Rayleigh–Ritz Variational Method

The Rayleigh–Ritz method turns a trial subspace of wavefunctions into a finite eigenproblem by making the energy stationary within that space. The variational principle gives an upper bound on the ground energy. Here we explain how to retain that bound when assembling and solving matrices, especially in a nonorthogonal basis.

Helpful background. Matrix diagonalization supplies finite-matrix residual and orthogonality tests.

Projecting the energy functional

Let $\{\phi_i\}_{i=1}^n$ be linearly independent functions in the quadratic-form domain of a self-adjoint Hamiltonian bounded below. Write $\psi=\sum_i c_i\phi_i$ and assemble

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

When only form-domain regularity is available, the second expression means the associated quadratic form, often evaluated after integration by parts with the appropriate boundary conditions. Then

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

Varying $c^\dagger$ under $c^\dagger Sc=1$ yields $Hc=ESc$. A coefficient vector of Euclidean norm one need not represent a normalized state. All overlaps, observables, and orthogonality constraints must respect $S$.

For a positive-definite overlap, factor $S=LL^\dagger$ and solve

$$\widetilde H y=Ey,\qquad \widetilde H=L^{-1}HL^{-\dagger},\qquad c=L^{-\dagger}y.$$

Use triangular solves rather than explicit inverses. Alternatively diagonalize $S=U\operatorname{diag}(s_i)U^\dagger$ and orthogonalize with retained columns $U_i/\sqrt{s_i}$.

Linear dependence and variational bounds

A very small $s_i$ signals a combination of basis functions with tiny norm. Scaling by $1/\sqrt{s_i}$ amplifies assembly noise. Removing directions below a stated threshold can improve stability, but it changes the trial space; vary that threshold as part of the calculation. A negative overlap eigenvalue larger than roundoff is an assembly or quadrature failure for an ordinary positive Hilbert-space metric.

For conforming nested spaces $V_n\subset V_{n+1}$, with the same Hamiltonian and exact matrix elements, the ordered Ritz values obey

$$E_k\leq E_k^{(n+1)}\leq E_k^{(n)}$$

for discrete eigenvalues covered by the min–max principle. Non-nested bases need not give a monotone sequence. Approximate quadrature, pseudopotential changes, or inconsistent boundary conditions can invalidate the claimed upper bound. A small algebraic residual only proves that the projected matrix problem was solved accurately; the residual outside the trial space may remain large.

A Gaussian oscillator calculation

In dimensionless units take $H=(-d^2/dx^2+x^2)/2$ and the normalized Gaussian

$$\phi_\alpha(x)=(\alpha/\pi)^{1/4}e^{-\alpha x^2/2}, \qquad \alpha>0.$$

Using $\langle x^2\rangle=1/(2\alpha)$ and $\langle p^2\rangle=\alpha/2$ gives

$$E(\alpha)=\frac14(\alpha+\alpha^{-1})\geq\frac12.$$

The minimum at $\alpha=1$ reaches the exact ground energy. With two widths, the off-diagonal overlap is

$$S_{\alpha\beta} =\left(\frac{2\sqrt{\alpha\beta}}{\alpha+\beta}\right)^{1/2}.$$

As $\beta\to\alpha$, the smaller eigenvalue of the two-function overlap tends to zero. More functions do not automatically mean more independent physical information. This example simultaneously checks variational minimization, overlap normalization, and linear-dependence handling.

Exercises

For two normalized functions with real overlap $s$, find the overlap eigenvalues and describe the limit $s\to1$.

Solution

$S=\begin{pmatrix}1&s\\s&1\end{pmatrix}$ has eigenvalues $1+s$ and $1-s$. The antisymmetric coefficient combination has norm proportional to $\sqrt{1-s}$. At $s=1$ the functions are identical and the nominal two-dimensional basis represents only one independent direction.

References

• Anderson, E., et al., LAPACK Users’ Guide, 3rd ed., SIAM, 1999, online guide — generalized Hermitian definite eigensystems.
• 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 — Rayleigh–Ritz projection and eigenvalue approximation.