Chemistry 301/ Mathematics 251Chapter 4

Eigen value problems

Eigen Value Problems

Ax x

Operator scalar

Matrix

Ax x

Differential equation

H E

Not all values of x or will work

called Eigen value

x called Eigen vector (function)

Matrix Eigen Value Problems

0

0

Ax x

Ax Ix

Ax Ix

A I x

Nontrivial solution

must be singular

0

A I

A I

Matrix Eigen Value Problems

11 12

21 22

a a

a a

A

11 12

21 22

11 12

21 22

11 22 12 21

211 22 11 22 12 21

00

0

0

0

0

a a

a a

a a

a a

a a a a

a a a a a a

Will give 2 Eigen values

# of Eigen values is same as order of matrix

Matrix Eigen Value Problems

2 2

2 1

A

2

2

2 20

2 1

2 1 4 0

2 1 2 4 0

6 0

2, 3

Eigen vectors?

1 1

2 2

1 2 1

1 2 2

2 22

2 1

2 2 2

2 2

x x

x x

x x x

x x x

Ax x

Substitute each into original Eigen value equation and determine vectors

For = 2

2 12x xBoth equations give

1

1

2c

1 1

2 2

1 2 1

1 2 2

2 23

2 1

2 2 3

2 3

x x

x x

x x x

x x x

For = -3

2 1½x xBoth equations give

2

2

1c

NormalizationFind constants such that dot product of each Eigen vector with itself is 1

For = 2

21

21

2 11 5

11 5

15

25

11 2 1

2

1 4 1

c

c

c

c

For = -3

22

22

2 12 5

12 5

25

15

22 1 1

1

4 1 1

c

c

c

c

Orthogonal vectorsConsider the dot product of two different Eigen vectors

1 2 1 2

21 2 2 2 0

1c c c c

Eigen vectors are orthogonal

Transformation matrixForm a transformation matrix where columns are normalized Eigen vectors

1 25 5

2 15 5

P

1 2 1 25 5 5 5

2 1 2 15 5 5 5

2 2

2 1

2 0

0 3

t

B P AP

Diagonalizes matrix and elements of diagonal matrix are Eigen values

Back to Inertia Tensor (Chapter 2)Water revisited

0

0

0 0

xx xy

xy xx

zz

I I

I I

I

I

2

2

2 2 2

2 2 2

0

0 0

0 0

0

0

2 0

2 4 4

2

xx xy

xy xx

zz

zz xx xx xy

zz

xx xx xy

xx xx xy

xx xx xx xy

xx xy

I I

I I

I

I I I I

I

I I I

I I I

I I I I

I I

Back to Inertia Tensor (Chapter 2)Eigen vectors

0

0

0 0

0

1

zz

xx xy

xy xx zz

zz

xx xy zz

xy xx zz

zz zz

I

I I x x

I I y I y

I z z

I x I y I x

I x I y I y

I z I z

x y

z

0 0

0 0

0 0

xx xy

xx xy

zz

I I

I I

I

I

0

0

0 0

0

12

xx xy

xx xy

xy xx xx xy

zz

xx xy xx xy

xy xx xx xy

zz xx xy

I I

I I x x

I I y I I y

I z z

I x I y I I x

I x I y I I y

I z I I z

z

y x

0

0

0 0

0

12

xx xy

xx xy

xy xx xx xy

zz

xx xy xx xy

xy xx xx xy

zz xx xy

I I

I I x x

I I y I I y

I z z

I x I y I I x

I x I y I I y

I z I I z

z

y x

1 12 2

1 12 2

0

0

0 0 1

P

Differential Eigen value ProblemsSchrödinger Eqn is an Eigen value problem

2

2

ˆ

ˆ , ,2

H E

H V x y zm

2

, , is what makes the problem difficult

constant (free particle - particle in a box,

particle on a ring, particle on a sphere)

½ (Harmonic Oscillator)

V x y z

V

V kx

Systems of 1st order Linear Equations

Consider systems with constant coefficients

1 11 1 12 2 1 1

2 21 1 22 2 2 2

1 1 2 2

n n

n n

n n n nn n n

x a x a x a x g t

x a x a x a x g t

x a x a x a x g t

1 2, , , nx t x t x twhere

In matrix form

X AX G 1

2

n

x

x

x

X

1

2

n

g

g

g

G

11 12 1

21 22 2

1 2

n

n

n n nn

a a a

a a a

a a a

A

Systems of 1st order Linear Equations

Special case

ti ix t e

xi must have the form

Why?

X AX

1 11 1 12 2 1

2 21 1 22 2 2

1 1 2 2

n n

n n

n n n nn n

x a x a x a x

x a x a x a x

x a x a x a x

Systems of 1st order Linear Equations

1

2t

n

e

X ξ ξ

X AX

te X ξ X

X AX

Eigen value problem

Systems of 1st order Linear Equations

1

2

1 1

4 1

x t

x t

X X X

2

1 10

4 1

1 4 0

1 2 0 or 1 2 0

1,3

Systems of 1st order Linear Equations

For 1

1

2

te

X

1 1

2 2

1 11

4 1

AX X

1 2 1

1 2 2

2 1

4

2

1

1

2tc e

X

1 1

2 2

1 13

4 1

AX X

1 2 1

1 2 2

2 1

3

4 3

2

32

1

2tc e

X

For 3

1 3

2

te

X

31 2

1 1

2 2t tc e c e

X

Systems of 1st order Linear Equations

f r

f r

k k A t A t

k k B t B t

AX X

A B

f r

f r

dAk A k B

dtdB

k A k Bdt

2

0

0

0

0

0,

f r

f r

f r r f

f r r f r f

f r

f r

k k

k k

k k k k

k k k k k k

k k

k k

Systems of 1st order Linear Equations

For 0

1

1

a

b

X

1

1

1 1

1 1

11

0

0

0

f r

f r

f r

f r

f

r

k k a

k k b

k a k b

k a k b

k ab

k

AX X

For f rk k

2

2

f rk k tae

b

X

2 2

2 2

2 2 2

2 2 2

2 1

f r

f rf r

f r f r

f r f r

k k a ak k

k k b b

k a k b k k a

k a k b k k b

b a

AX X

1

1

f

r

a kk

X 2

1

1f rk k t

a e

X

Systems of 1st order Linear Equations

1 2

11

1f rk k t

f

r

a a ekk

X 0 0@ 0 ,t A A B B

01 2

0

0 1 2

0 1 2

0 0 0 01 2 0

11

1

1 1f f

r r

f

r

f

r

k kk k

Aa ak

Bk

A a a

kB a ak

A B A Ba a A

0 0 0 00

11

11 1f r

f f

r r

k k t

k kfk k

r

A B A BA ek

k

X

Systems of 1st order Linear Equations

0if 0B

0 00

00

11

11 1

11

1

f r

f f

r r

f r

k k t

k kfk k

r

k k tfr

fr f r f

r

A AA ek

k

k Ak Aek

k k k kk

X

t

0

1eqm r

eqm feqm r f

r

A k AkB k k k

X

Systems of 1st order Linear Equations

0 0

0

0 0

a

a b

b

k A t A t

k k B t B t

k C t C t

AX XA B Ca bk k

a

a b

b

dAk A

dtdB

k A k BdtdC

k Bdt

0 0

0 0

0

00

0

0, ,

a

a b

b

ba

b

a b

a b

k

k k

k

kk

k

k k

k k

Systems of 1st order Linear Equations

For 0 1

1

1

a

b

c

X 1

1

1

1 1

1 1 1

1

1

1

0 0

0 0

0 0

0 0

0 0

0

is arbitrary

0

0

1

a

a b

b

a

a b

b

k a

k k b

k c

k a a

k a k b b

k b

c

c

AX X

X

Systems of 1st order Linear Equations

For ak

2

2

2

ak t

a

b e

c

X 2 2

2 2

2 2

2 2 2

2 2 2 2 2

2 2 2 2

2

0 0

0

0 0

is arbitrary

1

a

b a

b

a

a a

b a

b

b a

a

a b a

b

a a

ka b a k k

kb a k

k k tk k

kk k

k a a

k k b k b

k c c

k a k a a

k a k b k b b a

k b k c c b

a e

AX X

X

Systems of 1st order Linear Equations

For bk

3

3

3

bk t

a

b e

c

X 3 3

3 3

3 3

3 3 3

3 3 3 3

3 3 3 3

3

0 0

0

0 0

=0

is arbitrary

0

1

1

b

a

a b b

b

a b

a b b

b b

k t

k a a

k k b k b

k c c

k a k a a

k a k b k b b

k b k c c b

b e

AX X

X

Systems of 1st order Linear Equations

1 2 3

10 0

0 1

1 1

a a b

b a

b

b a

k k t k tk k

kk k

c a e b e

X

General Solution

At t=0

0 0

0

0 1 2 3

0

0 2

0 2 3 3 0 0

0 1 2 3 1 0 2 3 0 0 0

10 0

0 1

1 1

a

b a

b

b a

b a aa a

b a b a b a

b b

b a b a

kk k

kk k

B k k A kk kk k k k k k

k kk k k k

A

B c a b

C

A a

B a b b B A

C c a b c C a b A B C

X

Lotka-Volterra Mechanism

A+X 2X

X+Y 2Y

Y B

a

a b

b c

dAk AX

dtdX

k AX k XYdtdY

k XY k Ydt

System of nonlinear DE

Solve subject to A constant (replenished as needed)

0

0

0a

a b

b c

dAk A X W

dtdX

k A X k XYdtdY

k XY k Ydt

Lotka-Volterra MechanismApproximate solution

Find critical points

0dX dY

dt dt

0

0

0

0

0

0

a b

b c

a b

b c

dXk A X k XY

dtdY

k XY k YdtX k A k Y

Y k X k

00 0

0

aa b

b

cb c

b

k Ak A k Y Y

k

kk X k X

k

Lotka-Volterra MechanismClose to the critical points

0 0

00

0 0

0

00

0

0

0

c c aa b

b b b

b c

c b

c a ab c

c a ca

b b

a ac c

b b

b b

a

c

b

a

a

b a

b

c

k k k Aduk A u k u v

dt k k k

k uv k v

k v k uv

k k k Ak A

k k

k A k Ak k

k k A k Advk u v k v

dt k k

k A u k A

k

k

u

k v kk

uv k A u

k A u k uv

kv

0c a

b b

k k Adu dX dv dYu X v Y

k dt dt k dt dt

Lotka-Volterra Mechanism

0

0

0

0

0

0

c b

a b

c

a

duk k uvudt

k A k uvdv v

dt

k u

k A v

0

0

0c

a

k u u

k A v v

0

20

0

0

0

c

a

a c

a c

k

k A

k k A

i k k A i

Lotka-Volterra Mechanism

1 1

0 1 1

11 1 1

0

0c

a

cc

k a ai

k A b b

i ak b i a b

k

For i

1

1

iataue

bv

1

1iat

c

ua eiv k

Lotka-Volterra Mechanism

2 2

0 2 2

22 2 2

0

0c

a

cc

k a ai

k A b b

i ak b i a b

k

For i

2

2

iataue

bv

2

1iat

c

ua eiv k

Lotka-Volterra Mechanism

1 2

1 1iat iat

c c

ua e a ei iv k k

1 2

1 2

cos sin cos sin

cos sin cos sin

cos sin

cos sin

c

c

a t i t a t i tui a t i t a t i tv k

A t B t

B t A tk

1 2

1 2

A a a

B i a a

Lotka-Volterra Mechanism

time

Concentration X

Y

Xcritcial

Ycritical

Lotka-Volterra Mechanism

X

Y

Molecular Orbital (MO) Theory

A A B BC C

Linear Combination of Atomic Orbitals (LCAO)

, atomic orbital for atom A, and B, respectively

real, normalizedA B

2

2

ˆ

ˆ

ˆ

H E

H E E

H d E d

2

2 2

2 2 2 2

ˆ

ˆ ˆ ˆ ˆ

2A B

A A B B A A B B A A B B

A A A A B A B A B B A B B B

A B A B B A

C C H C C d E C C d

C H d C C H d C C H d C H d

E C d C d C C d

Molecular Orbital (MO) Theory

2 2 2 2

2 2 2 2

2

2 2

A AA A B AB A B BA B BB A B A B AB

A AA A B AB B BB A B A B AB

C H C C H C C H C H E C C C C S

C H C C H C H E C C C C S

Want to find the combination that gives the lowest energy

0A B

E E

C C

Gives

where

1= , ,

1AA AB A AB

AB BB B AB

E

H H C S

H H C S

HC SC

H C SEigen value problem with overlap

Eigen value problems with overlap

consider an transformation such that t

E

HC SC

α

α Sα I

Define

equation to solve becomes

t t

E

E

E

C αC

HαC SαC

α HαC α SαC

H C C

How do we determine ?α

Form a transformation matrix from the collection

of normalized Eigen vectorst

P

P SP Λ

Eigen value problems with overlap

SV V

Diagonal matrix of Eigen values

1 0

0 n

Λ

1½

0

0 n

Λ 1

1

1½ -½

1

0

0n

Λ Λ

Eigen value problems with overlap

½ ½

-½ -½ -½ ½ ½ -½

-½ -½

-½ -½

-½

t

t

tt

t

P SP Λ Λ

Λ P SPΛ Λ Λ Λ Λ I

P Λ SPΛ I

PΛ SPΛ I

α PΛ

Eigen value problems with overlap

1. Find Eigen values and vectors of S

2. Form 3. Transform H into H4. Find Eigen values and vectors of H5. Determine C

Molecular Orbital (MO) Theory

where

1= , ,

1AA AB A AB

AB BB B AB

E

H H C S

H H C S

HC SC

H C S

2 2

Eigen values of

10

1

1 0

1 1 0

1 ,1

AB

AB

AB

AB AB

AB AB

S

S

S

S S

S S

S SV V

Molecular Orbital (MO) Theory

12

For 1

11

1

AB

AB A AAB

AB B B

A AB B A AB A

B A

S

S V VS

S V V

V S V V S V

V V

12

For 1

11

1

AB

AB A AAB

AB B B

A AB B A AB A

B A

S

S V VS

S V V

V S V V S V

V V

11 112 2 -½

1 1 12 2 1

0

0AB

AB

S

S

P Λ

½ ½

½ ½

1 111 1 2 1 2 112 2-½

1 1 1 1 12 2 1 2 1 2 1

0

0

AB ABAB

AB AB AB

S SS

S S S

α PΛ

Molecular Orbital (MO) Theory

½ ½ ½ ½

½ ½ ½ ½

1 1 1 12 1 2 1 2 1 2 1

1 1 1 12 1 2 1 2 1 2 1

½ ½

½ ½

2

2 1 2 1 1

2

2 12 1 1

AB AB AB AB

AB AB AB AB

t

S S S SAA AB

AB BBS S S S

AA AB BB AA BB

AB AB AB

AA BB AA AB BB

ABAB AB

H H

H H

H H H H H

S S S

H H H H H

SS S

H α Hα

Molecular Orbital (MO) Theory

½ ½

½ ½

2

2 1 2 1 1

2

2 12 1 1

01

01

AB AB AB

ABAB AB

AB

AB

S S S

SS S

S

S

H

Special case A = B

AA BB

AB

H H

H

Molecular Orbital (MO) Theory

½ ½

½ ½

½

½

1 12 1 2 1

1 12 1 2 1

12 1

12 1

For 1

1

0

1

0

AB AB

AB AB

AB

AB

AB

S S

S S

S

S

ES

C

C αC

Special case A = B

½

2 1A B

AB

N

S

Molecular Orbital (MO) Theory

½ ½

½ ½

½

½

1 12 1 2 1

1 12 1 2 1

12 1

12 1

For 1

0

1

0

1

AB AB

AB AB

AB

AB

AB

S S

S S

S

S

ES

C

C αC

Special case A = B

½

2 1A B

AB

N

S

Molecular Orbital (MO) TheorySpecial case no overlap

AA A

BB B

AB

H

H

H

A A A

B B B

E

C CE

C C

HC C

2

2 2

2 2

2 22 2

0

0

0

0

4 4 4

2 2

A

B

A B

A B A B

A B A B

A B A B A B A B A B

E

E

E E

E E

E E

E

Molecular Orbital (MO) TheorySpecial case HF without overlap

13.6 eV (H)

18.6 eV (F)

1.0 eV

A

B

13.6 1.0

1.0 18.6A A

B B

E

C CE

C C

HC C

2

2

2

13.6 1.00

1.0 18.6

13.6 18.6 1.0 0

252.96 32.2 1.0 0

32.2 251.96 0

32.2 32.2 4 251.9613.4 eV, 18.8 eV

2

E

E

E E

E E

E E

E

Molecular Orbital (MO) TheorySpecial case HF without overlap

13.6 1.013.407

1.0 18.6

13.6 13.407

0.1926

0.1926 0.98 0.19

A A

B B

A B A

B A

H F H F

E

C C

C C

C C C

C C

N

HC C

13.6 1.018.793

1.0 18.6

13.6 18.793

5.193

5.193 0.19 0.98

A A

B B

A B A

B A

H F H F

C C

C C

C C C

C C

N

Molecular Orbital (MO) TheoryHomework

Find the allowed energies for the following

0

0

H

H