Many-electron atoms: Hartree-Fock self-consistent field and the Pauli exclusion principle

Intuition Beginner

The hydrogen atom has one electron, so its Schrodinger equation can be solved exactly. Every other atom has two or more electrons, and the electron-electron repulsion term $1/r_{ij}$ couples every pair of electron coordinates. This coupling makes exact solution impossible. No atom with more than one electron has a closed-form wave function.

The way forward is approximation. The most productive approximation treats each electron as moving in the average field created by the nucleus and all the other electrons. This is the self-consistent field idea. You guess a set of orbitals, compute the average repulsive field each electron feels from the others, solve for new orbitals in that field, and repeat until the orbitals stop changing. The result is a set of one-electron orbitals and energies that approximate the full many-electron problem.

The Pauli exclusion principle constrains how these orbitals fill. Because electrons are fermions, the total wave function must change sign when any two electrons swap positions. The mathematical object that guarantees this sign change is the Slater determinant. If two electrons try to occupy the same spin-orbital, two columns of the determinant become identical and the determinant vanishes. The exclusion principle is not an additional rule -- it follows from the antisymmetry requirement.

Visual Beginner

The self-consistent field procedure is iterative. A flow chart captures the logic:

1. Guess an initial set of orbitals (often hydrogen-like).
2. Build the effective potential from the current orbitals.
3. Solve the one-electron equations in that potential to get new orbitals and energies.
4. Test: have the orbital energies changed since the last cycle?
5. If yes, go back to step 2 with the new orbitals. If no, the field is self-consistent -- stop.

Each cycle refines the effective nuclear charge felt by each electron. For a lithium atom ($Z=3$), the two $1s$ electrons shield the $2s$ electron from the full $+3$ nuclear charge. The self-consistent result gives $Z_{\text{eff}}\approx 1.26$ for the $2s$ electron -- much reduced from 3, explaining why the outer electron is loosely bound and lithium is strongly electropositive.

Worked example Beginner

Estimate the effective nuclear charge for a valence electron in sodium ($Z=11$) using Slater's rules, and use it to explain why sodium has a low first ionisation energy.

Sodium has the electron configuration $1s^{2}2s^{2}2p^{6}3s^1$. The $3s$ valence electron is shielded by all ten inner electrons.

Slater's rules for a $3s$ electron: the 2 electrons in the $n=1$ shell contribute $0.85$ each, and the 8 electrons in the $n=2$ shell contribute $0.85$ each. Total shielding: $\sigma =2(0.85)+8(0.85)=8.50$. Effective nuclear charge: $Z_{\text{eff}}=11-8.50=2.50$.

The valence electron feels only a $+2.50$ pull from a nucleus that actually has charge $+11$. The ten core electrons screen most of the nuclear charge. This low $Z_{\text{eff}}$ explains why sodium loses its $3s$ electron easily, with a first ionisation energy of only $496\text{kJ/mol}$.

Check your understanding Beginner

Formal definition Intermediate+

For a $Z$-electron atom, the electronic Hamiltonian in atomic units ($\hslash =m_e=e=4\pi {\epsilon }_0=1$) is

$$\hat{H}=\sum _{i=1}^Z\hat{h}(i)+\sum _{i<j}^Z\frac{1}{r_{ij}},$$

where $\hat{h}(i)=-\frac{1}{2}{\nabla }_i^2-Z/r_i$ is the one-electron core Hamiltonian and $r_{ij}=|{\mathbf{r}}_i-{\mathbf{r}}_j|$ is the inter-electron distance. The electron-electron repulsion ${\sum }_{i<j}1/r_{ij}$ couples all electron coordinates, preventing separation of variables.

The Hartree-Fock approximation replaces the true many-electron wave function with a single Slater determinant

$$\Psi (1,2,\dots ,Z)=\frac{1}{\sqrt{Z!}}\left|\begin{array}{cccc}{\chi }_1(1)& {\chi }_2(1)& \cdots & {\chi }_Z(1)\\ {\chi }_1(2)& {\chi }_2(2)& \cdots & {\chi }_Z(2)\\ ⋮& ⋮& \ddots & ⋮\\ {\chi }_1(Z)& {\chi }_2(Z)& \cdots & {\chi }_Z(Z)\end{array}\right|,$$

where ${\chi }_a(i)={\varphi }_a({\mathbf{r}}_i){\sigma }_a(s_i)$ is a spin-orbital: a product of a spatial orbital ${\varphi }_a$ and a spin function ${\sigma }_a$ (spin-up $\alpha$ or spin-down $\beta$). The Slater determinant is automatically antisymmetric: swapping electrons $i$ and $j$ swaps rows $i$ and $j$, changing the sign. If any two spin-orbitals are identical, two columns are identical and the determinant is zero -- this is the Pauli exclusion principle.

The variational principle states that the expectation value of the Hamiltonian in any trial wave function is an upper bound to the exact ground-state energy: $⟨{\Psi }_{\text{trial}}|\hat{H}|{\Psi }_{\text{trial}}⟩\ge E_0$. Minimising the energy of the Slater determinant with respect to variations in the spin-orbitals (subject to orthonormality constraints) yields the Hartree-Fock equations

$$\hat{f}(i){\chi }_a(i)={\epsilon }_a{\chi }_a(i),$$

where the Fock operator is

$$\hat{f}(i)=\hat{h}(i)+\sum _{b=1}^Z\left[{\hat{J}}_b(i)-{\hat{K}}_b(i)\right].$$

The Coulomb operator ${\hat{J}}_b(i)$ represents the classical electrostatic repulsion from the charge distribution of electron $b$. The exchange operator ${\hat{K}}_b(i)$ is a non-classical term arising from the antisymmetry of the wave function. It has no classical analogue and is responsible for Hund's first rule.

The Hartree-Fock equations are nonlinear: the Fock operator depends on the orbitals it produces. They must be solved iteratively via the self-consistent field procedure: guess orbitals, construct $\hat{f}$, solve the eigenvalue problem, use the new orbitals to rebuild $\hat{f}$, and iterate until convergence.

Shielding, penetration, and the orbital energy ordering

The physical content of the Fock operator is captured by the effective nuclear charge $Z_{\text{eff}}$. An electron in orbital ${\varphi }_a$ moves in the field of a nucleus partially screened by the other $Z-1$ electrons:

$$Z_{\text{eff}}(a)=Z-\sigma (a),$$

where $\sigma (a)$ is the shielding constant. Slater's rules provide an empirical estimate of $\sigma$ based on orbital type and electron count. The key physics is penetration: $s$ orbitals ($l=0$) have nonzero probability density at the nucleus, so they spend more time close to $Z$, feeling less shielding and achieving lower energy. This is why $4s$ fills before $3d$: the $4s$ orbital penetrates the core more effectively than $3d$, experiencing a larger $Z_{\text{eff}}$ despite its higher principal quantum number.

The Pauli exclusion principle from the spin-statistics connection

Electrons are spin-$1/2$ particles (fermions). The spin-statistics theorem (Fierz 1939, Pauli 1940) requires that the wave function of a system of identical fermions be antisymmetric under exchange of any two particles. For the $Z$-electron atom:

$$\Psi (\dots ,{\mathbf{r}}_i,{\sigma }_i,\dots ,{\mathbf{r}}_j,{\sigma }_j,\dots )=-\Psi (\dots ,{\mathbf{r}}_j,{\sigma }_j,\dots ,{\mathbf{r}}_i,{\sigma }_i,\dots ).$$

Setting ${\mathbf{r}}_i={\mathbf{r}}_j$ and ${\sigma }_i={\sigma }_j$ gives $\Psi =-\Psi$, hence $\Psi =0$. Two electrons with the same spin cannot be found at the same point in space. The Slater determinant is the simplest wave function that satisfies this antisymmetry requirement for $Z$ electrons distributed among $Z$ spin-orbitals.

Counterexamples and common errors

• "The Hartree-Fock method gives the exact energy." No. HF includes electron-electron repulsion only in an average (mean-field) sense. The difference between the exact non-relativistic energy and the HF energy is the correlation energy $E_{\text{corr}}=E_{\text{exact}}-E_{\text{HF}}<0$. For helium, $E_{\text{corr}}\approx -0.042$ hartree ($\approx 1.1$ eV), which is comparable to chemical bond energies.

• "Shielding means inner electrons completely block the nuclear charge." Shielding is partial. The $2s$ electron in lithium does not shield the $1s$ electrons by a full unit of charge because the $2s$ orbital extends through the core region. Slater's rules give fractional shielding constants ($0.35$, $0.85$, etc.) that reflect this imperfect screening.

• "The Aufbau ordering comes from the Hartree-Fock equations." This is approximately true but not guaranteed. The HF orbital energies reproduce the Madelung ($n+l$) ordering for most neutral atoms, but the rule is empirical. Approximately 20 transition metals violate strict Aufbau ordering, and HF calculations reproduce these exceptions.

Key theorem with proof Intermediate+

Theorem (Variational principle for Hartree-Fock). Among all normalised single-determinant wave functions constructed from $Z$ orthonormal spin-orbitals, the one that minimises $⟨\Psi |\hat{H}|\Psi ⟩$ satisfies the Hartree-Fock equations $\hat{f}{\chi }_a={\epsilon }_a{\chi }_a$. The minimum energy is

$$E_{\text{HF}}=\sum _{a=1}^Z{h}_a+\frac{1}{2}\sum _{a,b=1}^Z\left(J_{ab}-K_{ab}\right),$$

where $h_a=⟨{\chi }_a|\hat{h}|{\chi }_a⟩$ is the one-electron core integral, $J_{ab}=⟨{\chi }_a{\chi }_b|r_{12}^{-1}|{\chi }_a{\chi }_b⟩$ is the Coulomb integral, and $K_{ab}=⟨{\chi }_a{\chi }_b|r_{12}^{-1}|{\chi }_b{\chi }_a⟩$ is the exchange integral.

Proof. The energy of the Slater determinant $\Psi =|{\chi }_1{\chi }_2\cdots {\chi }_Z|$ is evaluated by expanding the determinant and computing $⟨\Psi |\hat{H}|\Psi ⟩/⟨\Psi |\Psi ⟩$. The one-electron part gives ${\sum }_a{h}_a$. The two-electron part gives contributions from all pairs $(a,b)$:

$$\frac{1}{2}\sum _{a\ne b}\left(J_{ab}-K_{ab}\right)=\frac{1}{2}\sum _{a,b}\left(J_{ab}-K_{ab}\right),$$

where the second form uses the identity $J_{aa}=K_{aa}$ (self-interaction cancellation for real orbitals). The factor $1/2$ avoids double-counting electron pairs.

To minimise $E_{\text{HF}}$ subject to the orthonormality constraint $⟨{\chi }_a|{\chi }_b⟩={\delta }_{ab}$, introduce Lagrange multipliers ${\epsilon }_a$ and set the functional derivative to zero:

$$\frac{\delta }{\delta {\chi }_a^{\ast }}\left[E_{\text{HF}}-\sum _{a,b}{\epsilon }_{ab}\left(⟨{\chi }_a|{\chi }_b⟩-{\delta }_{ab}\right)\right]=0.$$

Carrying out the differentiation yields the Fock equations $\hat{f}{\chi }_a={\sum }_b{\epsilon }_{ab}{\chi }_b$. Choosing the unitary transformation that diagonalises the Lagrange multiplier matrix gives the canonical form $\hat{f}{\chi }_a={\epsilon }_a{\chi }_a$. The orbital energies are ${\epsilon }_a=h_a+{\sum }_b(J_{ab}-K_{ab})$. $\square$

Corollary (Koopmans' theorem). In the frozen-orbital approximation, $I{E}_a\approx -{\epsilon }_a$: the orbital energy approximates the negative of the ionisation energy for removing an electron from orbital $a$.

Exercises Intermediate+

Term symbols for light atoms Master

A term symbol ${}^{2S+1}{L}_J$ specifies the total spin $S$, total orbital angular momentum $L$ (encoded as $S,P,D,F,G,\dots$ for $L=0,1,2,3,4,\dots$), and total angular momentum $J$. Term symbols arise because the individual orbital angular momenta of electrons in an open shell couple to produce collective angular momentum states.

For a $p^2$ configuration (two electrons in three $p$ orbitals), the allowed term symbols are derived from the $\left(\genfrac{}{}{0ex}{}{6}{2}\right)=15$ microstates. Applying Hund's rules:

1. Maximise $S$: the maximum $M_S$ is $+1$ (both electrons spin-up in different orbitals), giving $S=1$ (triplet).
2. Maximise $L$ for that $S$: with both spins parallel, the electrons must occupy different $m_l$ orbitals. Maximum $M_L=+1+0=+1$, giving $L=1$ ($P$ term). The $m_l=+1,+1$ combination is forbidden by the Pauli principle for parallel spins.
3. Determine $J$: for less-than-half-filled shells ($p^2$ out of $p^6$), minimise $J=|L-S|=|1-1|=0$.

Ground-state term: ${}^3{P}_0$. The remaining microstates form ${}^1{D}_2$ (5 states) and ${}^1{S}_0$ (1 state). Check: $9+5+1=15$.

Russell-Saunders (LS) coupling holds when spin-orbit coupling is weak relative to the electrostatic interactions. This is valid for light atoms ($Z\lesssim 30$). For heavier atoms, spin-orbit coupling grows roughly as $Z^4$ and individual electrons couple their $l_i$ and $s_i$ first to give $j_i$, then couple $j_i$ values -- this is jj coupling. The transition between the two schemes is gradual and produces the complex spectroscopy of the lanthanides and actinides.

Spin-orbit coupling Master

The spin-orbit interaction couples an electron's intrinsic spin angular momentum to its orbital angular momentum. The operator is

$${\hat{H}}_{\text{SO}}=\frac{1}{2m_e^2{c}^2}\frac{1}{r}\frac{dV}{dr}\hat{\mathbf{L}}\cdot \hat{\mathbf{S}},$$

where $V(r)=-Z{e}^2/r$ is the nuclear potential. For a hydrogen-like atom, this splits each orbital with $l>0$ into two sub-levels: $j=l+1/2$ (lower energy for less-than-half-filled shells) and $j=l-1/2$. The splitting scales as $Z^4$ for hydrogen-like atoms and approximately $Z^2$ for many-electron atoms.

The splitting of the sodium $3p$ level into $3p_{1/2}$ and $3p_{3/2}$ produces the famous yellow doublet: $D_1$ at $589.6$ nm and $D_2$ at $589.0$ nm. The $0.6$ nm separation is small but easily resolved with a basic spectrometer. For heavy atoms like mercury, the $6p$ splitting reaches $1.3$ eV, dominating the fine structure.

In the LS-coupling limit, the spin-orbit splitting between adjacent $J$ levels within a term follows the Lande interval rule: $E(J)-E(J-1)=A\cdot J$, where $A$ is the spin-orbit coupling constant. Deviations from this rule quantify the breakdown of pure LS coupling.

Connections Master

• Atomic structure and electron configurations 14.01.01 provides the Aufbau principle, Hund's rules, and the empirical framework for electron configurations. The present unit supplies the quantitative Hartree-Fock machinery that underlies those empirical rules. The orbital energies computed by the SCF method reproduce the $n+l$ filling order and explain its exceptions through the balance of Coulomb and exchange contributions.

• Hydrogen atom quantum chemistry 14.04.01 gives the exact one-electron orbitals that serve as the starting point for the SCF iteration. The hydrogenic orbital classification ($s,p,d,f$) is inherited by the Hartree-Fock orbitals, which retain the same angular structure but acquire modified radial functions due to screening.

• Molecular orbital theory 14.05.02 generalises the Hartree-Fock method from atoms to molecules. The molecular Fock operator replaces the single-centre nuclear attraction $-Z/r$ with the multi-centre potential of a molecular geometry. The Slater determinant construction and the self-consistent field cycle carry over directly.

• Angular momentum operators and SU(2) 12.05.01 provides the algebraic framework for the term-symbol classification. The angular momentum addition theorems determine the allowed $L$, $S$, and $J$ values from open-shell configurations.

Historical & philosophical context Master

The self-consistent field method was introduced by Douglas Hartree in 1928. Hartree's original formulation used a product wave function (no antisymmetrisation). Vladimir Fock extended the method in 1930 to include exchange by using a Slater determinant, producing the coupled equations that bear both names. John Slater's 1929 paper on determinantal wave functions provided the practical algebraic framework, and his 1930 shielding rules made quantitative atomic structure calculations accessible without electronic computers.

The Pauli exclusion principle was originally an empirical postulate (Pauli, 1925) introduced to explain the periodic table and atomic spectra. Its derivation from the spin-statistics theorem (Fierz 1939, Pauli 1940) grounded the principle in the deeper structure of relativistic quantum field theory: half-integer spin particles must have antisymmetric wave functions. The philosophical shift from "no two electrons share the same quantum numbers" to "the many-electron wave function is antisymmetric" is substantial. The latter is more general -- it applies to all fermions, it constrains the full wave function rather than individual orbital assignments, and it produces physical effects (exchange energy, exchange hole) that have no analogue in the simpler "no sharing" formulation.

Hund's rules were formulated by Friedrich Hund in 1925--27 from spectroscopic regularities, before Hartree-Fock theory provided their theoretical justification. Hund's first rule (maximise $S$) is a consequence of the exchange integral: parallel-spin electrons are kept apart by the antisymmetric spatial wave function, reducing their Coulomb repulsion. Hund's second rule (maximise $L$) reflects the tendency of electrons to orbit in the same direction, maximising their spatial separation. Hund's third rule (minimise $J$ for less-than-half-filled, maximise $J$ for more-than-half-filled) comes from spin-orbit coupling. These rules, empirically discovered and later theoretically justified, remain the standard method for predicting ground-state term symbols.

The Hartree-Fock method was the dominant computational approach in quantum chemistry from the 1950s through the 1970s. Modern computational chemistry uses HF as the starting point for more accurate methods: Moller-Plesset perturbation theory, configuration interaction, coupled-cluster theory, and density functional theory. The HF orbitals define the reference state, and the correlation energy is recovered by systematic corrections.

Bibliography Master

• Hartree, D. R., "The Wave Mechanics of an Atom with a Non-Coulomb Central Field", Proc. Camb. Phil. Soc. 24 (1928), 89--132.
• Fock, V., "Naherungsmethode zur Losung des quantenmechanischen Mehrkorperproblems", Z. Physik 61 (1930), 126--148.
• Slater, J. C., "The Theory of Complex Spectra", Phys. Rev. 34 (1929), 1293--1322.
• Slater, J. C., "Atomic Shielding Constants", Phys. Rev. 36 (1930), 57--64.
• Koopmans, T., "Uber die Zuordnung von Wellenfunktionen und Eigenwerten zu den einzelnen Elektronen eines Atoms", Physica 1 (1934), 104--113.
• Szabo, A. & Ostlund, N. S., Modern Quantum Chemistry (Dover, 1996), Ch. 3.
• Bethe, H. A. & Salpeter, E. E., Quantum Mechanics of One- and Two-Electron Atoms (Springer, 1957).
• Levine, I. N., Quantum Chemistry, 7e (Pearson, 2014), Ch. 8.
• Atkins, P. & Friedman, R., Molecular Quantum Mechanics, 5e (Oxford, 2010), Ch. 4.
• Zumdahl, S. S. & DeCoste, D. J., Chemical Principles, 8e (Cengage, 2017), Ch. 7.5--7.9.
• Hund, F., "Zur Deutung der Molekelspektren", Z. Physik 40 (1927), 742--764.