Transcript TJ06_v21.ppt

BREAKING THE SYMMETRY IN JAHN-TELLER ACTIVE MOLECULES
BY ASYMMETRIC ISOTOPIC SUBSTITUTION:
SPLITTING THE ZERO-POINT VIBRONIC LEVEL
DMITRY G. MELNIK, JINJUN LIU AND TERRY A. MILLER
The Ohio State University, Dept. of Chemistry, Laser Spectroscopy Facility,
120 W. 18th Avenue, Columbus, Ohio 43210
ROBERT F. CURL,
Department of Chemistry and Rice Quantum Institute, Rice University, Houston,
Texas 77005
Problem outline:
Objective:
• analysis of the vibrational level structure of methoxy radicals
• study isotopic effects on vibrational energy level structure
Motivation:
• vibrational (vibronic) level structure samples molecular PES with
the isotopic substitution
• helps to study dynamics of unimolecular decomposition
• methoxy is a combustion intermediate, whose kinetics
and thermodynamics depend on its PES.
• test case for systems exhibiting conical intersection.
Summary of the vibrational analysis
Standard approach:
• formulate full Hamiltonian excluding rotational and translational parts in terms
of the parameters characterizing molecular PES
• formulate isotopic relationships
• set up the Hamiltonian in the appropriate basis set
• adjust the parameters of the Hamiltonian to adequately predict experimentally
observed energies.
Challenges (methoxy):
• complex interaction pattern due to competing spin-orbit and Jahn-Teller
interaction
• 9 vibrational modes makes the task computationally challenging
• rapidly increasing density of states makes vibrational assignment difficult
Spin-vibronic level structure of JT active molecule
(a)
Sym
Harmonic
v(E)=1
Sym
JT1
Sym
JT1+JT2
Asym
JT1+JT2
?
Asym
JT1+JT2 +SO
?
v=0
DE=E(A’)-E(A”)
(a) T. Barckholtz and T. A. Miller, Intl. Rev. Phys. Chem., 17, 435 (1998)
Vibronic Hamiltonian
H  H e0  TN  U (re ,R)
H e0  Te  U (re ,0)
R is a displacement vector in an arbitrary set of
internal coordinates spanning 3N-6 space. R=0
indicates a “reference” nuclear configuration.
 ex 
1   
1/ 2  1

2


Using the basis, 

   the vibronic Hamiltonian writes in matrix form

 i i    
 ey 
H ev  H  H e0 
1ˆ
1


I   PRT  G  PRT +R T  F  R   eˆyx   U1  R + R T U 2 R   ...  h.c.
2
2


Harmonic part
Where
1 0
Iˆ  

0 1
1  1 i 
eˆyx  

2  i 1
 R
G=BT M 1B, Bij   i
 d
 j
  2U (re , R) 
Fij =  
 R R 
i
j

R 0
Jahn-Teller coupling




 U (re , R) 
1
U 
 


 Ri
R 0
U
2

  2U (re , R) 
 

 R R 
i
j

R 0
At this point we disregard spin-orbit interaction, which can be trivially added later.
Hamiltonian reduction for the isotopic problem
H ev 
1ˆ
1


I   P T  P+QT  Λ  Q   eˆ1   U1  Q + QT U 2 Q   ...  h.c.
2
2


1
2
, U 
The values of , U 
can be measured experimentally. Suppose there
is an isotopologue for which all three matrices are known. We will call
it a “reference” isotopologue, and the corresponding values are labeled
1
2
with “(0)”:(0) , U (0)
 , U (0) 
1. “Potential reduction”: Hev can be rewritten for any isotopologue using unique sets
of normal coordinates
H ev( k ) 
1ˆ
1 T 2


I   P(Tk )  P(k) +QT(k)  Λ (k)  Q(k)   eˆyx   U1( k )   Q(k) + Q(k)
U ( k )  Q(k)   ...  h.c.
2
2


2. “Kinetic reduction”: Potential energy is unchanged upon isotopic substitution,
but kinetic energy is:
1ˆ
T
T
1/ 2
1/ 2
T
T

I   P(0)
  U (0)
G (0)
G ( k ) G (0
) U (0)   P(0) +Q (0)  Λ (0)  Q (0)  
2
1 T 2


eˆyx   U1(0)  Q(0) + Q(0)
U (0)  Q(0)   ...  h.c.
2


H ev( k ) 
Jahn-Teller interaction matrices
Bilinear JT coupling
A
modes
E
modes
Symmetric
to sxz
v1
v2
v3
v4 a
v5 a
v6 a
v4b
v5b
v6b
1
U (0)

Antisymmetric to sxz
 0 


0


 0 


k
 (0),4 
  k(0),5 


k
 (0),6 
 ik 
 (0),4 
 ik(0),5 
 ik 
 (0),6 
2
U (0)

0

0

 ...

 ...

bst


 ...

ibst



... ...
bst
ibst
0
...
...
igtt 
gtt 
...
...
...
igtt 
 gtt 
...
Linear JT coupling
Quadratic and crossquadratic JT coupling
Relationships to the experimentally observed values
k(0) m   2 D(0) m m3 
1/ 2
g (0) mm












... 
[for dimensionless K , e.g. B&M review ]
 (0) m(0) m K (0) mm

1/ 2


K (0) mm [for K in wavenumbers, e.g. Botschwina et al.]



 (0) m (0) m
Vibronic Hamiltonian for 3-mode (A+E) model (H3)
1. A model with a single pair of A and E modes.
y
2
1
R1
3
2.
R3
x
Normal modes are mass-weighted symmetry
modes.
3. Transforming internal coordinates to symmetry
coordinates,


 x 
  
Q   y  
z 
  


R2
1
3
2
3
0
1
3
1
6
1
2

1 
3 
 R1 
1  
   R2 
6  
R
1  3 


2
we write H ev(0) explicitly:
1 1 0
T
T
H ev(0)  
   P G (0) P+Q FQ  
2 0 1
1 0 
0 1
2
2



kx

bzx

g
x

y




   ky  bzy  2 gxy 


 0 1
1 0


Calculations of the vibronic level structure of isotopologues of H3
Parameters of Hev are chosen to approximately replicate fundamental
frequencies and strength of vibronic coupling in v3+v6 pair in CH3O
F33
4.0 105
G(0)

3 0

3
 0

2

0 0

F66
6.4 105

0

0

3

2
F66
6.4 105




G(1)  




5
2
1
8
1
8
5
4
0
0
k6
1.75 104

0


0

5

4
G(2)
g 66
b36
v3
v6
9 104 105 1096 980

 2

 1
 
8

 0



1
8
1
0

0


0

1


The Hamiltonian is set up in the vibrational basis vx,y,z = 0…10, resulting
in the vibronic matrix 2000x2000. Vibrational matrix elements:
1/ 4
G 
vi Qi vi  1   (0)ii 
 Fii 
 F
vi Pi vi  1  i  ii
G
 (0) ii
 vi  1 


 2 
1/ 4



1/ 2
 vi  1 


 2 
1/ 2
Calculations of the zero point splitting, DE = E(A’) – E(A”)
DE(b,g;k=-1.75*104)
DE(b,k;g=-9*104)
H2D
15
H2D
20
10
HD2
0
-5
-10
g, 1 4
0
-8
-6
-4
-2
-15
2
4
b, 10 4
6
8
10
DE  H2 D   DE  HD2 
0
-10
-20
0
2
Predicted by Scharf(a) and observed in CH3O(b) and C5H5(c)
(a) B. Scharf et al, J. Chem. Phys, 77, 2226 (1982)
(b) D. Melnik et al., J. Chem. Phys., submitted
(c) L. Yu et. al., J. Chem. Phys., 98, 2682, (1993)
HD2
|k|, 103
0
4
6
8
10
b, 1
04
5
DE, cm-1
E, cm-1
10
12
14
16
0
4
8
Isotopic problem for CH3O
Use the “potential reduction” to obtain fundamental frequencies and
Jahn-Teller parameters.
1. Fundamental frequencies are square roots of the eigenvalues of GF matrix.
2. Write potential energy as the function of the internal coordinates,
U  U  RCO , R j , q j , jij  
RCO
1
1
2
f
R

Rs0' Rs0" f bb Rb2   f sb Rs0 Rs Rb


ss s
2 s
2 b
sb
qj
Ri
j ij
Rs  RCO , R j
Rb  q j , jij
3. Fit f-parameters of the potential to the experimentally measured
deperturbed fundamental frequencies of CH3O an CD3O.
4. Diagonalize G(k)F for all isotopologues, obtain U(k) and fundamental
frequencies
Isotopic relationships for JT parameters in CH3O
Transformation between the normal coordinates of different isotopologues:
Tlm  UlG(l1/) 2G1/( m2) UTm
1/2
where U(m) is the unitary transformation which diagonalizes G1/2
(m) FG (m)
We can express parameters of one isotopologue (l), in terms those of another:
 l  TmlT  m Tml
U (1l )   TmlT U (1m ) 
U (2l )   TmlT U (2m )  Tml
Vibrational parameters of CHD2O
Fit fundamental frequencies
CH3O
mode
1
2
3
4
5
6
exp
1359
1051
2835
1417
1065
PES parameters (dyn/cm)
constant
Fit
WDC(a)
fCO
fCH
5.15(13) 105
4.31(5) 105
5.0  5.8 105
4.7  5.0 105
0
0
f HCO /( RCH
RCO
)
0
f HCH /( RCH
)2
0.49(2) 105
0.47(1) 105
0.57
0.46
0
fCO  HCO / RCO
0
fCH  HCO / RCH
0.94(12) 105
0.19(18) 105


0
fCH  HCH / RCO
0.55(9) 105

CD3O
pr. exp-pr exp
pr
exp-pr
2753

1961

1367
-8
995 1012
17
1030
21
1036 1025
11
2842
7
2100 2104
4
1443
26 1070 1040
30
1062
3
825 828
3
Predicted values of fundamental frequencies and JT coupling for CHD2O
LJT
mode
A’
A”
1
2
3
4
5
6
7
8
9
freq.
2810
2009
1279
1034
1026
892
2108
1271
835
D
0.013
0.005
0
0.1
0.017
0.27
0.02
0.16
0.21
QJT
1
2
3
4
5
6
7
8
9
-0.0002
0.0028 0.0033
0.0113 0.0046 0.0329
-0.0039 0.0078 -0.0129 0.0294
0.0059 0.0089 0.0165 0.0688 0.0473
0.0027 0.0158 0.0725 0.0733
0.095 0.1285
-0.0046 0.0013 0.0111 0.0034 0.0101 0.0021 0.0034
-0.0029 0.0035 0.059 0.009 0.0755 0.0311 0.0005 0.0227
0.0183 -0.0081 0.1123 0.0593 0.094 0.1067 0.0036 0.026 0.1498
(a) E. Wilson, J.D.Decius P.C.Cross, “Molecular Vibrations”
Vibrational level structure of CHD2O
v5+v6+v8+v9
2000
1500
E, cm
-1
1000
500
0
Experiment
v4+v6+v8+v9
Summary
• Vibronic calculations for a simplified 3-mode molecule show semi-quantitative
agreement with the observed splitting of the ground vibronic level.
• The ground vibronic level splitting is primarily cause by bilinear coupling,
hence the latter needs to be included in the vibronic analysis to correctly
analyze the vibrational level structure of JT-active moelcules. This necessitates
the multimode approach.
• To correctly interpret spectra of the asymmetrically substituted molecules,
all isotopologues need to be analyzed globally
• NEXT: set up multimode spin-vibronic problem and use nonlinear fit to obtain
parameters of the PES that correctly predict the observed spectra.
Acknowledgements
• Colleagues:
Dr. Gabriel Just,
Dr. Phillip Thomas,
Ming-Wei Chen,
Terrance Codd,
Neal Kline
Rabi Chhantyal-Pun
• OSU
• NSF