what is the potential in the equation above By isolating the non-interacting kinetic energy (Ts) and the Coulomb interaction (U) in our energy functional EVo(n) we get... 

Salvetti formula for the correlation energy in terms of the electron density and the non-interacting kinetic energy density, then replaces the latter by its second-order density-gradient expansion. (The Colle-Salvetti formula itself is derived from a number of theoretical approximations, and is fitted to the correlation energy of the helium atom.) The result is then cast into the GGA form of Eq. (59) via integration by parts[55]. 

For simplicity throughout this work we consider a spin-unpolarized closed-shell system where each (non degenerate) oceupied orbital is filled by n = 2 electrons with opposite spins. We can thus explicitly represent the non-interacting kinetic energy as an orbital-dependent functional ... 

Note that xc[p] contains contributions both from the electron-electron interaction and from the difference between the interacting kinetic energy (T[p]) and the non-interacting kinetic energy (Tni[p]). When energy functional is minimized (i.e. when p = Pg") we have... 

The functional derivative of the non-interacting kinetic energy with respect to the density can be obtained from Eq. (15) (for another derivation see Appendix A and Ref. 66) and thus the relation between the local Kohn-Sham potential r (r) and the external potential rext(r) is ... 

In this appendix we evaluate the functional derivative of the non interacting kinetic energy functional which is simple occupied-orbital dependent functional and thus can be treated with the chain-rule formalism developed... 

Next consider the non-interacting kinetic energy of (1.59). Scaling all the wave functions S in the constrained search as in (1.38) will scale the density as in (1.40) and scale each kinetic energy expectation value as in (1.43). Thus the constrained search for the unsealed density maps into the constrained... 

For example, the non-interacting kinetic energy is the sum of the separate kinetic energies of the spin-up and spin-down electrons ... 

The non-interacting kinetic energy Tg does not sample the spurious large-u part of the gradient expansion, so its gradient expansion (see (1.184) and (1.189)) works reasonably well even for realistic electron densities. In fact, we can use (1.79) to show that... 

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