The Hartree-Fock Self-Consistent-Field Equations

Derivation

The Hartree-Fock Self-Consistent-Field Equations

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Statement

For an \( N \)-electron system with Hamiltonian \( \hat{H}=\sum_i \hat{h}(i)+\tfrac12\sum_{i\ne j}\hat{v}(i,j) \), minimising the energy \( E=\langle\Psi|\hat{H}|\Psi\rangle \) over all single Slater determinants \( \Psi \) built from orthonormal spin orbitals \( \{\chi_a\} \) yields the coupled one-electron eigenvalue equations \( \hat{f}\,\chi_a=\varepsilon_a\chi_a \), where the Fock operator \( \hat{f}=\hat{h}+\sum_{b}\big(\hat{J}_b-\hat{K}_b\big) \) contains a local direct (Coulomb) potential \( \hat{J}_b \) and a nonlocal exchange potential \( \hat{K}_b \), both built self-consistently from the occupied orbitals.

Why it matters

Hartree–Fock is the point where the exact many-body Schrödinger problem, exponentially hard in the number of electrons, collapses into \( N \) coupled one-electron problems. It is the reference wavefunction of essentially all of quantum chemistry: the correlation energy is defined as the shortfall \( E_{\text{exact}}-E_{\text{HF}} \), and post-Hartree–Fock methods (configuration interaction, coupled cluster, Møller–Plesset) all build on the HF determinant.

Physically it is the cleanest realisation of a self-consistent mean field: each electron moves in the averaged Coulomb field of the others, plus a purely quantum exchange term that is forced on us by antisymmetry. That exchange term is the mathematical shadow of the Pauli principle — it keeps parallel-spin electrons apart, lowers the energy, and has no classical analogue. Understanding where it comes from is understanding why matter is stable and structured.

Assumptions

The wavefunction is a single Slater determinant of \( N \) orthonormal spin orbitals.This is the only correlation the ansatz captures — Fermi (exchange) correlation between parallel spins. Dropping it (allowing a superposition of determinants) is exactly what post-HF methods do to recover the missing Coulomb correlation; a single determinant cannot describe near-degenerate or strongly correlated states.

The electronic Hamiltonian is non-relativistic with clamped nuclei (Born–Oppenheimer).If dropped, spin–orbit and relativistic kinematics require the Dirac–Fock generalisation; for heavy atoms the non-relativistic orbital energies are quantitatively wrong.

The spin orbitals are constrained to be orthonormal, \( \langle\chi_a|\chi_b\rangle=\delta_{ab} \).Orthonormality is enforced by Lagrange multipliers; without it the variation gives a generalised eigenvalue problem \( \hat{f}\chi=\varepsilon\,\hat{S}\chi \) with an overlap operator, and the neat interpretation of \( \varepsilon_a \) is lost.

The variation is taken over the full one-electron Hilbert space (complete orbital space).In practice the orbitals are expanded in a finite basis, turning the integro-differential HF equations into the algebraic Roothaan–Hall equations \( \mathbf{F}\mathbf{C}=\mathbf{S}\mathbf{C}\boldsymbol{\varepsilon} \); the exact HF limit is only reached as the basis becomes complete.

The energy functional is stationary at a genuine minimum within the determinantal manifold.The HF equations are stationarity conditions only; a converged solution can be a saddle point (an unstable HF state). Dropping the minimum requirement means the "solution" may not be the lowest single-determinant energy, an issue diagnosed by stability analysis of the orbital Hessian.

Derivation

\[ \hat{H}=\sum_{i=1}^{N}\hat{h}(i)+\frac12\sum_{\substack{i,j=1\\ i\ne j}}^{N}\hat{v}(i,j),\qquad \hat{h}(i)=-\frac{\hbar^2}{2m}\nabla_i^2+v_{\text{ext}}(\mathbf{r}_i),\qquad \hat{v}(i,j)=\frac{e^2}{4\pi\varepsilon_0\,|\mathbf{r}_i-\mathbf{r}_j|} \]

Split the clamped-nucleus electronic Hamiltonian into one-body terms (kinetic energy plus the external nuclear attraction) and the symmetric pairwise Coulomb repulsion. A

\[ \Psi(\mathbf{x}_1,\dots,\mathbf{x}_N)=\frac{1}{\sqrt{N!}}\,\det\big[\chi_a(\mathbf{x}_b)\big],\qquad \mathbf{x}=(\mathbf{r},\omega),\qquad \langle\chi_a|\chi_b\rangle=\delta_{ab} \]

Adopt a single Slater determinant of orthonormal spin orbitals \( \chi_a(\mathbf{x}) \) (space \( \times \) spin). This is the most general fully antisymmetric product state built from a fixed orbital set, so it automatically satisfies the Pauli principle (prior results: exchange antisymmetry, exclusion). A

\[ E=\langle\Psi|\hat{H}|\Psi\rangle=\sum_{a=1}^{N}\langle a|\hat{h}|a\rangle+\frac12\sum_{a,b=1}^{N}\Big(\langle ab|ab\rangle-\langle ab|ba\rangle\Big) \]

Insert the determinant and apply the Slater–Condon rules: the \( N! \) permutations reduce, by orthonormality, to the identity plus single transpositions. Each electron pair contributes a direct integral \( \langle ab|ab\rangle \) and, from the antisymmetrising sign, a subtracted exchange integral \( \langle ab|ba\rangle \). The two-electron integral is \( \langle ab|cd\rangle=\iint \chi_a^*(1)\chi_b^*(2)\,\hat{v}_{12}\,\chi_c(1)\chi_d(2)\,d\mathbf{x}_1 d\mathbf{x}_2 \). C

\[ \langle aa|aa\rangle-\langle aa|aa\rangle=0 \quad\Longrightarrow\quad \text{the } a=b \text{ term vanishes identically} \]

The direct and exchange integrals coincide when \( a=b \), so the self-interaction cancels exactly. Hartree–Fock is therefore self-interaction free; the double sum may run over all \( a,b \) unrestricted. B

\[ \mathcal{L}[\{\chi\}]=E-\sum_{a,b=1}^{N}\varepsilon_{ba}\big(\langle a|b\rangle-\delta_{ab}\big) \]

To minimise \( E \) subject to orthonormality, introduce a Hermitian matrix of Lagrange multipliers \( \varepsilon_{ba} \) — one for each orthonormality constraint (prior result: constrained variation / variational principle). A

\[ \delta\mathcal{L}=\sum_a\Big[\langle\delta\chi_a|\hat{h}|\chi_a\rangle+\sum_b\big(\langle\delta\chi_a\,\chi_b|\chi_a\chi_b\rangle-\langle\delta\chi_a\,\chi_b|\chi_b\chi_a\rangle\big)-\sum_b\varepsilon_{ba}\langle\delta\chi_a|\chi_b\rangle\Big]=0 \]

Vary \( \chi_a\to\chi_a+\delta\chi_a \) and collect the terms linear in \( \delta\chi_a^* \). The factor \( \tfrac12 \) in \( E \) is cancelled by the two equal ways each index can be varied (the integrals are symmetric under \( 1\leftrightarrow2 \)). B

\[ \hat{h}\,\chi_a(1)+\sum_b\!\Big[\!\int\! |\chi_b(2)|^2\,\hat{v}_{12}\,d\mathbf{x}_2\Big]\chi_a(1)-\sum_b\!\Big[\!\int\! \chi_b^*(2)\,\chi_a(2)\,\hat{v}_{12}\,d\mathbf{x}_2\Big]\chi_b(1)=\sum_b\varepsilon_{ba}\,\chi_b(1) \]

Because \( \delta\chi_a^* \) is an arbitrary variation, its coefficient must vanish pointwise (fundamental lemma of the calculus of variations). This is the Hartree–Fock equation in its raw, non-canonical form. C

\[ \hat{J}_b(1)\,\chi_a(1)=\Big[\!\int\!\frac{|\chi_b(2)|^2\,e^2}{4\pi\varepsilon_0 r_{12}}\,d\mathbf{x}_2\Big]\chi_a(1),\qquad \hat{K}_b(1)\,\chi_a(1)=\Big[\!\int\!\frac{\chi_b^*(2)\,\chi_a(2)\,e^2}{4\pi\varepsilon_0 r_{12}}\,d\mathbf{x}_2\Big]\chi_b(1) \]

Name the two potentials. The Coulomb operator \( \hat{J}_b \) is local: it multiplies \( \chi_a(1) \) by the electrostatic potential of the charge cloud \( |\chi_b|^2 \). The exchange operator \( \hat{K}_b \) is nonlocal: it moves \( \chi_a \) under the integral and returns \( \chi_b(1) \), coupling the value of the orbital at one point to its value everywhere. B

\[ \hat{f}(1)\equiv\hat{h}(1)+\sum_{b=1}^{N}\big(\hat{J}_b(1)-\hat{K}_b(1)\big)\qquad\Longrightarrow\qquad \hat{f}\,\chi_a=\sum_b \varepsilon_{ba}\,\chi_b \]

Define the Fock operator as the one-electron operator inside the bracket. The HF equation now reads as an operator acting on \( \chi_a \) equalling a linear combination of the occupied orbitals. A

\[ \chi_a'=\sum_b U_{ba}\chi_b,\quad U\ \text{unitary}\ \Longrightarrow\ \Psi'=(\det U)\,\Psi,\quad \hat{f}'=\hat{f},\quad \boldsymbol{\varepsilon}'=U^{\dagger}\boldsymbol{\varepsilon}\,U \]

A unitary rotation among the occupied orbitals multiplies the determinant by a phase \( \det U \), leaving \( |\Psi| \) and \( E \) invariant. Crucially \( \hat{f} \) is also invariant, because \( \sum_b(\hat{J}_b-\hat{K}_b) \) depends only on the occupied-space projector \( \hat{\rho}=\sum_b|\chi_b\rangle\langle\chi_b| \), which no rotation among occupied orbitals can change. C

\[ \boldsymbol{\varepsilon}=\boldsymbol{\varepsilon}^{\dagger}\ \Longrightarrow\ \exists\,U:\ U^{\dagger}\boldsymbol{\varepsilon}\,U=\operatorname{diag}(\varepsilon_1,\dots,\varepsilon_N)\qquad\Longrightarrow\qquad \boxed{\ \hat{f}\,\chi_a=\varepsilon_a\,\chi_a\ } \]

Reality of \( \mathcal{L} \) makes \( \boldsymbol{\varepsilon} \) Hermitian, hence diagonalisable by a unitary \( U \). Choosing that \( U \) rotates to the canonical orbitals, in which the multiplier matrix is diagonal and each orbital satisfies a genuine eigenvalue equation. Since \( \hat{f} \) itself depends on all the \( \chi_b \), this must be solved self-consistently. C

Result

\[ \hat{f}\,\chi_a=\varepsilon_a\,\chi_a,\qquad \hat{f}=\hat{h}+\sum_{b=1}^{N}\big(\hat{J}_b-\hat{K}_b\big),\qquad E=\sum_a\langle a|\hat{h}|a\rangle+\frac12\sum_{a,b}\big(J_{ab}-K_{ab}\big) \]

Reading. Each electron obeys a one-body Schrödinger-like equation in an effective potential built from all the others: a classical Coulomb (Hartree) repulsion \( \hat{J}_b \) from the smeared-out charge of every occupied orbital, minus a quantum exchange \( \hat{K}_b \) that acts only between parallel spins and lowers the energy by keeping like-spin electrons apart. The equations are coupled and nonlinear — the potential depends on the very orbitals it determines — so they are solved by iteration: guess orbitals, build \( \hat{f} \), diagonalise, rebuild, repeat until the field stops changing. That loop is the self-consistent field (SCF). The canonical orbital energy \( \varepsilon_a \) equals \( \langle a|\hat{h}|a\rangle+\sum_b(J_{ab}-K_{ab}) \), so \( \sum_a\varepsilon_a=E+\tfrac12\sum_{ab}(J_{ab}-K_{ab}) \) double-counts the electron–electron energy: the total energy is not the sum of orbital energies.

Units check. Each spin orbital is normalised so \( |\chi|^2 \) has dimension \( L^{-3} \). In \( J_{ab}=\langle ab|ab\rangle \) the two densities give \( L^{-6} \), the measure \( d\mathbf{x}_1 d\mathbf{x}_2 \) gives \( L^{6} \), and \( e^2/(4\pi\varepsilon_0 r_{12}) \) is an energy; the product is an energy (J or eV). The Fock operator carries energy units, so its eigenvalues \( \varepsilon_a \) are energies, matching \( \hat{h} \). Every term in \( E \) is an energy, as required.

Limiting cases

One electron (\( N=1 \)): the \( a=b \) self-term vanishes, so \( \hat{f}=\hat{h} \) and the HF equation is the exact one-electron Schrödinger equation — HF is exact for hydrogen-like atoms.
Closed-shell two-electron atom (He): the two electrons share one spatial orbital with opposite spins, exchange between them vanishes, and \( E=2\langle h\rangle+J \) with a single Coulomb integral.
Vanishing interaction (\( e^2\to0 \)): \( \hat{J}_b,\hat{K}_b\to0 \), the orbitals become eigenfunctions of \( \hat{h} \) alone, and \( E\to\sum_a\varepsilon_a \) — the non-interacting limit where orbital energies do add up.
Homogeneous electron gas: plane-wave orbitals diagonalise \( \hat{f} \); the exchange term gives an energy \( \propto -k_F \) per electron, the origin of Slater's \( \rho^{1/3} \) exchange and of local-exchange density functionals.
Restricted vs unrestricted: forcing paired spins to share spatial orbitals (RHF) is exact for closed shells but biases open-shell and dissociating systems, where allowing \( \alpha \) and \( \beta \) orbitals to differ (UHF) lowers the energy.

Breaks when

Static (strong) correlation dominates. When two or more determinants are near-degenerate — stretched bonds, diradicals, transition-metal \( d \)-shells — a single determinant is qualitatively wrong. The textbook failure is H\(_2\) dissociation in RHF: forcing a closed shell drags the energy far above two isolated H atoms, giving a spurious ionic tail at large separation.
Dynamic correlation matters quantitatively. Even where HF is qualitatively right, the neglected instantaneous electron–electron avoidance (the Coulomb hole) is missing entirely. The correlation energy is only \( \sim1\% \) of the total but comparable to chemical bond and reaction energies, so bare HF is unreliable for thermochemistry.
Relativistic regimes. For heavy elements, spin–orbit coupling and relativistic contraction of inner shells are outside the non-relativistic Hamiltonian; HF orbital energies and even level orderings become wrong without the Dirac–Fock treatment.
Symmetry-broken / unstable solutions. The SCF can converge to a stationary point that is a saddle, not a minimum (an internal or external instability). Near such points the canonical orbitals and \( \varepsilon_a \) are unphysical until the broken-symmetry lower solution is found.

Failure modes

Adding orbital energies to get the total. Writing \( E=\sum_a\varepsilon_a \). This double-counts the electron–electron repulsion; the correct expression subtracts \( \tfrac12\sum_{ab}(J_{ab}-K_{ab}) \).
Letting exchange act between opposite spins. \( K_{ab} \) is nonzero only if \( \chi_a \) and \( \chi_b \) have the same spin; the spin integral kills antiparallel exchange. Students who apply exchange to every pair overcount the stabilisation.
Forgetting the factor of \( \tfrac12 \) or double-counting pairs. The pair sum \( \tfrac12\sum_{a\ne b} \) counts each pair once; dropping the \( \tfrac12 \) doubles the interaction energy.
Treating the HF equation as a fixed linear eigenproblem. Diagonalising \( \hat{f} \) once and stopping. \( \hat{f} \) is built from its own eigenvectors; without the SCF iteration the field is inconsistent with the orbitals.
Reading \( \varepsilon_a \) as "the energy of electron \( a \)". Orbital energies are Lagrange multipliers / Koopmans removal energies with frozen orbitals, not additive electron energies.
Believing HF has a self-interaction error. It does not — the \( a=b \) direct and exchange terms cancel exactly. (This is a genuine advantage over many approximate density functionals, which reintroduce self-interaction.)
Getting the exchange sign wrong. Exchange enters with a minus sign and lowers the energy; treating it as repulsive inverts the physics of the Fermi hole.

Discussion

The deep content of Hartree–Fock is that antisymmetry alone — before any dynamical correlation — already correlates electrons of parallel spin. The exchange term carves a "Fermi hole" around each electron: the probability of finding a same-spin electron nearby is suppressed, purely because the determinant must vanish when two like-spin coordinates coincide. This is the many-body meaning of the Pauli principle, and it is why the exchange integral \( K_{ab}\ge0 \) always lowers the energy. Opposite-spin electrons, by contrast, are left completely uncorrelated in HF; recovering their mutual avoidance (the Coulomb hole) is the entire business of post-HF theory.

The orbital energies carry real, if approximate, physical meaning through Koopmans' theorem: if the remaining orbitals are held frozen, the energy to remove an electron from occupied orbital \( a \) is \( -\varepsilon_a \), and the energy to add one to virtual orbital \( r \) is \( +\varepsilon_r \). The approximation neglects orbital relaxation (which lowers the ion) and correlation (which lowers the neutral); for many closed-shell molecules these errors partially cancel and \( -\varepsilon_{\text{HOMO}} \) is a fair first ionisation energy. A companion result, Brillouin's theorem, states that the HF ground determinant does not couple to any singly excited determinant, \( \langle\Psi_0|\hat{H}|\Psi_a^r\rangle=0 \) — which is precisely why the leading correlation correction (MP2) starts from double excitations.

Formally, the Fock operator is a functional of the one-body density matrix \( \hat{\rho}=\sum_b|\chi_b\rangle\langle\chi_b| \): \( \hat{f}[\hat{\rho}] \). The HF equations are the stationarity condition of the energy on the Grassmann manifold of rank-\( N \) projectors, and the nonlinearity is the statement \( [\hat{f}[\hat{\rho}],\hat{\rho}]=0 \) — the converged density commutes with the field it generates. This makes HF the mean-field truncation of the exact hierarchy of reduced density matrices, obtained by factorising the two-body density matrix into an antisymmetrised product of one-body density matrices. Kohn–Sham density-functional theory borrows the same one-determinant, self-consistent-field skeleton but replaces \( \sum_b(\hat{J}_b-\hat{K}_b) \) by \( \hat{J}[\rho]+\hat{v}_{\text{xc}}[\rho] \), trading exact nonlocal exchange for an approximate but local exchange-correlation potential.

Common misconceptions. Hartree–Fock is not the same as the older Hartree method: dropping the exchange term \( \hat{K}_b \) gives the Hartree equations, whose wavefunction is a simple product and violates antisymmetry. Nor does "self-consistent" mean "exact" — the SCF converges the mean field, not the correlation. And the canonical orbitals are not unique physical objects: any unitary rotation of the occupied set gives the same total energy and density, so localised and canonical orbitals describe identical HF states.

Worked examples

\[ \text{He, closed shell: } \chi_1=\phi\,\alpha,\ \chi_2=\phi\,\beta,\qquad \phi(r)=\Big(\frac{\zeta^3}{\pi}\Big)^{1/2}e^{-\zeta r},\qquad E=2\langle h\rangle+J \]

Both electrons occupy one hydrogen-like \( 1s \) orbital of variational exponent \( \zeta \), with opposite spins so their exchange vanishes; a one-parameter Slater ansatz. Atomic units (\( \hbar=m=e=4\pi\varepsilon_0=1 \), energy in hartree, \( 1\,\text{Ha}=27.211\,\text{eV} \)). A

\[ \langle h\rangle=\frac{\zeta^2}{2}-Z\zeta,\qquad J=\frac{5}{8}\zeta \quad\Longrightarrow\quad E(\zeta)=\zeta^2-2Z\zeta+\frac{5}{8}\zeta \]

Standard hydrogenic integrals: kinetic \( \zeta^2/2 \), nuclear attraction \( -Z\zeta \), and the \( 1s\)–\(1s \) Coulomb repulsion \( \tfrac58\zeta \). Set \( Z=2 \) for helium. B

\[ \frac{dE}{d\zeta}=2\zeta-2Z+\frac{5}{8}=0\quad\Longrightarrow\quad \zeta_\star=Z-\frac{5}{16}=2-0.3125=1.6875 \]

Minimise; the optimum exponent is the nuclear charge reduced by a screening constant \( 5/16 \) — each electron shields \( 0.3125 \) of a proton from the other. B

\[ E(\zeta_\star)=\zeta_\star^2-2\zeta_\star\cdot\zeta_\star=-\zeta_\star^2=-(1.6875)^2=-2.848\ \text{Ha} \]

Because \( 2Z-\tfrac58=2\zeta_\star \), the energy collapses to \( -\zeta_\star^2 \). A

\[ E_{\text{He}}=-2.848\ \text{Ha}=-77.5\ \text{eV} \]

Reading. This one-parameter SCF sits between the crude non-screened estimate and the true HF limit \( -2.862\,\text{Ha} \); the exact energy is \( -2.904\,\text{Ha} \), so the missing \( 0.042\,\text{Ha}\approx1.1\,\text{eV} \) is the (opposite-spin) correlation energy that no single determinant can reach. The screening constant \( 5/16 \) is the quantitative signature of the mean field.

\[ \varepsilon_{1s}=\langle h\rangle+J=\Big(\frac{\zeta_\star^2}{2}-Z\zeta_\star\Big)+\frac{5}{8}\zeta_\star \]

Continue the He solution: the Fock eigenvalue for the occupied \( 1s \) orbital is its one-electron energy plus the single Coulomb interaction with the other (opposite-spin) electron; no exchange contributes. Use \( \zeta_\star=1.6875,\ Z=2 \). A

\[ \varepsilon_{1s}=\big(1.4238-3.3750\big)+1.0547=-0.8965\ \text{Ha} \]

Numerically: \( \zeta_\star^2/2=1.4238 \), \( -Z\zeta_\star=-3.3750 \), \( J=\tfrac58(1.6875)=1.0547 \). B

\[ \text{Koopmans: } \text{IP}\approx-\varepsilon_{1s}=0.8965\ \text{Ha}\times27.211\ \tfrac{\text{eV}}{\text{Ha}}=24.4\ \text{eV} \]

The first ionisation energy is minus the highest occupied orbital energy, with the remaining orbital frozen. B

\[ E=2\varepsilon_{1s}-J=2(-0.8965)-1.0547=-2.848\ \text{Ha}\quad(\ne 2\varepsilon_{1s}=-1.793\ \text{Ha}) \]

Check the double-counting identity: the total energy is the sum of orbital energies minus the electron–electron energy \( V_{ee}=J \), not the bare sum. B

\[ \text{IP}_{\text{HF}}\approx24.4\ \text{eV}\quad(\text{experiment: }24.59\ \text{eV}) \]

Reading. Koopmans' theorem lands within \( 0.8\% \) of experiment because relaxation and correlation errors partly cancel. The parenthetical check makes the double-counting concrete: adding the two orbital energies gives \( -1.793\,\text{Ha} \), overcounting the repulsion \( J=1.055\,\text{Ha} \) that must be subtracted once to recover the correct \( -2.848\,\text{Ha} \).

Problems

Show that Hartree–Fock is free of self-interaction: prove that the \( a=b \) contribution to \( \tfrac12\sum_{ab}(J_{ab}-K_{ab}) \) vanishes for any normalised spin orbital, and explain physically why this is required.
Solution
The direct integral is \( J_{aa}=\langle aa|aa\rangle=\iint|\chi_a(1)|^2\,\hat{v}_{12}\,|\chi_a(2)|^2 \). The exchange integral is \( K_{aa}=\langle aa|aa\rangle=\iint\chi_a^*(1)\chi_a^*(2)\,\hat{v}_{12}\,\chi_a(2)\chi_a(1) \), which is the same integrand. Hence \( J_{aa}-K_{aa}=0 \). Physically, an electron must not repel its own charge cloud; antisymmetry enforces this exactly, so no spurious self-Coulomb energy survives.
Repeat the closed-shell variational calculation for the two-electron ion Li\(^+\) (\( Z=3 \)) using \( E(\zeta)=\zeta^2-2Z\zeta+\tfrac58\zeta \). Find the optimal exponent, the energy in Ha and eV, and the screening constant.
Solution
\( dE/d\zeta=2\zeta-2Z+\tfrac58=0\Rightarrow\zeta_\star=Z-\tfrac{5}{16}=3-0.3125=2.6875 \). Since \( 2Z-\tfrac58=2\zeta_\star \), \( E=-\zeta_\star^2=-(2.6875)^2=-7.223\,\text{Ha}=-7.223\times27.211=-196.5\,\text{eV} \). The screening constant is again \( 5/16=0.3125 \), independent of \( Z \) in this minimal model.
A closed-shell HF calculation gives two degenerate orbital energies \( \varepsilon_1=\varepsilon_2=-0.90\,\text{Ha} \) and an electron–electron energy \( V_{ee}=\tfrac12\sum_{ab}(J_{ab}-K_{ab})=1.05\,\text{Ha} \). Compute the total energy and state why it is not \( -1.80\,\text{Ha} \).
Solution
Using \( E=\sum_a\varepsilon_a-V_{ee} \): \( E=(-0.90-0.90)-1.05=-1.80-1.05=-2.85\,\text{Ha} \). The sum \( \sum_a\varepsilon_a=-1.80\,\text{Ha} \) counts the electron–electron repulsion twice (once in each orbital energy), so \( V_{ee} \) must be subtracted once to avoid double counting.
For a molecule the HF HOMO energy is \( \varepsilon_{\text{HOMO}}=-0.35\,\text{Ha} \) and the LUMO energy is \( \varepsilon_{\text{LUMO}}=+0.12\,\text{Ha} \). Using Koopmans' theorem, estimate the first ionisation energy and electron affinity in eV, and name one physical effect each estimate neglects.
Solution
\( \text{IP}\approx-\varepsilon_{\text{HOMO}}=0.35\times27.211=9.5\,\text{eV} \); \( \text{EA}\approx-\varepsilon_{\text{LUMO}}=-0.12\times27.211=-3.3\,\text{eV} \) (so the anion is predicted unbound at this level). Both neglect orbital relaxation of the remaining electrons; the IP additionally neglects the differential correlation energy between neutral and cation, while the EA (using a virtual orbital) is notoriously unreliable because virtual orbitals see the full \( N \)-electron field, not the \( N{+}1 \) system.
Derive the restricted closed-shell (RHF) total-energy formula. For \( N \) electrons in \( N/2 \) doubly occupied real spatial orbitals \( \{\phi_a\} \), perform the spin summation on \( E=\sum_i\langle i|\hat{h}|i\rangle+\tfrac12\sum_{ij}(J_{ij}-K_{ij}) \) and show \( E=2\sum_a h_{aa}+\sum_{ab}(2J_{ab}-K_{ab}) \), where the integrals are now over spatial orbitals.
Solution
Each spatial orbital \( \phi_a \) holds two spin orbitals \( \phi_a\alpha,\phi_a\beta \), so the one-body sum gives \( \sum_i h_{ii}=2\sum_a h_{aa} \). For the two-body sum, label spin orbitals by \( (a,\sigma) \). The Coulomb integral \( J \) is spin-independent, so summing over the two spins on each index gives \( \tfrac12\sum_{ab}\sum_{\sigma\sigma'}J_{ab}=\tfrac12\cdot4\sum_{ab}J_{ab}=2\sum_{ab}J_{ab} \). Exchange survives only for equal spins (\( \sigma=\sigma' \)), giving \( \tfrac12\sum_{ab}\sum_{\sigma}K_{ab}=\tfrac12\cdot2\sum_{ab}K_{ab}=\sum_{ab}K_{ab} \). Subtracting: \( E=2\sum_a h_{aa}+2\sum_{ab}J_{ab}-\sum_{ab}K_{ab}=2\sum_a h_{aa}+\sum_{ab}(2J_{ab}-K_{ab}) \), the standard RHF energy expression.