Variational principle exchange-correlation

Variational fitting enables a completely analytic treatment of the Xa exchange correlation functional. This practical first-principles AIMD method enables studies of detonations, photodissociation, and ultimately the even more complex chemical reactions that can be driven tribologicially. 

It is evident that the exchange-correlation potential operating on a electrons need not be the same as operating on [3 electrons for two possible reasons either (i) there can be different total numbers of a versus [3 electrons N, N ), or (ii) there can be differences in the local a versus [3 density (p , p ) whether or not the total numbers are different. Since the exchange-correlation holes are different for parallel vs. antiparallel spin electrons and more efficient for electrons of the same spin index, if there is sufficient freedom in the spin orbital density, these electron densities pa, pp) and corresponding potentials may separate into distinctive a and (3 parts. Let us consider a case where this is energetically favored in accordance with the variational principle and compare with the spin restricted case. 

There are lots of exchange-correlation potentials in the literature. There is an impression that their authors worried most about theory/experiment agreement. We can hardly admire this kind of science, but the alternative (i.e., the practice of ab initio methods with the intact and holy" Hamiltonian operator) has its own disadvantages. This is because finally we have to choose a given atomic basis set, and this influences the results. It is true that we have the variational principle at our disposal, and it is possible to tell which result is more accurate. But more and more often in quantum chemistry, we use some non-variational methods (cf. Chapter 10). Besides, the Hamiltonian holiness disappears when the theory becomes relativistic (cf. Chapter 3). 

From what has been said already with respect to the variational collapse and the minimax principle, it is clear from the beginning that the standard derivation of the Hohenberg-Kohn theorems [386], which are the fundamental theorems of nonrelativistic DFT and establish a variational principle, must be modified compared to nonrelativistic theory [383-385]. Also, we already know that the electron density is only the zeroth component of the 4-current, and we anticipate that the relativistic, i.e., the fundamental, version of DFT should rest on the 4-current and that different variants may be derived afterwards. The main issue of nonrelativistic DFT for practical applications is the choice of the exchange-correlation energy functional [387], an issue of equal importance in relativistic DFT [388,389] but beyond the scope of this book. 

---