# 7.2: Linear Variational Method and the Secular Determinant

- Page ID
- 13435

Learning Objectives

- Understand how the variational method can be expanded to include trial wavefunctions that are a linear combination of functions with coefficients that are the parameters to be varied.
- To be able to construct secular equations to solve the minimization procedure intrinsic to the variational method approach.
- To map the secular equations into the secular determinant
- To understand how the Linear Combination of Atomic Orbital (LCAO) approximation is a specific application of the linear variational method.

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.

Linear Variational Basis Functions

The set of functions {\(\phi_j\)} are called the 'linear variational' basis functions and are nothing more than members of a set of functions that are convenient to deal with. However, they are typically not arbitrary and are usually selected to address specific properties of the system:

- to obey all of the boundary conditions that the exact state \(| \psi _{trial} \rangle\) obeys,
- to be functions of the the same coordinates as \(| \psi _{trial} \rangle\),
- to be of the same symmetry as \(| \psi _{trial} \rangle\), and
- to be convenient to evaluate Hamiltonian terms elements \(\langle \phi_i|H|\phi_j \rangle\).

Beyond these conditions, nothing other than effort can limit the selection and number of such basis functions in the expansions in Equations \(\ref{Ex1}\) and \(\ref{Ex2}\).

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

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

Orthonormality

There is no explicit rule that the {\(\phi_j\)} functions have to be **orthogonal** or **normalized** functions, although they often are selected that way for convenience. Therefore, *a priori*, \(S_{ij}\) does not have to be \(\delta_{ij}\).

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**.

Hermitian Operators

Hermitian operators are operators that satisfy the general formula

\[ \langle \phi_i | \hat{A} | \phi_j \rangle = \langle \phi_j | \hat{A} | \phi_i \rangle \label{Herm1}\]

If that condition is met, then \(\hat{A}\) is a Hermitian operator. For any operator that generates a real eigenvalue (e.g., observables), then that operator is Hermitian. The Hamiltonian \(\hat{H}\) meets the condition of a Hermitian operator. Equation \ref{Herm1} can be rewriten as

\[A_{ij} =A_{ji}^*\]

where

\[A_{ij} = \langle \phi_i | \hat{A} | \phi_j \rangle\]

and

\[A_{ji} = \langle \phi_j | \hat{A} | \phi_i \rangle\]

Therefore, when applied to the Hamiltonian operator

\[H_{ij}^* =H_{ji}.\]

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*'.