Transcript Ohio_06
THEORETICAL STUDIES OF
HIGHLY EXCITED VIBRATIONAL
STATES USING VAN VLECK
PERTURBATION
THEORY
University of Wisconsin
Ned Sibert, Anne McCoy, Darin Burleigh,
Sai Ramesh, Xiaogang Wang, Marc
Joyeaux, Christoph Iung, and Christof
Jung
On σ-Type Doubling and Electron Spin in the Spectra of Diatomic
Molecules
J. H. Van Vleck
Department of Physics, University of Wisconsin
Received 31 January 1929
1958
BAND INTENSITIES IN TETRAHEDRAL COMPLEXES
BALLHAUSEN, C.J.; LIEHR,
ANDREW D.
1964
A PERTURBATION METHOD SUITABLE FOR HIGHER ORDER CALCULATIONS
Louck, James D.
1993
LARGE AMPLITUDE MOTIONS IN THE WATER MOLECULE
.Sage, M. L.
1991
Calculation of IR Intensities of Highly Excited Vibrational States in HCN Using Van Vleck
Perturbation Theory
McCOY, ANNE B.; SIBERT,
EDWIN L. III
1964
CONTRIBUTION OF ROTATION-VIBRATION INTERACTION TO THE SPIN-DOUBLET
SEPARATION IN $2\Pi$ DIATOMIC MOLECULES
James, Thomas C.
2003
A PRIMER ON DUNHAM'S APPROACH TO ANALYSIS OF SPECTRA OF FREE
DIATOMIC MOLECULES
OGILVIE, J.F.
1961
THEORETICALLY-CALCULATED VIBRATIONAL-ROTATIONAL SPECTRUM OF
$H_{2}$, HD, AND $D_{2}$
COOLEY, JAMES W.
1978
VAN VLECK TRANSFORMATION TO TENTH ORDER AS AN ALTERNATIVE TO THE
CONTACT TRANSFORMATION
THORVALD, PEDERSEN
1978
VAN VLECK TRANSFORMATION APPLIED TO THE BORN-OPPENHEIMER
SEPARATION
JORGENSEN, F.; THORVALD,
PEDERSEN
1994
DEFINITIVE EVALUATION OF NONADIABATIC VIBRATIONAL AND ADIABATIC
EFFECTS FROM VIBRATION-ROTATIONAL SPECTRA OF DIATOMIC MOLECULES
OGILVIE, J.F.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 next
Overview of Theoretical Spectroscopy
Potential Energy Surfaces
Hˆ ( N ) jN E Nj jN
Dynamics and Spectroscopy
Diagonalize H
] ]
T 1 HT K
H11 H12 H13 H14
K11
0
0
0
H21 H22 H23 H24
0
K22
0
0
H31 H32 H33 H34
0
0
K33
0
H41 H42 H43 H44
0
0
0
K44
Overview of Theoretical Spectroscopy
Potential Energy Surfaces
Hˆ ( N ) jN E Nj jN
Effective Hamiltonians
Dynamics and Spectroscopy
FE08
Effective Vibrational Rotational Hamiltonian in the Presence of a
Stark Field
B. Ram Prasad and Mangala Sunder Krishnan
The standard theory for analysing high resolution Vibrational-RotationalTorsional spectra of semi-rigid and non-rigid molecules is based on perturbation
theory, which leads to the concept of effective Hamiltonians. Though there are
several ways of obtaining molecular Hamiltonians, the method proposed by Van
Vleck has proven to be quite useful in the area of molecular spectroscopy.
Perturbation Theory
1
T HT K
H11 H12 H13 H14
K11
K12
K13
K14
H21 H22 H23 H24
K21
K22
K23
K24
H31 H32 H33 H34
K31
K32
K33
K34
H41 H42 H43 H44
K41
K42
K43
K44
This only works for when there are no degenerate states.
Wave functions for zero order Hamiltonian
Wave functions for full Hamiltonian
Solution to Vibrational Example if
w1 2w2
Solution to Vibrational Example if w1=2w2
Simple Solution to Vibrational Example if w1=2w2
|4,0>
|3,2>
|2,4>
|1,6>
|0,8>
Polyads
N=2n1+n2=constant=8
Simple Solution to Vibrational Example if w1=2w2
H=
Each block corresponds to
a polyad of states
Van Vleck Perturbation theory is a simple way to transform the
matrix to this form.
Visualizing Wave Functions
for SCCl2
Probability distributions of a
select eigenstate plotted as a
function of the Q5 and Q6
coordinates for increasing values
of Q1 going from (a)-(f).
Van Vleck Perturbation theory is a simple
way to transform H to a desired form.
If we write T as
1
T e
T HT e
i S
i S
,
He
then T is unitary if S is Hermitian.
i S
e
i [ S , ]
H K
One solves for S by expanding H and K in powers of .
e
in [ S ( n ) , ]
e
i2 [ S ( 2 ) , ] i [ S (1) , ]
e
H K
Result of 2nd order Van Vleck PT
CHBr3 normal mode CH stretch-bend states
Nt Polyad Labels
1
2
3
Ei
Ej
CHBr3 post Van Vleck CH stretch-bend states
Nt Labels
1
2
3
Ei
Ej
Comparison of 2nd, 4th, 6th, and 8th order Van Vleck
PT band origins (cm-1) for CH4
Xiao-Gang Wang and E. L. Sibert, JCP 124, 4510 (1999).
Comparison of 2nd, 4th, 6th, and 8th order Van Vleck
PT band origins (cm-1) for CH4
We are pushing the limits of CVPT so that it can be used to
accurately calculate highly excited states.
Rectilinear vs Curvilinear Approaches
Rectilinear CVPT results for H2CO vibrations
Curvilinear CVPT results for H2CO vibrations
Two Current Research Directions
1) Vibrations coupled to additional DOF
2) Creating good basis sets
Vibrations Coupled to Other
Degrees of Freedom
Many Hamiltonians can be written in the form
H H vib H x H cpl
CVPT allows one to simplify the analysis via the following transform.
K e
e
i S
i S
( H vib H x H cpl )e
H vib e
i S
Hx e
K vib H x K cpl
i S
i S
H cpl e
i S
Vibrations coupled to other DOF
Hamiltonians that can be written in the form
H H vib H x H cpl include:
Rotation-vibration of H2O, H2CO, SO2, CH4,
laser-molecule for HCN, H2O, H2CO, CHBr3, CHF3
Isomerization HCN
vibration-torsion CH3OH and deuterated analogues
condensed phase vibrational relaxation
Minimum energy positions vs torsion angle
Methanol
quartic expansion in
internal coordinates where
expansion coefficients are
functions of the torsion angle
calculated at the CCSD(T)
level using
the cc-pVTZ basis
Basic Approach
Comparison of experimental and theoretical
fundamentals and E-A splittings D (in cm-1 ).
Comparison of experimental and theoretical
fundamentals and E-A splittings D (in cm-1 ).
Creating Good Basis Sets
Molecular Physics, Vol. 103, (2005), 149–162.
SAI G. RAMESH and EDWIN L. SIBERT
Harmonic Basis
0
0
2000
1000
4000
Size of Vij
Ei
3000
4000
Ej
Post Van Vleck
0
0
2000
1000
4000
Size of Vij
Ei
3000
4000
Ej
CHF3 Results
We are using CVPT to
build good basis sets for
variational calculations.
C. Iung and F. Ribeiro, and E. Sibert
With these fundamental frequencies we were unable to identify
polyad quantum numbers.
HFCO results
of convergence tests.
Conclusions
Van Vleck perturbation theory remains an important tool for
treating molecular vibrations as well as vibrations coupled to
many other DOF.
A study of the vibrations of fluoroform with a sixth order nine-dimensional
potential: a combined perturbative-variational approach
SAI G. RAMESH and E. L. SIBERT, Mol. Phys, 103, 2005, 149–162
Quantum, semiclassical and classical dynamics of the bending modes of
acetylene
E. L. Sibert and Anne B. McCoy JCP (1996)
The Sibert Group