Vibration-rotation spectra
from first principles
Lecture 1: Variational nuclear motion calculations
Jonathan Tennyson
Department of Physics and Astronomy
University College London
OSU, February 2002
“(Variational calculations) will never displace the more
traditional perturbation theory approach to calculating …..
vibration-rotation spectra”
Carter, Mills and Handy, J. Chem. Phys., 99, 4379 (1993)
Rotation-vibration energy levels
The conventional view:
• Separate electronic and nuclear motion,
The Born-Oppenheimer approximation
• Vibrations have small amplitude
• Harmonic oscillations about equilibrium
• Rotate as a rigid body
• Rigid rotor model
Improved using perturbation theory
But
Small amplitude vibrations often poor approximation
What about dissociation?
Equilibrium not always a useful concept
What about multiple minima?
Perturbation theory may not converge
Diverges for J > 7 for water
For high accuracy need electron-nuclear coupling
Important at the 1 cm-1 level for H-containing molecules
Variational approaches: Ein > Ein+1
Internal coordinates: Eckart or Geometrically defined
• Exact nuclear kinetic energy operator
within the Born-Oppenheimer approximation
• Vibrational motion represented either by
Finite Basis Representation (FBR) or
Grid based Discrete Variable Representation (DVR)
• Solve problem using Variational Principle
• Potentials either ab initio or from fitting to spectra
Variational approaches
• Treats vibrations and rotations at the same time
• Interpret result in terms of potentials
• Only assume rigorous quantum numbers:
n, J, p, symmetry (eg ortho/para)
• Give spectra if dipole surface available
• Include all perturbations of energy levels and spectra
• Yield models that can be transferred between isotopomers
Provide a complete theoretical treatment with no assumptions
Internal coordinates:
Orthogonal coordinates for triatomics
Orthogonal coordinates have
diagonal kinetic energy operators.
Important for DVR approached
Hamiltonians for nuclear motion
Laboratory fixed:
3N coordinates
Translation, vibration, rotation
not separately identified
Space fixed: remove translation of centre-of-mass
3N-3 coordinates
Vibration and rotation not separately identified
Body fixed: fix (“embed”) axis system in molecule
3 rotational coordinates (2 also possible)
3N-6 vibrational coordinates (or 3N-5)
Hamiltonians for nuclear motion
Laboratory fixed:
Useless for variational calculations due to continuous
translational “spectrum”.
Used for Monte Carlo methods.
Space fixed:
Requires choice of internal coordinates.
Vibration and rotation not separately identified.
Widely used for Van der Molecules.
Same for J=0
Body fixed:
Requires choice of internal axis system.
Vibrational and rotational motion separately identified.
Singularities!
New Hamiltonian for each coordinate/axis system
Diatomic molecules: 1 vibrational mode
stretch
Hamiltonian:
Numerical solution: trivial on a pc
Eg LEVEL by R J Le Roy, University of Waterloo Chemical Physics
Research Report CP-642R (2001)
http://scienide.uwaterloo.ca/~leroy/level/
Triatomics: 3/4 vibrational mode
3 degrees of freedom (4 for linear molecules)
New mode: bend
Hamiltonian: many available, some general
Numerical solution: general programs available
Eg BOUND, DVR3D, TRIATOM
See CCP6 program library http://www.dl.ac.uk/CCP/CCP6/library.html
Tetratomics
6 vibrational degrees of freedom
New mode: torsion
New mode: umbrella
Hamiltonian: available for special cases
Numerical solution: results for low energies
No published general programs
Pentatomics
12 degrees of freedom
New modes:
book, ring puckering, wag, deformation, etc
Hamiltonian: for very few special cases
eg XY4 systems, polyspherical coordinates
(polyspherical coordinates are orthogonal coordinates
formed by any combination of Radau and Jacobi coordinates)
Numerical solution: almost none (CH4)
Vibrating molecules with N atoms
3N-6 degrees of freedom
Modes: all different types
Hamiltonian: not generally available but see
J. Pesonen, Vibration-rotation kinetic energy operators: A geometric
algebra approach, J. Chem. Phys., 114, 10598 (2001).
Numerical solution: awaited for full problem
But MULTIMODE by S Carter & JM Bowman gives solutions for
semi-rigid systems using SCF & CI methods plus approximations
http://www.emory.edu/CHEMISTRY/faculty/bowman/multimode/
Triatomics:
general form of the Born-Oppenheimer Hamiltonian
KV vibrational kinetic energy operator
KVR vibration-rotation kinetic energy operator
(null if J=0)
V the electronic potential energy surface
Steps in a calculation: choose…
1. …a potential (determines accuracy)
2. …coordinates (defines H)
3. …basis functions for vibrational motion
Effective Hamiltonian after intergration
over angular and rotational coordinates.
Case where z is along r1
Vibrational KE
Vibrational KE
Non-orthogonal coordinates only
Rotational & Coriolis terms
Rotational & Coriolis terms
Non-orthogonal coordinates only
Reduced masses
(g1,g2) define coordinates
General coordinates
r2
r1
q
Choice of g1 and g2 defines coordinates
r2 embedding
Body-fixed axes:
Embeddings implemented in
DVR3D
r1 embedding
bisector embedding
(d) NEW!
z-perpendicular embedding
Basis functions.
General functions:
Floating spherical Gaussians
Stretch functions:
Morse oscillator (like)
Harmonic oscillators
Spherical oscillators, etc
Non-orthogonal
Must be complete set
Problems as R
0
Bending functions:
Associate Legendre functions Coupling to rotational function
ensures correct behaviour at linearity
Jacobi polynomials
Rotational functions:
Spherical top functions, DJMK Complete set of (2J+1) functions
Performing a Variational Calculation:
1. Construct individual matrix elements
2. Construct full Hamiltonian matrix
3. Diagonalize Hamiltonian: get Ei and
Matrix elements
Hnm = < n | T + V | m >
Can often obtain matrix elements over
Kinetic Energy operator analytically in closed form
For general potential function, V,
need to obtain matrix elements using numerical
For Polynomial basis functions, Pn, use
M-point Gaussian quadrature to give
Points, xi, Weights, wi
quadrature
< n | V | m > = Si wi Pn(xi) Pm(xi) V(xi)
Scales badly (~MN) with number of modes, N
Grid based methods
Discrete Variable Representation (DVR) uses
points and weights of Gaussian quadrature.
Wavefunction obtained at grid of points,
not as a continuous function.
DVR is isomorphic to an FBR
DVR versus FBR
DVR advantages
• Diagonal in the potential (quadrature approximation)
< a| V | b > = dab V(xa)
• Sparse Hamiltonian matrix
• Optimal truncation and diagonalization
based on adiabatic separation
• Can select points to avoid singularities
DVR disadvantages
• Not strictly variational (difficult to do small calculation)
• Problems with coupled basis sets
• Inefficient for non-orthogonal coordinate systems
Transformation between DVR and FBR quick & simple
Matrix diagonalization
• Matrices usually real symmetric
• Diagonalization step rate limiting for triatomics, a N3.
• Intermediate diagonalization and truncation
major aid to efficiency.
Iterative versus full matrix diagonalizer
• Is matrix sparse?
• How many eigenvalues required?
• Are eigenvectors needed?
• Is matrix too large to store?
Rotational excitation
2J+1 spherical top functions, DJkM, form a complete set.
Rotational parity, p, divides problem in two:
Two step variational procedure essential for treating high J:
• First step: diagonalize J+1 “vibrational” problems assuming
k, projection of J along z axis, is good quantum number.
• Second step: diagonalize full Coriolis coupled problem
using truncated basis set.
Can also compute rotational constants directly as expectation values.
Transition intensities
Compute linestrength as
Sij = |St < i | mt | j >|2
where m is dipole surface (not derivatives)
| i > and | j > are variational wavefunctions
• Rotational and vibrational spectra at same time
• Only rigorous selection rules:
DJ = +/- 1, p = p’
DJ = 0,
p = 1- p’
(ortho  ortho, para  para).
All weak transitions automatically included.
• Best done in DVR
• Expensive (time & disk) for large calculation
• More accurate than experiment?
The DVR3D program suite: triatomic vibration-rotation spectra
Potential energy
Surface, V(r1,r2,q)
J Tennyson, NG Fulton &
Dipole
function
m(r1,r2,q)
JR Henderson, Computer
Phys. Comm., 86, 175 (1995).
Why calculate VR spectra?
• Test potential energy surfaces
construct potentials
• Predict assign spectra
lab, astronomy, etc
• Calculate transition intensities
physical data from observed spectra eg n, T,…..
atmospheric studies, astrophysics, combustion ….
• Generate bulk data
partition functions  specific heats, opacities
JANAF, astrophysics, etc
• Link with reaction dynamics
eg HCN  HNC
H3+ + hn  H2 + H+
• Quantum ``chaology''
Classical dynamics of highly excited molecules is chaotic
Potentials:
Ab initio
or
Spectroscopically determined
Linelist and assignments
• Linelist : theoretically calculated transitions including :
1- transition frequencies : Eupper - Elower
2- Intensities
3- Eupper and Elower and quantum numbers
• Spectra : Measured set of transitions in a molecule at given T o
and p
• Assignment : Identify the quantum numbers of the lower
and upper levels.
What for ?
Temperature dependency
Pressure broadening

M-Dwarf Stars
Oxygen rich, cool stars: T = 2000 – 4000 K
Spectra dominated by molecular absorptions
H2O, TiO, CO most important
Water opacity
Viti & Tennyson computed VT2 linelist:
All vibration-rotation levels up to 30,000 cm-1
Giving ~ 7 x 108 transitions
Absorption (cm-1 atm-1 at STP)
Absorption by steam at T = 3000 K
Ludwig
Hitran
linelist
3.0
2.0
1.0
0.0
0
500
1000
Frequency (cm-1)
JH Schryber, S Miller & J Tennyson, JQSRT, 53, 373 (1995)
The Sun: T = 5760 K
Molecules on the Sun
Sunspots
T=3200K
H2, H2O,
CO, SiO
T=5760K
Diatomics
H2, CO,
CH, OH,
CN, etc
Sunspots Image from SOHO : 29 March 2001
Sunspot, T ~ 3200 K
Penumbra, T ~ 4000 K
Sunspot: N-band spectrum
Sunspot
lab
L Wallace, P Bernath et al, Science, 268, 1155 (1995)
Assigning a spectrum with 50 lines per cm-1
1. Make ‘trivial’ assignments
(ones for which both upper and lower level known experimentally)
2. Unzip spectrum by intensity
6 – 8 % absorption strong lines
Only strong/medium lines assigned so far
4 – 6 % absorption medium
2 – 4 % absorption weak
< 2 % absorption grass (but not noise)
3. Variational calculations using ab initio potential
Partridge & Schwenke, J. Chem. Phys., 106, 4618 (1997)
+ adiabatic & non-adiabatic corrections for Born-Oppenheimer approximation
4. Follow branches using ab initio predictions
branches are similar transitions defined by
J – K a = na
or
J – Kc = nc,
n constant
OL Polyansky, NF Zobov, S Viti, J Tennyson, PF Bernath & L Wallace, Science, 277, 346 (1997).
Sunspot: N-band spectrum
Sunspot
Assignments
lab
L-band & K-band spectra also assigned
Variational calculations:
Assignments using branches
Spectroscopically
Determined potential
Error / cm-1
Accurate but extrapolate poorly
Ab initio potential
Less accurate but extrapolate well
J
The Future:
PDVR3D:
DVR3D program for parallel computers,
Eg Cray-T3E or IBM SP2
H 2O
• All J = 0 states to dissociation (> 1000 states)
20 minutes wallclock on 64 Cray T3E processors
• All J > 0 up to dissociation. Scales as (J+1).
Needs reliable potentials!
HY Mussa and J Tennyson, J. Chem. Phys., 109, 10885 (1998).