Lecture 26Molecular orbital theory II

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.

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.

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?

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

Variational theorem

Any wave function can be expanded (exactly) as a linear combination of eigenfunctions of H (completeness).

True ground state WF

Ground state energy

Normalization

Orthonormality

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.

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.

Minimization At a minimum of a function, its first derivative

(gradient) must be zero (a necessary but not a sufficient condition).

0; 0A B

E E

c c

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.

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:

0; 0A B

E E

c c

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:

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

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.

The first equation

The second equation

Matrix eigenvalue equation

Resonance integral(negative)

Coulomb integralOverlap integral

Matrix eigenvalue equation

Use approximation S = 0 to simplify

A A A

B B B

c c

c c

1 1 0

1 0 1A A A

B B B

c c cS

c c cS

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

Matrix eigenvalue equation

H E Operator eigenvalue equation

A A A

B B B

c c

c c

Matrix eigenvalue equation

eigenfunctioneigenvectoreigenvalue

Hamiltonianoperator ormatrix

What is λ? This correspondence suggests that λ

(undetermined) actually represents energy!

A

B

c

c

The left-hand side becomes

+)

We find that in fact λ = E !

The right-hand side becomes

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.

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.

1

1A A A

B B B

c cSE

c cS

Matrix eigenvalue equation

0

0A A

B B

E ES c

ES E c

1

1A A A

B B B

c cSE

c cS

10

0A A

B B

c E ES

c ES E

If this matrix has an inverse, we invariably obtain a trivial, nonphysical solution:

Inverse matrix

a b e f ae bg af bh

c d g h ce dg cf dh

11 0

0 1

a b a b

c d c d

Unit matrixDefinition of inverse

11a b d b

c d c aad bc

We have verified that this is indeed the inverse

Inverse matrix

11a b d b

c d c aad bc

determinant

If the determinant is zero, the inverse does not exist

Inverse matrix0

0A A

B B

E ES c

ES E c

trivialnonphysicalsolution

physicalsolution

Determinant is nonzero

Determinant is zero

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.