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In 2014, 50 years following the introduction of density functional theory (DFT), a rigorous understanding of it was published [AIP Advances, 4, 127,104 (2014)]. This understanding includes two features that complete the theory in practice, inasmuch as they are necessary for its correct application in electronic structure calculations; this understanding elucidates what appears to have been the crucial misunderstanding for 50 years, namely, the confusion between a stationary solution, attainable with most basis sets, following self-consistent iterations, with the ground state solution. The latter is obtained by a calculation that employs the well-defined optimal basis set for the system. The aim of this work is to review the above understanding and to extend it to the relativistic generalization of density functional theory by Rajagopal and Callaway [Phys. Rev. B7, 1912 (1973)]. This extension straightforwardly follows similar steps taken in the non-relativistic case, with the four-component current density, in the former, replacing the electronic charge density, in the latter. This new understanding, which completes relativistic DFT in practice, is expected to be needed for the study of heavy atoms and of materials (from molecules to solids) containing them—as is the case for some high temperature superconductors.

From its introduction by Hohenberg and Kohn [

Self-interaction [

Steps taken to remedy the referenced limitations ascribed to DFT have partly consisted of the introduction of non-local versions of the exchange correlation potential, mostly in the form of generalized gradient approximations (GGA) [

In 1998, our group presented exceptionally accurate, electronic and related properties of barium titanate (BaTiO_{3}) [_{3}. Our calculated electronic and related properties for wurtzite GaN, C, and Si were also in general agreement with experiment [

Despite the clear success of the BZW and BZW-EF method in describing or predicting electronic and related properties of materials, as amply illustrated below, our alternative explanation of and solution to the band gap problem have mostly been ignored up to 2014. The argument has been that, without the self-interaction correction and the addition of the derivative discontinuity of the exchange correlation energy to the band gap obtained with a DFT potential, one is not expected to get an agreement with experiment. In 2014, we presented a new understanding of DFT that rigorously adheres to conditions inherent to it; i.e., we proved that these conditions have to be met by a self-consistent calculation before its results can possess the full physical content of DFT. We provide a brief review of this proof [

We presented [

The initial, non-relativistic form of density functional theory, as introduced by Hohenberg and Kohn, rests on two theorems the authors proved. The first one of these theorems states that, except for an additive constant, the external potential

and T and U as the kinetic and electron-electron interaction operators, the energy

where

is a unique functional of the charge density.

where the summation is over occupied states only.

The second theorem of Hohenberg and Kohn [

meaning that the total number of particles is kept constant.

Hohenberg and Kohn [

where

The above steps in the derivation of density functional theory (DFT) directly lead to its rigorous, mathematical and physical understanding articulated by Bagayoko [

It is crucial to underscore here what appears to have been the key misunderstanding of DFT in many calculations, namely, the confusion between a stationary solution and the ground state solution; the former is obtained with most basis sets, following self-consistency iterations, while the latter is found after several successive, self-consistent calculations, with increasing, embedded basis sets, as explained below in connection with our computational method (BZW and BZW-EF). Embedding here signifies that, except for the first calculation with a small basis set, each of the other calculations employs the basis set of the one preceding it plus one orbital. Depending on the s, p, d, or f character of the added orbital, the size of the basis set increases by 2, 6, 10, or 14, respectively, taking the spin into account.

The above new understanding of DFT requires that electronic structure calculations search for and attain the absolute minima of the occupied energies for the systems under study. Such an attainment signifies that the ground state of the system under study has been reached. As alluded to above, a self-consistent calculation with a single basis set produces a stationary solution among a potentially infinite number of such solutions. Given that such a calculation cannot claim to have attained the minimum of the energy content of the Hamiltonian in Equation (2), a unique functional of the charge density, its results should not be expected to possess the full, physical content of DFT and they generally do not, as attested to by the recalcitrance of the band gap problem. To reach the ground state energy, the use of our computational method (BZW and BZW-EF) or similar ones appears to be necessary. This method is fully described in several publications [

The first of these features of our method is the performance of several self-consistent calculations with increasing, embedded basis sets. This implementation begins with a relatively small basis set that must be large enough to account for all the electrons in the atomic or ionic species in the molecules or solids under study. Successive, self-consistent calculations are subsequently performed, where each calculation employs the basis set of the one preceding it plus one additional orbital. Graphical and numerical comparisons of the occupied energies from two consecutive calculations, upon setting the Fermi energy to zero, show the lowering of some or of all the occupied energies as the basis set is augmented. The successive calculations stop when three consecutive ones lead to the same occupied energies, within our computational uncertainties of 0.005 eV. Three calculations are needed, given that instances were found where two consecutive calculations gave the same occupied energies, but the calculation following the second one led to some occupied energies lower than their counterparts from the two calculations. Clearly, these two calculations led to a local minimum that is not to be confused with the ground state for which absolute minima of the occupied energies are required. Further calculations, with much larger, augmented basis sets do not change the charge density. Hence, they do not change the content of the Hamiltonian, even though the corresponding matrices have larger dimensions. This feature of the BZW or BZW-EF method completes DFT in practice, inasmuch as it enables the required search and attainment of the ground state.

The above implementation of the linear combination of atomic orbitals (LCAO) is known as the Bagayoko, Zhao, and Williams (BZW) method if the added orbitals are in the order of increasing energies for the excited states in the atomic or ionic species that form the molecule or solid under study. In the BZW-EF enhancement, orbitals are not necessarily added in the order of increasing energies; rather, for a given principal quantum number n, on an atomic or ionic site, p, d, and f orbitals, if applicable, are added before the corresponding s orbital. This counter-intuitive approach recognizes the fact that, for the valence states, polarization has primacy over spherical symmetry. Even though LDA and GGA calculations with the BZW method led to accurate descriptions of several semiconductors, including predictions that have been confirmed by experiments for cubic Si_{3}N_{4} [

The second feature of the method that completes DFT in practice consists of the determination, among the potential infinite number of calculations that produce the same occupied energies, of the one that provides the DFT description of the material under study. Among the first three consecutive calculations that produce the same occupied energies, the first one, with the smallest of the three basis sets, provides the DFT description of the system. The corresponding basis set is called the optimal basis set. This choice, introduced by the BZW method and maintained in the BZW-EF enhancement, is based on the Rayleigh theorem for eigenvalues [

Clearly, upon the attainment of the absolute minima of the occupied energies, a further lowering of an unoccupied energy is not due to any interaction in the Hamiltonian which does not change from its value obtained with the optimal basis set. Hence, such a lowering is a mathematical artifact stemming from the Rayleigh theorem. This artifact is the well-defined basis set and variational effect noted in the introduction. It can be invoked only after the occupied energies have reached their absolute minima; before that, the lowering of occupied and unoccupied energies is ascribed to physical interactions contained in the Hamiltonian that changes from one calculation to the next. In principle, there is an infinite number of augmented basis sets that are larger than the optimal one, some of them lower some unoccupied energies. The above referenced basis set and variational effect is not to be confused with the rather ill-defined basis set effect. In 2014, Bagayoko [

Indeed, the first calculation to reach the absolute minima of the occupied energies is the one providing the true DFT description of the material under study. Calculations with basis sets obtained by augmenting that of this calculation (i.e., the optimal basis set) do not change the charge density. So, as already noted, they do not change the content of the Hamiltonian. The energy content of this Hamiltonian, as given in Equation (2), is a unique functional of the charge density, according to the first Hohenberg-Kohn theorem. Another way of stating this fact is that the spectrum of the Hamiltonian is a unique functional of the charge density. Consequently, the lowered, unoccupied energies, while the charge density does not change, do not belong to the true DFT spectrum of the Hamiltonian. Such lowered, unoccupied energies, while the occupied ones do not change, explain the band gap underestimation by calculations that do not deliberately search for and attain the absolute minima of the occupied energies. The above second feature of the BZW and BZW-EF method completes DFT in practice, inasmuch as it enables the identification of the optimal basis set, i.e., that of the calculation that provides the true DFT description of the material under study. In doing so, this feature of our method avoids the destruction of the DFT or physical content of the lowest, unoccupied energies with the use of over-complete basis sets.

Bagayoko discussed the cases of nine semiconductors [_{2} [

While the above agreements attest to the correct nature of our understanding of DFT and of our completion of it in practice, the term confirmation is more appropriate for the predictions we have made and that have been later verified by experiment. Specifically, for c-Si_{3}N_{4}, LDA-BZW calculations of Bagayoko and Zhao [

In 2005, Bagayoko and Franklin [

Our prediction of the indirect nature of the band gap of rutile TiO_{2} [_{2}, and the smallest of the direct gaps of 3.05 eV, we felt compelled to publish these results. For us, it was a matter of informing the condensed matter theory community of a very rare instance where DFT-BZW-EF calculations appear to have led to an erroneous result for the band gap of a material. In 2014, Santara et al. [_{2}.

The above confirmation of the accuracy of LDA or GGA BZW and BZW-EF calculations for the description and prediction of electronic and related properties of materials is a motivation for extending our understanding of non-relativistic DFT to the relativistic generalization of DFT. This understanding and the accompanying completion of relativistic DFT are expected to be needed for theoretical studies of heavy elements (i.e., Z of 57 and higher) and of many systems that include them. Several high temperature superconductors, for which there remains much theoretical work to do, are examples of such materials. Depending on the degree of accuracy desired, this relativistic DFT can be very useful for the description of most atoms, molecules, semiconductors, insulators, or metals.

As we have done in the non-relativistic case, we closely follow below the derivation of the relativistic generalization of DFT, by Rajagopal and Callaway [

subject to the four-vector potential

namics by Schweber [

where the Hamiltonian is

The terms on right hand side of the above expression follow.

Using the radiation gauge, the authors wrote the Coulomb interaction between electrons explicitly as

The last term on the right hand side of Equation (8),

The above assumption of fixed nuclei consists of neglecting the motion of nuclei, as compared to that of electrons. This assumption therefore requires the classic Born-Oppenheimer approximation, i.e., that the motion of nuclei and that of electrons can be separated. Rajagopal and Callaway [

where

The continuity equation, with the four-current density in the ground state above, is

After defining the terms above, Rajagopal and Callaway proceeded to show that the external potential,

where

Following the approach of Hohenberg and Kohn, Rajagopal and Callaway [

Equation (18) is the expression of the requirement to have the total number of particles kept constant.

With

with

The above inequality, in Equation (19) above, expresses the relativistic counterpart of the second Hohenberg- Kohn theorem.

As in the case of non-relativistic DFT, the use of the relativistic generalization of DFT for electronic structure calculations has the following requirements, if the results are to possess the physical content of relativistic DFT: (a) the number of particles has to be kept constant and (b.1) one has to utilize the correct ground state current density or (b.2) one has to search for and reach the absolute minima of the occupied energies. As in the non-re- lativistic case, the four-current density corresponding to the absolute minima of the occupied energies is that of the ground state. Again, Equation (19) clearly shows that the results of calculations with a single four-component spinor basis set cannot be expected to provide the correct relativistic DFT description of a material. Such results are merely from an arbitrary, stationary solution out of a practically infinite number of such solution.

From the above understanding of the non-relativistic and relativistic density functional theory (DFT), it follows that failures of single basis set calculations to produce the correct description of materials stems from their non- adherence to one condition inherent to the validity of DFT. Indeed, in the absence of a known, exact ground state charge or four-current density, the search and attainment of the absolute minima of the occupied energies is simply required for results of electronic structure calculations to possess the full, physical content of relativistic or non-relativistic DFT. The Bagayoko, Zhao, and Williams (BZW) method, as enhanced by Ekuma and Franklin (BZW-EF) leads to highly accurate descriptions and predictions of electronic and related properties of materials, with DFT potentials. It does so without invoking the derivative discontinuity of the exchange correlation energy or self-interaction correction. Further, with this method, one does not seem to need ad hoc potentials to obtain or to approach results in agreement with experiment. The above understanding of DFT and the BZW-EF method open the way for highly accurate descriptions and predictions of electronic and related properties of materials. In so doing, they enable theory to inform and to guide the design and fabrication of material-based devices.

This work was funded in part the US National Science Foundation [NSF, Award Nos. EPS-1003897, NSF (2010- 2015)-RII-SUBR, and HRD-1002541], the US Department of Energy, National Nuclear Security Administration (NNSA, Award No. DE-NA0002630), LaSPACE, and LONI-SUBR. The author thanks Ms. Lashounda Franklin for her assistance in editing this article.

Diola Bagayoko, (2016) Understanding the Relativistic Generalization of Density Functional Theory (DFT) and Completing It in Practice. Journal of Modern Physics,07,911-919. doi: 10.4236/jmp.2016.79083