Spectral, Variational, and Optimization Algorithms

Stationary quantum calculations contain several different problems: representing a Hamiltonian, solving its spectrum, optimizing a restricted state, and sometimes making the Hamiltonian self-consistent. This chapter explains how to solve each problem and check the answer. Keeping these steps distinct helps you decide whether a small residual means that the desired physical quantity is accurate.

Enter this chapter

Begin with matrix diagonalization and Rayleigh–Ritz. The former solves a finite matrix; the latter explains how a trial space creates that matrix and its overlap metric. Follow with basis-set convergence before interpreting extra digits as improved physical accuracy.

For a few states in a large matrix, use sparse eigensolvers to select the spectral target, then Lanczos or Davidson methods according to the available operator action and preconditioner. Excited-state targeting covers orthogonality, degeneracy, and tracking through parameter changes.

Choose the numerical question

Question	Method entry	Decisive check
All levels of a moderate Hermitian matrix	Dense diagonalization	Residuals and orthogonality
A few extremal or interior levels	Sparse eigensolvers	Original-problem residuals and target completeness
Best state in a parameterized family	Variational optimization	Gradients, constraints, and ansatz enlargement
Lowest state with a good initial overlap	Ground-state projection	Filter ratio, overlap, and independent cutoff limits
A nonlinear mean-field branch	Self-consistency	Field residual and physical stability
A pulse meeting a state or gate objective	Optimal control	Gradient check and independently refined propagation

Imaginary-time evolution gives a specific exponential projector, including normalization and gap-dependent convergence. Hamiltonian truncation explains how discarded states affect energies and why products of truncated operators require care.

The electronic-structure sequence

Start with basis sets and molecular integrals, which define the represented electronic Hamiltonian. Then solve Hartree–Fock or a density-functional workflow with explicit density, overlap, and energy conventions.

Configuration interaction introduces linear determinant expansions and finite-space exactness. Coupled cluster introduces nonlinear projected equations and a different treatment of excitation products. Neither an accurate eigensolver nor a small amplitude residual removes the chosen physical approximation. Electronic reduced density matrices supplies independent contraction and energy checks.

For molecular motion, construct Born–Oppenheimer surfaces, verify derivatives and branch identity, and then compute vibrational levels. Quantum-chemistry benchmarks provides exact fixtures and a reproducible comparison hierarchy. Pairing problems use BdG numerics, with explicit Nambu redundancy and a distinction between fermionic and bosonic eigenproblems.

What an eigenpair residual tells you

For a normalized approximate eigenvector $x$, define $r=Hx-\theta x$ with $\theta=x^\dagger Hx$. Hermiticity implies

$$\operatorname{dist}(\theta,\operatorname{spec}H)\leq\|r\|.$$

This certifies proximity to the spectrum of the supplied matrix. It does not certify the intended state, the continuum limit, the correlation approximation, or a nonlinear global minimum. A useful result therefore includes target identity, algebraic residual, representation refinement, and the observable actually requested.

As a simple checkpoint, compare a two-level calculation with an exact diagonalization, then enlarge a physical basis while holding the solver tolerance well below the observed changes. The first comparison tests the algorithm; the second tests the represented physics. Near degeneracy, compare eigenspace projectors rather than arbitrary eigenvector phases.

Exercises

A calculation reports an eigenpair residual of $10^{-12}$ while doubling the basis changes the energy by $10^{-4}$. Which change should be investigated next?

Solution

The basis change dominates the observed error scale. Further tightening the eigensolver is unlikely to improve the result meaningfully. Continue basis refinement, check its asymptotic behavior, and converge the target observable separately.

References

• Bai, Z., J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, eds., Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM, 2000, online text — eigenproblem method selection.
• Bartlett, R. J., and M. Musiał, “Coupled-cluster theory in quantum chemistry,” Reviews of Modern Physics 79, 291–352 (2007), doi:10.1103/RevModPhys.79.291 — electronic correlation methods.