Density-Functional Theory
5. MATHEMATICAL CONSIDERATIONS
N
vðrÞdr
ð117Þ
¼ E½v; N
R
qðrÞvðrÞdr:
The q-ensemble is most directly analogous to the isothermal–isobaric ensemble in classical statistical thermodynamics.
5. MATHEMATICAL CONSIDERATIONS
In the preceding discussion, several subtle mathematical points have been overlooked.
Of these, the most important is the v-representability problem: the Hohenberg–Kohn theorems indicate that a ground-state electron density uniquely determines its asso-ciated external potential and thus all the properties of the system, including the exchange-correlation energy. However, the Hohenberg–Kohn treatment does not ad-dress how one can tell whether a given density is a ground-state electron density for some system, that is, whether a given electron density is v-representable. For a long time, it was suspected that every reasonable electron density might be v-representable;
however, Levy [2], Lieb [83], and Englisch and Englisch [3] have demonstrated that this is not the case. Fortunately, no essential difficulties arise; one may define the exchange-correlation energy (as well as all of the other properties of a system) for non-v-representable densities in such a way as to preserve the variational principle, Eq. (23) [83–86].
Related to the v-representability problem is the Kohn–Sham v-representability problem. That is, given a system of interest, can one always find an internal potential, w(r), such that the ground-state electron density of the Kohn–Sham model system is the same as that of the state of interest? Again, the answer seems to be no [87], but if one allows fractional occupation numbers of the Kohn–Sham orbitals, then no essential difficulties arise [3,88,89]. We note that in this case, the idempotency constraint, Eq.
(71), is no longer appropriate, and the less stringent Eq. (70) should be used instead.
Finally, there is the matter of fractional numbers of electrons; in Eqs. (30) and (31) and throughout Sec. 4, we found it convenient to consider systems with noninteger numbers of electrons. Since no such systems exist in nature, the properties of these systems must be defined in an appropriate way. Several different arguments converge on the same result: the properties of a system with N+e electrons (0 V e V 1) should be taken as the appropriate weighted average of the properties of the systems with integer numbers of electrons [5,90]:
Q½vðrÞ; N þ euQ½vðrÞ; N þ eðQ½vðrÞ; N þ 1 Q½vðrÞ; NÞ: ð118Þ Among the unpleasant consequences of this result is that changes in properties of a system due to changes in the number of electrons (or nonnumber conserving changes in the electron density) cannot be computed without first explicitly specifying whether the number of electrons is increasing or decreasing. Again, however, this difficulty is not insurmountable and merely requires that derivatives with respect to the number of electrons and functional derivatives with respect to the electron density carry an additional notation as to whether the change in question increases or decreases the number of electrons in the system.
6. SUMMARY
Over the course of the last decade, density-functional theory and, in particular, the Kohn–Sham method have become the methods of choice for modeling the electronic structure and chemical reactivity of large systems. This is due, in large part, to the development of accurate density functionals for the exchange-correlation energy and availability of efficient computational implementations; progress in both directions has been facilitated by the simplifications obtained by using the ground-state electron density, instead of the many-electron wave function, as the descriptor of molecular states. In addition to providing accurate quantitative descriptions of most chemical systems, density-functional theory provides qualitative descriptors that elucidate the factors driving chemical reactions.
The present review has been very selective, stressing the rationale behind density-functional methods above their applications and excluding many important topics (both theoretical and computational). The interested reader may refer to anyone of the many books [91–93] or review articles [94–101] on density-functional theory for more details. Of special importance is the extension of density-functional theory to time-dependent external potentials [102–105], as this enables the dynamical behavior of molecules, including electronic excitation, to be addressed in the context of DFT [106–
108]. As they are particularly relevant to the present discussion, we cite several articles related to the formal foundations of density-functional theory [85,100,109–111], linear-scaling methods [63,112–116], exchange-correlation energy functionals [25, 117–122], and qualitative tools for describing chemical reactions [123–126,126–132].
ACKNOWLEDGMENTS
P.W.A. acknowledges financial support from a NIH postdoctoral fellowship and W.
Y. acknowledges support from the National Science Foundation and the National Institutes of Health.
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