1. Lecture 26 Molecular orbital theory II

2. Numerical aspects of MO theory
We learn and carry out a mathematical procedure to determine the MO coefficients, which is based on variational theorem, Lagrange’s undetermined multiplier method, and matrix eigenvalue equation.
These are mathematical concepts of fundamental importance and their use and benefit go far beyond quantum chemistry.

3. What is the best wave function?
We have not discussed how to determine the expansion coefficients of LCAO MO’s (except when they are determined by symmetry).
To find the “best” LCAO MO’s with optimized coefficients, we must have a mathematical criterion by which to identify the “best” approximate wave function for a given Hamiltonian.

4. Variational theorem
A general mathematical technique applicable to many problems of chemistry, physics, and mathematics (an important tool for many-body problems complementary to perturbation theory).
We learn its application to quantum mechanics.
Question: we have N normalized wave functions that may or may not be the eigenfunctions of the Hamiltonian H. What is the best wave function for the ground state?

5. Variational theorem
Answer: evaluate the expectation values of energy: and pick the one with the lowest expectation value. That is the best wave function. Why? ΨA ΨB ΨC ΨD

6. Variational theorem
Any wave function can be expanded (exactly) as a linear combination of eigenfunctions of H (completeness).
True ground state WF
Orthonormality
Ground state energy
Normalization

7. Variational theorem
The closer the wave function is to the true ground-state one (which is measured by how close c0 is to unity), the lower the energy gets. Equality holds when and only when the wave function is the true wave function.
We can vary (hence the name variational theorem) a wave function’s shape (while maintaining normalization) to minimize its energy expectation value to systematically improve the approximation.

8. Variational determination of MO’s
The MO is a linear combination of two AO’s:
We find coefficients that minimize the energy (assuming real orbitals and coefficients),
Under the normalization condition.

9. Minimization
At a minimum of a function, its first derivative (gradient) must be zero (a necessary but not a sufficient condition).

10. Constrained minimization
Question: How can we incorporate this constraint during minimization (without it, E is not bounded from below and a minimum does not exist)?
Answer: Lagrange’s undetermined multiplier method.

11. Lagrange’s undetermined multiplier method
We need to minimize E by varying cA and cB,
With the constraint that the wave function remains normalized,
Step 1: We write the constraint into a “... = 0” form:

12. Lagrange’s undetermined multiplier method
Step 2: We define a new quantity L(Lagrangian) to minimize, where we also introduce an additional parameter λ (undetermined multiplier)
Step 3: Minimize L by varying cA and cB as well as λ with no constraint:

13. Lagrange’s undetermined multiplier method
The 3rd equation reduces to the constraint, which we must satisfy
Once the 3rd equation (constraint) is satisfied, the 1st and 2nd equations reduce to minimization of E by varying cA and cB.
Minimization of E wrt two parameters with one constraint is equivalent to minimization of L wrt three parameters

14. Lagrange’s undetermined multiplier as a buy-one-get-one-free
Please minimize this function.
Wait … let me add this constraint. Don’t worry – I only added zero.

15. The first equation

16. The second equation

17. Matrix eigenvalue equation
Overlap integral
Coulomb integral
Resonance integral
(negative)

18. Matrix eigenvalue equation
Use approximation S = 0 to simplify
This matrix corresponds to 1 in arithmetic: a unit matrix
Matrix acts on a vector and returns the same vector, apart from a constant factor λ

19. Matrix eigenvalue equation
Operator eigenvalue equation
Hamiltonian
operator or
matrix
eigenfunction
eigenvector
eigenvalue
Matrix eigenvalue equation

20. What is λ?
This correspondence suggests that λ (undetermined) actually represents energy! +)
The left-hand side becomes
The right-hand side becomes
We find that in fact λ = E !

21. Lagrange’s undetermined multiplier
Many chemistry and physics problems are recast into constrained minimization or maximization.
Lagrange’s method can convert them into unconstrained minimization or maximization with slightly increased dimension.
The undetermined multiplier ends up in representing an important physical quantity.

22. Matrix eigenvalue equation
How do we solve this equation?
Observation: two equations for three unknowns (cA, cB, and E) – indeterminate?
The indeterminacy is removed by the normalization constraint.

23. Matrix eigenvalue equation
If this matrix has an inverse, we invariably obtain a trivial, nonphysical solution:

24. Inverse matrix Definition of inverse Unit matrix
We have verified that this is indeed the inverse

25. Inverse matrix determinant
If the determinant is zero, the inverse does not exist

26. Determinant is nonzero
Inverse matrix
Determinant is nonzero
trivial
nonphysical
solution
physical
solution
Determinant is zero

27. Summary
Variational theorem gives us a way to identify the best approximate wave function for the ground state; it is the one with the lowest energy expectation value.
Variational theorem therefore leads to minimization of expectation value with the constraint that the wave function is normalized.
Constrained minimization can be converted to unconstrained minimization with slightly higher dimension by Lagrange’s undetermined multiplier method.
With these, optimization of LCAO MO’s becomes a matrix eigenvalue equation.