---
title: The Pauli Principle, Atoms, and the Periodic Table
module: Identical Particles
moduleNumber: 9
lessonNumber: 2
order: 902
summary: >
  Antisymmetry packaged as a Slater determinant turns the exclusion principle
  into an operating rule for building atoms. Helium shows the machinery in full:
  the electron-electron repulsion splits into a direct Coulomb integral and an
  exchange integral, and the exchange term alone pushes the spin-triplet
  (orthohelium) below the spin-singlet (parahelium) with no magnetic interaction
  in sight. Screening, the aufbau order, and Hund's rules then assemble the whole
  periodic table from the same antisymmetry.
topics: [Identical Particles]
sources:
  - book: Griffiths & Schroeter
    ref: "Ch. 5 — Identical Particles; §5.2 Atoms, §5.2.1 Helium, §5.2.2 The Periodic Table"
  - book: Cohen-Tannoudji, Diu & Laloë
    ref: "Complement A_XIV — The Slater determinant; Ch. XIV Systems of identical particles"
  - book: Tipler & Llewellyn
    ref: "Ch. 7 — Atomic Physics; §7-4 Ground States and the Periodic Table, §7-5 Excited States and Spectra"
draft: false
---

The [exchange symmetry](/quantum-mechanics/identical-particles/identical-particles-and-exchange-symmetry)
of the previous lesson fixes the form of every multi-electron state:
antisymmetric under the interchange of any two electrons. Turning that
constraint into the structure of atoms takes one construction, the Slater
determinant, and one calculation, the splitting of the electron-electron
repulsion into a direct and an exchange piece. Helium is the smallest atom where
both electrons and their repulsion appear, and it displays the whole mechanism.
Screening, the filling order of subshells, and Hund's rules then extend the same
antisymmetry across the periodic table.

## Slater determinants

A single electron in an atom occupies a **spin-orbital**, a product of a spatial
orbital and a spin state,

$$
\phi_\mu(\vec x) = \psi_{n\ell m}(\vec r)\,\chi_{m_s}(s),
\qquad \vec x = (\vec r, s),
$$

where the composite label $\mu = (n,\ell,m,m_s)$ collects every quantum number.
An $N$-electron state must be totally antisymmetric in the $N$ coordinates
$\vec x_1,\dots,\vec x_N$. The antisymmetrized product of $N$ distinct
spin-orbitals is the **Slater determinant**,

$$
\Psi(\vec x_1,\dots,\vec x_N)
= \frac{1}{\sqrt{N!}}
\begin{vmatrix}
\phi_{\mu_1}(\vec x_1) & \phi_{\mu_2}(\vec x_1) & \cdots & \phi_{\mu_N}(\vec x_1) \\
\phi_{\mu_1}(\vec x_2) & \phi_{\mu_2}(\vec x_2) & \cdots & \phi_{\mu_N}(\vec x_2) \\
\vdots & \vdots & \ddots & \vdots \\
\phi_{\mu_1}(\vec x_N) & \phi_{\mu_2}(\vec x_N) & \cdots & \phi_{\mu_N}(\vec x_N)
\end{vmatrix}.
$$

Rows are labelled by electrons, columns by spin-orbitals. Two determinant
identities carry the physics.[^ct-slater]

- **Antisymmetry.** Interchanging two electrons swaps two rows, and a determinant
  changes sign under a row swap. The state is antisymmetric by construction.
- **Exclusion.** If two spin-orbitals coincide, two columns are equal and the
  determinant vanishes. No two electrons share all four quantum numbers.

$$
% caption: The Slater determinant for two electrons, normalized by $1/\sqrt2$.
% Each entry is the spin-orbital $\phi$ (columns: orbital $a$ or $b$) evaluated at
% an electron coordinate (rows). Swapping the electron rows flips the sign; equal
% columns (identical spin-orbitals) make the determinant vanish.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % matrix frame
  \draw[thick] (0,0) rectangle (4.6,2.2);
  \node at (1.15,1.6) {$a(1)$};
  \node at (3.45,1.6) {$b(1)$};
  \node at (1.15,0.6) {$a(2)$};
  \node at (3.45,0.6) {$b(2)$};
  % row / column labels
  \node[black, anchor=east, font=\scriptsize] at (-0.15,1.6) {electron 1};
  \node[black, anchor=east, font=\scriptsize] at (-0.15,0.6) {electron 2};
  \node[acc, anchor=south, font=\scriptsize] at (1.15,2.3) {orbital $a$};
  \node[acc, anchor=south, font=\scriptsize] at (3.45,2.3) {orbital $b$};
\end{tikzpicture}
$$

For two electrons the determinant expands to the antisymmetric combination of the
previous lesson,

$$
\Psi(\vec x_1,\vec x_2)
= \frac{1}{\sqrt 2}\Bigl[\phi_a(\vec x_1)\phi_b(\vec x_2) - \phi_a(\vec x_2)\phi_b(\vec x_1)\Bigr].
$$

> **Definition (Slater determinant).** The totally antisymmetric $N$-electron
> state formed as the determinant of the matrix $\phi_{\mu_j}(\vec x_i)$ of
> single-particle spin-orbitals, normalized by $1/\sqrt{N!}$. It is the standard
> building block of atomic and molecular structure.

## Helium: the direct and exchange integrals

Helium has a nucleus of charge $Z=2$ and two electrons. Its Hamiltonian is

$$
\hat H = \underbrace{\left(-\frac{\hbar^2}{2m}\nabla_1^2 - \frac{Ze^2}{4\pi\varepsilon_0 r_1}\right)
       + \left(-\frac{\hbar^2}{2m}\nabla_2^2 - \frac{Ze^2}{4\pi\varepsilon_0 r_2}\right)}_{\hat H_0}
       + \underbrace{\frac{e^2}{4\pi\varepsilon_0 |\vec r_1 - \vec r_2|}}_{\hat V_{ee}}.
$$

Drop $\hat V_{ee}$ and each electron sits in a hydrogenic potential with charge
$Z=2$. The [hydrogenic energies](/quantum-mechanics/central-potentials/the-hydrogen-atom)
scale as $Z^2$, so the ground state, both electrons in the $1s$ orbital, has
zeroth-order energy

$$
E_0 = 2 \times \bigl(-Z^2 \times 13.6\ \text{eV}\bigr)
= -8 \times 13.6\ \text{eV} = -108.8\ \text{eV}.
$$

The measured ground-state energy is $-79.0\ \text{eV}$ (the sum of the two
ionization energies, $24.6 + 54.4\ \text{eV}$).[^codata] The $30\ \text{eV}$
discrepancy is the electron-electron repulsion, dropped in $\hat H_0$. Treat
$\hat V_{ee}$ in first-order perturbation theory. For the ground configuration both
electrons occupy $1s$, the spatial state is symmetric, and the spins form the
singlet. The first-order energy shift is the expectation of $\hat V_{ee}$ in the
$1s^2$ spatial state,

$$
\langle \hat V_{ee}\rangle
= \frac{e^2}{4\pi\varepsilon_0}\int\!\!\int
\frac{|\psi_{1s}(\vec r_1)|^2\,|\psi_{1s}(\vec r_2)|^2}{|\vec r_1 - \vec r_2|}\,
\d^3 r_1 \, \d^3 r_2
= \frac{5}{4}\,Z \times 13.6\ \text{eV}.
$$

With $Z=2$ this is $34.0\ \text{eV}$, raising the estimate to
$-108.8 + 34.0 = -74.8\ \text{eV}$, within a few electron-volts of the measured
$-79.0\ \text{eV}$. The remaining gap closes once the electrons are allowed to
screen one another; the [variational method](/quantum-mechanics/approximation-methods/the-variational-method)
with an effective charge $Z_{\text{eff}} = Z - \tfrac{5}{16} = 1.69$ gives
$-77.5\ \text{eV}$.[^gs-helium]

### Exchange splitting in the excited states

The instructive structure appears in the excited configurations $1s\,n\ell$,
where one electron stays in $1s$ and the other occupies a higher orbital. Now the
two spatial orbitals differ, $\psi_a = \psi_{1s}$ and $\psi_b = \psi_{n\ell}$, and
the spatial state may be symmetric or antisymmetric,

$$
\psi_\pm(\vec r_1,\vec r_2) = \frac{1}{\sqrt 2}\Bigl[\psi_a(\vec r_1)\psi_b(\vec r_2) \pm \psi_a(\vec r_2)\psi_b(\vec r_1)\Bigr].
$$

Antisymmetry of the total state ties $\psi_+$ to the spin singlet ($S=0$,
**parahelium**) and $\psi_-$ to the spin triplet ($S=1$, **orthohelium**).
Evaluate $\langle \hat V_{ee}\rangle$ in $\psi_\pm$. The cross terms of the
symmetrized state generate two integrals,

$$
\langle \hat V_{ee}\rangle_\pm = J \pm K,
$$

with the **direct** (Coulomb) integral

$$
J = \frac{e^2}{4\pi\varepsilon_0}\int\!\!\int
\frac{|\psi_a(\vec r_1)|^2\,|\psi_b(\vec r_2)|^2}{|\vec r_1 - \vec r_2|}\,\d^3 r_1\,\d^3 r_2
$$

and the **exchange** integral

$$
K = \frac{e^2}{4\pi\varepsilon_0}\int\!\!\int
\frac{\psi_a^{\ast}(\vec r_1)\psi_b^{\ast}(\vec r_2)\,\psi_a(\vec r_2)\psi_b(\vec r_1)}{|\vec r_1 - \vec r_2|}\,\d^3 r_1\,\d^3 r_2 .
$$

$J$ is the classical electrostatic energy of the two charge clouds
$|\psi_a|^2$ and $|\psi_b|^2$. $K$ has no classical analog: it comes from the
interference of the two ways to assign electrons to orbitals, and it is positive
for the Coulomb interaction. The singlet (parahelium) lies at $J+K$ and the
triplet (orthohelium) at $J-K$, so

$$
E_{\text{triplet}} - E_{\text{singlet}} = -2K < 0 .
$$

The triplet lies below the singlet by $2K$. Nothing magnetic entered the
calculation; the interaction was pure electrostatic Coulomb. The triplet has an
antisymmetric spatial state, its electrons avoid one another (the fermionic
exchange correlation), their repulsion is reduced, and the energy drops. This is
the physical content of the exchange integral: aligning the spins forces spatial
antisymmetry, which lowers the Coulomb energy.[^gs-para]

$$
% caption: Each excited helium configuration $1s\,n\ell$ shifts up by the direct
% integral $J$, then splits into a lower triplet (orthohelium, spins parallel,
% antisymmetric space, energy $J-K$) and a higher singlet (parahelium, spins
% paired, symmetric space, energy $J+K$), separated by $2K$ from the exchange
% integral $K$.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % unperturbed level
  \draw[black, thick, dashed] (0,1.4) -- (2.6,1.4);
  \node[black, anchor=east, font=\scriptsize] at (-0.1,1.4) {$1s\,nl$};
  \node[black, anchor=south, font=\scriptsize] at (1.3,1.45) {no repulsion};
  % shifted by J
  \draw[black, thick] (3.2,1.9) -- (5.2,1.9);
  \node[black, anchor=south, font=\scriptsize] at (4.2,1.95) {shift $+J$};
  % split into singlet (up) and triplet (down)
  \draw[black, thick] (6.0,2.6) -- (8.4,2.6);
  \draw[acc, thick] (6.0,1.2) -- (8.4,1.2);
  \node[black, anchor=west, font=\scriptsize] at (8.5,2.6) {singlet (para)};
  \node[acc, anchor=west, font=\scriptsize] at (8.5,1.2) {triplet (ortho)};
  % connectors
  \draw[black] (5.2,1.9) -- (6.0,2.6);
  \draw[black] (5.2,1.9) -- (6.0,1.2);
  % 2K brace (hand-drawn)
  \draw[black] (7.2,1.2) -- (7.6,1.2);
  \draw[black] (7.4,1.2) -- (7.4,2.6);
  \draw[black] (7.2,2.6) -- (7.6,2.6);
  \node[black, anchor=west, font=\scriptsize] at (7.45,1.9) {$2K$};
\end{tikzpicture}
$$

## Screening and effective charge

Beyond helium, an electron does not feel the bare nuclear charge $Z$. The inner
electrons partly cancel it, so an outer electron sees a reduced **effective
charge** $Z_{\text{eff}} < Z$. The cancellation depends on how far the electron
penetrates toward the nucleus, and penetration depends on orbital angular
momentum.

A low-$\ell$ orbital has a spatial density that reaches closer to the nucleus,
past the screening cloud of inner electrons, and feels a larger $Z_{\text{eff}}$.
A high-$\ell$ orbital is held out by the centrifugal barrier and is screened more
completely. Within a shell of fixed $n$, energy therefore rises with $\ell$,

$$
E_{ns} < E_{np} < E_{nd} < E_{nf},
$$

lifting the exact hydrogenic degeneracy in $\ell$. In hydrogen all $\ell$ at a
given $n$ share an energy; screening in a multi-electron atom removes that
accident.[^gs-screen]

$$
% caption: A penetrating $2s$ orbital has inner amplitude that samples the
% unscreened nucleus, so it feels a larger effective charge and sits below the
% $2p$ orbital, which the centrifugal barrier holds outside the screening cloud.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-0.2,0) -- (5.6,0) node[right, font=\scriptsize] {$r$};
  \draw[->, black] (0,-0.2) -- (0,3.0) node[above, font=\scriptsize] {radial density};
  % 2s: small inner (penetrating) lobe, node, large outer lobe
  \draw[acc, thick, domain=0:5.3, samples=160]
    plot (\x, {1.4*(1.3*\x)*(1.3*\x)*(2-1.3*\x)*(2-1.3*\x)*exp(-1.3*\x)});
  % 2p: single outer lobe, no inner amplitude
  \draw[black, thick, dashed, domain=0:5.3, samples=160]
    plot (\x, {0.42*(1.3*\x)*(1.3*\x)*(1.3*\x)*(1.3*\x)*exp(-1.3*\x)});
  \node[acc, anchor=west, font=\scriptsize] at (0.35,2.55) {$2s$ (penetrating)};
  \node[black, anchor=west, font=\scriptsize] at (3.15,2.25) {$2p$ (screened)};
\end{tikzpicture}
$$

## The aufbau order and electron configurations

Screening sets the order in which subshells fill. The empirical **Madelung rule**
orders subshells by increasing $n+\ell$, and by increasing $n$ where $n+\ell$
ties. Filling in that order (the **aufbau principle**), placing electrons one at a
time into the lowest available spin-orbital, generates the ground-state
configuration of each element.

$$
% caption: The aufbau (Madelung) filling order: subshells fill by increasing
% $n+\ell$, and by increasing $n$ within a tie. Reading the diagonals gives
% $1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p,\dots$.
\begin{tikzpicture}[>=stealth, font=\footnotesize,
  cell/.style={draw, minimum size=8.5mm, font=\scriptsize}]
  \definecolor{acc}{HTML}{4A6FA5}
  % rows n=1..4 (top to bottom), columns l = s,p,d,f (left to right)
  \node[cell] at (0,3.0) {$1s$};
  \node[cell] at (0,2.0) {$2s$};
  \node[cell] at (0,1.0) {$3s$};
  \node[cell] at (0,0.0) {$4s$};
  \node[cell] at (1.1,2.0) {$2p$};
  \node[cell] at (1.1,1.0) {$3p$};
  \node[cell] at (1.1,0.0) {$4p$};
  \node[cell] at (2.2,1.0) {$3d$};
  \node[cell] at (2.2,0.0) {$4d$};
  \node[cell] at (3.3,0.0) {$4f$};
  % direction legend in the empty upper-right (arrows point down-left along
  % constant n+l diagonals; the caption states the n+l rule)
  \draw[->, acc, thick] (3.6,3.3) -- (2.6,2.4);
  \draw[->, acc, thick] (3.1,3.3) -- (2.1,2.4);
  \draw[->, acc, thick] (2.6,3.3) -- (1.6,2.4);
  \node[acc, anchor=west, font=\scriptsize] at (1.0,3.65) {aufbau order};
\end{tikzpicture}
$$

The rule is a good approximation, not a law: $4s$ fills before $3d$ because
$4s$ ($n+\ell=4$) beats $3d$ ($n+\ell=5$), which puts potassium and calcium ahead
of the transition metals. Half-filled and filled subshells carry extra stability,
producing the well-known anomalies where the tabulated configuration departs from
the naive filling.

| Element | Naive filling | Actual configuration | Reason |
| --- | --- | --- | --- |
| Chromium (24) | $[\text{Ar}]\,4s^2 3d^4$ | $[\text{Ar}]\,4s^1 3d^5$ | half-filled $3d^5$ stability |
| Copper (29) | $[\text{Ar}]\,4s^2 3d^9$ | $[\text{Ar}]\,4s^1 3d^{10}$ | filled $3d^{10}$ stability |

## Hund's rules

A partly filled subshell has many ways to distribute its electrons among the
degenerate orbitals. **Hund's rules** select the ground-state term, and the first
two are direct consequences of the exchange physics worked out for helium.

> **Rule (Hund's rules).** For the ground term of a configuration:
> 1. Maximize the total spin $S$. Parallel spins force an antisymmetric spatial
>    state, which lowers the Coulomb repulsion by the exchange energy.
> 2. For the largest $S$, maximize the total orbital angular momentum $L$.
>    Electrons orbiting the same way avoid one another and again lower repulsion.
> 3. The total angular momentum is $J = |L-S|$ for a subshell less than half
>    filled and $J = L+S$ for one more than half filled (spin-orbit coupling).

The first rule is the helium triplet generalized: aligning spins forces spatial
antisymmetry, and the exchange integral pays for it. Applying the rules to the
$2p$ series shows the pattern, with term symbols written $^{2S+1}L_J$.

$$
% caption: Filling the three $2p$ orbitals by Hund's first rule: electrons enter
% singly with parallel spins before any orbital is doubly occupied, maximizing
% the total spin.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % three boxes for carbon 2p^2
  \foreach \i in {0,1,2} {
    \draw[black, thick] (\i*0.9,0) rectangle (\i*0.9+0.7,0.7);
  }
  \draw[->, acc, thick] (0.35,0.12) -- (0.35,0.58);
  \draw[->, acc, thick] (1.25,0.12) -- (1.25,0.58);
  \node[anchor=north, font=\scriptsize] at (1.15,-0.15) {carbon $2p^2$: $^{3}P$};
  % nitrogen 2p^3
  \begin{scope}[xshift=4.3cm]
    \foreach \i in {0,1,2} {
      \draw[black, thick] (\i*0.9,0) rectangle (\i*0.9+0.7,0.7);
    }
    \draw[->, acc, thick] (0.35,0.12) -- (0.35,0.58);
    \draw[->, acc, thick] (1.25,0.12) -- (1.25,0.58);
    \draw[->, acc, thick] (2.15,0.12) -- (2.15,0.58);
    \node[anchor=north, font=\scriptsize] at (1.15,-0.15) {nitrogen $2p^3$: $^{4}S$};
  \end{scope}
  % oxygen 2p^4
  \begin{scope}[xshift=8.6cm]
    \foreach \i in {0,1,2} {
      \draw[black, thick] (\i*0.9,0) rectangle (\i*0.9+0.7,0.7);
    }
    \draw[->, acc, thick] (0.25,0.12) -- (0.25,0.58);
    \draw[<-, acc, thick] (0.45,0.12) -- (0.45,0.58);
    \draw[->, acc, thick] (1.25,0.12) -- (1.25,0.58);
    \draw[->, acc, thick] (2.15,0.12) -- (2.15,0.58);
    \node[anchor=north, font=\scriptsize] at (1.15,-0.15) {oxygen $2p^4$: $^{3}P$};
  \end{scope}
\end{tikzpicture}
$$

Carbon ($2p^2$) puts both electrons in separate orbitals with parallel spins:
$S=1$, and the largest compatible $L$ is $1$, giving a $^3P$ term; less than half
filled, so $J=|L-S|=0$ and the ground term is $^3P_0$. Nitrogen ($2p^3$) is
half-filled with all three spins parallel: $S=\tfrac32$, and the only way to keep
all $m_\ell$ distinct forces $L=0$, giving $^4S_{3/2}$. Oxygen ($2p^4$) has one
doubly-occupied orbital: $S=1$, $L=1$, and now more than half filled, so
$J=L+S=2$ and the term is $^3P_2$.

| Configuration | $S$ | $L$ | Filling | $J$ | Ground term |
| --- | --- | --- | --- | --- | --- |
| Carbon $2p^2$ | $1$ | $1$ | $<$ half | $0$ | $^3P_0$ |
| Nitrogen $2p^3$ | $\tfrac32$ | $0$ | half | $\tfrac32$ | $^4S_{3/2}$ |
| Oxygen $2p^4$ | $1$ | $1$ | $>$ half | $2$ | $^3P_2$ |

## The shape of the periodic table

The blocks of the periodic table are the subshells being filled. The two columns
on the left fill an $s$ subshell, the six on the right fill a $p$ subshell, the ten
transition-metal columns fill a $d$ subshell, and the fourteen lanthanide and
actinide columns fill an $f$ subshell. Column count is $2(2\ell+1)$, the number of
spin-orbitals in the subshell: exclusion fixes the widths.

$$
% caption: The periodic table divided into blocks by the subshell being filled.
% Block widths $2(2\ell+1)$ are the spin-orbital counts fixed by the exclusion
% principle: $s$ two, $p$ six, $d$ ten, $f$ fourteen.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % s block (2 wide)
  \draw[thick] (0,0) rectangle (1.0,3.5);
  \node at (0.5,1.75) {$s$};
  \node[anchor=north, font=\scriptsize] at (0.5,-0.1) {$2$};
  % d block (10 wide)
  \draw[thick] (1.2,0.8) rectangle (6.2,2.7);
  \node at (3.7,1.75) {$d$};
  \node[anchor=north, font=\scriptsize] at (3.7,0.7) {$10$};
  % p block (6 wide)
  \draw[thick] (6.4,0) rectangle (9.4,3.5);
  \node at (7.9,1.75) {$p$};
  \node[anchor=north, font=\scriptsize] at (7.9,-0.1) {$6$};
  % f block (14 wide, detached below)
  \draw[thick] (1.2,-1.6) rectangle (8.2,-0.6);
  \node at (4.7,-1.1) {$f$};
  \node[anchor=north, font=\scriptsize] at (4.7,-1.7) {$14$};
\end{tikzpicture}
$$

Two facts about ordinary matter descend directly from the antisymmetry of the
electron wavefunction. Atoms have shell structure and chemistry because electrons
cannot all fall into the $1s$ orbital; they stack into successive subshells, and
the outermost, partly filled subshell sets an element's chemical behavior. And
bulk matter resists compression because pressing atoms together forces their
electrons toward common states that the exclusion principle forbids, producing the
degeneracy pressure that holds up white dwarfs and neutron stars. Both are the
Pauli principle, read at the scale of a single atom and at the scale of a star.

[^ct-slater]: **Cohen-Tannoudji, Diu & Laloë**, _Quantum Mechanics_ (Wiley, 1977), Complement A_XIV — the Slater determinant, its antisymmetry under electron exchange, and the vanishing of the determinant when two spin-orbitals coincide. https://onlinelibrary.wiley.com/doi/book/10.1002/9783527617203.
[^codata]: Ionization energies of helium: first $24.587\ \text{eV}$, second $54.418\ \text{eV}$, summing to a ground-state binding of $-79.005\ \text{eV}$. NIST Atomic Spectra Database, https://physics.nist.gov/PhysRefData/ASD/. The Rydberg energy $13.606\ \text{eV}$ is the CODATA value, https://physics.nist.gov/cuu/Constants/.
[^gs-helium]: **Griffiths & Schroeter**, _Introduction to Quantum Mechanics_ (3rd ed., Cambridge, 2018), §5.2.1 — the helium ground state: the $Z^2$ zeroth-order energy $-108.8\ \text{eV}$, the first-order repulsion $\tfrac54 Z\times 13.6\ \text{eV}=34\ \text{eV}$, and the variational improvement with $Z_{\text{eff}}=Z-\tfrac{5}{16}$. Cambridge listing: https://www.cambridge.org/highereducation/books/introduction-to-quantum-mechanics/990799CA07A83FC5312402AF6860311E.
[^gs-para]: **Griffiths & Schroeter**, §5.2.1 — orthohelium and parahelium: the direct integral $J$, the exchange integral $K$, the energy $J\pm K$ for singlet/triplet, and the triplet lying $2K$ below the singlet from the purely electrostatic exchange term. See also **Cohen-Tannoudji**, Complement B_XIV.
[^gs-screen]: **Griffiths & Schroeter**, §5.2.2 — atoms and the periodic table: screening and the effective charge, the lifting of the $\ell$-degeneracy by penetration, the aufbau/Madelung filling order, and Hund's rules. Comparison treatment: **Tipler & Llewellyn**, _Modern Physics_ (6th ed.), §7-4–§7-5.