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5.80 Small-Molecule Spectroscopy and Dynamics
Fall 2008
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5.80 Lecture #1
Fall, 2008
Page 1
Lecture #1: Matrices are Useful in Spectroscopic Theory
For next time, read handout on the Van Vleck transformation and look at the notes on coupled harmonic
oscillators.
Outline
*
Correspondences
bra
operator
*
*
row column
N × 1
column vector
⎛ ⎞
ψi → i = ⎜ ⎟
⎝ ⎠
ket
*
ψi → i =
1 × N
 →O
O
row vector
N × N square matrix
Solution of Schrödinger Equation corresponds to solving a determinantal secular equation for the
{Ek}
ψ
;
The {Ek} are the H kk
in the special “diagonal” representation of H
*
The {ψk} are obtained from the basis states {φk} as columns of the unitary matrix T from the
similarity transformation T†HφT = Hψ that “diagonalizes” Hφ;
*
Matrix representation of arbitrary f(Q).
Usual procedure to obtain a fit model.
 a
exact H
differential
operator
a complete
set of basis
functions
Approximate H perturbation
Fit model
theory
an infinite
* finite
matrix
simplified
e.g. Born- expressed Van Vleck
a finite matrix
matrix
Oppenheimer in terms of transformation expressed in
combine
OR
and normal structure
↓
linearly
terms of
* algebraic
mode
dependent
parameters introduces
microscopic
formulas
separations e.g. fij force many small
model
molecular
expressed in
of variables constants
parameters
constants
terms of
terms
effective
not necessarily
molecular
the same even
constants
if they have
same name
Dunham
constants
ωe, ωexe
exact H
truncate H
5.80 Lecture #1
Fall, 2008
Page 2
Matrices are useful because:
*
they display all necessary information;
*
can be “read” and simplified by perturbation theory “order sorting” via
*
labor saving tricks for avoiding the evaluation of unnecessary integrals, such as formulas
 in the Harmonic Oscillator basis set.
 and P
for all matrix elements of Q
No integrals actually evaluated.
No functions actually looked at.
H ij
;
E − E oj
o
i
Quantum Mechanics Operators follow the rules of matrix multiplication.
B
 ) ψ dτ =
e.g. ∫ ψ*i ( A
∑
j
( ∫ ψ Aψ dτ ) ( ∫ ψ Bψ dτ )
k
*
i
k
*
k
j
completeness
of {ψ}
= ∑ A ik Bkj = (AB) ij
k
This is very useful because we can generate many matrices by simple operations on one matrix.
E.g.
Q = R – Re
V(Q) = ∑ cn Qn
n
n
matrix of Q
(Qn) = (Q)n
Q × Q × … Q
so instead of evaluating Q1, Q2, Q3, etc. we just evaluate Q and derive all the rest by matrix operations.
↑ formulas, not integrals!
⎡* shortcuts to selection rules
There is even some diagrammatic insight ⎢* calculations of a specific element of Qn
⎣
5.80 Lecture #1
Fall, 2008
0
0
0
Q=
0
0
0
Page 3
0
0
0
0
0
0
0
(Q 2 ) = 0
0
0
0
0
0
v = ±1 non-zero matrix elements of Q
0
0
v = 0, ±2 selection rules
At the end of this lecture we will see that we are
not restricted to integer powers of Q.
etc.
Suppose we have a convenient and complete (orthonormal) basis set {i}.
) can be expanded in terms of the {}.
Any arbitrary function (including an eigenfunction of H
wavefunction picture
N
k = a i
can be k
i
*i k d = a ki
i=1
matrix picture
*k = U
U is a transformation that converts {} into {}.
0
0
k k = 1 1
i
=
i
0 0 k = 0…1…0 N 1
*
i i = 0…1…0 column matrix
5.80 Lecture #1
Fall, 2008
Page 4
N
We know ψk = ∑ a ki φ i
i=1
We want ψ = Uφ where U is N × N matrix that transforms φ into ψ.
By this we mean
⎛
0
⎞
⎛
U k1 ⎞
⎛
0
⎞
⎛
0
⎞
⎜ ⎟
⎜
⎟ ⎜
⎟
⎜
⎟
⎜
⎟
⎜ 0
⎟ ⎜ U k 2 ⎟
⎜
⎟
ψk =
⎜1
⎟ =
∑
U ki φ i =
⎜
⎟ +
⎜
⎟ + …
⎜
⎟
⎜ ⎟
⎜
⎟ ⎜
⎟
⎜
⎟
i
⎜
⎟
⎜
⎟ ⎜
⎟
⎜
⎟
⎜ ⎟
⎜
⎟ ⎜
⎟
⎜
⎟
0
0
0
U
⎝
⎠
ψ
⎝
⎠
φ
⎝
⎠
φ
⎝
kN ⎠
φ
k-th row of U
⎛
0
⎞
⎛
U k1 ⎞
⎜ ⎟ ⎜
⎟
⎜
⎟ ⎜
⎟
⎜1
⎟ =
⎜
⎟
⎜ ⎟ ⎜
⎟
⎜
⎟ ⎜
⎟
⎜ ⎟ ⎜
⎟
0
U
⎝
⎠
ψ
⎝
kN ⎠
φ
U ki = a ki = ∫ φ*iψk dτ
We can go in the opposite direction
U–1ψ = φ
U −ij1 = U*ji
and show that U–1 = U†
φi is the i-th row of U–1 or the i-th column of U*.
5.80 Lecture #1
Fall, 2008
Page 5
Derivation of Secular Determinant
Now for every operator there is a matrix representation.
⎛
O11  O1N ⎞
⎟
 → O =
⎜ 
O
 ⎟
⎜
⎜
⎟
⎝
O N1  O NN ⎠
φ
φ dτ
O ij =
∫
φ*i O
j
Schrödinger Equation in ψ eigenbasis picture
ψ = E ψ
H
k
k k
mixing coefficient
∞
ψk = ∑ a ki φ i
basis function - convenient
usually defined as
eigenfunctions of a part
 called H
°
of H
 °φ = E °φ
H
i
i i
i=1
Matrix notation:
H φij =
∫
φ dτ =
φ*i H
j
φ
iH j
φ
5.80 Lecture #1
Fall, 2008
Page 6
φ
φ ⎞
⎛
k ⎞
⎛
H11
⎛
a1k ⎞

H1N
a1
⎜ φ
⎟ ⎜ ⎟
⎜ ⎟
⎜ H 21
⎟ ⎜ ⎟ = E ⎜ ⎟
k⎜
⎜
⎟ ⎜ ⎟
 ⎟
⎜
φ
⎟
⎜
k ⎟
⎜ k ⎟
⎝
H N1
⎠
φ ⎝
a N ⎠
φ
⎝
a N ⎠
φ
convenient
representation
⎛
φ
⎜∑
H1i
⎜ i
⎜
⎜∑
H 2 i
LHS =
⎜ i
⎜

⎜
⎜
⎜∑
H Ni
⎝
i
initially unknown
mixing coefficients initially unknown
energies
⎞
a ki ⎟
⎟
⎛
a1k ⎞
⎟
⎜ ⎟
a ki ⎟
⎜
⎟
⎟ = Ek ⎜ ⎟

⎟
⎜
k ⎟
⎟
⎝
a N ⎠
⎟
a ki ⎟
⎠
move everything into one column matrix
⎛
⎞
k
k
⎞
⎜ ∑
( H1i a i − δ1i E k a i ) ⎟ ⎛
k
⎜ i
⎟ ⎜∑
a i (
H1i − δ1i E k )
⎟ ⎛
0
⎞
⎟ ⎜ 0
⎟
⎜
⎟ ⎜ i
k
k
⎜
⎟ =
⎜ ⎟
⎜∑
( H 2i a i − δ 2i E k a i ) ⎟ =
etc.
⎜
⎟ ⎜ 
⎟
⎜ i
⎟
⎜
⎟ ⎜
⎟
⎜
⎟
etc.
⎜
⎟ ⎝
0
⎠
⎜⎜
⎠
⎟⎟ ⎝
⎝
⎠
A system of N linear homogeneous equations in N unknowns ( a ki i = 1…N ) . A nontrivial solution
exists if the determinant of coefficients of the unknown {a ki } is zero.
H11 − E k
0 = H 21
H12
H 22 − E k
H1N
unit matrix
φ
= |H –1E k|
HNN − E k
5.80 Lecture #1
Fall, 2008
Page 7
This is the secular determinant. Must solve for special values of Ek which satisfy the requirement
0 = |Hφ – 1Ek|.
 . {E } k = 1, 2, … N
These are eigenvalues of H
k
COMPUTERS!
1.
start with complete set {φ}
2.
compute all Hφ matrix elements
3.
“diagonalize” Hφ
time required ∝ N3
4.
solve for {a ki } : one set of N coefficients for each of the N eigenvalues.
How to “diagonalize” Hφ? Seek a similarity transformation T.
ψ
⎛
H11
⎞
⎜
⎟
ψ
H 22
⎜
⎟
−1 φ
φ
T H T=H =
⎜
⎟

⎜
⎟
ψ
H NN
⎝
⎠
0
0
H iiψ ≡ E k are the eigenvalues we seek.
Computer generates T iteratively (Jacobi Rotations, see handout)
Special properties:
*
all Quantum Mechanical operators (that correspond to real observables) are “Hermitian” (real
eigenvalues) which means that their matrix representations have the property
O = O†
Oij = O*ji
*
T is “unitary”
T†T = 1
(i.e., T† = T–1)
5.80 Lecture #1
Fall, 2008
Now I’ll show that T contains the information we seek about the {a ki } .
⎛
0 ⎞
⎛
0 ⎞
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
ψ⎜ ⎟
† φ
φ
T H T = H where H 1 = E k ⎜1 ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎝
0 ⎠
ψ
⎝ 0
⎠
ψ
ψk
left multiply by T
T†HφTψk = Ekψk
TT† = 1
TT †H φ Tψk = TE k ψk
H φ ( Tψk ) = E k ( Tψk )
an eigenvector of Hφ
an eigenvalue equation
⎛
0
⎞ ⎛ T1k ⎞ ⎛ a1k ⎞
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟ ⎜
⎟
Tψk = T ⎜1 ⎟ =
⎜ ⎟ =
⎜ ⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜
⎟ ⎜
⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ k ⎟
⎝
0
⎠
ψ
⎝
TNk ⎠ψ
⎝ a N ⎠
φ
kth column of T
The columns of T are the eigenvectors of H in the φ representation.
φ = Tψ
ψ = T†φ
Page 8
5.80 Lecture #1
Fall, 2008
Page 9
Special example of finding matrix representation of any general function of a matrix.
Suppose we want Q3/2
*
Generate Qφ in the convenient φ basis
*
diagonalize Qφ [not the representation that diagonalizes H]
θ
⎛
Q11
0
0
⎞
⎜
⎟
T †Qφ T = Qθ =
⎜ 0  0 ⎟
⎜
⎟
0
QθNN ⎠
⎝
0
*
*
( Qθ )
3/2
⎛
θ 3/2
0
⎜(
Q11 )
=
⎜ 0

⎜
⎜ 0
0
⎝
⎞
⎟
⎟
⎟
3/2
θ
(
QNN ) ⎠⎟
0
0
transform back to φ basis
T(Qθ)3/2T† = [Q3/2]φ
The only reason why this is not as wonderful and general as it seems is that it is
impossible to diagonalize an infinite matrix and all Harmonic Oscillator basis sets
are infinite. Still, truncation at a finite (large) dimension gives accurate results for
the lowest few eigenstates. Accuracy can be tested by doing calculation twice,
once for N × N and once for (N + 1) × (N + 1) and looking for stability of results
for the lowest levels of interest.