Derivative discontinuity

In Sect. 1.3, our density functionals were defined as constrained searches over wave functions. Because all wave functions searched have the same electron number, there is no way to make a number-nonconserving density variation Jn(r). The functional derivatives are defined only up to an arbitrary constant, which has no effect on (1.50) when Jd r Jn(r) = 0. 

To complete the definition of the functional derivatives and of the chemical potential /x, we extend the constrained search from wavefunctions to ensembles [49,50]. An ensemble or mixed state is a set of wavefunctions or pure states and their respective probabilities. By including wavefunctions with different electron numbers in the same ensemble, we can develop a density functional theory for non-integer particle number. Fractional particle mun-bers can arise in an open system that shares electrons with its environment, and in which the electron number fluctuates between integers. 

The upshot is that the ground-state energy E N) varies linearly between two adjacent integers, and has a derivative discontinuity at each integer. This discontinuity arises in part from the exchange-correlation energy (and entirely so in cases for which the integer does not fall on the boimdary of an electronic shell or subshell, e.g., for = 6 in the carbon atom but not for AT = 10 in the neon atom). 

By Janak s theorem [51], the highest partly-occupied Kohn-Sham eigenvalue ho equals dE/dN = fj, and so changes discontinuously [49,50] at an integer Z  

Since the asymptotic decay of the density of a finite system with Z electrons is controlled by Iz, we can show that the exchange-correlation potential tends to zero as r — c [52]  

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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... 

An alloy is said to be of Type II if neither the AC nor the BC component has the structure a as its stable crystal form at the temperature range T]. Instead, another phase (P) is stable at T, whereas the a-phase does exist in the phase diagram of the constituents at some different temperature range. It then appears that the alloy environment stabilizes the high-temperature phase of the constituent binary systems. Type II alloys exhibit a a P phase transition at some critical composition Xc, which generally depends on the preparation conditions and temperature. Correspondingly, the alloy properties (e.g., lattice constant, band gaps) often show a derivative discontinuity at Xc. 

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. 

By the way, through ensemble theory with unequal weights, Ref. [68] identifies an effective potential derivative discontinuity that links physical excitation energies to excited Kohn-Sham orbital energies from a ground-state calculation.)... 

Indeed, there is such an approach to DFT that gives a physical justification to the above assumption of continuity with the only complication involved being the non-differentiability of Eo n) at an integer number of electrons n = N, a phenomenon known as DFT derivative discontinuity . The approach is based on an extension of the original Hohenberg-Kohn theorem [20] to the grand canonical ensemble first given by Mermin [21]. It... 

In a recent paper [25] Axe = 0 was claimed, meaning that the overall derivative discontinuity is entirely due to its Kohn-Sham component but this issue is still under intense debate. 

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. 

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],... 

Prediction of derivative discontinuities in the magnetic moment versus strain curves of Fe monolayers [88] and other transition metals [89]... 

The B-spline has a continuous first derivative because we require this but it has discontinuities in the second derivative. Another desideratum that we can therefore employ is that the sum of squares of these second-derivative discontinuities be as small as possible. It turns out that the twin goals of fitting the data and maximizing the smoothness of the first derivative in this way compete with each other. This is a problem that will be with us in all higher dimensions. 

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