---
title: The Central-Field Approximation and the Self-Consistent Field
module: Many-Electron Atoms
moduleNumber: 5
lessonNumber: 2
order: 502
summary: >
  The N-electron Hamiltonian does not separate because every pair of electrons
  repels. The central-field approximation replaces that pairwise repulsion with
  an averaged spherical potential each electron feels, restoring hydrogen-like
  orbitals labelled by n and ℓ. The Thomas-Fermi statistical model fixes the
  shape of the screened charge from Fermi-gas thermodynamics; the Hartree
  self-consistent field determines it exactly by iterating orbitals against the
  potential they generate until the two agree.
topics: [Many-Electron Atoms]
sources:
  - book: Bransden & Joachain
    ref: "Ch. 7 — Many-Electron Atoms; §7.1–7.2 The Central-Field Approximation; Ch. 8 The Thomas-Fermi Model and the Self-Consistent Field"
  - book: Foot
    ref: "Ch. 4 — The Alkalis and the Central-Field Approximation; §4.1–4.4"
  - book: Demtröder
    ref: "Ch. 6 — Atoms with Many Electrons"
draft: false
---

The [periodic table](/atomic-physics/many-electron-atoms/periodic-table-atomic-spectra)
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
$\vec r_1, \ldots, \vec r_N$, the non-relativistic Hamiltonian is[^bj-71]

$$
H = \sum_{i=1}^{N}\left(-\frac{\hbar^2}{2m}\nabla_i^2 - \frac{Ze^2}{4\pi\varepsilon_0\,r_i}\right)
\; + \sum_{i<j}\frac{e^2}{4\pi\varepsilon_0\,|\vec r_i - \vec r_j|}.
$$

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
$e^2/4\pi\varepsilon_0|\vec r_i - \vec r_j|$ 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.

$$
% caption: The N-electron atom. One-body attraction to the nucleus (blue) sums
% over electrons and separates; the pairwise repulsion (dashed) couples every
% pair and blocks separation.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% nucleus
\fill[black!70] (0,0) circle (3.2pt);
\node[black!70, anchor=west] at (0.14,-0.22) {$Ze$};
% electrons
\coordinate (e1) at (-1.9,1.1);
\coordinate (e2) at (2.0,0.7);
\coordinate (e3) at (0.2,-1.8);
\foreach \p in {(e1),(e2),(e3)}{\fill[black] \p circle (2.6pt);}
% attraction lines
\draw[acc, thick] (0,0) -- (e1);
\draw[acc, thick] (0,0) -- (e2);
\draw[acc, thick] (0,0) -- (e3);
% repulsion (dashed) between each pair
\draw[black, thick, dashed] (e1) -- (e2);
\draw[black, thick, dashed] (e2) -- (e3);
\draw[black, thick, dashed] (e1) -- (e3);
% legend in the clear lower-left corner
\draw[acc, thick] (-3.0,-1.4) -- (-2.4,-1.4);
\node[acc, anchor=west] at (-2.35,-1.4) {attraction};
\draw[black, thick, dashed] (-3.0,-1.95) -- (-2.4,-1.95);
\node[black, anchor=west] at (-2.35,-1.95) {repulsion};
\end{tikzpicture}
$$

## 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:[^bj-71]

$$
H = \underbrace{\sum_{i=1}^{N}\left(-\frac{\hbar^2}{2m}\nabla_i^2 + U(r_i)\right)}_{H_0}
\; + \underbrace{\left[\sum_{i<j}\frac{e^2}{4\pi\varepsilon_0\,|\vec r_i-\vec r_j|}
- \sum_{i=1}^{N}\left(U(r_i)+\frac{Ze^2}{4\pi\varepsilon_0\,r_i}\right)\right]}_{H_1}.
$$

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

$$
\left(-\frac{\hbar^2}{2m}\nabla^2 + U(r)\right)u_{n\ell m}(\vec r)
= \varepsilon_{n\ell}\,u_{n\ell m}(\vec r).
$$

Because $U(r)$ is central, the orbital separates exactly as in hydrogen into a
radial factor and a spherical harmonic, $u_{n\ell m} = R_{n\ell}(r)\,Y_{\ell m}(\theta,\phi)$,
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 $\varepsilon_{n\ell}$ depends on
$\ell$ as well as $n$. That single change reorganizes the periodic table.

> **Definition (Central-field approximation).** The replacement of the true,
> non-separable $N$-electron problem by $N$ independent electrons, each moving in
> one common spherically symmetric potential $U(r)$ that models the nucleus plus
> the averaged repulsion of all the others. The exact eigenstates are recovered
> only in the limit that the residual interaction $H_1$ vanishes; in practice
> $U(r)$ is chosen to make $H_1$ small, and $H_1$ is then treated by perturbation
> theory.

The potential $U(r)$ interpolates between two exact limits set by
[penetration and shielding](/atomic-physics/many-electron-atoms/periodic-table-atomic-spectra).
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:

$$
U(r) \xrightarrow{\;r\to 0\;} -\frac{Ze^2}{4\pi\varepsilon_0\,r}.
$$

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:

$$
U(r) \xrightarrow{\;r\to\infty\;} -\frac{(Z-N+1)e^2}{4\pi\varepsilon_0\,r}
= -\frac{e^2}{4\pi\varepsilon_0\,r}\quad(N=Z).
$$

Writing $U(r) = -Z_{\text{eff}}(r)\,e^2/4\pi\varepsilon_0 r$ defines an effective
charge that runs from $Z$ at the origin down to $1$ at infinity.

$$
% caption: The screened effective potential U(r) (solid) sits between the bare
% nuclear Coulomb well −Ze²/4πε₀r (steep, dashed) and the fully screened
% −e²/4πε₀r (shallow, dashed) it approaches far out.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,-3.0) -- (0,1.2) node[above, black!70] {$U$};
\draw[->, black] (0,0) -- (6.4,0) node[right, black!70] {$r$};
% bare -Z/r (steep)
\draw[black, thick, dashed] (0.42,-2.9) .. controls (0.7,-1.7) and (1.1,-1.05) .. (1.8,-0.72)
  .. controls (3.0,-0.42) and (4.6,-0.24) .. (6.1,-0.18);
\node[black, anchor=west] at (0.7,-2.55) {bare nucleus};
% fully screened -1/r (shallow)
\draw[black, thick, dashed] (1.0,-0.95) .. controls (1.8,-0.55) and (3.2,-0.28) .. (6.1,-0.13);
\node[black, anchor=west] at (4.5,-0.5) {screened tail};
% actual U(r): starts like -Z/r, crosses to -1/r
\draw[acc, very thick] (0.42,-2.9) .. controls (0.75,-1.55) and (1.3,-1.2) .. (2.0,-1.0)
  .. controls (3.0,-0.72) and (4.4,-0.4) .. (6.1,-0.2);
\node[acc, anchor=west] at (2.3,-1.15) {net potential $U(r)$};
\end{tikzpicture}
$$

## 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.[^bj-8] Locally the electrons fill momentum states up to a
Fermi momentum $p_{\!F}(\vec r)$, and a spin-$\tfrac12$ gas packs

$$
n(\vec r) = \frac{1}{3\pi^2}\left(\frac{p_{\!F}(\vec r)}{\hbar}\right)^{3}
$$

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 $\phi(r)$ and the neutral-atom boundary
$\phi\to 0$ as $r\to\infty$,

$$
\frac{p_{\!F}^2(r)}{2m} - e\phi(r) = 0
\quad\Longrightarrow\quad
p_{\!F}(r) = \sqrt{2m e\phi(r)}.
$$

Combining the two relations expresses the density through the potential,

$$
n(r) = \frac{1}{3\pi^2\hbar^3}\big(2m e\phi\big)^{3/2},
$$

and Poisson's equation $\nabla^2\phi = e\,n/\varepsilon_0$ (for $r>0$) closes the
system into one nonlinear differential equation for $\phi$. Substituting
$\phi(r) = \dfrac{Ze}{4\pi\varepsilon_0\,r}\,\chi(x)$ with the scaled radius
$x = r/b$ reduces it to a universal, parameter-free form:[^bj-8]

$$
\frac{\d^2\chi}{\d x^2} = \frac{\chi^{3/2}}{x^{1/2}},
\qquad
\chi(0)=1,\quad \chi(\infty)=0,
$$

with the Thomas-Fermi length

$$
b = \frac{1}{2}\left(\frac{3\pi}{4}\right)^{2/3}\frac{a_0}{Z^{1/3}}
\approx 0.885\,\frac{a_0}{Z^{1/3}}.
$$

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.

$$
% caption: The universal Thomas-Fermi function χ(x). It starts at χ(0)=1 with
% slope −1.588 and decays monotonically, setting the screened charge Zχ(r/b) for
% every neutral atom.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (6.4,0) node[right, black!70] {$x$};
\draw[->, black] (0,0) -- (0,3.4) node[above, black!70] {screening};
\node[black, anchor=east] at (-0.05,3.0) {$1$};
\draw[black, dashed] (0,3.0) -- (0.2,3.0);
% chi(x): convex decreasing curve from (0,3) toward 0
\draw[acc, very thick] (0,3.0) .. controls (0.5,2.05) and (1.0,1.5) .. (1.6,1.12)
  .. controls (2.4,0.72) and (3.4,0.42) .. (4.6,0.24)
  .. controls (5.3,0.16) and (5.9,0.12) .. (6.2,0.1);
% initial slope guide
\draw[black, dashed] (0,3.0) -- (1.9,0.0);
\node[black, anchor=west] at (2.05,0.5) {tangent at origin};
\end{tikzpicture}
$$

## 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 $-e\,|u_j(\vec r')|^2$, and the potential
energy it contributes to electron $i$ is the Coulomb integral of that density.
Requiring $\Psi$ to minimize $\langle H\rangle$ (a variational condition on each
orbital) yields the **Hartree equations**:[^bj-72]

$$
\left[-\frac{\hbar^2}{2m}\nabla^2 - \frac{Ze^2}{4\pi\varepsilon_0\,r}
+ \sum_{j\ne i}\int \frac{e^2\,|u_j(\vec r')|^2}{4\pi\varepsilon_0\,|\vec r-\vec r'|}\,\d^3 r'\right]
u_i(\vec r) = \varepsilon_i\,u_i(\vec r).
$$

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**.

> **Definition (Self-consistent field).** A potential $U(r)$ and a set of orbitals
> $\{u_i\}$ such that solving the one-electron Schrödinger equation in $U(r)$
> reproduces the orbitals whose averaged charge generates $U(r)$. It is found by
> iteration: guess a potential, solve for orbitals, rebuild the potential from
> their charge, and repeat until the potential stops changing.

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.

$$
% caption: The Hartree self-consistent-field cycle. A trial potential yields
% orbitals; their averaged charge builds a new potential; the loop repeats until
% input and output agree.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth,
  box/.style={draw, minimum width=27mm, minimum height=11mm, align=center}]
\definecolor{acc}{HTML}{4A6FA5}
\node[box, draw=acc, text=acc, thick] (guess) at (0,2.4) {guess $U(r)$};
\node[box] (solve) at (5.0,2.4) {solve for orbitals $u_i$};
\node[box] (dens) at (5.0,0) {build charge density};
\node[box] (newu) at (0,0) {new potential $U(r)$};
\draw[->, thick] (guess) -- (solve);
\draw[->, thick] (solve) -- (dens);
\draw[->, thick] (dens) -- (newu);
\draw[->, thick] (newu) -- (guess);
\node[black, anchor=east] at (-0.2,1.2) {converged?};
\node[black, anchor=west] at (5.5,1.2) {repeat};
\end{tikzpicture}
$$

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

```algorithm
caption: $\textsc{Hartree-SCF}(Z, N)$ — self-consistent field by iteration
initialize $U(r)$ from the Thomas-Fermi profile
repeat
  for each occupied orbital $i$ do
    solve $[-\tfrac{\hbar^2}{2m}\nabla^2 + U(r)]\,u_i = \varepsilon_i u_i$
  end for
  form the charge density $\rho(r) = -e\sum_i |u_i(r)|^2$
  spherically average and rebuild $U_{\text{new}}(r)$ from $\rho$ and the nucleus
  $U(r) \gets (1-\alpha)\,U(r) + \alpha\,U_{\text{new}}(r)$ // damped update
until $\max_r |U_{\text{new}}(r) - U(r)| < \text{tol}$
return $\{u_i, \varepsilon_i\}$
```

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 $\varepsilon_i$ 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 $\varepsilon_i$ over occupied orbitals therefore
counts each electron pair **twice**. The correct total energy subtracts the
double count:[^bj-72]

$$
E = \sum_{i}\varepsilon_i - \sum_{i<j} J_{ij},
\qquad
J_{ij} = \int\!\!\int \frac{e^2\,|u_i(\vec r)|^2\,|u_j(\vec r')|^2}
{4\pi\varepsilon_0\,|\vec r-\vec r'|}\,\d^3r\,\d^3r',
$$

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
$\varepsilon_i$ and $\varepsilon_j$.

By Koopmans' approximation, the orbital energy $-\varepsilon_i$ estimates the
energy to remove that electron — the ionization energy from orbital $i$ — provided
the remaining orbitals do not relax. That connects the computed $\varepsilon_{n\ell}$
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:

$$
\varepsilon_{n\ell} = -\frac{\text{Ry}}{(n-\delta_\ell)^2}
= -\frac{\text{Ry}}{n^{\ast 2}},
\qquad
n^\ast = n-\delta_\ell.
$$

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](/atomic-physics/quantum-hydrogen-atom/quantum-defects-alkali-spectra);
here it is the natural label the self-consistent field attaches to each
single-particle level.

$$
% caption: Effective charge Z_eff(r) as an electron moves outward through the
% shells. It steps down from Z near the nucleus toward 1 outside the last closed
% shell, each plateau a screening region.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (6.6,0) node[right, black!70] {$r$};
\draw[->, black] (0,0) -- (0,3.4) node[above, black!70] {screened charge};
\node[black, anchor=east] at (-0.05,3.0) {$Z$};
\node[black, anchor=east] at (-0.05,0.6) {$1$};
% descending staircase (smoothed): Z near 0, plateaus, then 1
\draw[acc, very thick]
  (0.15,3.0) -- (1.0,2.85)
  (1.0,2.85) .. controls (1.3,2.4) and (1.5,1.95) .. (1.9,1.9)
  (1.9,1.9) -- (2.7,1.82)
  (2.7,1.82) .. controls (3.0,1.4) and (3.2,1.0) .. (3.7,0.95)
  (3.7,0.95) -- (4.6,0.9)
  (4.6,0.9) .. controls (4.9,0.75) and (5.2,0.62) .. (5.7,0.6)
  (5.7,0.6) -- (6.4,0.6);
\draw[black, dashed] (0,0.6) -- (6.4,0.6);
\node[black, anchor=south] at (0.6,2.9) {K};
\node[black, anchor=south] at (2.25,1.88) {L};
\node[black, anchor=south] at (4.1,0.96) {M ...};
\end{tikzpicture}
$$

## 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](/atomic-physics/many-electron-atoms/ls-jj-coupling-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](/atomic-physics/many-electron-atoms/identical-particles-hartree-fock)
method supplies next.

[^bj-71]: **Bransden & Joachain**, _Physics of Atoms and Molecules_, 2nd ed., §7.1 — the $N$-electron Hamiltonian, the central-field approximation, the add-and-subtract split into $H_0 + H_1$, and the small-$r$/large-$r$ limits of the effective potential. <https://www.pearson.com/en-gb/subject-catalog/p/physics-of-atoms-and-molecules/P200000005386>
[^bj-8]: **Bransden & Joachain**, _Physics of Atoms and Molecules_, 2nd ed., Ch. 8 — the Thomas-Fermi statistical model: the degenerate-gas density, the constant-chemical-potential condition, the universal dimensionless equation $\chi'' = \chi^{3/2}/x^{1/2}$, and the scale length $b \approx 0.885\,a_0 Z^{-1/3}$.
[^bj-72]: **Bransden & Joachain**, _Physics of Atoms and Molecules_, 2nd ed., §7.2 — the Hartree product ansatz, the variational derivation of the Hartree equations, the self-consistent-field iteration, and the total energy $E = \sum_i\varepsilon_i - \sum_{i<j}J_{ij}$ correcting the double-counted repulsion.