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configuration interaction retaining a reference determinant and its single and double excitations. Conventional matrix-vector iterations have leading $O(o^2{v}^4)$ work for occupied and virtual dimensions $o,v$, hence $O(N^6)$ when both grow with system size. Truncated CI is not generally size extensive.

A correlation between many-electron states classified by total spin and spatial symmetry along a nuclear path. Orbital occupations, spin adaptation and configuration interaction are needed; one-electron energy curves alone are not a state diagram.

In LS coupling, first add the individual orbital angular momenta and spins separately, then add their totals:The Coulomb interaction between electrons must separate different atomic terms more strongly than the spin-orbit interaction mixes them. Then $L,S,J,M_J$ are useful quantum numbers, with $J=|L-S|,\dots ,L+S$. This approximation is usually better for lighter chemical atoms and well-separated atomic terms; strong configuration interaction or strong spin-orbit coupling can spoil it. The term symbol is ${}^{2S+1}{L}_J$.

The general electric-dipole selection rules, for isolated chemical atoms with definite parity and total electronic angular momentum, are a change of parity, $\Delta J=0,\pm 1$ excluding $J=0\leftrightarrow 0$, and $\Delta M_J=0,\pm 1$, with the accessible magnetic sublevels determined by the light's polarization. In pure LS coupling, the additional selection rules are $\Delta S=0$ and $\Delta L=0,\pm 1$, excluding $L=0\leftrightarrow 0$. In a one-electron description the electric dipole operator also requires $\Delta l=\pm 1$ for the active electron. These latter statements can fail as exact labels when atomic terms mix; the general parity and total-angular momentum restrictions still apply. Unresolved hyperfine structure is neglected here.

For strontium, the closed core contributes $L=S=0$. The Pauli exclusion principle makes the two equivalent $5s$ electrons a singlet state. Different radial orbitals permit both a singlet state and a triplet state. Thus the complete lists of atomic fine-structure levels areThe $5s5p$ electron configuration has odd parity; the other two electron configurations have even parity.

The $4f^7$ shell carries a large spin (its ground atomic term under Hund's rules has $L=0,S=7/2$), and the $6s$ electron penetrates the inner shells. Its nonzero probability density at the atomic nucleus supplies a potentially large Fermi contact interaction with the nuclear magnetic dipole moment. The $6s$ shell here is singly occupied, so the obligatory paired-$6s^2$ cancellation is absent. Both stable europium isotopes have nonzero nuclear spin, as the spectrum below also establishes.

The term symbol ${}^8{D}_{11/2}$ specifies $S=7/2$, $L=2$, $J=11/2$. Its large $J$, combined with appreciable nuclear spin, permits a wide range of total angular momentum $F$ and hence a substantial total hyperfine structure span. For $I=5/2$, $F=3,\dots ,8$ and the span is $30|a|$. The penetrating $6s$ contribution and the available range of $F$ explain the large hyperfine structure; the term symbol alone does not determine its constant, which also depends on electronic recoupling, radial wavefunctions and configuration interaction.

Restricted Hartree–Fock incorrectly retains ionic configurations when a homonuclear bond dissociates. As overlap vanishes, its symmetry-preserving occupied orbital approaches $({\varphi }_A+{\varphi }_B)/\sqrt{2}$. The doubly occupied spatial product containsThe first and last terms place both electrons on one separated atom. Their combined probability remains one half, while the correct neutral singlet has one electron on each atom. The resulting spurious Coulomb repulsion raises the restricted dissociation limit. This is missing static electron correlation, not a failure of the Born-Oppenheimer approximation.

A two-configuration expansion using both ${\sigma }_g^2$ and ${\sigma }_u^2$ can cancel the ionic terms and recover the Heitler–London separated-atom singlet. Configuration interaction in the appropriate active space or a multiconfigurational self-consistent treatment supplies this remedy. The Unrestricted Hartree-Fock method can instead localize opposite spins on different atoms and give the correct separated-atom energy, but its single determinant breaks spin symmetry and is not the pure singlet at intermediate separation. The distinction between restricted and symmetry-broken descriptions of stretched hydrogen is treated in the original analysis of Coulson and Fischer.^[ref]

For conventional CISD, the leading matrix-vector contraction scales as $O(o^2{v}^4)$ in occupied and virtual orbital dimensions. If both grow proportionally to a system-size parameter $N$, the familiar estimate is $O(N^6)$ work per iteration; the iteration count and integral-storage strategy add implementation-dependent costs.

Calculated potential energy surfaces can predict equilibrium geometries and harmonic force constants, permitting comparison with diffraction-derived structures, rotational constants and vibrational spectra. Calculating several roots or different charge states can also estimate excitation or ionization energies. Such predictions depend on basis and truncation quality; truncated CI is not generally size extensive and is not automatically reliable for strongly multiconfigurational bond breaking. The quoted polynomial scaling is for CISD, not full configuration interaction, whose determinant space grows combinatorially.