Functional derivative discontinuity

Perdew J P, Parr R G, Levy M and Balduz J L Jr 1982 Density-functional theory for fractional particle number derivative discontinuities of the energy Phys. Rev. Lett. 49 1691-4... 

There are different ways of implementing the cut-off approximation. The simplest is to neglect all contributions if the distance is larger than the cut-off. This is in general not a very good method as the energy function becomes discontinuous. Derivatives of the energy function also become discontinuous, which causes problems in optimization... 

Perdew, J. P., Parr, R. G., Levy, M., Balduz, J. L., Jr., 1982, Density Functional Theory for Fractional Particle Number Derivative Discontinuities of the Energy , Phys. Rev. Lett., 49, 1691. 

In carrying out analytical or numerical optimization you will find it preferable and more convenient to work with continuous functions of one or more variables than with functions containing discontinuities. Functions having continuous derivatives are also preferred. Case A in Figure 4.1 shows a discontinuous function. Is case B also discontinuous ... 

Perdew JP, Parr RG, Levy M, Balduz Jr JL (1982) Density-functional theory for fractional particle number Derivative discontinuities of the energy, Phys Rev Lett, 49 1691-1694... 

Density-Functional Theory for Fractional Particle Number Derivative Discontinuities of the Energy. 

The Coulombic potential becomes infinitely negative when an electron and a nucleus coalesce and, because of this, the state function for an atom or molecule must exhibit a cusp at a nuclear position. That is, as shown by Kato (1957), the first derivative of the function is discontinuous at the position of a nucleus. Thus, while the charge density is a maximum at the position of a nucleus, this point is not a true critical point because Vp, like is discontinuous there. However, as discussed in Election E2.1, this is not a problem of practical import and the nuclear positions behave topologically as do (3, — 3) critical points in the charge distribution and hereafter they will be referred to as such. 

Here S(R) is the overlap integral / abdr and a and b are taken as normalized to unity. While S is a smooth function of Ry the CF calculations show that a derivative discontinuity exists in A(R) at R = 1.6RequmMum- Simple approximations in density functional theory do not reproduce this type of bond-breaking behaviour. 

Kurth S, Stefanucci G, Khosravi E, Verdozzi C, Gross E (2010) Dynamical Coulomb blockade and the derivative discontinuity of time-dependent density functional theory. Phys Rev Lett 104(23) 236801... 

One of the most intriguing properties of the exact functional, which has resisted all attempts of describing it in local or semilocal approximations, is the derivative discontinuity of the xc functional with respect to the total particle number [50, 58, 59],... 

The energy gap obtained in such band-structure calculations is the one called HOMO-LUMO gap in molecular calculations, i.e., the difference between the energies of the highest occupied and the lowest unoccupied singleparticle states. Neglect of the derivative discontinuity A, defined in Eq. (65), by standard local and semilocal xc functionals leads to an underestimate of the gap (the so-called band-gap problem ), which is most severe in transition-metal oxides and other strongly correlated systems. Self-interaction corrections provide a partial remedy for this problem [71, 72, 73, 74],... 

We note that this relation is not the same as Eq. (31). The electronic motion is thus correlated also for the HF wave function. A deeper analysis shows that for electrons with opposite spin will the second term (the exchange term) disappear when we integrate over the spin variable. This is not the case for electrons with the same spin (compare the exchange term in the HF energy expression). The electrons thus create a hole around themselves were other electrons with the same spin are forbidden. We call this Fermi correlation. Such a hole is not created for electrons with the opposite spin. There is a finite probability to find them in the same point in space but the magnitude of the exact wave function has a minimum for ri2 = 0 (ri2 being the distance between a pair of electrons). The derivative of the exact wave function is discontinuous at this point. We call this behavior a cusp. The wave function has the form... 

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