Rayleigh-Ritz method

In quantum mechanics, the Rayleigh-Ritz method, also known as the linear variation method, is a method to obtain (approximate) solutions of the time-independent Schrödinger equation. In numerical analysis, it is a method of solving differential equations with boundary conditions and eigenvalue problems. In the latter field it is sometimes called the Rayleigh-Ritz-Galerkin procedure.

The method

We will describe the method from a quantum mechanical point of view and consider as the basic problem to be solved the eigenvalue problem of a Hamilton operator H, which is Hermitian and contains second derivatives. The results can be transferred fairly easily to similar non-quantum mechanical eigenvalue problems and certain differential equations. The essential point of the Rayleigh-Ritz method is the rephrasing of the original problem into another, equivalent, problem, which is the determination of stationary point(s) (usually the minimum) of a certain functional.

So, our aim is to find an approximate method for solving the eigenvalue equation

$H\Phi =E\Phi \,.$

We will show that this equation can be replaced by the problem of finding a stationary point of the expectation value of H with respect to Φ

$E[\Phi ]\equiv {\frac {\langle \Phi |H|\Phi \rangle }{\langle \Phi |\Phi \rangle }}.$

Note that this is a map of the function Φ onto the number E[Φ], i.e., we have here a functional. The bra-ket notation implies the integration over a configuration space, which in electronic structure theory usually is $\scriptstyle \mathbb {R} ^{3N}$ with N being the number of particles of the system described by the Hamiltonian H. We assume boundary conditions on Φ and integration limits such that E[Φ] is finite and the Hamiltonian H is Hermitian. In that case E[Φ] is a real number. The boundary condition in electronic structure theory is the vanishing of Φ at infinite interparticle distances. The bra-ket is a 3N fold integral over all of $\scriptstyle \mathbb {R} ^{3N}$, i.e., integrals in Cartesian coordinates from minus to plus infinity. Bounded configuration spaces with periodic boundary conditions also occur in quantum mechanics and variational Rayleigh-Ritz theory, as outlined below, applies to the latter cases as well.

The variational principle

Most non-relativistic Hamiltionians in quantum mechanics have a lower bound, that is there is a finite real number B such that

$E[\Phi ]>B\,$

for all admissible Φ. The function Φ is admissible when the functional E[Φ] is well-defined (for instance Φ must be twice differentiable) and Φ has the correct boundary conditions. We assume from now on that H has a lower bound.

The highest lower bound is the lowest eigenvalue E₀ of H. To show this we assume that the eigenvalues and eigenvectors of H are known and moreover that the eigenvectors are orthonormal and complete (form an orthonormal basis of Hilbert space),

$H\phi _{k}=E_{k}\phi _{k},\quad E_{0}<E_{1}\leq E_{2},\dots$

where for convenience sake we assume the ground state to be non-degenerate. Because H is bounded from below, E₀ ≡ E[φ₀] > B is finite. Completeness and orthonormality gives the resolution of the identity

$1=\sum _{k=0}|\phi _{k}\rangle \langle \phi _{k}|.$

Inserting this resolution into the functional at appropriate places, and using that the Hamilton matrix on basis of eigenvectors is diagonal with E_k on the diagonal, yields

$E[\Phi ]={\frac {\sum _{k}\langle \Phi |\phi _{k}\rangle E_{k}\langle \phi _{k}|\Phi \rangle }{\sum _{k}\langle \Phi |\phi _{k}\rangle \langle \phi _{k}|\Phi \rangle }}={\frac {\sum _{k}|c_{k}|^{2}E_{k}}{\sum _{k}|c_{k}|^{2}}}\geq {\frac {E_{0}\sum _{k}|c_{k}|^{2}}{\sum _{k}|c_{k}|^{2}}}=E_{0},$

where $\scriptstyle c_{k}\equiv \langle \phi _{k}|\Phi \rangle$. This result states that any expectation value is larger than or equal to the lowest eigenvalue E₀. The lowest expectation value E[Φ] of H is obtained if we take Φ = φ₀, because it is then equal to E₀. If, conversely, we find a Φ such that E[Φ] = E₀, it can be concluded that Φ is the exact eigenfunction of lowest energy, because supposing the opposite gives a contradiction:

$H\Phi \neq E_{0}\Phi \quad \Longrightarrow \quad {\frac {\langle \Phi |H|\Phi \rangle }{\langle \Phi |\Phi \rangle }}\neq E_{0}\quad \Longrightarrow E[\Phi ]\neq E_{0}.$

This result is referred to as the variational principle.

Matrix eigenvalue problem

It will be convenient to rewrite the functional in the Lagrange form of undetermined multipliers. That is, we redefine the functional as

$E[\Phi ]=\langle \Phi |H|\Phi \rangle -\lambda \langle \Phi |\Phi \rangle ,$

where λ is an undetermined multiplier. It can be shown that this functional has the same stationary values for the same functions Φ as the original functional.

Take now an arbitrary, non-complete, expansion set χ_k, and approximating Φ by

$\Phi \approx \sum _{k=1}^{M}c_{k}\chi _{k},$

we get the approximation for the functional

$E[\Phi ]\approx \sum _{k,m=1}^{M}\left[c_{k}^{*}c_{m}H_{km}-\lambda c_{k}^{*}c_{m}S_{km}\right]=\mathbf {c} ^{\dagger }\mathbf {H} \mathbf {c} -\lambda \mathbf {c} ^{\dagger }\mathbf {S} \mathbf {c}$

with the M × M matrices

$H_{km}\equiv \langle \chi _{k}|H|\chi _{m}\rangle ,\quad {\hbox{and}}\quad S_{km}\equiv \langle \chi _{k}|\chi _{m}\rangle .$

Stationary points—such as minima—of the functional are obtained from equating derivatives to zero,

${\frac {\partial E[\Phi ]}{\partial c_{k}^{*}}}=0\quad {\hbox{and}}\quad {\frac {\partial E[\Phi ]}{\partial c_{k}}}=0.$

Letting k run from 1 to M, these equations give rise to

$\mathbf {H} \mathbf {c} =\lambda \mathbf {S} \mathbf {c} \quad {\hbox{and}}\quad \mathbf {c} ^{\dagger }\mathbf {H} =\lambda \mathbf {c} ^{\dagger }\mathbf {S} ,$

which, however, are the same equations because the matrices H and S are Hermitian and λ is therefore real. Hence, a stationary point of the approximate functional is obtained by solution of the generalized matrix eigenvalue equation,

$\mathbf {H} \mathbf {c} =E\mathbf {S} \mathbf {c} \qquad \qquad \qquad \qquad \qquad \qquad (1)$

where we changed the notation from λ to E.

Make now two assumptions: the set χ_k is complete and the set is orthonormal. (The second assumption is non-essential, but simplifies the notation). In an orthonormal basis the following simple expression holds for the expansion coefficients,

$c_{k}=\langle \chi _{k}|\Phi \rangle$

Note, parenthetically, that in general completeness of the expansion requires infinite M. Introduce the resolution of identity

$1=\sum _{k=1}^{\infty }|\chi _{k}\rangle \langle \chi _{k}|$

into

$\sum _{k}\langle \chi _{m}|H|\chi _{k}\rangle \langle \chi _{k}|\Phi \rangle =E\sum _{k}\delta _{mk}\langle \chi _{k}|\Phi \rangle \quad \Longrightarrow \quad \langle \chi _{m}|H|\Phi \rangle =\langle \chi _{m}|\Phi \rangle ,\quad m=1,2,\ldots ,\infty .$

Multiply on the left by $\scriptstyle |\chi _{m}\rangle$, sum over m, and use the resolution of the identity again, then we find the result that we set out to prove: The determination of stationary points of the functional E[Φ] in a complete basis is equivalent to solving

$H\Phi =E\Phi \,.$

Ritz's important insight was that a function expanded in a non-complete basis χ_k that minimizes the expectation value of H, gives an approximation to the lowest eigenvector of H.

Reiterating and summarizing the above, we see that the Rayleigh-Ritz variational method starts by choosing an expansion basis χ_k of dimension M. Then this expansion is inserted into the expectation value of the Hamiltonian, whereupon the solution of the generalized matrix eigenvalue (1) yields stationary points (usually minima). The lowest eigenvector of this M dimensional matrix problem approximates the lowest exact solution of the Schrödinger equation. Clearly, the success of the method relies to a large extent on the quality of the expansion basis. When the expansion basis nears completeness, the solution vector nears the exact solution.

History

In the older quantum mechanics literature the method is known as the Ritz method, called after the mathematical physicist Walter Ritz, who first devised it. In prewar quantum mechanics texts it was customary to follow the highly influential book by Courant and Hilbert, who were contemporaries of Ritz and write of the Ritz procedure (Ritzsches Verfahren). It is parenthetically amusing to note that the majority of the old quantum mechanics texts quote the wrong year for the appearance of Ritz's work, 1909 instead of 1908, an error first made in the Courant-Hilbert treatise.

In the numerical analysis literature one usually prefixes the name of Lord Rayleigh to the method, and lately this has become common in quantum mechanics, too. Leissa recently became intrigued by the name and after reading the original sources, he discovered that the methods of the two workers differ considerably, although Rayleigh himself believed that the methods were very similar and that his own method predated the one of Ritz by several decades. However, according to Leissa's convincing conclusion, Rayleigh was mistaken and the method now known as Rayleigh-Ritz method is solely due to Ritz. Leissa states: Therefore, the present writer concludes that Rayleigh’s name should not be attached to the Ritz method; that is, the Rayleigh–Ritz method is an improper designation.