Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 2A / Section A / 19

configuration interaction with singles and doubles (CISD) uses a linear combination of the reference Slater determinant and singly and doubly excited determinants. Its many-electron trial state is therefore not restricted to one determinant; the orbitals may be held fixed while the configuration coefficients are varied.

Kohn-Sham equations in density functional theory use an auxiliary one-electron Hamiltonian whose effective potential depends on the density. Its orbitals construct a density representing an interacting electronic system; this is distinct from explicitly diagonalizing a large many-electron determinant Hamiltonian.^[ref]

molecular mechanics evaluates a parametrized potential of atomic positions, with terms such as bond stretching, angle bending, torsion and nonbonded interactions. electrons are represented implicitly through the fitted force field, rather than by electronic orbitals, a density or an electronic wavefunction. This example is different from both preceding electronic methods.

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.