Editing Density functional theory (section)

==Derivation and formalism==
As usual in many-body electronic structure calculations, the nuclei of the treated molecules or clusters are seen as fixed (the [[Born–Oppenheimer approximation]]), generating a static external potential {{mvar|V}}, in which the electrons are moving. A [[Stationary state|stationary electronic state]] is then described by a [[wavefunction]] {{math|Ψ('''r'''<sub>1</sub>, …, '''r'''<sub>''N''</sub>)}} satisfying the many-electron time-independent [[Schrödinger equation]]

: <math> \hat H \Psi = \left[\hat T + \hat V + \hat U\right]\Psi = \left[\sum_{i=1}^N \left(-\frac{\hbar^2}{2m_i} \nabla_i^2\right) + \sum_{i=1}^N V(\mathbf r_i) + \sum_{i<j}^N U\left(\mathbf r_i, \mathbf r_j\right)\right] \Psi = E \Psi, </math>

where, for the {{mvar|N}}-electron system, {{mvar|Ĥ}} is the [[Hamiltonian (quantum mechanics)|Hamiltonian]], {{mvar|E}} is the total energy, <math>\hat T</math> is the kinetic energy, <math>\hat V</math> is the potential energy from the external field due to positively charged nuclei, and {{mvar|Û}} is the electron–electron interaction energy. The operators <math>\hat T</math> and {{mvar|Û}} are called universal operators, as they are the same for any {{mvar|N}}-electron system, while <math>\hat V</math> is system-dependent. This complicated many-particle equation is not separable into simpler single-particle equations because of the interaction term {{mvar|Û}}.

There are many sophisticated methods for solving the many-body Schrödinger equation based on the expansion of the wavefunction in [[Slater determinant]]s. While the simplest one is the [[Hartree–Fock]] method, more sophisticated approaches are usually categorized as [[post-Hartree–Fock]] methods. However, the problem with these methods is the huge computational effort, which makes it virtually impossible to apply them efficiently to larger, more complex systems.

Here DFT provides an appealing alternative, being much more versatile, as it provides a way to systematically map the many-body problem, with {{mvar|Û}}, onto a single-body problem without {{mvar|Û}}. In DFT the key variable is the electron density {{math|''n''('''r''')}}, which for a [[Normalisable wave function|normalized]] {{math|Ψ}} is given by

: <math>n(\mathbf r) = N \int{\mathrm d}^3 \mathbf r_2 \cdots \int{\mathrm d}^3 \mathbf r_N \, \Psi^*(\mathbf r, \mathbf r_2, \dots, \mathbf r_N) \Psi(\mathbf r, \mathbf r_2, \dots, \mathbf r_N).</math>

This relation can be reversed, i.e., for a given ground-state density {{math|''n''<sub>0</sub>('''r''')}} it is possible, in principle, to calculate the corresponding ground-state wavefunction {{math|Ψ<sub>0</sub>('''r'''<sub>1</sub>, …, '''r'''<sub>''N''</sub>)}}. In other words, {{math|Ψ}} is a unique [[functional (mathematics)|functional]] of {{math|''n''<sub>0</sub>}},<ref name='Hohenberg1964' />

: <math>\Psi_0 = \Psi[n_0],</math>

and consequently the ground-state [[Expectation value (quantum mechanics)|expectation value]] of an observable {{mvar|Ô}} is also a functional of {{math|''n''<sub>0</sub>}}:

: <math> O[n_0] = \big\langle \Psi[n_0] \big| \hat O \big| \Psi[n_0] \big\rangle.</math>

In particular, the ground-state energy is a functional of {{math|''n''<sub>0</sub>}}:

: <math>E_0 = E[n_0] = \big\langle \Psi[n_0] \big| \hat T + \hat V + \hat U \big| \Psi[n_0] \big\rangle,</math>

where the contribution of the external potential <math>\big\langle \Psi[n_0] \big| \hat V \big| \Psi[n_0] \big\rangle</math> can be written explicitly in terms of the ground-state density <math>n_0</math>:

: <math>V[n_0] = \int V(\mathbf r) n_0(\mathbf r) \,\mathrm d^3 \mathbf r.</math>

More generally, the contribution of the external potential <math>\big\langle \Psi \big| \hat V \big| \Psi \big\rangle</math> can be written explicitly in terms of the density <math>n</math>:

: <math>V[n] = \int V(\mathbf r) n(\mathbf r) \,\mathrm d^3 \mathbf r.</math>

The functionals {{math|''T''[''n'']}} and {{math|''U''[''n'']}} are called universal functionals, while {{math|''V''[''n'']}} is called a non-universal functional, as it depends on the system under study. Having specified a system, i.e., having specified <math>\hat V</math>, one then has to minimize the functional

: <math>E[n] = T[n] + U[n] + \int V(\mathbf r) n(\mathbf r) \,\mathrm d^3 \mathbf r</math>

with respect to {{math|''n''('''r''')}}, assuming one has reliable expressions for {{math|''T''[''n'']}} and {{math|''U''[''n'']}}. A successful minimization of the energy functional will yield the ground-state density {{math|''n''<sub>0</sub>}} and thus all other ground-state observables.

The variational problems of minimizing the energy functional {{math|''E''[''n'']}} can be solved by applying the [[Lagrange multiplier|Lagrangian method of undetermined multipliers]].<ref name='Kohn1965'>{{cite journal |title=Self-consistent equations including exchange and correlation effects |journal=Physical Review |year=1965 |first1=W. |last1=Kohn |last2=Sham |first2=L. J. |volume=140 |issue=4A |pages=A1133–A1138 |doi=10.1103/PhysRev.140.A1133 |bibcode = 1965PhRv..140.1133K |doi-access=free}}</ref> First, one considers an energy functional that does not explicitly have an electron–electron interaction energy term,

: <math>E_s[n] = \big\langle \Psi_\text{s}[n] \big| \hat T + \hat V_\text{s} \big| \Psi_\text{s}[n] \big\rangle,</math>

where <math>\hat{T}</math> denotes the kinetic-energy operator, and <math>\hat{V}_\text{s}</math> is an effective potential in which the particles are moving. Based on <math>E_s</math>, [[Kohn–Sham equations]] of this auxiliary noninteracting system can be derived:

: <math>\left[-\frac{\hbar^2}{2m} \nabla^2 + V_\text{s}(\mathbf r)\right] \varphi_i(\mathbf r) = \varepsilon_i \varphi_i(\mathbf r),</math>

which yields the [[molecular orbital|orbitals]] {{mvar|φ<sub>i</sub>}} that reproduce the density {{math|''n''('''r''')}} of the original many-body system

: <math>n(\mathbf r ) = \sum_{i=1}^N \big|\varphi_i(\mathbf r)\big|^2.</math>

The effective single-particle potential can be written as

: <math>V_\text{s}(\mathbf r) = V(\mathbf r) + \int \frac{n(\mathbf r')}{|\mathbf r - \mathbf r'|} \,\mathrm d^3 \mathbf r' + V_\text{XC}[n(\mathbf r)],</math>

where <math>V(\mathbf r)</math> is the external potential, the second term is the Hartree term describing the electron–electron [[coulombic force|Coulomb repulsion]], and the last term {{math|''V''<sub>XC</sub>}} is the exchange–correlation potential. Here, {{math|''V''<sub>XC</sub>}} includes all the many-particle interactions. Since the Hartree term and {{math|''V''<sub>XC</sub>}} depend on {{math|''n''('''r''')}}, which depends on the {{mvar|φ<sub>i</sub>}}, which in turn depend on {{math|''V''<sub>s</sub>}}, the problem of solving the Kohn–Sham equation has to be done in a self-consistent (i.e., [[iteration|iterative]]) way. Usually one starts with an initial guess for {{math|''n''('''r''')}}, then calculates the corresponding {{math|''V''<sub>s</sub>}} and solves the Kohn–Sham equations for the {{mvar|φ<sub>i</sub>}}. From these one calculates a new density and starts again. This procedure is then repeated until convergence is reached. A non-iterative approximate formulation called [[Harris functional]] DFT is an alternative approach to this.

;Notes
# The one-to-one correspondence between electron density and single-particle potential is not so smooth. It contains kinds of non-analytic structure. {{math|''E''<sub>s</sub>[''n'']}} contains kinds of singularities, cuts and branches. This may indicate a limitation of our hope for representing exchange–correlation functional in a simple analytic form.
# It is possible to extend the DFT idea to the case of the [[Green's function (many-body theory)|Green function]] {{mvar|G}} instead of the density {{mvar|n}}. It is called as [[Luttinger–Ward functional]] (or kinds of similar functionals), written as {{math|''E''[''G'']}}. However, {{mvar|G}} is determined not as its minimum, but as its extremum. Thus we may have some theoretical and practical difficulties.
# There is no one-to-one correspondence between one-body [[density matrix]] {{math|''n''('''r''', '''r'''′)}} and the one-body potential {{math|''V''('''r''', '''r'''′)}}. (All the eigenvalues of {{math|''n''('''r''', '''r'''′)}} are 1.) In other words, it ends up with a theory similar to the Hartree–Fock (or hybrid) theory.