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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. “( Variational calculations) will never displace the more

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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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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

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  • 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

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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.


New Hamiltonian for each coordinate/axis system

Same for J=0

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Diatomic molecules: 1 vibrational mode



Numerical solution: trivial on a pc

Eg LEVEL by R J Le Roy, University of Waterloo Chemical Physics Research Report CP-642R (2001)

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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


See CCP6 program library

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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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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)

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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

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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…

  • …a potential (determines accuracy)

  • …coordinates (defines H)

  • …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




Choice of g1 and g2 defines coordinates

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Body-fixed axes: Embeddings implemented in DVR3D

r2 embedding

r1 embedding

bisector embedding


z-perpendicular embedding

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Basis functions.

General functions:

Floating spherical Gaussians


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 (2J+1) functions

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Performing a Variational Calculation:

  • Construct individual matrix elements

  • Construct full Hamiltonian matrix

  • Diagonalize Hamiltonian: get Ei and

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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 quadrature

For Polynomial basis functions, Pn, use

M-point Gaussian quadrature to give

Points, xi, Weights, wi

< n | V | m > = 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.

Wavefunction 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)

  • < 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

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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

  • 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.

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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?

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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


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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 initioorSpectroscopically 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






Absorption (cm-1 atm-1 at STP)



0 500 1000

Frequency (cm-1)

JH Schryber, S Miller & J Tennyson, JQSRT, 53, 373 (1995)

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H2, H2O,




H2, CO,


CN, etc

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



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

  • < 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




L-band & K-band spectra also assigned

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Variational calculations:

Assignments using branches



Accurate but extrapolate poorly

Error / cm-1

Ab initio potential

Less accurate but extrapolate well


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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 (> 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).