The Rayleigh-Ritz variational method (RR) is one of the approximate methods most commonly used in the study of the electronic structure of atoms and molecules [1,2]. One of its main advantages is that it provides increasingly accurate upper bounds to all the eigenvalues of the Hamiltonian operator of the system [3,4]. In this paper we provide a comprehensible overview of the approach and illustrate some of its relevant points by means of a simple problem.

The starting point of our analysis is a linearly independent set of vectors $\mathcal{V}=$$\left\{f_1,f_2,\dots \right\}$. Clearly, the only solution to the vector equation is $a_i=0$ for all $i=1,2,\dots ,N$. If we apply the bras $\left.〈f_j\right|$, $j=1,2,\dots ,N$, to this equation from the left we obtain where $S_{ij}=\left.〈f_i\right|f_j〉$. We have an homogeneous system of N linear equations with N unknowns $a_i$ with the only solution $a_i=0$. Consequently, $\left|\mathbf{S}\right|\ne 0$ where $\mathbf{S}={\left(S_{ij}\right)}_{i,j=1}^N$ is an $N\times N$ Hermitian matrix and $\left|...\right|$ stands for determinant. Note that $S_{ij}=S_{ji}^{*}$ so that ${\mathbf{S}}^{†}=\mathbf{S}$ where † stands for adjoint. The matrix $\mathbf{S}$ is commonly called overlap matrix[1].

Let $\mathbf{v}$ be an eigenvector of $\mathbf{S}$ with eigenvalue s, $\mathbf{Sv}=s\mathbf{v}$, then ${\mathbf{v}}^{†}\mathbf{Sv}=s{\mathbf{v}}^{†}\mathbf{v}$. If $v_i$, $i=1,2,\dots ,N$, are the elements of the $N\times 1$ column vector $\mathbf{v}$ then and we conclude that $s>0.$ In other words, the overlap matrix $\mathbf{S}$ is positive definite.

We are interested in the eigenvalue equation for an Hermitian operator H. In order to solve it approximately we propose and ansatz of the form where $\mathcal{V}=\left\{f_1,f_2,\dots \right\}$ is not only assumed to be linearly independent but also complete.

The RR variational method consists of minimizing the integral with respect to the expansion coefficients $c_j$

This equation leads to the so-called secular equation[1,2] where, $H_{ij}=\left.〈f_i\right|H\left|f_j\right.〉$. There are nontrivial solutions $c_j$, $j=1,2,\dots ,N$, provided that the secular determinant vanishes where $\mathbf{H}={\left(H_{ij}\right)}_{i,j=1}^N$ is an $N\times N$ Hermitian matrix.

For each of the roots of the secular determinant (9), $W_1\le W_2\le \dots \le W_N$, we derive an approximate solution; for example, when $W=W_k$ we have and the secular equation (8) can be rewritten

If we define the $N\times N$ matrices $\mathbf{W}={\left(W_i{\delta }_{ij}\right)}_{i,j=1}^N$ and $\mathbf{C}={\left(c_{ij}\right)}_{i,j=1}^N$then this equation can be rewritten in matrix form as which is equivalent to and the procedure reduces to the diagonalization of the matrix ${\mathbf{S}}^{-1}\mathbf{H}$ by means of the invertible matrix $\mathbf{C}$. Note that ${\mathbf{S}}^{-1}$ exists because $\mathbf{S}$ is positive definite as argued above.

In order to determine the coefficients $c_{jk}$ completely, we require that $〈{\phi }_i\left|{\phi }_j\right.〉={\delta }_{ij}$ that leads to that in matrix form reads where $\mathbf{I}$ is the $N\times N$ identity matrix. It follows from equations (15) and (12) that

It is clear that there exists an invertible matrix ($\mathbf{C}$) that transforms two Hermitian matrices ($\mathbf{H}$ and $\mathbf{S}$), one of them positive definite ($\mathbf{S}$), into diagonal form. This procedure is well known in the mathematical literature[5]. However, it is most important to note that equations (15) and (16) are not what we commonly know as matrix diagonalization. In fact, the eigenvalues of $\mathbf{S}$ are not unity and the eigenvalues of $\mathbf{H}$ are not the RR eigenvalues $W_i$. We will illustrate this point in Section 3 by means of a simple example. It is also worth noting that that we cannot obtain $\mathbf{C}$ neither from (15) or (16). One obtains the matrix $\mathbf{C}$ in the process of diagonalizing ${\mathbf{S}}^{-1}\mathbf{H}$ as in equation (13) and the remaining undefined matrix elements $c_{ij}$ from equation (15).

Since $\mathbf{S}$ is positive definite, we can define ${\mathbf{S}}^{1/2}$. The matrix $\mathbf{U}={\mathbf{S}}^{1/2}\mathbf{C}$ is unitary as shown by

On substituting $\mathbf{C}={\mathbf{S}}^{-1/2}\mathbf{U}$ into equation (16) we obtain

This equation is just the standard diagonalization of the Hermitian matrix ${\mathbf{S}}^{-\mathbf{1}/\mathbf{2}}{\mathbf{HS}}^{-1/2}$.

If the basis set is orthonormal, $\left.〈f_i\right|f_j〉={\delta }_{ij}$, then $\mathbf{S}=\mathbf{I}$, ${\mathbf{C}}^{†}={\mathbf{C}}^{-1}$ and the secular equation (13) becomes

In this particular case, the eigenvalues of the matrix $\mathbf{H}$ are the RR eigenvalues $W_i$. Note that equations (16) and (19) look identical but were derived under different assumptions (they agree only when $\mathbf{S}=\mathbf{I}$).

As a simple example we consider the dimensionless eigenvalue equation

In order to illustrate the RR variational method with a non-orthogonal basis set we choose $f_i\left(x\right)=x^i(1-x)$, $i=1,2,\dots$, that satisfy the boundary conditions at $x=0$ and $x=1$.

A straightforward calculation shows that and

Table 1 and Table 2 show the RR eigenvalues $W_i$, $i=1,2,3,4$, for $\lambda =0$ and $\lambda =1$, respectively. We appreciate that the approximate eigenvalues converge from above as expected[3,4].

In what follows, we illustrate some of the general results of Section 2 for the simplest case $N=2$ when $\lambda =0$. The matrices are and we obtain

One can easily verify that these matrices already satisfy equations (15) and (16). On the other hand, the symmetric matrices $\mathbf{S}$ and $\mathbf{H}$ can be diagonalized in the usual way by orthogonal matrices that we call ${\mathbf{U}}_S$ and ${\mathbf{U}}_H$, respectively.

We clearly see that the eigenvalues of $\mathbf{S}$ are not unity and those of $\mathbf{H}$ are not the RR eigenvalues $W_i$ as argued in Section 2.

Using equation (25) one can easily obtain

We have shown that the main equations of the Rayleigh-Ritz variational method [1,2] lead to the mathematical problem of diagonalization of two Hermitian matrices[5]. Although equations (15) and (16) are discussed in some textbooks on quantum chemistry, the latter does not appear to be correctly interpreted[1].