Question

Q1

What is the difference between the linear variational method and more "traditional" variational method for approximating eigenstate

and energies. How does this affect the approach to minimize the variational energy?

Q2

What are the Secular Equations? What is the Secular determinant? What is the difference? Does solving one affect the need to solve the

other?

Answer 1

1. The variational method is the other main approximate method used in quantum mechanics. Compared to perturbation theory, the variational method can be more robust in situations where it's hard to determine a good unperturbed Hamiltonian (i.e., one which makes the perturbation small but is still solvable). On the other hand, in cases where there is a good unperturbed Hamiltonian, perturbation theory can be more efficient than the variational method. The basic idea of the variational method is to guess a ``trial'' wavefunction for the problem, which consists of some adjustable parameters called ``variational parameters.'' These parameters are adjusted until the energy of the trial wavefunction is minimized. The resulting trial wavefunction and its corresponding energy are variational method approximations to the exact wavefunction and energy.

2. Following equations form the secular equations:

These equation have a "trivial" and useless solution c₁= c₂ = 0.  The condition that there should exist a nontrivial solution of these equations is that the secular determinant should be zero: (Following is under same determinant)

Everything in this equation is a known number except E.  The equation is therefore an equation for E.  In the present case where the molecular orbital was a linear combination of just two atomic orbitals it is a quadratic equation and has two solutions for E.  These two values of E are the molecular orbital energies.  We always get the same number of molecular orbitals as atomic orbitals we start with.