Vibration-rotation spectra from first principles

Lecture 1: Variational nuclear motion calculationsLecture 2: Rotational motion, Spectra, PropertiesLecture 3: ApplicationsLecture 4: Calculations of spectroscopic accuracy

Jonathan TennysonDepartment of Physics and AstronomyUniversity College London

QUASAAR Winter School,Grenoble, Jan/Feb 2006

Rotational basis functions.

Eg bending functions: Associate Legendre functionsCoupled functions Pjk(θ) DJ

Mk(α,β,γ) = Yjk(θ,γ) YJM(α,β)

Spherical top functions, DJMk (α,β,γ)

• M is projection of J onto lab z axis• k is projection of J onto body-fixed z axis• Complete set of (2J+1) functions• (α,β,γ) are “Euler” angles used for embedding

Coupling rotational and functions ensures correct behaviour at linearity

Vibrational KE

Vibrational KENon-orthogonal coordinates only

Rotational & Coriolis terms

Rotational & Coriolis termsNon-orthogonal coordinates only

Effective Hamiltonian after intergrationover angular and rotational coordinates.Case where z is along r1

Reduced masses (g1,g2) define coordinates

General coordinates

r2

r1θ

Choice of g1 and g2 defines coordinates

Body-fixed axes: Embeddings implemented in DVR3D

r2 embedding

r1 embedding

bisector embedding

(d) z-perpendicular embedding

y

Rotational excitation2J+1 spherical top functions, DJ

kM, 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 of the appropriate inverse moment of inertias.

Transition intensitiesCompute linestrength as

Sij = |Σt < i | µt | j >|2where µ is dipole surface (not derivatives)

| i > and | j > are variational wavefunctions• Rotational and vibrational spectra at same time• Only rigorous selection rules:

∆J = +/− 1, p = p’∆J = 0, p = 1− p’(orthoßà ortho, paraßà para).All weak transitions automatically included.

• Best done in DVR• Expensive (time & disk) for large calculation• More accurate than experiment?

Transition intensities

Einstein A coefficient, Aif, units s-1

Absorbtion intensity

Second radiation coefficient

Internal partition function, assuming Qelec = 1

The DVR3D program suite: triatomic vibration-rotation spectra

Potential energySurface,V(r1,r2,θ)

Dipole functionµ(r1,r2,θ)

J Tennyson, MA Kostin, P Barletta, GJ Harris

OL Polyansky, J Ramanlal & NF Zobov

Computer Phys. Comm. 163, 85 (2004).

www.tampa.phys.ucl.ac.uk/ftp/vr/cpc03

Nuclear spin statisticsPauli Principle for interchange of identical particles Ψ12 = σ Ψ21

σ = +1 for Bosons (integer spin) σ = −1 for Fermions (half-integer spin)

For molecules need to consider Ψ = ψnuc ψelec ψvib ψrot

Eg H2, H has i = ½ ie a Fermion I = ½ + ½ = 0,1 Like electronic states of He, I = 0 is anti-symmetric wrt interchange I = 1 is symmetric wrt interchange ψelec and ψvib both symmetric wrt interchange So I=0 gives ψrot even ie J = 0, 2, 4, etc. g = (2I+1) = 1 Para I=1gives ψrot odd ie J = 1, 3, 5, etc and g = (2I+1) = 3 Ortho

Water: H2OLike H2 except • v3 vibrational mode is odd• Rotational levels have substructure (J,Ka,Kc)

So Ka-Kc+ v3 even is para g = 1So Ka-Kc+ v3 odd is ortho g = 3

What about HDO ?? g = ?

For general discussion and more complicated moleculesSee: PR Bunker & P Jensen, Molecular Symmetry & Spectroscopy, 2nd edition

Two conventions

• Astronomers’:For water (& H2)g = ¼ for para and g = ¾ for orthoIn this convention HDO has g = 1

• Physicists’ and Chemists’For water (& H2)g = 1 for para and g = 3 for orthoIn this convention HDO has g = 6 (remember D has I = 1)

Either OK if used consistently

Fractionation reactionH2O + HD ßà HDO + D2

Equilibrium constant

K(T) = Q(HDO) Q(D2) exp(∆E/kt)Q(H2O) Q(HD)

∆E = zpe(HDO) + zpe(D2) − zpe(H2O) – zpe(HD) < 0

Nuclear spin degeneracy factors, g, must be consistent

Potentials: Ab initio or Spectroscopically determined

Spectroscopically determined potentials

Assume V has linear form

Hellman-Feyman Theorem

For case of dx = dci

Means can compute derivates wrt to all constantsusing a single set of wavefunctions