# 7.2: Linear Variational Method and the Secular Determinant

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A special type of variation widely used in the study of molecules is the so-called linear variation function, where the trial wavefunction is a linear combination of \(N\) linearly independent functions (often atomic orbitals) that not the eigenvalues of the Hamiltonian (since they are not known). For example

\[| \psi_{trial} \rangle = \sum_{j=1}^N a_j |\phi_j \rangle \label{Ex1}\]

and

\[ \langle \psi_{trial} | = \sum_{j=1}^N a_j^* \langle \phi_j | \label{Ex2}\]

In these cases, one says that a 'linear variational' calculation is being performed.

As discussed in Section 7.1, the variational energy for a generalized trial wavefunction is

\[ E_{trial} = \dfrac{ \langle \psi _{trial}| \hat {H} | \psi _{trial} \rangle}{\langle \psi _{trial} | \psi _{trial} \rangle} \label{7.1.8}\]

Substituting Equations \ref{Ex1} and \ref{Ex2} into Equation \ref{7.1.8} involves addressing the numerator and denominator individually. For the numerator, the integral can be expanded thusly:

\[\begin{align} \langle\psi_{trial} |H| \psi_{trial} \rangle &= \sum_{i}^{N} \sum_{j} ^{N}a_i^{*} a_j \langle \phi_i|H|\phi_j \rangle. \\[4pt] &= \sum_{i,\,j} ^{N,\,N}a_i^{*} a_j \langle \phi_i|H|\phi_j \rangle. \label{MatrixElement}\end{align}\]

We can rewrite the following integral in Equation \ref{MatrixElement} as a function of the basis elements (not the trial wavefunction) as

\[ H_{ij} = \langle \phi_i|H|\phi_j \rangle\]

So the numerator of the right side of Equation \ref{7.1.8} becomes

\[\langle\psi_{trial} |H| \psi_{trial} \rangle = \sum_{i,\,j} ^{N,\,N}a_i^{*} a_j H_{ij} \label{numerator}\]

Similarly, the denominator of the right side of Equation \ref{7.1.8} can be expanded

\[\langle \psi_{trial}|\psi_{trial} \rangle = \sum_{i,\,j} ^{N,\,N}a_i^{*} a_j \langle \phi_i | \phi_j \rangle \label{overlap}\]

We often simplify the integrals on the right side of Equation \ref{overlap} as

\[ S_{ij} = \langle \phi_i|\phi_j \rangle \]

where \(S_{ij}\) are **overlap integrals** between the different {\(\phi_j\)} basis functions. Equation \ref{overlap} is thus expressed as

\[\langle \psi_{trial}|\psi_{trial} \rangle = \sum_{i,\,j} ^{N,\,N}a_i^{*} a_j S_{ij} \label{denominator}\]

Substituting Equations \ref{numerator} and \ref{denominator} into the variational energy formula (Equation \ref{7.1.8}) results in

\[ E_{trial} = \dfrac{ \displaystyle \sum_{i,\,j} ^{N,\,N}a_i^{*} a_j H_{ij} }{ \displaystyle \sum_{i,\,j} ^{N,\,N}a_i^{*} a_j S_{ij} } \label{Var}\]

For such a trial wavefunction as Equation \ref{Ex1}, the variational energy depends quadratically on the 'linear variational' \(a_j\) coefficients. These coefficients can be varied just like the parameters in the trial functions of Section 7.1 to find the optimized trial wavefunction (\(| \psi_{trial} \rangle\)) that approximates the true wavefunction (\(| \psi \rangle\)) that we cannot analytically solve for.

## Minimizing the Variational Energy

The expression for variational energy (Equation \ref{Var}) can be rearranged

\[E_{trial} \sum_{i,\,j} ^{N,\,N} a_i^*a_j S_{ij} = \sum_{i,\,j} ^{N,\,N} a_i^* a_j H_{ij} \label{7.2.9}\]

The optimum coefficients are found by searching for minima in the variational energy landscape spanned by varying the \(\{a_i\}\) coefficients (Figure \(\PageIndex{1}\)).

We want to minimize the energy with respect to the linear coefficients \(\{a_i\}\), which requires that

\[\dfrac{\partial E_{trial}}{\partial a_i}= 0\]

for all \(i\).

Differentiating both sides of Equation \(\ref{7.2.9}\) for the \(k^{th}\) coefficient gives,

\[ \dfrac{\partial E_{trial}}{\partial a_k} \sum_{i,\,j} ^{N,\,N} a_i^*a_j S_{ij}+ E_{trial} \sum_i \sum_j \left[ \dfrac{ \partial a_i^*}{\partial a_k} a_j + \dfrac {\partial a_j}{\partial a_k} a_i^* \right ]S_{ij} = \sum_{i,\,j} ^{N,\,N} \left [ \dfrac{\partial a_i^*}{\partial a_k} a_j + \dfrac{ \partial a_j}{\partial a_k}a_i^* \right] H_{ij} \label{7.2.10}\]

Since the coefficients are independent

\[\dfrac{\partial a_i^*}{ \partial a_k} = \delta_{ik}\]

and

\[S_{ij} = S_{ji}\]

and also since the Hamiltonian is a Hermitian Operator (see below)

\[H_{ij} =H_{ji}\]

then Equation \(\ref{7.2.10}\) simplifies to

\[ \dfrac{\partial E_{trial}}{\partial a_k} \sum_i \sum_j a_i^*a_j S_{ij}+ 2E_{trial} \sum_i a_i S_{ik} = 2 \sum_i a_i H_{ik} \label{7.2.11}\]

At the minimum variational energy, when

\[\dfrac{\partial E_{trial}}{\partial a_k} = 0\]

then Equation \(\ref{7.2.11}\) gives

\[ {\sum _i^N a_i (H_{ik}–E_{trial} S_{ik}) = 0} \label{7.2.12}\]

for all \(k\). The equations in \(\ref{7.2.12}\) are call the** Secular Equations**.

If the functions \(\{|\phi_j\rangle \}\) are orthonormal, then the overlap matrix \(S\) reduces to the unit matrix (one on the diagonal and zero every where else) and the Secular Equations in Equation \ref{7.2.12} reduces to the more familiar Eigenvalue form:

\[ \sum\limits_i^N H_{ij}a_j = E_{trial} a_i .\label{seceq2}\]

Hence, the secular equation, in either form, have as many eigenvalues \(E_i\) and eigenvectors {\(C_{ij}\)} as the dimension of the \(H_{ij}\) matrix as the functions in \(| \psi_{trail} \rangle\) (Example \ref{Ex1}). It can also be shown that between successive pairs of the eigenvalues obtained by solving the secular problem at least one exact eigenvalue must occur (i.e.,\( E_{i+1} > E_{exact} > E_i\) , for all i). This observation is referred to as '*the bracketing theorem*'.

## The Secular Determinant

From the secular equations with an orthonormal functions (Equation \ref{seceq2}), we have \(k\) simultaneous secular equations in \(k\) unknowns. These equations can also be written in matrix notation, and for a non-trivial solution (i.e. \(c_i \neq 0\) for all \(i\)), the determinant of the secular matrix must be equal to zero.

\[ { | H_{ik}–ES_{ik}| = 0} \label{7.2.13}\]

To solve Equation \ref{7.2.13}, the determinate should be expanded and then set to zero. That generates a polynomial (called a **characteristic equation**) that can be directly solved with linear algebra methods or numerically.

Equation \(\ref{7.2.13}\) can be solved to obtain the energies \(E\). When arranged in order of increasing energy, these provide approximations to the energies of the first \(k\) states (each having an energy higher than the true energy of the state by virtue of the variation theorem). To find the energies of a larger number of states we simply use a** greater number** of basis functions \(\{\phi_i\}\) in the trial wavefunction (Example \ref{Ex1}). To obtain the approximate wavefunction for a particular state, we substitute the appropriate energy into the secular equations and solve for the coefficients \(a_i\).

The linear variational method is used extensively in molecular orbitals of molecules and further examples will be postponed until that discussion in Chapters 9.

## Contributors

Claire Vallance (University of Oxford)