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7.2: Linear Variational Method and the Secular Determinant

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

\[| Φ \rangle = \sum_J^N C_J |Φ_J \rangle \label{Ex1}\]

and

\[ \langle Φ | = \sum_J^N C_J* \langle Φ_J |  \label{Ex2}\]

In these cases, one says that a 'linear variational' calculation is being performed. The set of functions {\(Φ_J\)} are usually constructed to obey all of the boundary conditions that the exact state \(\Psi\) obeys, to be functions of the the same coordinates as \(Ψ\), and to be of the same spatial and spin symmetry as Ψ. Beyond these conditions, the {\(Φ_J\)} are nothing more than members of a set of functions that are convenient to deal with (e.g., convenient to evaluate Hamiltonian matrix elements \(\langle Φ_I|H|Φ_J \rangle\) that can, in principle, be made complete if more and more such functions are included in the expansion in Equations \(\ref{Ex1}\) and \(\ref{Ex2}\) (i.e., increase \(N\)).

For such a trial wavefunction, the variational energy depends quadratically on the 'linear variational' \(C_J\) coefficients:

\[\langle Φ |H| Φ \rangle = \sum_{I,J} ^{N,N}C_IC_J \langle Φ_Ι|H|Φ_J \rangle.\]

Minimization of this energy with the constraint that \(|Φ \rangle\) remain normalized, i.e.,

\[\langle Φ|Φ \rangle = \sum\limits_{IJ} C_IC_J \langle Φ_I | Φ_J \rangle= 1\]

gives rise to a so-called secular or eigenvalue-eigenvector problem:

\[\sum\limits_J [\langle Φ_I|H|Φ_J \rangle - E \langle Φ_I|Φ_J \rangle] C_J = \sum\limits_J [H_{IJ} - E S_{IJ} ]C_J = 0. \]

If the functions \(\{|Φ_J\rangle \}\) are orthonormal, then the overlap matrix \(S\) reduces to the unit matrix and the above generalized eigenvalue problem reduces to the more familiar form:

\[ \sum\limits_J^N H_{IJ}C_J = E C_I .\]

The secular problem, in either form, has as many eigenvalues \(E_i\) and eigenvectors {\(C_{iJ}\)} as the dimension of the \(H_{IJ}\) matrix as \(Φ\). 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'.

Variational methods, in particular the linear variational method, are the most widely used approximation techniques in quantum chemistry. To implement such a method one needs to know the Hamiltonian \(H\) whose energy levels are sought and one needs to construct a trial wavefunction in which some 'flexibility' exists (e.g., as in the linear variational method where the \(C_J\) coefficients can be varied). This tool will be used to develop several of the most commonly used and powerful molecular orbital methods in chemistry.

Application to \(N=2\) Expansion

The goal is to solve for the set of all \(c\) values that minimize the energy \(E_{\phi}\). Evaluating \(\int \psi^* \hat{H} \psi d\tau\) and \(\int \psi^* \psi d\tau\) gives \(E_{\phi}\) by

\[E_{\phi} = \dfrac {\int \psi^* \hat{H} \psi d\tau}{\int \psi^* \psi d\tau}\label{7B}\]

Evaluation of Numerator with linear superposition wavefunctions

\[\int \psi^* \hat{H} \psi d\tau = \int (c_1 f_1 + c_2 f_2)^* \hat{H} (c_1 f_1 + c_2 f_2)d\tau\]

\[\int \psi^* \hat{H} \psi d\tau= c_1^2 \int f_1^* \hat{H} f_1 d\tau + c_1 c_2 \int f_1^* \hat{H} f_2 d\tau + c_1 c_2 \int f_2^* \hat{H} f_1 d\tau + c_2^2 \int f_2^* \hat{H} f_2 d\tau\]

\[\int \psi^* \hat{H} \psi d\tau = c_1^2 H_{11} + 2c_1 c_2 H_{12} + c_2^2 H_{22}\]

where

\[ \color{ref} H_{ij} = \int f_{i}^* \hat{H} f_{j} d\tau\]

and  since \(H\) is a Hermitian operator.

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

Evaluation of Denominator with linear superposition wavefunctions

\[\int \psi^* \psi d\tau = c_1^2 S_{11} + 2c_1 c_2 S_{12} + c_2^2 S_{22}\]

where

\[ \color{red} S_{ij} = \int f_i^* f_j d\tau\]

Plugging in the evaluations of the two integrals into Equation \(7B\) gives

\[E_{\phi} = \dfrac{c_1^2 H_{11} + 2c_1 c_2 H_{12} + c_2^2 H_{22}}{c_1^2 S_{11} + 2c_1 c_2 S_{12} + c_2^2 S_{22}}\label{8B}\]

Rearrange the equation to

\[(c_1^2 S_{11} + 2c_1 c_2 S_{12} + c_2^2 S_{22})E_{\phi}) = c_1^2 H_{11} 2c_1 c_2 H_{12} C_2^2 H_{22}\]

Take the derivative of the previous equation with respect to \(c_1\) and set it equal to \(0\). After some algebra and rearranging, you will wind up with

\[c_1(H_{11} - ES_{11}) + c_2(H_{12} - ES_{12}) = 0\]

Following the same Variational Method procedure by taking \(\dfrac {d}{dc_2}\) and setting it equal to \(0\) gives

\[c_1(H_{12} - ES_{12}) + c_2(H_{22} - ES_{22}) = 0\]

Writing this set of homogeneous linear equations in matrix form gives

\[\begin{pmatrix} H_{11} - ES_{11} & H_{12} - ES_{12} \\ H_{12} - ES_{12} & H_{22} - ES_{22} \end{pmatrix} \begin{pmatrix} c_1 \\ c_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \label{21.3}\]

For the equations to have a solution, the determinant of the matrix must be equal to zero. Writing out the determinant will give us a polynomial equation in \(E\) that we can solve to obtain the orbital energies in terms of the Hamiltonian matrix elements \(H_{ij}\) and overlap integrals \(S_{ij}\). The number of energies obtained by ‘solving the secular determinant’ in this way is equal to the order of the matrix, in this case two. 

The secular determinant for Equation \(\ref{21.3}\) is (noting that \(S_{11}\) = \(S_{22} = 1\) since the SALCs are normalized)

\[(H_{11} - E)(H_{22} - E) - (H_{12} - ES_{12})^2 = 0 \label{21.4}\]

Expanding and collecting terms in \(E\) gives

\[E^2(1-S_{12}^2) + E(2H_{12}S_{12} - H_{11} - H_{22}) + (H_{11}H_{22} - H_{12}^2) = 0 \label{21.5}\]

which can be solved using the quadratic formula to give the energies of the two molecular orbitals.

\[E_\pm = \dfrac{-(2H_{12}S_{12} - H_{11} - H_{22}) \pm \sqrt{(2H_{12}S_{12} - H_{11} - H_{22})^2 - 4(1-S_{12}^2)(H_{11}H_{22} - H_{12}^2)}}{2(1-S_{12}^2)} \label{21.6}\]

To obtain numerical values for the energies, we need to evaluate the integrals \(H_{11}\), \(H_{22}\), \(H_{12}\), and \(S_{12}\). This would be quite a challenge to do analytically, but luckily there are a number of computer programs that can be used to calculate the integrals.

This gives linear equations in \(c_1\) and \(c_2\) terms, and linear algebra says that the determinant is \(0\) by

\[\color{red} \begin{vmatrix} H_{11}-ES_{11}&H_{12}-ES_{12} \\ H_{12}-ES_{12}&H_{22}-ES_{22}\end{vmatrix}=0\label{9B}\]

At this point, the goal is to find the ratio between \(c_1\) and \(c_2\), which is done by solving for \(E\) and putting the value into

\[c_1^2 H_{11} + 2c_1 c_2 H_{12} + C_2^2 H_{22}=0\]

and 

\[c_1(H_{12} - ES_{12}) + c_2(H_{22} - ES_{22}) = 0\]

Solving for the ratio \(c_1 / c_2\) gives you value \(A\), and you can rewrite \(c_2\) in terms of \(c_1\) and \(A\) by \(c_2 = c_1/A\). By setting \(\phi = c_1 f_1 + c_1/A f_2\) gives the value for \(c_1\) and solves for the best wavefunction \(\phi\). 

Hermitian Operators

Hermitian operators are operators that satisfy the general formula

\[\int f^* \hat{A} f d\tau = \int f \hat{A} f^* d\tau\]

where \(f^*\) is the complex conjugate of \(f\). If that condition is met, then \(\hat{A}\) is a Hermitian operator. In this case, \(\hat{H}\) is meets the condition, so \(\hat{H}\) is a Hermitian operator.

Example \(\PageIndex{1}\): Linear Combination of Atomic Orbitals (LCAO) Approximation

Trial wavefunctions that consist of linear combinations of simple functions

\[ | \psi(r) \rangle = \sum_i c_i | \phi_i(r) \rangle \nonumber\]

form the basis of the Linear Combination of Atomic Orbitals (LCAO) method introduced by Lennard-Jones and others to compute the energies and wavefunctions of atoms and molecules. The functions \(\{| \phi_i \rangle \}\) are selected so that matrix elements can be evaluated analytically.

Slater orbitals using Hydrogen-like wavefunctions

\[ | \phi_i \rangle = Y_{lm}(\theta,\phi) e ^{-\alpha r} \nonumber\]

and Gaussian orbitals of the form

\[ | \phi_i \rangle = Y_{lm}(\theta,\phi) e ^{-\alpha r^2} \nonumber\]

are the most widely used forms. Gaussian orbitals form the basis of many quantum chemistry computer codes. Because Slater orbitals give exact results for Hydrogen, we will use Gaussian orbitals to test the LCAO method on Hydrogen, following S.F. Boys, Proc. Roy. Soc. A 200, 542 (1950) and W.R. Ditchfield, W.J. Hehre and J.A. Pople, J. Chem. Phys. Rev. 52, 5001 (1970) with the basis set. Because products of Gaussians are also Gaussian, the required matrix elements are easily computed.

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