Electron Configurations

An electron configuration records how the electrons of an atom or ion are distributed among a chosen set of one-electron orbitals or subshells. A familiar example is

$$1s^2\,2s^2\,2p^2.$$

This notation is powerful because it compresses electron counting, shell structure, parity, and the starting point for angular-momentum coupling into one line. It is also easy to overread. A configuration is generally not a complete wavefunction, not an observable partition into named electrons, and not necessarily an exact label of an energy eigenstate.

The safest interpretation is:

A configuration is an occupation pattern relative to a declared one-electron basis and coupling convention.

That basis may come from a central-field model, Hartree–Fock calculation, relativistic self-consistent field, or another orbital construction. Residual interactions can mix several configurations into one physical level, and rotating the orbitals can change the reported configuration weights without changing the many-electron state.

Required background. The multi-electron introduction motivates interacting atoms; central fields define the subshell basis, orbital interpretations distinguish basis labels from observables, and Pauli exclusion limits each spin-orbital occupation.

Hund’s rules rank the allowed terms after a configuration has been chosen. The broader energetic trends across elements are discussed in the periodic table.

Species, Charge, and Electron Count

Begin with the number of electrons. For nuclear charge $Z$ and ionic charge $Qe$, where $Q>0$ denotes a positive ion,

$$N=Z-Q.$$

Every proposed configuration must contain exactly $N$ electrons. In atomic spectroscopy, a Roman numeral convention is also common:

Spectroscopic label	Charge	Example
I	$Q=0$	C I is neutral carbon
II	$Q=+1$	C II is singly ionized carbon
III	$Q=+2$	C III is doubly ionized carbon

Thus the Roman numeral is the ion charge plus one. It does not count valence electrons or specify an excitation level.

Electron counting is elementary but diagnostic. A configuration with the wrong total occupation cannot be rescued by a term label, coupling scheme, or correlation correction.

Shells, Subshells, and Spin-Orbitals

In a nonrelativistic central field, a spatial orbital can be labeled

$$\phi_{n\ell m_\ell}(\mathbf r),$$

with principal or radial label $n$, orbital angular momentum $\ell$, and projection $m_\ell=-\ell,\ldots,\ell$. Including a spin basis $\omega_{m_s}(\sigma)$ gives a spin-orbital

$$\begin{aligned} \chi_{n\ell m_\ell m_s}(x) &=\phi_{n\ell m_\ell}(\mathbf r) \times\omega_{m_s}(\sigma), \\ m_s&=\pm\frac12. \end{aligned}$$

The coordinate $x=(\mathbf r,\sigma)$ includes space and spin. A subshell groups all spin-orbitals with the same $n$ and $\ell$. It contains

$$g_\ell=2(2\ell+1)$$

spin-orbitals, so its occupation $q$ obeys

$$0\leq q\leq 2(2\ell+1).$$

The familiar capacities follow immediately:

Subshell	$\ell$	Spatial orbitals	Spin-orbitals	Maximum occupation
$s$	0	1	2	2
$p$	1	3	6	6
$d$	2	5	10	10
$f$	3	7	14	14

These are exact dimension counts for the declared one-electron basis. They do not determine the energetic order of the subshells.

Shell language is context dependent

For the Coulomb one-electron problem, all orbitals with the same $n$ share an energy before fine structure and radiative corrections. It is then natural to call the complete set of $\ell=0,\ldots,n-1$ states the $n$th shell, whose capacity is

$$\sum_{\ell=0}^{n-1}2(2\ell+1)=2n^2.$$

In a multi-electron central field, screening and penetration lift the hydrogenic degeneracy between different $\ell$ values. Atomic spectroscopy therefore often uses shell loosely for groups such as $n=3$, while actual calculations organize orbitals by subshell or relativistic spinor. The word should not be used to imply an exact degeneracy that the Hamiltonian does not possess.

Relativistic subshells

When spin–orbit structure is built into the one-electron basis, a subshell is commonly labeled $n\ell_j$ or by the Dirac angular quantum number $\kappa$. A fixed-$j$ subshell contains

$$2j+1$$

magnetic spinors. Thus a nonrelativistic $p$ subshell separates into $p_{1/2}$ and $p_{3/2}$ subshells of capacities $2$ and $4$. Configuration labels must therefore be read together with the coupling convention used to define them.

jj Coupling shows how these occupations are antisymmetrized, coupled to total $J$, and tested as approximate labels of real atomic levels.

Occupation Notation

For subshells indexed by $a$, a nonrelativistic configuration may be written schematically as

$$\mathcal C =\prod_a(n_a\ell_a)^{q_a}, \qquad \sum_a q_a=N.$$

The product sign is symbolic: it lists occupations and is not multiplication of wavefunctions. In a chosen spin-orbital basis, the same subshell occupation is

$$q_{n\ell} =\sum_{m_\ell=-\ell}^{\ell} \sum_{m_s=\pm1/2} n_{n\ell m_\ell m_s},$$

where every fermionic mode has $n_{n\ell m_\ell m_s}=0$ or $1$. The subshell exponent may exceed one because it sums distinct complete spin-orbitals.

Full and compact forms

Neutral carbon has six electrons. Its commonly used ground-configuration label is

$$1s^2\,2s^2\,2p^2 =[\mathrm{He}]\,2s^2\,2p^2.$$

The bracketed helium symbol abbreviates the filled $1s^2$ core. It does not insert a literal helium atom inside carbon. Similarly,

$$[\mathrm{Ar}] =1s^2\,2s^2\,2p^6\,3s^2\,3p^6.$$

An exponent of one is often omitted, so $[\mathrm{Ar}]\,3d^5\,4s$ means $[\mathrm{Ar}]\,3d^5\,4s^1$. The order in which subshells are printed is conventional. It may emphasize principal shell, spectroscopic ancestry, or an Aufbau mnemonic; it is not by itself a measured ordering of orbital energies.

What a configuration fixes

A declared configuration fixes:

• the total number of electrons;
• the occupation of each listed subshell;
• the maximum determinant space compatible with those occupations;
• the configuration parity in a parity-adapted central-field basis.

Its parity is

$$\pi_{\mathcal C} =(-1)^{\sum_a q_a\ell_a}.$$

Closed subshells contribute even parity, so only open subshells need to be inspected in most practical examples.

What a configuration omits

A configuration does not generally fix:

• which $m_\ell$ and $m_s$ spin-orbitals are occupied in a determinant;
• how determinants are combined to enforce total $L$, $S$, $J$, or parity;
• which term or fine-structure level is intended;
• the radial form of the orbitals;
• the coefficients of other configurations in the physical state.

For example, $2p^2$ is not one state. The $p$ subshell contains six spin-orbitals, and choosing two gives

$$\binom{6}{2}=15$$

determinants before symmetry adaptation. Their decomposition into allowed terms is developed in Atomic Term Symbols.

More generally, if occupations in distinct subshells are $q_a$, the number of spin-orbital determinants associated with the configuration is

$$D_{\mathcal C} =\prod_a\binom{g_a}{q_a}.$$

This count is a basis dimension, not an energy degeneracy after all interactions are included.

Determinants, Configuration-State Functions, and Levels

Three layers are often compressed into one informal phrase.

Slater Determinants in Atoms develops the transformation among these layers in detail, including determinant phases and symmetry adaptation.

Determinant

A Slater determinant specifies occupation of complete spin-orbitals. In creation-operator notation,

$$|D_I\rangle =a_{i_1}^{\dagger}a_{i_2}^{\dagger}\cdots a_{i_N}^{\dagger}|0\rangle,$$

with a fixed ordering convention. It is antisymmetric by construction. An open-subshell configuration usually generates many such determinants.

Configuration-state function

A configuration-state function, or CSF, is a symmetry-adapted linear combination of determinants:

$$|\Phi_{I\Gamma}\rangle =\sum_D d_{ID}^{(\Gamma)}|D\rangle.$$

The label $\Gamma$ collects exact symmetries imposed in the calculation. For a field-free rotationally invariant atom it commonly includes $J$ and parity. In a nonrelativistic electrostatic calculation it may also include separately conserved $L$ and $S$.

Atomic level

An approximate eigenstate in the retained CSF space is

$$|\Psi_{k\Gamma}\rangle =\sum_I c_{Ik}^{(\Gamma)} |\Phi_{I\Gamma}\rangle.$$

Several CSFs may come from one configuration, and several configurations may contribute CSFs with the same $\Gamma$. A level label such as a principal configuration and term identifies the dominant ancestry of this eigenvector; it does not replace the eigenvector.

The Aufbau Principle as a Conditional Result

The German word Aufbau means building up. In its careful form, the principle says that a reference configuration can be constructed by occupying low-energy one-electron states subject to Pauli exclusion. This statement is exact for a fixed noninteracting reference Hamiltonian and only heuristic for an interacting self-consistent atom.

Consider

$$H_0=\sum_p\varepsilon_p a_p^\dagger a_p.$$

An occupation-number eigenstate has energy

$$\begin{aligned} E_0(\{n_p\}) &=\sum_p\varepsilon_p n_p,\\ n_p&\in\{0,1\},\\ \sum_p n_p&=N. \end{aligned}$$

Suppose $\varepsilon_i<\varepsilon_j$, mode $j$ is occupied, and mode $i$ is empty. Exchanging their occupations changes the energy by

$$\Delta E_0 =\varepsilon_i-\varepsilon_j<0.$$

Therefore the ground state of this fixed $H_0$ occupies the $N$ lowest spin-orbitals. That exchange argument is the precise theorem behind the elementary filling rule.

Why the theorem does not settle real atoms

The electronic Hamiltonian also contains two-electron interactions:

$$H =\sum_{pq}h_{pq}a_p^\dagger a_q +\frac12 \sum_{pqrs}v_{pqrs} a_p^\dagger a_q^\dagger a_s a_r.$$

In a self-consistent model, even the orbitals and their $\varepsilon_p$ values depend on which states are occupied. The pair interaction adds direct and exchange energies that are not a sum of fixed one-electron numbers, and the residual Hamiltonian mixes configurations. Relativistic shifts, core relaxation, and correlation add further state dependence.

Consequently, these are different questions:

1. Which orbital is lower in a chosen one-electron reference potential?
2. Which configuration minimizes a self-consistent total energy for a specified atom or ion?
3. Which electron-removal channel gives the lowest ionization threshold?
4. Which configuration is the largest component of the exact or calculated eigenstate?

They need not have the same answer.

The Madelung Ordering and Its Limits

The common $n+\ell$ or Madelung rule orders subshells by increasing $n+\ell$, breaking ties by lower $n$. It gives the mnemonic sequence beginning

$$1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s,\ldots$$

For many neutral ground configurations this is an efficient memory aid. Its qualitative basis is that penetration and screening split subshells that would be degenerate in a hydrogenic model. It is not an eigenvalue theorem for the many-electron Coulomb Hamiltonian.

Several limitations matter:

• The order depends on nuclear charge, ionic charge, and occupation.
• Nearby $ns$ and $(n-1)d$ subshells can reorder as electrons are added or removed.
• Direct Coulomb, exchange, and radial-relaxation energies belong to the total configuration, not to a universal list of orbital boxes.
• Spin–orbit and other relativistic effects become increasingly important for heavy atoms.
• Near-degenerate configurations may mix so strongly that one filling string is an incomplete description.

Neutral chromium and copper are familiar warnings. Their commonly assigned leading ground configurations are

$$\begin{aligned} \mathrm{Cr\ I}:&\quad [\mathrm{Ar}]\,3d^5\,4s,\\ \mathrm{Cu\ I}:&\quad [\mathrm{Ar}]\,3d^{10}\,4s, \end{aligned}$$

rather than the simplest uncorrected Madelung predictions. Saying only that half-filled or filled subshells are “especially stable” is not a calculation. The energy balance includes all direct, exchange, relaxation, relativistic, and correlation contributions for the species under study.

Ionization supplies another warning. In first-row transition atoms, $4s$ can be occupied before substantial $3d$ filling in the neutral sequence, yet $4s$ occupation is commonly lost first upon ionization. The reference potential and orbital ordering have changed. “Last filled” is not a general theorem for “first removed.”

For actual assignments, use evaluated level data rather than repairing the mnemonic with a longer exception list. The NIST Atomic Spectra Database separates ground shells, principal configurations, levels, energies, and uncertainties for atoms and ions.

Closed Subshells and Closed-Shell States

A nonrelativistic subshell is closed when

$$q=g_\ell=2(2\ell+1).$$

Every $m_\ell$ value then occurs with both spin projections. The filled determinant is invariant under rotations within the complete subshell, so it carries

$$L=0, \qquad S=0, \qquad J=0.$$

Its parity is even because

$$(-1)^{q\ell} =(-1)^{2(2\ell+1)\ell}=+1.$$

A configuration composed entirely of closed nonrelativistic subshells therefore supports a $^1S_0$ state. In a relativistic description, a completely filled $j$ subshell likewise contributes total $J=0$.

Closed subshell is not synonymous with noble gas

The phrases closed subshell, closed shell, and noble-gas core answer different questions.

• $2p^6$ is a closed subshell.
• $[\mathrm{Ne}]$ abbreviates a particular collection of closed subshells.
• $[\mathrm{Ar}]\,4s^2$ contains a closed $4s$ subshell but is not a noble-gas configuration.
• A “closed-shell core” is a modeling partition between orbitals treated as core and those treated as active or valence.

Closed angular-momentum coupling does not make a core physically inert. Core orbitals can relax between states, polarize in the field of a valence electron, contribute to exchange, and participate through virtual or real core excitations.

Valence Electrons and Active Spaces

In the simplest atomic language, valence electrons occupy subshells outside a compact closed core. Alkali atoms illustrate the idea:

$$[\text{closed core}]\,ns.$$

The low-lying spectrum is then often dominated by motion of one valence electron, as developed in Alkali Atoms. The electron is not distinguishable from the core electrons, and a proper state remains antisymmetric under exchange of all electrons.

For open-$d$ and open-$f$ atoms, “valence” is more model dependent. Several near-degenerate subshells may have to be included in the active space even when some lie inside the outermost principal shell. A useful computational partition is

$$\mathcal H_{\mathrm{orb}} =\mathcal H_{\mathrm{core}} \oplus\mathcal H_{\mathrm{active}} \oplus\mathcal H_{\mathrm{virtual}}.$$

Here:

• core orbitals are constrained to remain occupied in the chosen model;
• active orbitals are allowed to change occupation among retained configurations;
• virtual orbitals are unoccupied in the reference but available for excitations.

This is a truncation strategy, not a fundamental division of the atom. A frozen-core calculation should state which orbitals were frozen and test whether core polarization or core–valence correlation matters for the observable of interest.

Hole notation

An almost filled subshell can be organized in terms of holes. If $g_\ell$ is the capacity, $(n\ell)^{g_\ell-1}$ contains one hole relative to the closed subshell. Hole language simplifies angular-momentum counting because the missing state carries a complementary angular structure. It does not turn the many-electron atom into a literal positive particle moving independently of the others.

Configurations Versus Exact Eigenstates

Let $\hat N_a$ be the occupation operator for subshell $a$:

$$\hat N_a =\sum_{p\in a}a_p^\dagger a_p.$$

A configuration basis state is an eigenstate of all chosen $\hat N_a$. The full atomic Hamiltonian generally does not commute with each of these basis-dependent operators:

$$[H,\hat N_a]\neq0.$$

Off-diagonal one- and two-electron matrix elements can transfer occupation between subshells while preserving exact symmetries and total electron number. Configuration is therefore usually not a conserved quantum number.

By contrast, an isolated rotationally invariant atom can have exact labels such as total $J$ and parity $\pi$. For a spin-independent nonrelativistic Coulomb Hamiltonian, $L$ and $S$ are separately conserved as well. Spin–orbit interactions generally preserve $J$ and parity while making $L$ and $S$ approximate. External fields can reduce the symmetry further.

The distinction can be summarized as follows:

Label	Origin	Usually exact for an isolated atom?
electron number $N$	conserved charge	yes
total $J$ and parity	rotational and inversion symmetry	yes, for the usual field-free Hamiltonian
$L$ and $S$	nonrelativistic electrostatic symmetry	approximate once spin-dependent terms matter
configuration $\mathcal C$	chosen orbital occupations	generally no
principal configuration	largest component in a stated analysis	assignment, not a conserved quantity

A principal label is not a basis restriction

The exact helium ground state is conventionally labeled $1s^2$, but it is not confined to the single paired determinant made from one fixed $1s$ orbital. Its radial and angular correlation requires other configurations in a fixed orbital expansion. Conversely, an open-shell CSF can already combine several determinants to enforce total spin and angular momentum. Neither the printed principal label nor the number of CSFs alone establishes a single-determinant approximation.

Configuration Mixing

Choose orthonormal CSFs $|\Phi_{I\Gamma}\rangle$ with the same exact symmetry $\Gamma$. The Hamiltonian matrix is

$$H_{IJ}^{(\Gamma)} =\langle\Phi_{I\Gamma}|H|\Phi_{J\Gamma}\rangle.$$

If an off-diagonal element is nonzero, diagonalization mixes the two basis functions. Exact symmetry blocks the matrix: states of different conserved $J$, parity, or other exact labels cannot mix under that Hamiltonian.

Two-configuration model

For two basis states, choose phases so that the coupling $V$ is real and nonnegative:

$$\begin{gathered} H= \begin{pmatrix} E_A & V\\ V & E_B \end{pmatrix}, \\ \Delta=E_B-E_A>0. \end{gathered}$$

The eigenvalues are

$$E_{\pm} =\frac{E_A+E_B}{2} \pm \sqrt{ \left(\frac{\Delta}{2}\right)^2+V^2 }.$$

Writing the lower state as

$$|\Psi_-\rangle =\cos\theta\,|A\rangle -\sin\theta\,|B\rangle,$$

one may choose $0\leq\theta<\pi/4$ with

$$\tan(2\theta)=\frac{2V}{\Delta}.$$

When $|V|\ll\Delta$, the admixture amplitude is of order $V/\Delta$ and its weight is of order $(V/\Delta)^2$. At degeneracy, $\Delta=0$, the eigenstates are equal mixtures and the splitting is $2|V|$. This is why small interactions can produce large mixing near an accidental or systematic near-degeneracy.

Level repulsion and avoided crossings

If an external parameter changes $E_A-E_B$ while the two states retain the same exact symmetry and $V\neq0$, the eigenvalues avoid crossing. For a parameter-independent $V$, their minimum separation is $2|V|$, attained at $E_A=E_B$. If the coupling itself varies, the minimum need not occur there. A true crossing can remain when symmetry forces $V=0$.

Avoided crossings are not merely graphical curiosities. Across the crossing, the dominant configuration character transfers from one eigenvalue branch to the other. Following only the energy order can therefore give a different state label from following the eigenvector continuously.

Transition-amplitude borrowing

Mixing also changes observables. For a transition operator $T$ and final state $|f\rangle$,

$$\langle f|T|\Psi_-\rangle ={}\cos\theta\,\langle f|T|A\rangle -\sin\theta\,\langle f|T|B\rangle.$$

A nominally weak configuration can borrow transition amplitude from an allowed component. Conversely, two contributions can cancel. Energies alone may look well converged while branching ratios, lifetimes, hyperfine constants, or polarizabilities remain sensitive to small admixtures.

Configuration Interaction and Correlation

Configuration interaction, or CI, diagonalizes the Hamiltonian in a selected determinant or CSF space. Schematically,

$$\mathbf H\mathbf c_k =E_k\mathbf c_k.$$

Within that finite space, diagonalization is exact. The physical approximation lies in the orbital basis, Hamiltonian, frozen-core choice, and retained configuration space. Convergence should be tested by enlarging those ingredients in a controlled way.

Configuration mixing and electron correlation overlap but are not synonyms:

• Configuration mixing describes nonzero coefficients of several configuration basis functions.
• Near-degeneracy or static correlation requires several important reference configurations.
• Dynamic correlation is often represented by many individually small excitations that improve short-range electron avoidance.
• Hartree–Fock exchange is already present in one optimized determinant and should not be relabeled as post-Hartree–Fock correlation.

A different orbital basis can compress or spread the expansion. Full CI includes every determinant in a fixed finite one-electron space. Rotating orbitals within that space preserves its many-electron span and predicted state, although expansion coefficients change. A truncated CI space need not be invariant under the same rotation.

Configuration weights

For orthonormal CSFs grouped by configuration $\mathcal C$, one may report

$$w_{\mathcal C} =\sum_{I\in\mathcal C}|c_I|^2, \qquad \sum_{\mathcal C}w_{\mathcal C}=1$$

within the normalized retained expansion. This is the expectation of the projector onto the declared configuration subspace. The orbital definition and retained space must therefore accompany it.

It is a probability for that specified projector, not a basis-independent intrinsic property of the atom. Changing the orbital subspaces changes the question being asked.

Reading Evaluated Spectroscopic Data

Configurations are inferred by combining theory and experiment. Energy positions, total angular momentum, parity, Landé factors, transition strengths, isotope shifts, hyperfine patterns, and systematic trends across an isoelectronic sequence all help constrain an assignment. No detector directly returns a complete occupation string.

The NIST Atomic Spectra Database makes this distinction explicit. Its ground-configuration field identifies the largest component in a calculated eigenvector, and its documentation warns that assignments can be uncertain for complex heavy atoms and ions. Level searches can also report leading percentages when available.

In complex spectra, compare the assignment with energies, magnetic factors, and transition strengths; a principal configuration is an interpretation of an eigenvector, not a direct detector readout. Bracket notation also only abbreviates a core: freezing that core is a separate computational approximation.

Exercises

Exercise 1: Count and classify a configuration

Consider the configuration $[\mathrm{Ne}]\,3s^2\,3p^4$ in a nonrelativistic central-field basis.

1. Find the total electron number.
2. Identify the open subshell.
3. Find the configuration parity.
4. Count the spin-orbital determinants associated with the open subshell.
5. Explain why the answers do not determine a unique term.

Solution

The neon core contains ten electrons, and the displayed occupations add six, so

$$N=10+2+4=16.$$

The $3s$ subshell has capacity two and is closed. The $3p$ subshell has capacity six and is open. Its parity contribution is

$$(-1)^{q\ell} =(-1)^{4\cdot1}=+1,$$

while all closed subshells also contribute even parity. The number of determinants is

$$\binom{6}{4}=15.$$

Those determinants have different $M_L$ and $M_S$ values and combine into several symmetry-adapted terms. Occupation alone does not specify which combination or energy level is intended.

Exercise 2: The fixed-Hamiltonian Aufbau theorem

Let $H_0=\sum_p\varepsilon_p a_p^\dagger a_p$ have nondegenerate one-electron energies in increasing order. Prove that its $N$-fermion ground state occupies the $N$ lowest modes. Name two assumptions that fail when this result is promoted to an exact rule for a real atom.

Solution

Suppose a candidate ground state leaves a lower mode $i$ empty while occupying a higher mode $j$, with $\varepsilon_i<\varepsilon_j$. Swapping the two occupations preserves particle number and Pauli exclusion but changes the energy by

$$\Delta E_0=\varepsilon_i-\varepsilon_j<0.$$

The candidate was therefore not a ground state. Repeating this exchange removes every inversion, leaving precisely the $N$ lowest modes occupied.

For a real atom, the Hamiltonian is not a fixed sum of independent one-electron number operators: electron–electron interactions add direct, exchange, and off-diagonal matrix elements. In a self-consistent description the orbitals and their energies also depend on the occupations. Either failure is enough to invalidate a universal filling order.

Exercise 3: Closed-subshell checks

For a filled $(n\ell)^{2(2\ell+1)}$ subshell, verify that the sums of $m_\ell$ and $m_s$ vanish. Explain why complete filling gives $L=S=0$, and show that the parity is even.

Solution

Each $m_\ell=-\ell,\ldots,\ell$ occurs once with $m_s=+1/2$ and once with $m_s=-1/2$. Hence

$$\begin{aligned} M_L &=2\sum_{m_\ell=-\ell}^{\ell}m_\ell\\ &=0,\\ M_S &=(2\ell+1) \left(\frac12-\frac12\right)\\ &=0. \end{aligned}$$

Vanishing projections alone would not prove $L=S=0$ for an arbitrary state. Here the determinant fills the complete orbital and spin multiplets and is invariant, up to its overall phase, under their rotations. It is therefore a scalar with $L=S=0$. Its parity is

$$(-1)^{2(2\ell+1)\ell}=+1.$$

Exercise 4: Quantify two-configuration mixing

Take $E_A=0$, $E_B=0.20\ \mathrm{eV}$, and $V=0.05\ \mathrm{eV}$ in the two-configuration Hamiltonian. Find the eigenvalues and the weight of $|B\rangle$ in the lower eigenstate.

Solution

Here $\Delta=0.20\ \mathrm{eV}$. The eigenvalues are

$$\begin{aligned} E_{\pm} &=0.10\ \mathrm{eV} \pm \sqrt{ (0.10\ \mathrm{eV})^2 +(0.05\ \mathrm{eV})^2 } \\ &=0.10\ \mathrm{eV} \pm0.1118\ \mathrm{eV}. \end{aligned}$$

Thus

$$\begin{gathered} E_-\simeq-0.0118\ \mathrm{eV}, \\ E_+\simeq0.2118\ \mathrm{eV}. \end{gathered}$$

The mixing angle obeys

$$\tan(2\theta)=\frac{0.10}{0.20}=0.5,$$

so $\theta\simeq0.2318$ radians. The $|B\rangle$ weight in $|\Psi_-\rangle$ is

$$\sin^2\theta\simeq0.0528.$$

A coupling one quarter of the unperturbed separation already produces about a $5.3\%$ secondary weight and shifts both energies outward.

Exercise 5: Show that configuration expansions depend on orbitals

Let $a$ and $b$ be orthonormal spatial orbitals, and define

$$c=\frac{a+b}{\sqrt2}, \qquad d=\frac{a-b}{\sqrt2}.$$

Write the paired determinant $|a^2\rangle=a_\alpha^\dagger a_\beta^\dagger|0\rangle$ in the $c,d$ basis. Use the normalized open-shell singlet

$$|cd;S=0\rangle =\frac{1}{\sqrt2} \left( c_\alpha^\dagger d_\beta^\dagger -c_\beta^\dagger d_\alpha^\dagger \right)|0\rangle.$$

What does the result show?

Solution

The inverse transformation is $a=(c+d)/\sqrt2$. Expanding the pair creation operator and putting creation operators in a consistent order gives

$$|a^2\rangle =\frac12|c^2\rangle +\frac{1}{\sqrt2}|cd;S=0\rangle +\frac12|d^2\rangle.$$

The squared coefficients add to one:

$$\frac14+\frac12+\frac14=1.$$

A state represented by one doubly occupied configuration in the $a,b$ basis requires three configuration components in the rotated $c,d$ basis. The physical state did not change; only its orbital representation did. This is why configuration counts and weights require a declared basis.

References

• B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules, 2nd ed., Pearson, 2003.
• E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, Cambridge University Press, 1935.
• R. D. Cowan, The Theory of Atomic Structure and Spectra, University of California Press, 1981.
• C. Froese Fischer, T. Brage, and P. Jönsson, Computational Atomic Structure: An MCHF Approach, Institute of Physics Publishing, 1997.
• I. P. Grant, Relativistic Quantum Theory of Atoms and Molecules: Theory and Computation, Springer, 2007.
• International Union of Pure and Applied Chemistry, “Configuration”, Compendium of Chemical Terminology, 5th ed., online version 5.0.0, 2025.
• W. R. Johnson, Atomic Structure Theory: Lectures on Atomic Physics, Springer, 2007.
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• W. C. Martin and W. L. Wiese, “Atomic Spectroscopy: An Introduction”, in G. W. F. Drake, ed., Atomic, Molecular, and Optical Physics Handbook, AIP Press, 1996; NIST online revision.
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