Kohn–Sham DFT

The foundation the rest of the methods branch stands on: the reformulation of the interacting many-electron problem as a self-consistent one-particle theory of the ground-state density $n(\mathbf{r})$. Density-functional theory replaces the $3N$-dimensional wavefunction with a three-dimensional density as the basic variable, and the Kohn–Sham construction makes that exact in principle and tractable in practice. Every correction in the beyond-DFT ladder is repairing one specific failure of the approximate piece introduced here.

Thomas–Fermi: the density-functional precursor

The original density-only idea predates the theorems. Thomas and Fermi modelled the kinetic energy of an electron gas as a local functional of the density, taken from the uniform gas point by point,

$$T_{\mathrm{TF}}[n]=C_{\mathrm{F}}\int n(\mathbf{r}{)}^{5/3}d\mathbf{r},C_{\mathrm{F}}=\frac{3}{10}(3{\pi }^2{)}^{2/3},$$

and added the classical electrostatic (Hartree) and external-potential energies. As a quantitative theory it fails qualitatively: with a purely local kinetic functional and no shell structure, Teller’s theorem shows that molecules do not bind in Thomas–Fermi theory — the total energy of any molecule is higher than that of the separated atoms, so chemistry is absent. The lesson is that the kinetic energy is exactly the term a crude density functional gets disastrously wrong, and it motivates the Kohn–Sham device of computing $T$ from orbitals instead.

The Hohenberg–Kohn theorems

Hohenberg and Kohn put the density on rigorous footing with two theorems for a system of interacting electrons in an external potential $v_{\mathrm{ext}}(\mathbf{r})$ — the clamped-nuclei potential of the Born–Oppenheimer PES.

HK1 (the mapping). The ground-state density $n_0(\mathbf{r})$ determines the external potential $v_{\mathrm{ext}}$ uniquely up to an additive constant. The proof is a two-line reductio: two potentials differing by more than a constant cannot share a ground-state density without violating the Rayleigh–Ritz variational principle. Since $v_{\mathrm{ext}}$ fixes the Hamiltonian, and the Hamiltonian fixes everything, the map $n_0\mapsto v_{\mathrm{ext}}\mapsto \hat{H}\mapsto {\Psi }_0\mapsto (\text{all observables})$ is a bijection on the set of $v$-representable densities: the ground-state density carries the full information of the many-body ground state.

HK2 (the variational principle). Define the total-energy functional $E_{v_{\mathrm{ext}}}[n]=F[n]+\int v_{\mathrm{ext}}(\mathbf{r})n(\mathbf{r})d\mathbf{r}$. Then for the true external potential, $E_{v_{\mathrm{ext}}}[n]\ge E_0$ for any admissible trial density, with equality iff $n=n_0$. The ground-state energy and density follow from minimizing a density functional — a variational principle over a 3D field rather than a $3N$-dimensional wavefunction.

The universal functional is the internal energy,

$$F[n]=⟨\Psi [n]|\hat{T}+{\hat{V}}_{ee}|\Psi [n]⟩,$$

universal in that it contains no reference to $v_{\mathrm{ext}}$ — the same $F[n]$ serves every system. The catch is its domain: the original definition requires $n$ to be $v$-representable (the ground-state density of some potential), a condition with no simple characterization. The Levy–Lieb constrained search repairs this, defining

$$F_{\mathrm{LL}}[n]=\underset{\Psi \to n}{\min }⟨\Psi |\hat{T}+{\hat{V}}_{ee}|\Psi ⟩,$$

the minimum of $⟨\hat{T}+{\hat{V}}_{ee}⟩$ over all antisymmetric wavefunctions yielding the density $n$. This needs only $N$-representability (trivially satisfiable), sidesteps $v$-representability entirely, and makes $F[n]$ well-defined on a clean domain.

The Kohn–Sham non-interacting mapping

The functional $F[n]$ is exact but unknown — and, as Thomas–Fermi showed, its kinetic part is the treacherous term. Kohn and Sham’s device: introduce an auxiliary non-interacting system of electrons, in an effective potential $v_{\mathrm{KS}}(\mathbf{r})$ chosen so that its ground-state density equals the exact interacting density $n(\mathbf{r})$. The auxiliary system is a single determinant of Kohn–Sham orbitals $\{{\psi }_i\}$, so its kinetic energy is computed exactly from orbitals,

$$T_s[n]=\sum _i^{\mathrm{occ}}⟨{\psi }_i|-\frac{1}{2}{\nabla }^2|{\psi }_i⟩,n(\mathbf{r})=\sum _i^{\mathrm{occ}}|{\psi }_i(\mathbf{r}){|}^2.$$

Now partition $F[n]$ as

$$F[n]=T_s[n]+E_{\mathrm{H}}[n]+E_{\mathrm{xc}}[n],E_{\mathrm{H}}[n]=\frac{1}{2}\iint \frac{n(\mathbf{r})n({\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}d\mathbf{r}d{\mathbf{r}}^{\prime },$$

which defines the exchange–correlation functional $E_{\mathrm{xc}}[n]$ as everything left over:

$$\boxed{E_{\mathrm{xc}}[n]\equiv (T[n]-T_s[n])+(V_{ee}[n]-E_{\mathrm{H}}[n])}$$

— the sum of the kinetic-correlation correction (interacting minus non-interacting kinetic energy) and the non-classical electron–electron energy (exchange plus Coulomb correlation, beyond the Hartree mean field). Everything hard and unknown is now isolated in this one term.

The Kohn–Sham equations

Applying the HK2 variational principle to the partitioned energy, the auxiliary orbitals satisfy the Kohn–Sham equations — a Schrödinger-like one-particle problem,

$$(-\frac{1}{2}{\nabla }^2+v_{\mathrm{KS}}[n](\mathbf{r})){\psi }_i={\epsilon }_i{\psi }_i,v_{\mathrm{KS}}=v_{\mathrm{ext}}+v_{\mathrm{H}}+v_{\mathrm{xc}},$$

with the Hartree potential $v_{\mathrm{H}}(\mathbf{r})=\int n({\mathbf{r}}^{\prime })|\mathbf{r}-{\mathbf{r}}^{\prime }{|}^{-1}d{\mathbf{r}}^{\prime }$ and the exchange–correlation potential the functional derivative

$$v_{\mathrm{xc}}(\mathbf{r})=\frac{\delta E_{\mathrm{xc}}[n]}{\delta n(\mathbf{r})}.$$

Because $v_{\mathrm{KS}}$ depends on the density it produces, the equations are solved self-consistently, exactly as in Hartree–Fock — indeed KS-DFT inherits the SCF machinery and, in a finite basis, the same Roothaan-type $\mathbf{FC}=\mathbf{SC}\mathbf{\epsilon }$ matrix problem. The distinction from HF is sharp: HF treats exchange exactly and correlation not at all with a non-local exchange operator; KS-DFT carries both exchange and correlation in $E_{\mathrm{xc}}$ but through a multiplicative potential $v_{\mathrm{xc}}(\mathbf{r})$, and is exact in principle. The price is that the exact $E_{\mathrm{xc}}$ is unknown and must be approximated — the subject of Jacob’s ladder.

A caution on interpretation: the KS orbitals and eigenvalues are auxiliary constructs of the fictitious system, not quasiparticle states. Only the highest occupied eigenvalue has a rigorous meaning (it equals $-\mathrm{IP}$ for the exact functional, by the asymptotic decay of the density); the KS gap is not the fundamental gap, missing the derivative discontinuity that GW supplies.

The point: $E_{\mathrm{xc}}$ is small but decisive

It is tempting to read the partition as relegating the unknown to a minor correction, since $E_{\mathrm{xc}}$ is typically only a few percent of the total electronic energy. That reading is wrong, and it is the central fact of practical DFT. The total energy is dominated by $T_s$ and the electrostatic terms, which are large and largely system-independent — they cancel out of the energy differences that physics actually cares about. What does not cancel is $E_{\mathrm{xc}}$: binding energies, reaction barriers, lattice constants, magnetic exchange, and band gaps are governed almost entirely by the errors in the approximate $E_{\mathrm{xc}}$. A 1% error in the total energy can be a 30% error in a binding energy. So $E_{\mathrm{xc}}$ must never be called negligible — it is small in magnitude and the source of essentially all the error, which is exactly why the entire beyond-DFT ladder and the self-interaction error hub exist.

Prerequisites

Born–Oppenheimer PES · second quantization

Builds toward: the XC ladder · self-interaction error · Hartree–Fock (the wavefunction companion)

Key references

• Thomas–Fermi theory — L. H. Thomas, Proc. Camb. Phil. Soc. 23, 542 (1927); E. Fermi, Z. Phys. 48, 73 (1928).
• No binding in Thomas–Fermi — E. Teller, Rev. Mod. Phys. 34, 627 (1962).
• The Hohenberg–Kohn theorems — P. Hohenberg & W. Kohn, Phys. Rev. 136, B864 (1964).
• The Kohn–Sham equations — W. Kohn & L. J. Sham, Phys. Rev. 140, A1133 (1965).
• Constrained search (rigorous $F[n]$) — M. Levy, Proc. Natl. Acad. Sci. USA 76, 6062 (1979); E. H. Lieb, Int. J. Quantum Chem. 24, 243 (1983).