Vibration-rotation spectra from first principles

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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
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(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)
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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
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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 gt 7 for water
For high accuracy need electron-nuclear
coupling Important at the 1 cm-1 level for
H-containing molecules
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• Variational approaches Ein gt Ein1
• 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

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• 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
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Internal coordinates
Orthogonal coordinates for triatomics
Orthogonal coordinates have diagonal kinetic
energy operators. Important for DVR approached
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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)
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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. Body fixed Requires
choice of internal axis system. Vibrational and
rotational motion separately identified. Singulari
ties! New Hamiltonian for each coordinate/axis
system
Same for J0
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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.uwa
terloo.ca/leroy/level/
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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

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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
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Pentatomics
12 degrees of freedom
New modes book, ring puckering, wag,
deformation, etc
Hamiltonian for very few special cases eg
XY4 systems, polyspherical coordinates (polyspheri
cal coordinates are orthogonal coordinates
formed by any combination of Radau and Jacobi
coordinates)
Numerical solution almost none (CH4)
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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/
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Triatomics general form of the Born-Oppenheimer
Hamiltonian
KV vibrational kinetic energy operator KVR
vibration-rotation kinetic energy operator
(null if J0) 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

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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
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General coordinates
r2
q
r1
Choice of g1 and g2 defines coordinates
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Body-fixed axes Embeddings implemented in DVR3D
r2 embedding
r1 embedding
bisector embedding
(d) NEW! z-perpendicular embedding
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Basis functions.
General functions Floating spherical Gaussians
Non-orthogonal
Stretch functions Morse oscillator
(like) Harmonic oscillators Spherical
oscillators, etc
Must be complete set Problems as R 0
Bending functions Associate Legendre
functions Jacobi polynomials
Coupling to rotational function ensures correct
behaviour at linearity
Rotational functions Spherical top functions,
DJMK
Complete set of (2J1) functions
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Performing a Variational Calculation

1. Construct individual matrix elements
2. Construct full Hamiltonian matrix
3. Diagonalize Hamiltonian get Ei and

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Matrix elements
Hnm lt n T V m gt
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 quadrature
For Polynomial basis functions, Pn, use M-point
Gaussian quadrature to give Points, xi, Weights,
wi
lt n V m gt Si wi Pn(xi) Pm(xi) V(xi)
Scales badly (MN) with number of modes, N
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Grid based methods
Discrete Variable Representation (DVR) uses
points and weights of Gaussian quadrature. Wavefun
ction obtained at grid of points, not as a
continuous function.
DVR is isomorphic to an FBR
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DVR versus FBR

• DVR advantages
• Diagonal in the potential (quadrature
  approximation)
• lt a V b gt 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
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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?

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Rotational excitation

• 2J1 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 J1 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.

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Transition intensities

• Compute linestrength as
• Sij St lt i mt j gt2
• where m is dipole surface (not derivatives)
• i gt and j gt 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?

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The DVR3D program suite triatomic
vibration-rotation spectra
Potential energy Surface, V(r1,r2,q)
J Tennyson, NG Fulton JR Henderson, Computer
Phys. Comm., 86, 175 (1995).
Dipole function m(r1,r2,q)
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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

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Potentials Ab initio or
Spectroscopically determined
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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
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Absorption by steam at T 3000 K
Ludwig
3.0
Hitran
linelist
2.0
Absorption (cm-1 atm-1 at STP)
1.0
0.0
0 500
1000
Frequency (cm-1)
JH Schryber, S Miller J Tennyson, JQSRT, 53,
373 (1995)
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The Sun T 5760 K
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Molecules on the Sun
Sunspots Image from SOHO 29 March 2001
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Sunspot, T 3200 K
Penumbra, T 4000 K
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Sunspot N-band spectrum
Sunspot
lab
L Wallace, P Bernath et al, Science, 268, 1155
(1995)
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Assigning a spectrum with 50 lines per cm-1

• Make trivial assignments
• (ones for which both upper and lower level
  known experimentally)
• 2. Unzip spectrum by intensity
• 6 8 absorption strong lines
• 4 6 absorption medium
• 2 4 absorption weak
• lt 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 Ka na or J Kc nc,
  n constant

Only strong/medium lines assigned so far
OL Polyansky, NF Zobov, S Viti, J Tennyson, PF
Bernath L Wallace, Science, 277, 346 (1997).
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Sunspot N-band spectrum
Sunspot
Assignments
lab
L-band K-band spectra also assigned
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Variational calculations
Assignments using branches
Spectroscopically
Determined potential
Accurate but extrapolate poorly
Error / cm-1
Ab initio potential Less accurate but extrapolate
well
J
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The Future

• PDVR3D
• DVR3D program for parallel computers,
• Eg Cray-T3E or IBM SP2
• H2O
• All J 0 states to dissociation (gt 1000 states)
• 20 minutes wallclock on 64 Cray T3E processors
• All J gt 0 up to dissociation. Scales as (J1).
• Needs reliable potentials!

HY Mussa and J Tennyson, J. Chem. Phys., 109,
10885 (1998).