Rayleigh-Ritz Method

The Rayleigh-Ritz method is the finite-dimensional form of the variational principle. Choose a trial subspace, represent the Hamiltonian on that subspace, and diagonalize the resulting matrix. The lowest eigenvalue is a variational upper bound to the true ground-state energy.

This is the basic structure behind basis-set quantum mechanics, configuration interaction, oscillator-basis truncations, finite-element discretizations, and many numerical diagonalization methods.

The broader error ledger and method-selection roadmap are developed in Variational and Bound Methods.

When basis functions themselves carry nonlinear widths, exponents, or centers, their outer optimization and the required overlap-matrix derivative belong to Variational Parameters.

Trial Subspaces

Let $\mathcal V_N$ be an $N$-dimensional subspace spanned by trial vectors

$$\mathcal V_N = \operatorname{span} \{ \lvert\phi_1\rangle,\ldots,\lvert\phi_N\rangle \}.$$

A general trial vector in this subspace has the form

$$\lvert\psi\rangle = \sum_{j=1}^N c_j\lvert\phi_j\rangle.$$

Rayleigh-Ritz minimizes the energy quotient over all coefficient vectors $c_j$ in the chosen subspace:

$$E_N = \min_{\psi\in\mathcal V_N,\ \psi\ne0} \frac{\langle\psi|H|\psi\rangle}{\langle\psi|\psi\rangle}.$$

By the variational principle,

$$E_N\ge E_0.$$

Orthonormal Basis

If the basis vectors are orthonormal, define

$$H_{ij} = \langle\phi_i|H|\phi_j\rangle.$$

Then

$$\langle\psi|H|\psi\rangle = \sum_{i,j}c_i^*H_{ij}c_j, \qquad \langle\psi|\psi\rangle = \sum_i |c_i|^2.$$

Minimizing the quotient gives the finite-dimensional eigenvalue problem

$$\sum_{j=1}^N H_{ij}c_j = E c_i, \qquad i=1,\ldots,N.$$

The lowest eigenvalue of this matrix is the best variational energy inside $\mathcal V_N$. The corresponding eigenvector gives the best trial state in that subspace.

Nonorthogonal Basis

Many useful basis functions are not orthogonal. Define both the Hamiltonian matrix and overlap matrix:

$$H_{ij} = \langle\phi_i|H|\phi_j\rangle, \qquad S_{ij} = \langle\phi_i|\phi_j\rangle.$$

The variational condition becomes the generalized eigenvalue problem

$$\sum_{j=1}^N H_{ij}c_j = E \sum_{j=1}^N S_{ij}c_j.$$

Equivalently,

$$Hc=ESc.$$

The overlap matrix must be positive definite on the chosen span. If two basis functions are nearly linearly dependent, $S$ becomes ill-conditioned and the numerical eigenvalues may become unstable.

Why Diagonalization Appears

The quotient

$$R(c) = \frac{c^\dagger Hc}{c^\dagger Sc}$$

is stationary when variations with respect to $c_i^*$ vanish. For the nonorthogonal case,

$$\delta \left( c^\dagger Hc - E c^\dagger Sc \right) = 0$$

gives

$$Hc=ESc.$$

Thus matrix diagonalization is not an extra approximation after choosing the subspace. It is the exact minimization problem inside that subspace.

Convergence with Basis Size

If the subspaces are nested,

$$\mathcal V_1\subset\mathcal V_2\subset\cdots,$$

then the corresponding lowest Ritz values satisfy

$$E_1\ge E_2\ge\cdots\ge E_0.$$

The sequence decreases because each larger subspace contains all previous trial states. Under suitable completeness and domain conditions, the sequence can converge to the exact ground-state energy.

For practical calculations, convergence should be checked by increasing the basis size, changing basis families, or comparing with independent approximations such as perturbation theory or WKB when those are available.

Excited-State Estimates

The Rayleigh-Ritz spectrum also gives approximations to excited states. For a self-adjoint Hamiltonian with discrete low-lying spectrum, the min–max principle implies that the $k$th Ritz value in a sufficiently rich trial subspace is an upper bound to the $k$th exact eigenvalue when states are ordered consistently. The precise theorem, its proof, and its limits at essential spectrum are given in Upper Bounds and the Min–Max Principle.

In practice, excited-state estimates are more delicate than the ground-state estimate. One must preserve symmetry sectors, avoid variational collapse, and monitor convergence of eigenvectors as well as eigenvalues. A computed state with the wrong nodal structure or wrong symmetry may approximate the wrong physical level.

Choosing a Basis

A good basis should make the important physics cheap to represent. Common choices include:

• exact eigenstates of a nearby solvable Hamiltonian,
• harmonic oscillator functions for confined or approximately quadratic problems,
• plane waves for translation-invariant or periodic systems,
• localized finite elements or grid functions for spatially inhomogeneous problems,
• symmetry-adapted states for angular momentum, parity, or internal symmetries.

The best choice depends on the Hamiltonian, boundary conditions, target energy range, and required observables. The anharmonic oscillator is a standard test case because a harmonic oscillator basis gives a transparent comparison between perturbative and variational calculations.

Matrix Elements

The method is only as reliable as the matrix elements. For a coordinate-space Hamiltonian

$$H = - \frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V(x),$$

one computes

$$H_{ij} = \int dx\, \phi_i^*(x) \left[ - \frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V(x) \right] \phi_j(x).$$

If the basis functions obey the boundary conditions, integration by parts often gives a symmetric kinetic-energy matrix:

$$T_{ij} = \frac{\hbar^2}{2m} \int dx\, \frac{d\phi_i^*}{dx} \frac{d\phi_j}{dx},$$

with boundary terms vanishing. This form is often numerically better because it makes positivity of the kinetic energy explicit.

Common Mistakes

• Forgetting the overlap matrix for a nonorthogonal basis.
• Comparing eigenvalues from unrelated nonnested basis sets as if they must be monotone.
• Using a basis that violates boundary conditions or symmetry constraints.
• Trusting high excited states near the truncation edge. They are usually dominated by cutoff artifacts.
• Treating diagonalization error and basis-truncation error as the same thing. A matrix can be diagonalized accurately while still representing a poor trial subspace.

Exercises

1. In an orthonormal two-state basis, let

$$H = \begin{pmatrix} A & B \\ B^* & C \end{pmatrix}.$$

Find the two Ritz values.

Solution

The characteristic equation is

$$(A-E)(C-E)-|B|^2=0.$$

Therefore

$$E_\pm = \frac{A+C}{2} \pm \sqrt{ \left(\frac{A-C}{2}\right)^2 + |B|^2 }.$$

The lower value $E_-$ is the best variational energy inside the two-dimensional subspace.

2. Show that enlarging a trial subspace cannot increase the lowest Ritz value.

Solution

If $\mathcal V_N\subset\mathcal V_{N+1}$, then the minimization over $\mathcal V_{N+1}$ includes every state that was available in $\mathcal V_N$. A minimum over a larger set cannot be larger:

$$\min_{\psi\in\mathcal V_{N+1}}E[\psi] \le \min_{\psi\in\mathcal V_N}E[\psi].$$

Hence $E_{N+1}\le E_N$.

3. Explain why a nonorthogonal basis with nearly dependent vectors can cause numerical trouble even if the variational principle remains mathematically true.

Solution

Nearly dependent vectors make the overlap matrix $S$ nearly singular. Then small errors in matrix elements, arithmetic, or basis construction can produce large changes in the generalized eigenvectors and sometimes spurious eigenvalues. The variational statement assumes exact matrix elements in the exact trial subspace; numerical computation also needs a well-conditioned representation of that subspace.

Cross-Links

• VQE applies the Rayleigh–Ritz bound to parameterized quantum states and sampled Hamiltonian measurements.

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• B. N. Parlett, The Symmetric Eigenvalue Problem, SIAM, 1998.
• G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins University Press, 2013.