Hartree–Fock theory

The variational mean-field theory of the interacting electron problem: the best single Slater determinant. It is the wavefunction-side companion to Kohn–Sham DFT — both are self-consistent one-particle theories — but where KS-DFT folds exchange and correlation into an approximate density functional, Hartree–Fock treats exchange exactly and correlation not at all. That trade is the whole character of the method, and it makes HF the one mean-field theory that is rigorously free of one-electron self-interaction.

The single-determinant ansatz

Restrict the trial wavefunction of $N$ electrons to a single antisymmetrized product of orthonormal spin-orbitals $\{{\varphi }_i\}$,

$${\Psi }_{\mathrm{HF}}({\mathbf{x}}_1,\dots ,{\mathbf{x}}_N)=\frac{1}{\sqrt{N!}}\det [{\varphi }_i({\mathbf{x}}_j)]=\frac{1}{\sqrt{N!}}\left|\begin{array}{ccc}{\varphi }_1({\mathbf{x}}_1)& \cdots & {\varphi }_N({\mathbf{x}}_1)\\ ⋮& \ddots & ⋮\\ {\varphi }_1({\mathbf{x}}_N)& \cdots & {\varphi }_N({\mathbf{x}}_N)\end{array}\right|,$$

with $\mathbf{x}=(\mathbf{r},\sigma )$ a combined space–spin coordinate. The determinant enforces fermionic antisymmetry by construction (it is the second-quantized state $|\Phi ⟩={\hat{a}}_1^{†}\cdots {\hat{a}}_N^{†}|0⟩$; see second quantization). Minimizing the electronic Hamiltonian

$$\hat{H}=\sum _i\hat{h}(i)+\frac{1}{2}\sum _{i\ne j}\frac{1}{|{\mathbf{r}}_i-{\mathbf{r}}_j|},\hat{h}=-\frac{1}{2}{\nabla }^2+v_{\mathrm{ext}},$$

over the orbitals — the external potential $v_{\mathrm{ext}}$ being the clamped-nuclei field of the Born–Oppenheimer PES — gives the Hartree–Fock energy

$$E_{\mathrm{HF}}=\sum _i⟨i|\hat{h}|i⟩+\frac{1}{2}\sum _{ij}(⟨ij|ij⟩-⟨ij|ji⟩),$$

where $⟨ij|kl⟩=\iint {\varphi }_i^{\ast }(\mathbf{x}){\varphi }_j^{\ast }({\mathbf{x}}^{\prime })|\mathbf{r}-{\mathbf{r}}^{\prime }{|}^{-1}{\varphi }_k(\mathbf{x}){\varphi }_l({\mathbf{x}}^{\prime })d\mathbf{x}d{\mathbf{x}}^{\prime }$. The first two-electron term is the direct (Coulomb) energy $J_{ij}=⟨ij|ij⟩$; the second, the exchange energy $K_{ij}=⟨ij|ji⟩$, is a pure consequence of antisymmetry with no classical analogue, and is nonzero only between like-spin orbitals.

The Fock operator and the self-consistent field

Stationarity of $E_{\mathrm{HF}}$ under orthonormality-constrained variations $\delta {\varphi }_i$ yields the canonical Hartree–Fock equations, a set of one-electron eigenvalue problems

$$\hat{f}{\varphi }_i={\epsilon }_i{\varphi }_i,\hat{f}=\hat{h}+\sum _j({\hat{J}}_j-{\hat{K}}_j),$$

with the orbital energies ${\epsilon }_i$ the Lagrange multipliers of the constraints. The Fock operator $\hat{f}$ is an effective one-electron Hamiltonian: every electron moves in the average (mean) field of all the others. The Coulomb operator is multiplicative,

$${\hat{J}}_j(\mathbf{r}){\varphi }_i(\mathbf{r})=[\int \frac{|{\varphi }_j({\mathbf{r}}^{\prime }){|}^2}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}d{\mathbf{r}}^{\prime }]{\varphi }_i(\mathbf{r}),$$

the electrostatic potential of the charge density of orbital $j$; the exchange operator is non-local (integral, state-dependent),

$${\hat{K}}_j(\mathbf{r}){\varphi }_i(\mathbf{r})=[\int \frac{{\varphi }_j^{\ast }({\mathbf{r}}^{\prime }){\varphi }_i({\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}d{\mathbf{r}}^{\prime }]{\varphi }_j(\mathbf{r}).$$

Because $\hat{f}$ depends on its own eigenfunctions through $\hat{J}$ and $\hat{K}$, the equations are solved by iteration to self-consistency — the original self-consistent-field (SCF) procedure. The non-locality of $\hat{K}$ is exactly what KS-DFT trades away for a multiplicative $v_{\mathrm{xc}}(\mathbf{r})$, and is the structural reason the HF self-energy ${\Sigma }_{\mathrm{HF}}(\mathbf{r},{\mathbf{r}}^{\prime })$ is spatially non-local but frequency-independent.

Exact cancellation of the Hartree self-interaction

The diagonal terms tell the decisive story. In the Coulomb sum the $i=j$ term is present — $J_{ii}$ is the spurious electrostatic interaction of orbital $i$ with itself. But the exchange sum contains the same diagonal term $K_{ii}=⟨ii|ii⟩=J_{ii}$, and it enters with the opposite sign. The self-Coulomb and self-exchange contributions cancel identically, term by term,

$$J_{ii}-K_{ii}=0\text{for every }i,$$

so no electron interacts with its own charge density. Hartree–Fock is therefore rigorously one-electron self-interaction-free: it satisfies exactly the condition $E_{\mathrm{H}}[n_i]+E_{\mathrm{X}}^{\mathrm{exact}}[n_i]=0$ — for a one-electron density $n_i$ — that approximate semilocal functionals violate. This is precisely why HF supplies the reference against which the self-interaction error of density functionals is defined, and why admixing a fraction of HF (exact) exchange is the mechanism by which hybrid functionals partially undo it. The flip side, also part of the SIE story: HF errs in the opposite direction on the many-electron problem — its $E(N)$ curve is concave (over-localizing), the mirror of the convex semilocal error.

Koopmans’ theorem

If one assumes the orbitals of the $N$- and $(N-1)$-electron systems are identical — the frozen-orbital approximation, neglecting relaxation — then removing an electron from occupied orbital $i$ costs exactly $-{\epsilon }_i$. Hence Koopmans’ theorem:

$${\epsilon }_i\approx -{\mathrm{IP}}_i,{\epsilon }_a\approx -{\mathrm{EA}}_a,$$

the occupied orbital energy approximates the (negative) ionization potential and the virtual orbital energy the (negative) electron affinity. The approximation benefits from a cancellation of errors: orbital relaxation upon ionization lowers the true IP relative to $-{\epsilon }_i$, while the correlation energy that HF omits is larger in the $N$-electron system and raises it, the two partially offsetting. This is unrelated to the Kohn–Sham “Koopmans / linearity” condition of the delocalization-error note, which is an exact condition on $\partial E/\partial N$ rather than a frozen-orbital estimate.

Correlation, defined by what HF misses

Because the determinant is the exact ground state only of a non-interacting (or mean-field) Hamiltonian, $E_{\mathrm{HF}}$ is an upper bound to the true energy. Löwdin defines the correlation energy as the remainder,

$$E_{\mathrm{corr}}\equiv E_{\mathrm{exact}}-E_{\mathrm{HF}}<0,$$

i.e. everything beyond the best single determinant within a given one-particle basis. It is small in magnitude (a percent or so of the total energy) but, as always, dominates the chemically relevant energy differences. Recovering it is the business of post-HF methods (MP2, configuration interaction, coupled cluster) and, by a different route, of the correlation functional in KS-DFT. Note the terminology clash: “correlation” here is the Löwdin (single-determinant) definition; the KS-DFT $E_{\mathrm{c}}$ is referenced to the Kohn–Sham determinant and additionally absorbs the difference in kinetic energy, so the two are not numerically the same quantity.

Roothaan–Hall: the practical SCF

Expanding the spatial orbitals in a finite atom-centred basis $\{{\chi }_{\mu }\}$, ${\varphi }_i={\sum }_{\mu }{C}_{\mu i}{\chi }_{\mu }$ (LCAO), turns the integro-differential HF equations into the Roothaan–Hall generalized matrix eigenvalue problem

$$\mathbf{F}\mathbf{C}=\mathbf{S}\mathbf{C}\mathbf{\epsilon },$$

with $\mathbf{F}$ the Fock matrix in the basis, $\mathbf{C}$ the MO coefficients, and $\mathbf{S}$ the (non-orthogonal) overlap matrix $S_{\mu \nu }=⟨{\chi }_{\mu }|{\chi }_{\nu }⟩$. Because $\mathbf{F}$ depends on $\mathbf{C}$ through the two-electron integrals, this is iterated to self-consistency. Two spin treatments: restricted HF (RHF) forces $\alpha$ and $\beta$ electrons to share the same spatial orbitals (correct for closed shells); unrestricted HF (UHF) gives the two spins separate spatial orbitals, lowering the energy for open shells and bond dissociation at the cost of spin contamination ($⟨{\hat{S}}^2⟩$ no longer exact). The matrix structure here is the same that the Kohn–Sham SCF inherits with $\mathbf{F}$ replaced by the KS matrix — the basis-mobility of $\{{\chi }_{\mu }\}$ is also what generates the Pulay force of Hellmann–Feynman forces.

Prerequisites

Born–Oppenheimer PES · second quantization & Slater determinants

Builds toward: Kohn–Sham DFT (the density-functional companion) · hybrid functionals (admixed exact exchange) · GW (${\Sigma }_{\mathrm{HF}}=iGv$ as the unscreened, no-$W$ limit) · self-interaction error

Key references

• The self-consistent field — D. R. Hartree, Proc. Camb. Phil. Soc. 24, 89 (1928); V. Fock, Z. Phys. 61, 126 (1930).
• The determinantal (antisymmetric) ansatz — J. C. Slater, Phys. Rev. 35, 210 (1930).
• Orbital energies as ionization potentials — T. Koopmans, Physica 1, 104 (1934).
• LCAO matrix SCF — C. C. J. Roothaan, Rev. Mod. Phys. 23, 69 (1951).
• Definition of correlation energy — P.-O. Löwdin, Adv. Chem. Phys. 2, 207 (1959).