Theoretical Modelling of the Water Dimer: Progress and Current Direction Ross E. A. Kelly, Matt Barber, & Jonathan Tennyson Department of Physics & Astronomy.

Theoretical Modelling of Theoretical Modelling of the Water Dimer: the Water Dimer:

Progress and Current DirectionProgress and Current Direction

Ross E. A. Kelly, Matt Barber, & Jonathan TennysonRoss E. A. Kelly, Matt Barber, & Jonathan TennysonDepartment of Physics & AstronomyDepartment of Physics & Astronomy

University College LondonUniversity College London

Gerrit C. Groenenboom & Ad van der Avoird Gerrit C. Groenenboom & Ad van der Avoird Theoretical Chemistry, Institute for Molecules & Materials, Theoretical Chemistry, Institute for Molecules & Materials,

Radboud University, Nijmegen.Radboud University, Nijmegen.

NPL, June 2008NPL, June 2008

ContentsContents

I. Review of previous workI. Review of previous work II. Characterising more statesII. Characterising more states III. New Potential Energy SurfaceIII. New Potential Energy Surface IV. Franck-Condon Type Approach IV. Franck-Condon Type Approach V. Vibrational Averaging of the PESV. Vibrational Averaging of the PES

I.1. Brocks I.1. Brocks et al. et al. HamiltonianHamiltonian

Water Dimer Water Dimer Vibration-Rotation TunnelingVibration-Rotation Tunneling ((VRTVRT) levels from the ) levels from the Rigid Dimer Hamiltonian Rigid Dimer Hamiltonian by Brocks by Brocks et al. et al. [1].[1].

Only for the Only for the Intermolecular modesIntermolecular modesUsed for water dimer previously, detailed Used for water dimer previously, detailed

account [2].account [2].Dependent on V (6D). We used new 12D Dependent on V (6D). We used new 12D

Potential Energy Surface (PES). Potential Energy Surface (PES). Compared with Low temperature high-resolution Compared with Low temperature high-resolution

Tetrahertz Spectroscopy (prepared in Tetrahertz Spectroscopy (prepared in supersonic molecular beams), around 5 K.supersonic molecular beams), around 5 K.

[1] G. Brocks, A. van der Avoird, B. T. Sutcliffe, J. Tennyson, Mol. Phys. 50, 1025 (1983).

[2] G. C. Groenenboom, et al., JCP 113, 6702 (2000).

Tunnelling between equivalent states in the PES is Tunnelling between equivalent states in the PES is feasible!feasible!

Acceptor Tunnelling:Acceptor Tunnelling: No bond breaking hereNo bond breaking here Lowest tunnelling barrier Lowest tunnelling barrier

Also, by breaking the Hydrogen bond, other tunnelling Also, by breaking the Hydrogen bond, other tunnelling paths possible: paths possible: Donor-Acceptor interchangeDonor-Acceptor interchange Donor Bifurcation TunnellingDonor Bifurcation Tunnelling

I.2. Vibration-Rotation TunnellingI.2. Vibration-Rotation Tunnelling

I.3. Labelling Water Dimer StatesI.3. Labelling Water Dimer States Can be represented by Permutation-Inversion Group GCan be represented by Permutation-Inversion Group G

1616..

1 1

1 1

5 5

5 5

2 2

2 2

6 6

6 6

6 6

6 6

5

5 5

5

4

4

4

4

3

3

3

3

3 3

3 3

4

4

4

4

1 1

1 1

2

2 2

2

Isomorphic to D4h

with Irreducible Elements:

A1

+, A2

+, A1

-, A2

-, B1

+, B2

+, B1

-, B2

-, E+, E-

-> Water Dimer Spectroscopic Labels

I.4. Ground State VRT Levels for HI.4. Ground State VRT Levels for H44OO22

[1] X. Huang, B. J. Braams, J. M. Bowman, R. E. A. Kelly, J. Tennyson, G. C. Groenenboom, A. van der Avoird, J. Chem. Phys. 128,

034312 (2008).

Very good agreement:Very good agreement: Ground State Tunnelling Ground State Tunnelling splittingssplittings Rotational ConstantsRotational Constants

Not so good agreement:Not so good agreement: Acceptor TunnellingAcceptor Tunnelling

II. Characterising States up to 60cm-1II. Characterising States up to 60cm-1

J=0,…,8, K=0,..,J.J=0,…,8, K=0,..,J. J=0,…,20, K=0,1,2.J=0,…,20, K=0,1,2. E states are not included because they are very E states are not included because they are very

large calculations – UCL LEGION facility.large calculations – UCL LEGION facility. Actually many more states included, should be Actually many more states included, should be

relatively simple to go up to say 100-200cm-1.relatively simple to go up to say 100-200cm-1.

Helped with a new 64GB RAM computer.Helped with a new 64GB RAM computer.– Large Hamiltonians can be stored in memory.Large Hamiltonians can be stored in memory.

III. Modified water dimer PESIII. Modified water dimer PES

)],,(),,([)](2)()([),( 21212121 eeecorr VVVVVV aaaaaaaaa

),(),,(),,(),,( 21212121 aaaaraaraa corrcorr VVVV

New 12D Huang et al. PES seems to work well for low-level New 12D Huang et al. PES seems to work well for low-level dimer VRT statesdimer VRT states

Not so well for Monomer Modes.Not so well for Monomer Modes. Correction for monomer modes:Correction for monomer modes:

New Potential Expression:New Potential Expression:

Tests for PotentialTests for Potential Revaluation of the saddle points.Revaluation of the saddle points. Revaluation of the dimer VRT states.Revaluation of the dimer VRT states.

Picture from: X. Huang, B. J. Braams, J. M. Bowman, J. Phys. Chem. A 110, 445 (2006).

Dimer Absorption ModelDimer Absorption Model

to calculate water dimer absorption to calculate water dimer absorption throughout visible and IR region in the throughout visible and IR region in the atmosphere atmosphere ab initioab initio..

Direct Computation impossible!Direct Computation impossible! We have developed a new model.We have developed a new model.

IV. Franck-Condon Type IV. Franck-Condon Type ApproximationApproximation

Recap – Recap – FC approxFC approx::

BO approx:BO approx:

Assume Transition is vertical:Assume Transition is vertical:

fvr

fei

ivr

iei

2

fiI

2fvr

fe

ivr

ieI

22fvr

ivr

fe

ieI

Franck-Condon FactorFranck-Condon Factor

(square of overlap integral)(square of overlap integral)

Electronic Band Electronic Band intensityintensity

Adiabatic Separation of Vibrational Adiabatic Separation of Vibrational ModesModes

dmm 21

Separate intermolecular and intramolecular modes.Separate intermolecular and intramolecular modes.

mm11 = water monomer 1 Vibrational Wavefunction = water monomer 1 Vibrational Wavefunction

mm22 = water monomer 2 Vibrational Wavefunction = water monomer 2 Vibrational Wavefunction

d = dimer Vibration-Rotation Wavefunctiond = dimer Vibration-Rotation Wavefunction

Transition:Transition:

Approximation:Approximation:(Franck-Condon type).(Franck-Condon type).00thth Order Model Order Model

2

2121

2fffiii

fi dmmdmmI

fi mm

mExcite

22

1

22

1122

fifi

mmddmmfi

=1=1

Franck-Condon FactorFranck-Condon Factor Monomer Vibrational Monomer Vibrational

Band IntensityBand Intensity

IV. Franck-Condon Type ApproxIV. Franck-Condon Type Approx

Comp realisationComp realisation

Monomer Vibrational Band intensitiesMonomer Vibrational Band intensities– > Matt.> Matt.

Franck-Condon factors:Franck-Condon factors:– Overlap between dimer states on adiabatic Overlap between dimer states on adiabatic

potential energy surfaces for water monomer potential energy surfaces for water monomer initial and final statesinitial and final states

V. Vibrational AveragingV. Vibrational Averaging Modify van der Avoird Modify van der Avoird et alet al. methodology to implement 12D . methodology to implement 12D

flexibility for VRT levels.flexibility for VRT levels. Since only 6D code.Since only 6D code. Separate intermolecular and intramolecular modes.Separate intermolecular and intramolecular modes. For each monomer state and calculate VRT levels.For each monomer state and calculate VRT levels. We want to vibrationally average the potential for monomer We want to vibrationally average the potential for monomer

modes.modes.

In this way, we can create a 12D effective PES.In this way, we can create a 12D effective PES.

1| 2|

);,()()(

)()(|);,(|)()(22 rQQQQ

QQrQQQQ

BAAABB

AABBBABBAA

V

V

V. Vibrational AveragingV. Vibrational Averaging

Very many potential energy points need to be evaluated.Very many potential energy points need to be evaluated. Example: Example:

– typical number of DVR points:typical number of DVR points:

– {28, 28, 44} gives {28, 28, 44} gives 17,864 points for monomer17,864 points for monomer

– 17,86417,86422 = 319,122,496 points for the dimer = 319,122,496 points for the dimer

– 319,122,496 * 2,894,301 intermolecular points 319,122,496 * 2,894,301 intermolecular points

= 923,349,349,048,896 points - one bad headache! = 923,349,349,048,896 points - one bad headache!

Energies up to 16,000 cm-1 sufficient.Energies up to 16,000 cm-1 sufficient. Use simpler monomer wavefunctions.Use simpler monomer wavefunctions. Easier Computation:Easier Computation:

– typical number of DVR points with different Morse Parameters:typical number of DVR points with different Morse Parameters:

– {9,9,24} gives 1,080 points for monomer (cf. 17,864){9,9,24} gives 1,080 points for monomer (cf. 17,864)

– 1,0801,08022 = 1,166,400 points for the dimer (cf. 319,122,496) = 1,166,400 points for the dimer (cf. 319,122,496)

– 1,166,400 * 2,894,301 intermolecular points 1,166,400 * 2,894,301 intermolecular points

= 3,374,862,926,400 points - one not so bad headache! = 3,374,862,926,400 points - one not so bad headache!

(cf. 923,349,349,048,896) (cf. 923,349,349,048,896)

V. Vibrational AveragingV. Vibrational Averaging

Calculations to be done on UCL condor service (pool of 1,400 UCL computers)Calculations to be done on UCL condor service (pool of 1,400 UCL computers)

DiscussionDiscussion

Work in Progress.Work in Progress. Comments welcome.Comments welcome.