Editing Density functional theory (section)

==Relativistic formulation (ab initio functional forms)==
The same theorems can be proven in the case of relativistic electrons, thereby providing generalization of DFT for the relativistic case. Unlike the nonrelativistic theory, in the relativistic case it is possible to derive a few exact and explicit formulas for the relativistic density functional.

Let one consider an electron in the [[Hydrogen-like atom|hydrogen-like ion]] obeying the relativistic [[Dirac equation]]. The Hamiltonian {{mvar|H}} for a relativistic electron moving in the Coulomb potential can be chosen in the following form ([[atomic units]] are used):

: <math>H= c (\boldsymbol \alpha \cdot \mathbf p) + eV + mc^2\beta,</math>

where {{math|''V'' {{=}} −''eZ''/''r''}} is the Coulomb potential of a pointlike nucleus, {{math|'''p'''}} is a momentum operator of the electron, and {{mvar|e}}, {{mvar|m}} and {{mvar|c}} are the [[elementary charge]], [[electron mass]] and the [[speed of light]] respectively, and finally {{math|'''α'''}} and {{mvar|β}} are a set of [[Gamma matrices|Dirac 2&nbsp;×&nbsp;2 matrices]]:

:<math>\begin{align}
 \boldsymbol\alpha &= \begin{pmatrix} 0 & \boldsymbol\sigma \\ \boldsymbol\sigma & 0 \end{pmatrix}, \\ 
 \beta &= \begin{pmatrix} I & 0 \\ 0 & -I \end{pmatrix}.
\end{align}</math>

To find out the [[eigenfunction]]s and corresponding energies, one solves the eigenfunction equation

: <math>H\Psi = E\Psi,</math>

where {{math|Ψ {{=}} (Ψ(1), Ψ(2), Ψ(3), Ψ(4))<sup>T</sup>}} is a four-component [[wavefunction]], and {{mvar|E}} is the associated eigenenergy. It is demonstrated in Brack (1983)<ref>{{cite journal |title=Virial theorems for relativistic spin-1/2 and spin-0 particles |first=M. |last=Brack |journal=Physical Review D |volume=27 |issue=8 |page=1950 |year=1983 |doi=10.1103/physrevd.27.1950 |bibcode=1983PhRvD..27.1950B |url=https://epub.uni-regensburg.de/12408/1/46.pdf}}</ref> that application of the [[virial theorem]] to the eigenfunction equation produces the following formula for the eigenenergy of any bound state:

: <math>E = mc^2 \langle \Psi | \beta | \Psi \rangle
= mc^2 \int \big|\Psi(1)\big|^2 + \big|\Psi(2)\big|^2 - \big|\Psi(3)\big|^2 - \big|\Psi(4)\big|^2 \,\mathrm{d}\tau,</math>

and analogously, the virial theorem applied to the eigenfunction equation with the square of the Hamiltonian yields

: <math>E^2 = m^2 c^4 + emc^2 \langle \Psi | V\beta | \Psi \rangle.</math>

It is easy to see that both of the above formulae represent density functionals. The former formula can be easily generalized for the multi-electron case.

One may observe that both of the functionals written above do not have extremals, of course, if a reasonably wide set of functions is allowed for variation. Nevertheless, it is possible to design a density functional with desired extremal properties out of those ones. Let us make it in the following way:

: <math>F[n] = \frac{1}{mc^2} \left(mc^2 \int n \,d\tau - \sqrt{m^2 c^4 + emc^2 \int Vn \,d\tau} \right)^2 + \delta_{n, n_e} mc^2 \int n \,d\tau,</math>

where {{math|''n<sub>e</sub>''}} in [[Kronecker delta]] symbol of the second term denotes any extremal for the functional represented by the first term of the functional {{mvar|F}}. The second term amounts to zero for any function that is not an extremal for the first term of functional {{mvar|F}}. To proceed further we'd like to find Lagrange equation for this functional. In order to do this, we should allocate a linear part of functional increment when the argument function is altered:

: <math>F[n_e + \delta n] = \frac{1}{mc^2} \left(mc^2 \int (n_e + \delta n) \,d\tau - \sqrt{m^2 c^4 + emc^2 \int V(n_e + \delta n) \,d\tau} \right)^2.</math>

Deploying written above equation, it is easy to find the following formula for functional derivative:

: <math>\frac{\delta F[n_e]}{\delta n} = 2A - \frac{2B^2 + AeV(\tau_0)}{B} + eV(\tau_0),</math>

where {{math|''A'' {{=}} ''mc''<sup>2</sup>∫ ''n<sub>e</sub>'' d''τ''}}, and {{math|''B'' {{=}} {{sqrt|''m''<sup>2</sup>''c''<sup>4</sup> + ''emc''<sup>2</sup>∫''Vn<sub>e</sub>'' d''τ''}}}}, and {{math|''V''(''τ''<sub>0</sub>)}} is a value of potential at some point, specified by support of variation function {{math|''δn''}}, which is supposed to be infinitesimal. To advance toward Lagrange equation, we equate functional derivative to zero and after simple algebraic manipulations arrive to the following equation:

: <math>2B(A - B) = eV(\tau_0)(A - B).</math>

Apparently, this equation could have solution only if {{mvar|''A'' {{=}} ''B''}}. This last condition provides us with Lagrange
equation for functional {{mvar|F}}, which could be finally written down in the following form:

: <math>\left(mc^2 \int n \,d\tau \right)^2 = m^2 c^4 + emc^2 \int Vn \,d\tau.</math>

Solutions of this equation represent extremals for functional {{mvar|F}}. It's easy to see that all real densities,
that is, densities corresponding to the bound states of the system in question, are solutions of written above equation, which could be called the Kohn–Sham equation in this particular case. Looking back onto the definition of the functional {{mvar|F}}, we clearly see that the functional produces energy of the system for appropriate density, because the first term amounts to zero for such density and the second one delivers the energy value.