On the Raleigh-Ritz Variational Method. Non-orthogonal Basis Set

1. Introduction

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.

2. The Rayleigh-Ritz Variational Method

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

$$\sum _{i=1}^N{a}_i{f}_i=0,$$

(1)

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

$$\sum _{i=1}^N{S}_{ji}{a}_i=0,j=1,2,\dots ,N,$$

(2)

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

$${\mathbf{v}}^{†}\mathbf{Sv}=\left.〈\sum _{i=1}^N{v}_i{f}_i\right|\sum _{j=1}^N{v}_j{f}_j〉>0,$$

(3)

and we conclude that $s>0.$ In other words, the overlap matrix $\mathbf{S}$ is positive definite.

We are interested in the eigenvalue equation

$$\begin{array}{ccc}\hfill H{\psi }_n& =& E_n{\psi }_n,n=1,2,\dots ,\hfill \\ \hfill E_1& \le & E_2\le \dots ,〈{\psi }_i\left|{\psi }_j\right.〉={\delta }_{ij},\hfill \end{array}$$

(4)

for an Hermitian operator H. In order to solve it approximately we propose and ansatz of the form

$$\phi =\sum _{j=1}^N{c}_j{f}_j,$$

(5)

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

$$W=\frac{\left.〈\phi \right|H\left|\phi \right.〉}{\left.〈\phi \right|\phi 〉},$$

(6)

with respect to the expansion coefficients $c_j$

$$\frac{\partial W}{\partial c_j}=0,j=1,2,\dots ,N.$$

(7)

This equation leads to the so-called secular equation[1,2]

$$\sum _{j=1}^N\left(H_{ij}-W{S}_{ij}\right)c_j=0,i=1,2,\dots ,N,$$

(8)

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

$$\left|\mathbf{H}-W\mathbf{S}\right|=0,$$

(9)

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

$${\phi }_k=\sum _{j=1}^N{c}_{jk}{f}_j,$$

(10)

and the secular equation (8) can be rewritten

$$\sum _{j=1}^N{H}_{ij}{c}_{jk}=\sum _{j=1}^N{W}_k{S}_{ij}{c}_{jk}=\sum _{j=1}^N\sum _{m=1}^N{S}_{ij}{W}_m{\delta }_{mk}{c}_{jm}.$$

(11)

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

$$\mathbf{HC}=\mathbf{SCW},$$

(12)

which is equivalent to

$${\mathbf{C}}^{-1}{\mathbf{S}}^{-1}\mathbf{HC}=\mathbf{W},$$

(13)

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

$$〈{\phi }_i\left|{\phi }_j\right.〉=\sum _{k=1}^N\sum _{m=1}^N{c}_{ki}^{*}{c}_{mj}\left.〈f_k\right|f_m〉={\delta }_{ij},$$

(14)

that in matrix form reads

$${\mathbf{C}}^{†}\mathbf{SC}=\mathbf{I},$$

(15)

where $\mathbf{I}$ is the $N\times N$ identity matrix. It follows from equations (15) and (12) that

$${\mathbf{C}}^{†}\mathbf{HC}=\mathbf{W}.$$

(16)

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

$${\mathbf{U}}^{†}\mathbf{U}={\mathbf{C}}^{†}{\mathbf{S}}^{\mathbf{1}/\mathbf{2}}{\mathbf{S}}^{1/2}\mathbf{C}=\mathbf{I}.$$

(17)

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

$${\mathbf{U}}^{†}{\mathbf{S}}^{-1/2}{\mathbf{HS}}^{-1/2}\mathbf{U}=\mathbf{W}.$$

(18)

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

$${\mathbf{C}}^{†}\mathbf{HC}=\mathbf{W}.$$

(19)

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 $\mathbf{S}=\mathbf{I}$).

3. Simple Example

As a simple example we consider the dimensionless eigenvalue equation

$$H\psi =E\psi ,H=-\frac{1}{2}\frac{d^2}{d{x}^2}+\lambda x,\psi \left(0\right)=\psi \left(1\right)=0.$$

(20)

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

$$S_{ij}=\frac{2}{\left(i+j+1\right)\left(i+j+2\right)\left(i+j+3\right)},$$

(21)

and

$$H_{ij}=\frac{ij}{\left(i+j\right)\left(i+j+1\right)\left(i+j-1\right)}+\frac{2\lambda }{\left(i+j+2\right)\left(i+j+3\right)\left(i+j+4\right)}.$$

(22)

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

$$\mathbf{S}=\frac{1}{60}\left(\begin{array}{cc}2\hfill & 1\hfill \\ 1\hfill & \frac{4}{7}\hfill \end{array}\right),\mathbf{H}=\frac{1}{12}\left(\begin{array}{cc}2\hfill & 1\hfill \\ 1\hfill & \frac{4}{5}\hfill \end{array}\right),$$

(23)

and we obtain

$${\mathbf{C}}^{-1}{\mathbf{S}}^{-1}\mathbf{HC}=\mathbf{W}=\left(\begin{array}{cc}5\hfill & 0\hfill \\ 0\hfill & 21\hfill \end{array}\right),\mathbf{C}=\sqrt{30}\left(\begin{array}{cc}1\hfill & \sqrt{7}\hfill \\ 0\hfill & -2\sqrt{7}\hfill \end{array}\right).$$

(24)

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.

$$\begin{array}{ccc}\hfill {\mathbf{U}}_S^{†}{\mathbf{SU}}_S& =& \frac{1}{420}\left(\begin{array}{cc}9-\sqrt{74}\hfill & 0\hfill \\ 0\hfill & 9+\sqrt{74}\hfill \end{array}\right),\hfill \\ \hfill {\mathbf{U}}_S& =& \left(\begin{array}{cc}\sqrt{\frac{1}{2}-\frac{5\sqrt{174}}{148}}\hfill & \sqrt{\frac{1}{2}+\frac{5\sqrt{174}}{148}}\hfill \\ -\sqrt{\frac{1}{2}+\frac{5\sqrt{174}}{148}}\hfill & \sqrt{\frac{1}{2}-\frac{5\sqrt{174}}{148}}\hfill \end{array}\right),\hfill \\ \hfill {\mathbf{U}}_H^{†}{\mathbf{HU}}_H& =& \frac{1}{60}\left(\begin{array}{cc}7-\sqrt{34}\hfill & 0\hfill \\ 0\hfill & 70+\sqrt{34}\hfill \end{array}\right),\hfill \\ \hfill {\mathbf{U}}_H& =& \left(\begin{array}{cc}\sqrt{\frac{1}{2}-\frac{3\sqrt{34}}{68}}\hfill & \sqrt{\frac{1}{2}+\frac{3\sqrt{34}}{68}}\hfill \\ -\sqrt{\frac{1}{2}+\frac{3\sqrt{34}}{68}}\hfill & \sqrt{\frac{1}{2}-\frac{3\sqrt{34}}{68}}\hfill \end{array}\right)\hfill \end{array}$$

(25)

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

$${\mathbf{S}}^{1/2}=\left(\begin{array}{cc}\sqrt{\frac{233}{8880}+\frac{7\sqrt{7}}{8880}}\hfill & \sqrt{\frac{21}{2960}-\frac{7\sqrt{7}}{8880}}\hfill \\ \sqrt{\frac{21}{2960}-\frac{7\sqrt{7}}{8880}}\hfill & \sqrt{\frac{151}{62160}+\frac{7\sqrt{7}}{8880}}\hfill \end{array}\right).$$

(26)

4. Conclusions

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].