The Central-Field Approximation and the Self-Consistent Field

The periodic table followed from two rules applied to hydrogen-like orbitals: fill in order of energy, obey the exclusion principle. That picture smuggled in an assumption it never justified — that a well-defined single-electron orbital, labelled by $n$ and $\ell$, exists at all once electrons repel one another. It does not exist exactly. The task here is to build the effective one-electron problem whose orbitals the filling rules use, and to determine the screened potential that defines them. Two constructions do this: the Thomas-Fermi model treats the electrons as a statistical fluid and gets the potential from Fermi-gas thermodynamics; the Hartree self-consistent field gets it exactly by iteration.

The N-electron Hamiltonian

For a nucleus of charge $Ze$ fixed at the origin and $N$ electrons at positions ${\vec{r}}_1,\dots ,{\vec{r}}_N$, the non-relativistic Hamiltonian is¹

The one-body terms — kinetic energy and the electron-nucleus attraction — are a sum over electrons, and by themselves would separate into $N$ independent hydrogenic problems. The obstruction is the last term. The pair repulsion $e^2/4\pi {\epsilon }_0|{\vec{r}}_i-{\vec{r}}_j|$ couples every pair of coordinates, so the wave function does not factor into a product of single-electron functions and the Schrödinger equation admits no closed solution for $N\ge 2$. The repulsion is not small: for the two electrons of helium the mean repulsion is about $30$ eV, comparable to the $54$ eV each electron feels from the bare nucleus. It cannot be treated as a perturbation on independent electrons.

The central-field idea

Each electron moves in the field of the nucleus and the smeared-out cloud of the others. If that cloud is close to spherically symmetric — a good approximation for a closed-shell core and not a bad one otherwise — then each electron sees, to leading order, a central (radially symmetric) potential $U(r)$. Add and subtract this quantity inside the Hamiltonian:¹

The point of the split is to choose $U(r)$ so that $H_1$, the residual interaction, is as small as possible. When it is small, $H_0$ is a sum of identical one-body operators and its eigenfunctions are antisymmetrized products of single-particle orbitals $u_{n\ell m}(\vec{r})$, each obeying

Because $U(r)$ is central, the orbital separates exactly as in hydrogen into a radial factor and a spherical harmonic, $u_{n\ell m}=R_{n\ell }(r)Y_{\ell m}(\theta ,\varphi )$, so $\ell$, $m_{\ell }$, and (with spin) $m_s$ survive as good quantum numbers. What does not survive is the accidental $\ell$-degeneracy of hydrogen: because $U(r)$ is no longer a pure $1/r$, the energy ${\epsilon }_{n\ell }$ depends on $\ell$ as well as $n$. That single change reorganizes the periodic table.

The potential $U(r)$ interpolates between two exact limits set by penetration and shielding. Very close to the nucleus an electron is inside all the others, which contribute no field (a spherical shell exerts no interior force), so it feels the full nuclear charge:

Very far out it sees the nucleus screened by the other $N-1$ electrons, a net charge $Z-(N-1)$; for a neutral atom, $N=Z$, that is a single unit:

Writing $U(r)=-Z_{\text{eff}}(r)e^2/4\pi {\epsilon }_{0}r$ defines an effective charge that runs from $Z$ at the origin down to $1$ at infinity.

The Thomas-Fermi statistical model

Before the potential is found orbital by orbital, a statistical estimate fixes its scale and shape. The Thomas-Fermi model treats the electrons as a degenerate Fermi gas whose density $n(\vec{r})$ varies slowly on the scale of the local de Broglie wavelength.² Locally the electrons fill momentum states up to a Fermi momentum $p_F(\vec{r})$, and a spin-$\frac{1}{2}$ gas packs

electrons per unit volume. The highest occupied level must have the same total energy everywhere — otherwise electrons would flow from high chemical potential to low — so with electrostatic potential $\varphi (r)$ and the neutral-atom boundary $\varphi \to 0$ as $r\to \infty$,

Combining the two relations expresses the density through the potential,

and Poisson's equation ${\nabla }^2\varphi =en/{\epsilon }_0$ (for $r>0$) closes the system into one nonlinear differential equation for $\varphi$. Substituting $\varphi (r)=\frac{Ze}{4\pi {\epsilon }_{0}r}\chi (x)$ with the scaled radius $x=r/b$ reduces it to a universal, parameter-free form:²

with the Thomas-Fermi length

Every neutral atom shares the same function $\chi (x)$; only the length scale $b$ carries $Z$. The result is compact and its consequences are exact within the model:

• Universality. All neutral atoms have the same electron-density profile once lengths are measured in units of $b$.
• Shrinking scale. Because $b\propto Z^{-1/3}$, the bulk of the electron cloud contracts as heavier nuclei pull it in; the inner shells lie deep and compact.
• No shells. The statistical model returns a smooth monotone density with no shell oscillations. It captures the mean field, not the quantized structure, so it seeds an iteration rather than ending one.

The Hartree self-consistent field

The Thomas-Fermi density is an average; the Hartree method keeps the individual orbitals. Suppose the $N$-electron wave function is a simple product of orbitals, $\Psi =u_1({\vec{r}}_1)u_2({\vec{r}}_2)\cdots u_N({\vec{r}}_N)$. Each electron then moves in the electrostatic potential produced by the smeared charge of all the others. The charge density from electron $j$ is $-e|u_j({\vec{r}}^{\prime }){|}^2$, and the potential energy it contributes to electron $i$ is the Coulomb integral of that density. Requiring $\Psi$ to minimize $⟨H⟩$ (a variational condition on each orbital) yields the Hartree equations:³

The bracket is a one-electron Hamiltonian, but its potential depends on the very orbitals $u_j$ being solved for. That circularity is the defining feature: the field determines the orbitals, and the orbitals determine the field. A solution must be self-consistent.

The direct potential is spherically averaged before the next iteration, which keeps $U(r)$ central and preserves $\ell$ and $m_{\ell }$ as good quantum numbers. The loop terminates when the input and output potentials agree to a set tolerance.

The procedure is a fixed-point iteration on the potential.

The damping factor $\alpha \in (0,1]$ mixes old and new potentials to stop the iteration oscillating between two configurations, a routine numerical safeguard.

Orbital energies and the total energy

The eigenvalue ${\epsilon }_i$ is the energy of orbital $i$ moving in the field of all the others, so it already contains the full interaction of electron $i$ with every other electron. Summing ${\epsilon }_i$ over occupied orbitals therefore counts each electron pair twice. The correct total energy subtracts the double count:³

with $J_{ij}$ the classical Coulomb repulsion between the two charge clouds. The lesson is that atomic binding energies are not the sum of orbital energies; the electron-electron repulsion must be removed once to avoid counting it in both ${\epsilon }_i$ and ${\epsilon }_j$.

By Koopmans' approximation, the orbital energy $-{\epsilon }_i$ estimates the energy to remove that electron — the ionization energy from orbital $i$ — provided the remaining orbitals do not relax. That connects the computed ${\epsilon }_{n\ell }$ directly to measured ionization energies and to the alkali term diagrams.

Effective quantum numbers

Far from the core the outer electron of an alkali atom sees the screened $-1/r$ tail, so its bound energies revert to a hydrogen-like form with a shifted principal quantum number:

The quantum defect ${\delta }_{\ell }$ measures how far the orbital penetrates the core: low-$\ell$ orbitals dive into the unscreened region, feel more charge, bind tighter, and carry a larger defect. The full treatment of ${\delta }_{\ell }$ and the alkali spectra is the subject of the quantum-defect lesson; here it is the natural label the self-consistent field attaches to each single-particle level.

The residual interaction

The central-field approximation is only the zeroth step. Its neglected part, $H_1$, splits into a spherically symmetric remainder — absorbed by a better choice of $U(r)$ — and a genuinely non-central piece: the anisotropic part of the electron-electron repulsion together with the spin-orbit interaction. Those two pieces, small compared with the central field, decide the fine structure of a configuration:

• the residual electrostatic repulsion splits a configuration into terms of different total $L$ and $S$;
• the spin-orbit coupling then splits each term into levels of different $J$.

Which of the two dominates sets the coupling scheme — LS or jj — and the resulting term symbols. The exchange piece of the residual repulsion, absent from the Hartree product ansatz because it ignores antisymmetry, is the missing ingredient the Hartree-Fock method supplies next.