9. CI and multireference active spaces
Position in the course: Lesson 9 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
Prerequisites: Determinants, variation and second quantization
1. Linear variation over determinants
Configuration interaction writes \(|\Psi\rangle=\sum_I c_I|\Phi_I\rangle\) in orthonormal determinants made from a chosen orbital basis. Varying \(c^\dagger Hc\) subject to \(c^\dagger c=1\) gives \(Hc=Ec\). Full CI includes all determinants for the chosen electrons and spin-orbital space. It is exact for that finite-basis Hamiltonian, not for an unapproximated physical molecule. Basis incompleteness, relativistic omissions and the Born-Oppenheimer approximation remain.
For \(N\) electrons in \(M\) spin orbitals the unrestricted count is \(\binom{M}{N}\) before symmetry restrictions. This combinatorial growth explains why FCI is a benchmark for small spaces rather than a routine solution for large molecules.
2. Worked avoided crossing
Choose two normalized configurations with diagonal energies \(E_A,E_B\) and coupling \(v\). Their low eigenvalue is
At equal diagonals, the eigenstates are equal-weight combinations and the gap is \(2|v|\). Far from equality, the lower diagonal configuration dominates. A single reference switches character poorly near this region; a two-configuration treatment remains finite. With a vanishing symmetry-allowed coupling, a true crossing can remain. This toy matrix explains near-degeneracy but does not by itself specify a molecular reaction coordinate.
3. Bond dissociation and static correlation
In a minimal hydrogen-like two-center model, a restricted doubly occupied bonding orbital contains both covalent and ionic configurations. At large separation, an appropriate combination with the antibonding double occupation can cancel unwanted ionic character and describe one electron on each center. Near equilibrium a dominant configuration plus many small corrections often suffices; near dissociation several configurations acquire comparable importance. These are useful dynamic/static correlation distinctions rather than sharply separated observables.
Natural orbital occupations are eigenvalues of a one-particle density matrix. For spin-summed spatial orbitals, values range from zero to two. Occupations near one in a pair can indicate a need for multiple configurations, but thresholds depend on the system and state; no single diagnostic decides every case.
4. Why truncated CI is not size consistent
Suppose each isolated fragment needs a reference and one double substitution. Multiplying the two fragment wavefunctions creates a term with a double on each fragment: a global quadruple. A dimer CISD expansion omits it, though independent fragment CISD calculations contain both factors. Thus \(E_{A+B}^{CISD}\) need not equal \(E_A^{CISD}+E_B^{CISD}\) at infinite separation. Variationality and size consistency are different properties.
5. Active spaces and orbital optimization
A complete active space distributes a selected number of electrons among selected active orbitals, while inactive orbitals stay doubly occupied and external orbitals stay empty. CASCI solves the configuration problem with fixed orbitals; CASSCF also optimizes orbitals. Include all essential bonding/antibonding partners and relevant near-degenerate shells. Compare active spaces, inspect occupations and follow orbital character along geometry. A small CAS captures selected static correlation but leaves much external dynamic correlation untreated.
6. Exercises with explained solutions
Count unrestricted determinants for two electrons in four spin orbitals. There are \(\binom42=6\). Fixing spin projection or using spin-adapted combinations reduces the calculation blocks, not the original combinatorial fact.
Does adding determinants raise a lowest variational CI root? For nested spaces and exact diagonalization, no; the old vector remains admissible. This monotonic bound does not repair the missing disconnected terms responsible for size inconsistency.
Why keep an antibonding orbital active even when empty near equilibrium? Its occupation can become essential during bond stretching. An active space chosen solely by initial occupied orbitals may fail along the path.
7. References and study connections
- DePrince group academic HF tutorial
- DePrince group academic tutorial collection
- Psi4 official coupled-cluster documentation
- Helgaker, Jørgensen and Olsen: Molecular Electronic-Structure Theory
The derivations and toy arithmetic are original teaching constructions. No molecular simulation is reported here.
8. Related theory and practice
Quantum mechanics · Density functional theory · Quantum Monte Carlo · Gaussian