Chapter 8 Approximation Methods, Hueckel Theoryscienide2.uwaterloo.ca/.../Ch_8.pdf · Chapter 8 – Approximation Methods, Hueckel Theory Approximation Methods A) The variational

Winter 2013 Chem 356: Introductory Quantum Mechanics

108

Chapter 8 – Approximation Methods, Hueckel Theory ............................................................................ 108

Approximation Methods ....................................................................................................................... 108

The Linear Variational Principle ............................................................................................................ 111

Example Linear Variations..................................................................................................................... 113

Chapter 8 – Approximation Methods, Hueckel Theory

Approximation Methods

A) The variational principle

For any normalized wave function , the expectation value of , the exact

groundstate energy.

Proof: with

If we would measure the energy we would find with probability

( )

This argument is a bit shaky when has degenerate eigenvalues. You will do a correct proof in the

assignments.

If would not be normalized we can calculate

and then

or in the final form: The variational principle

where is the exact ground state energy

0ˆ ˆ,H H E

n n

n

c ˆn n nH E

nE2

n nP c

ˆ ˆ*( ) n n

n

H H d P E

0 0n n n

n n

P E E P E 0nE E

H

2 *( ) ( )N d

02

1H E

N

(1)

0 0

ˆ*( ) ( )

* ( ) ( )

Domain

Domain

H dE E

d

0E


Chapter 8 – Approximation Methods, Hueckel Theory 109

The variational energy , is exact when , the exact ground state wavefunction.

Examples: use a trial wavefunction that depends on one or more parameters … Then minimize the

trial energy

Some simple (trivial) examples:

Take trial wavefunction of the type

Or

Q: what is ? What is ?

A: The exact wavefunction has the form ,

by minimizing the energy we should get ,

Q: Take the Hamiltonian for the Hydrogen atom, and an state

Take the trial wavefunction

- What are the integrals to evaluate?

- What is the optimal value for ?

- What is the value for ?

A:

Minimizing :

again: exact

You can do those problems yourself and see if you get the correct answer

Nontrivial example:

Take the Hamiltonian for H-atom, s-orbital, and use the trial wavefunction

0E 0( ) ( )

,

,E

22

. 2

1 1ˆ2 2

h u

dH x

dx

2 /2( ) xx e 2 /2xNe

0E

2 /2xNe 0

1

2E

1 0

1

2E

0l 2 2

2

2

0

ˆ42

d d eH r

dr dr rr

re

0E

2

0

02

0

ˆ4( )

4

r r

r r

r e He drE

r e e dr

0 ( )0

E

0

1opt

a

2

0 0

0 0

1

2 4

eE E

a

2

0

0 2

4a

me

2re



now, since the trial wavefunction cannot be exact for any

Note: Gaussian trial orbitals (basis sets) are widely used in electronic structure programs. This is because

integrals are easily evaluated over Gaussians.

This is the origin of the name for the Gaussian Program: It uses Gaussian basic functions!

2 2

2 2

2

0

02

0

ˆ4( )

4

r r

r r

r e He drE

r e e dr

...

1

2 2 2

3/2

0

3 2

2 (2 )e

e

m

2 2

0

3/2 1/2

0

30

2 (2 )e

E e

m

1 2

23/2 2

0

2

3(2 )

em e

2 4

3 2 4

018

e

opt

m e

4

0 2 2

0

4( )

3 16

e

opt

m eE

2

0

0 0

( ) 0.4244

opt

eE

a

2

0

0 0

1

2 4

eE

a

0 0E E



The Linear Variational Principle

Consider a trial wave function

Let us assume for simplicity

- Real coefficients, functions,

- Orthonormal expansion functions:

Then we can try to optimize the coefficients

,

( ) ( )n n m m

n m

D c f c f d

Then

Or

( ) ( )n n

n

c f

nf

*( ) ( )n m nmDomain

f f d

0

*( ) ( )( )

* ( ) ( )

H d NE c

Dd

( ) ( )n n m mN c f Hc f d

,

ˆ( ) ( )n m n m

n m

c c f Hf d

,

n nm m

n m

c H c

,

( ) ( )n m n m

n m

c c f f d 2

,

,

n m n m n

n m n

c c c

0k

E

c

k

2

0k k

k

N DD N

c cN

c D D

k k

N N D

c D c

k k

N DE

c c

km m n nk

m nk

NH c c H

c

2 2n k

nk k

Dc c

c c



Since , this is twice the same equation.

This has the form of a matrix eigenvalue equation!

m

kmk H

H and c c E

H

We can also write 0kmE c H

This has the form of a linear equation.

, with A E H 1

This type of equation only has a solution which .

Hence det 0E H 1 equation for “secular determinant”

Let us discuss examples later. For now I want to draw the analogy:

Schrodinger equation

If we make a basis expansion

Then we get a matrix type Schrodinger equation

c EcH

With ˆ*( ) ( )nm n mH H d H

Such an eigenvalue equation has solutions for an matrix. They represent approximations

to the ground and excited states.

If the basis is not orthonormal the define and c cEH S (see MQ)

Or det 0SE H eigenvalues .

0km m k n nk k

m n

H c Ec c H Ec

nk knH H

0km m kH c Ec

km m k

m

H c c E

0Ac

det 0A

E

H E

n n

n

c f |n m nmf f

M M M

*( ) ( )nm n mS H d E



Example Linear Variations

Consider particle in the box

Now add to a linear potential

Use as a trial wave function

2 2 2

22n

nE

ma

The eigenvalues of this Hamiltonian are

This happens to be pretty good solution, especially as is small

2 2

22

dH

m dx

2( ) sinn

n xx

n a

H 0Vx

a

1 2

2 2 2sin sin

x xc c

a a a a

, 1,2i j 2 2

0

2

2sin sin

2ij

Vi x d j xH x

a a m a adx

0 0

1 2

0 0

22

16

2 9

16

29

V VE

V VE

2

0 02 2

2maV

0 02 2 2

2

0 0

2

161

2 9

1624

29

Hmn

2 2

22n

ma

1/22

0 0

2

5 3219

2 2 9n

0

0

0

15 3 2

2 2

42

V

V

0V



Other instructive example: consider particle on the ring

, with the degenerate solutions

Now apply a magnetic field, which adds to the Hamiltonian ˆzL i

Under the influence of the perturbation the levels split. Calculate the energy splitting.

I used wrong formula; sign is wrong

H E 1 =

2 2

2 22mR

1m

1cos

1sin

2

1 22E

mR

ˆ ˆ2

e z z

e

eB L L

m

2 e

e

m

0 zH H L

1 2

1 1cos sinc c

1 1ˆ cos sinz zL i

1 1ˆ sin coszL i

2

2

2

2

2

2

E imR

i EmR

2

22 2

2det 0

2H EZ E

mR

2

22E

mR

2

22E

mR



Can I find eigenfunctions?

What are the eigenfunctions then?

(we normalize, factor)

Bottom line: We can indicate perturbation

Diagonalize over degenerate states.

Examples: H-atom:

Diagonalize over 2p orbitals … , eigenfunctions from diagonalizing

Hamiltonian. Everything comes out by brute force.

Example 2: add in additional magnetic interaction

diagonalize over 2p and 2s orbitals

all the splitting from diagonalization

The linear variational principle is a very powerful tool to calculate approximate eigenfunctions

2

22

22

2

1 12

2

2

imR

i imRi

mR

2

22

22

2

1 12

2

2

imR

i imRi

mR

1cos sin ~ ii e

i

1cos sin ~ ii e

i

1

2

2 2 2

2 2 22 2

i ii e emR mR

2 2 2

2 2 22 2

i ii e emR mR

0H H V

H

( )

0 ( )neH H g v L S

H 3

2

2 p 1

2

2 p 6 6

( )

0

2 ˆ( )2 2

ne

z z z z

e e

e eH H g v L S B L B S

m m

H



It is widely used to calculate the splitting of energies in a degenerate manifold, when adding a

perturbation.

When the energies of a Hamiltonian are not degenerate, one can get a good estimate of the energy

correction due to a perturbation , by calculating .

Hence if , then eigenvalues of are given to first approximation by

These are just the diagonal elements of the Hamiltonian matrix = First order Perturbation Theory:

If we go back to box + linear field

all energies are shifted by

If zero-order states are degenerate, first-order perturbation theory is useless. Instead use linear

variational principle

Example in sin( )mx basis

choose other basis: results

always diagonalize over zeroth-order states: degenerate first-order perturbation theory

Another example of : Hückel -electron theory

In organic chemistry, many molecules are essentially planar. The plane contains “ ” carbon, oxygen,

nitrogen atoms. The out of plane -orbitals constitute the -orbitals. The molecule’s - orbitals are

0H

V V

(0)

0 n n nH E ˆ ˆ ˆeH H V

0ˆ *( ) ( )n n n nH H V d

(0) *( ) ( )n n nE V d

2 2

0

2

2

VdH x

m adx

0H v

2sinn

n x

a a

2 2(0)

22nE

ma

0

0

2ˆ sina

n n

V n xV xdx

a a a

0 02

2

V Va

a a a

(0) 0

n n

VE E

a 0V

a

ˆzL

ime ime

Hc cE

2sp

zP



linear combinations of the atomic -orbitals. One can parameterize a one-electron effective

Hamiltonian matrix as follows. Let us restrict ourselves to carbons.

D

Rule: on diagonal for any two adjacent atoms connected by a -bond

Following the variational principle we

a) Diagonalize the Hamiltonian orbital energies , eigenvectors

b) Fill up orbital levels from the bottom up putting an and a electron in each level.

Occupy as many levels as you have -electrons

c) If levels are degenerate, fill them up with -electrons first, then add additional -

electrons

d) Total energy:

e) Density Matrix ,

kl k l

occupied

D c c

(see McQuarrie)

zP

2sp

H

0 0

0

0

0 0

H

0 0

0 0

0 0

H

0

0

0

0

H

(1) (2)0 ~ Ne

z zP V P d

kE kc

occupiedorbitals

E

kl kl

kl

E H D



This procedure would work fine in MathCad

How do we do it on paper? Take det EH 1

ethylene

- electrons

Using and is a bit tedious for larger problems

divide each column by and define

Or

Always:

is double solution.

ok

0E

E

2 2 0E

E

E

2

E

x

E x

21

1 01

xx

x 1x E

1 1

1 1 0

1 1

x

x

x

1 1

1 1

x

x

3 31 1 3 2 0x x x x x x

1x 1 3 2 0

2x 8 6 2 0

0i

i

x TrN

1x

2 21 2 2 1 2x x x x x

3 3 2x x



3 -electrons (4-fold degenerate)

Triplet (3-fold degenerate)

Singlet

What if we would look at the singlet state of the anion?

This is not a stable structure, the molecule would distort

Jahn-Teller Distortion!

I might ask questions of determinant too hard to solve

k kE x

4*4



I would give you the solution

You show that is your secular determinant

You can guess the orbitals (phases) from symmetry arguments: The orbitals are always

symmetric or antisymmetric with respect to plane or axis of symmetry

If you know value of you can solve

:

orthogonal combinations

:

1 2 3 4, , ,x x x x

1 2 3 4( )( )( )( )x x x x x x x x

x

1 1

1 1 0

1 1

x a

x b

x c

1x 0a b c

(1 -1 0)

1 1 -2

2x

2 1 1 1 0

1 2 1 1 0

1 1 2 1 0

Documents

Chapter 8 Approximation Methods, Hueckel Theoryscienide2.uwaterloo.ca/.../Ch_8.pdf · Chapter 8 – Approximation Methods, Hueckel Theory Approximation Methods A) The variational