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