Quantum Dynamics Studies of Anomalous Isotope Effects

Dmitri Babikov

Marquette University, Chemistry Department

Milwaukee, Wisconsin, USA.

Outline

What we learned about the MIF from studies of the anomalous isotope effect in O3?

What remains a challenge for theory and experiment on ozone?

What has already been done on the S-containing species?

What can be done in the near future on the S-containing species?

(Mauersberger, Geophys. Res. Lett. 8, 935, 1981)

(Heidenreich & Thiemens, JCP 78, 892, 1983)

In the laboratory studies: equal in 17O and 18O “mass independent”

Discovery of Enrichments in O3In the atmosphere:“anomalous”

O + O2 + M O3 + M

(Mauersberger and co., J. Geophys. Res. 95, D1, 901, 1990).

Rates can differ by more then 50%,remarkably large isotope effect taking into account a small change of mass!

Traced to the three-body recombination reaction which forms ozone:

First Round of Theoretical Research on Anomalous Isotope Effect

Year Method Author Comments

1981 Experimental Discovery Mauersberger  Stratosphere

1983 Statistics Kaye, Strobel Depletion?

1986 -1992

9 papers Bates Failed on rates

1996 SIKIE Gellene Contradictsexp. results

1997 Classical Trajectories Gross, Billingalso Schatz

Tiny effect, in wrong direction

This is not a complete list…

State of the Problem before the Y 2000

“Currently, in spite of 15 years of intensive experimental and theoretical investigation,.. the mechanism that is responsible remains unidentified.”

“Despite the progress that has been made during the past 10 years, a convincing physical explanation of the process that results in enrichment is still missing.”

M. H. Thiemens, Science 283, 341 (1999) K. Mauersberger, Science 283, 370 (1999)

- Theory was not particularly successful.

- Experiments results were not complete and were not summarized in the form useful to theoreticians.

- Situation changed dramatically in the XXI century!

Relative Rates (Exp.) (Mauersberger and co., PCCP 3, 4718, 2001)

Very strong isotope effects.

No any obvious correlation with O3 masses.

Experimental Rates vs. DZPE

18→16 16→1817→16 16→1718→17 17→18

16→1617→1718→18

(Mauersberger and co., PCCP 3, 4718, 2001)

xO + yOzO → (xOyOzO)* → xOyO + zO

+ M → O3

Their explanation of the correlation:

ZPEs of O2 on the reactant and product sides are different. The atom exchange reaction is slightly exothermic or endothermic. This affects rate of the reaction, and lifetime of the intermediates.

This paper stimulated new round of theoretical work.

Year Method Author Comments

1981 Experimental Discovery Mauersberger   Stratosphere

2001 -2004

RRKM + Non-statistical effect + Tuning parameters

Gao, Marcus,Hathorn

No PES, emp., unphys. values

2002 Reduced Dimensionality + Sudden Approx.

Charlo, Clary Small, mostly in wrong direction

Second Round of Theoretical Research on Anomalous Isotope Effect

2003 Quantum Reactive Scattering

Babikov, Walker, Kendrick, Pack

J = 0 only

2004 Classical Trajectories + adhoc ZPE

Schinke and co.

Empirical

2005 Inelastic Scattering + Sudden Approx.

Xie, Bowman b = 0 only, two orientations

2009 Resonance Lifetimes (J > 0) + Strong Collision Approx.

Grebenshikov, Schinke

Symmetric Top Approx.

Statistical Theory of Anomalous Isotope Effect (Marcus)

- Densities of states are computed from models (free rotor; hindered rotor).

- Simple models for deactivation are assumed (strong collision; step-ladder; exponential).

h =1

h =1.18

DE =210 cm-1

Marcus and co., JCP 117, 1536 (2002).

Findings:

- Effects of the Ya,b cancel in the “scrambled”

conditions and have no effect on enrichments.

- The h -effect is essential for enrichments.

RRKM

Ya,b – partitioning factor ( due to DZPE ); h – symmetry factor ( in either w

or r ).

- Parameters DE and h are tuned to fit expe- rimental results for 16+1818 and 18+1616.

O3

Na† Nb

†

16+1818

1618+18

The First Quantum Dynamics Treatment (Clary)

● Vibrational motion of O3 is described by the wavefunction:

● A reduced dimensionality approximation is employedin which the bending angle in Jackoby coordinates is fixed:

This wave function is represented numerically by a 2D-grid of points (140x140)

),,( Rrv

);(8.136 const );,( Rrv ),(3O RrV

O3 bound states

resonances O3* ~ 90 bound states

~ 60 resonances

Charlo and Clary, JCP 117, 1660 (2002).

● Bound states and scattering resonances (metastable states) are computed for J = 0 by solving the TISE using the stabilization method:

),(),(]ˆ[33 OO RrERrVT vvv

The collision of O3 + M is treated by introducing spherical polar coordinates for M:

The scattering of M is described by a multi-dimensional wave function in the form:

The Coupled-Channel Approach

),(),;(

),,,,(M

MMMMMM Rr

R

RfRrR v

v

v

Sudden collision approximation is used (O3 cant rotate during the collision): - dependence is parametric (essentially 1D). Separate

calculations are performed for a large number of fixed values of angles (10x24). Final results are obtained by averaging over the angles.

),;( MMM Rf

Next approximation neglects the ro-vibrational couplings for O3 states (IOS).

333 OOOtot2M

M2M

2

M

2

ˆ)(2

)1(

2ˆ VTVV

R

llR

RRH

same as in O3

Result of the calculations is a scattering matrix for state-to-state transitions:

~ 100 open, ~ 20 closed channels. Solved on a grid of points in RM (~150):

Substitution into the TISE leads to a system of coupled-channel equations:

The Coupled-Channel Approach

)(2

)()1(

M2M2

2M

2M

2

RfVRfkR

ll

R vv

vvvv

vvvv VVV 3Otot

- potential coupling matrix

22 /)(2 vv EEk - wave vector in channel v

0)0( vf

)2/()2/(M

MMM1

)( lRkivv

lRkivv

v

Rv

vv eSek

Rf

)()()()( recstab TkTkEES vvvv

Done for ~ 30 total energies E, separately for ℓ = 0 to 150:

The First Quantum Results (Clary)

● Results of this very nice work were not particularly encouraging: only one reaction showed large isotope effect in the right direction. In all other cases the isotope effect was either small, or in wrong direction.

● No any correlation was observed. No clear mechanism proposed.

● Due to a number of approximations used it was somewhat hard to figure out the problem.

Note: This method was successful in the treatment of HCN + Ar energy transfer.

What possibly could be a problem in O3 case?

My guess is: Unlucky combination of the PES features + reduced dimensionality implemented using Jackobi coordinates.

),( RrV

Role of Quantum Scattering Resonances (Babikov)

● Focus on energies and lifetimes of O3* states.

● Vibrational wave functions are full dimensional (3D):

- the adiabatically-adjusting principal-axes hyper-spherical

coordinates (APH) are employed.

● Accurate ab initio global PES was used.

● Only the J = 0 case was considered.

● Coupled-Channel treatment of O + O2 scattering is employed.

),,( v

spectrum lifetime)()( EQES jjjj

O + O2 O3*

O3* + M O3 + M

Babikov et al, J. Chem. Phys. 118, 6298 (2003).

j =1

j =3

j =5

j =7

j =1

j =0

j =2

j =3

j =4

j =5

18O18O 16O18O

DZ

PE

En

erg

y (

eV

)

Lifetime (ps)

16O18O18O J = 0

O3

PES

ZPEO2

O2

0.Threshold

EOO

The Lifetime Spectrum

j =1

j =3

j =5

j =7

j =1

j =0

j =2

j =3

j =4

j =5

18O18O 16O18O

DZ

PE

En

erg

y (

eV

)

Lifetime (ps)

16O18O18O

En

erg

y (

eV

)

Lifetime (ps)

16O16O18O

j =0

j =1

j =5

j =6

j =3

j =1

j =2

j =3

j =4

j =5

16O18O 16O16O

DZ

PE

Babikov et alCPL 372, 686 (2003).

16+1818

PES

1618+18

161818

D ZPE

Stable O3

ZPE1818

ZPE1618

Mechanism of DZPE Isotope Effect : 16O18O18O

16+1818

D ZPE

Stable O3

Metastable O3*

PES

ZPE1818

ZPE1618

Babikov et alCPL 372, 686 (2003).

1618+18

161818

Rate: 1.50 Rate: 0.92

Mechanism of DZPE Isotope Effect : 16O18O18O

Babikov et alCPL 372, 686 (2003).

“Background”

Rate: 1.45Rate: 0.92

16+1618

D ZPE

Stable O3

Metastable O3*

PES

ZPE1618

ZPE1616

1616+18

161618

• Quantum effect; (classical trajectories could not reproduce).• General effect.

Mechanism of DZPE Isotope Effect : 16O16O18O

3) correct order of magnitude.

D. Babikov et al, J. Chem. Phys. 119, 2577, 2003

1618

1818

1616

53.30181618

181816

J

33.00161816

161618

J

1) source of a very large isotope effect.

56.1EXP181618

181816

73.0EXP161816

161618

2) is always in the right direction.

2~EXP0J

2

1~EXP

0J

232 O OOO O )( AA

k

k

ik

kBB

Afi

Adi

Bdi

Bfi

E

M M )( O O 33 sik

iE}/exp{)( EETk i

si

A simple model:

Mechanism of DZPE Isotope Effect

Channel-specific rate coefficients:

i

Ai

A si

Ais

idi

di

diA

i kKkkk

kBA

A

][M

i

Bi

B si

Bis

idi

di

diB

i kKkkk

kBA

B

][M;

;

Introducing DZPE into Classical Trajectories (Schinke)

Although it is impossible to introduce rigorously the quantum ZPE into classical trajectory simulations, it appears feasible to mimic the DZPE effect using a simple trick:

- DZPE is introduced into the PES ad hoc;- The time classical trajectories for O + O2 spend in the O3* region is determined;

Schinke and Fleurat-Lessard, JCP 122, 94317 (2005).

),()(1)( TkeP T

Adjustable parameters describe stabilization and are used to fit the experimental data:

E

- symmetry effect (postulated).

- energy transfer efficiency (model),

- stabilizing collision frequency ( ~ P ),

Smooth dependence,no resonances.

- Stabilization probability is defined as:

3D Coupled-Channel Treatment (Bowman)

● Same approach as Clary, but O3 wave functions are full dimensional (3D), as the PES. A basis set of 100 Legendre polynomials is used for the bending motion.

● Calculations were carried out only for ℓ = 0(zero impact parameter), single value of collision energy, and three orientations of O3*(head, tail, perpendicular).

● The focus was on the DZPE effect and the role of van der Waals states.

Xie and Bowman, Chem. Phys. Lett. 412, 131 (2005).

Results: - Confirmed importance of the DZPE range;- Proposed that the vdW states are

important.

J > 0 Calculations of the DZPE Effect (Grebenshchikov)

● Centrifugally Sudden Approximation for rotation and the symmetric top rotor model are used.

● PES is simplified by removing the vdW part, leaving only the covalent well.

● Narrow resonances (G ≤ 1 cm-1) are determined for J ≤ 40, K ≤ 10.

● First order perturbation theory is used to determine the branching ratios for two channels.

● Stabilization is not treated, simple model is used (strong collision assumption).

Grebenshchikov and Shinke, JCP 131, 181103 (2009).

The bottom-line:

Several different authors/methods show importance of the DZPE range. Although not yet modeled with full rigor, this effect appears to be fairly well understood at this point.

2010 O + O2 inelastic scattering H. Guo exc. agreement with K. Boering

New Round of Theoretical and Experimental Research

2010 Energy transfer in Ar + O3

(rotational sudden CC)Ivanov and Schinke

no symmetryeffect found!

Year Method Author Comments

1981 Experimental Discovery Mauersberger   Stratosphere

2011 Energy transfer in Ar + O3

(mixed quantum-classical)Ivanov and Babikov

in progress

Focus on: - Explaining very detailed experimental results on O + O2 scattering studied in the molecular beam conditions;

- Finding origin of the symmetry effect (h-factor).

Search for Origin of the Symmetry Effect (Schinke)

Red – 686Black – 866

Ivanov and Schinke, Mol. Phys. 108, 259 (2010).

● Sudden Collision Approximation for energy transfer in the Ar + O3 collisions; IOSA and the Coupled-Channel formalism.

● Similar to Clary/Bowman, but only the bound states (below dissociation threshold) are taken into consideration.

Q.: Inaccurate near threshold?

● The wave functions and the PES are full dimensional, but the PES is simplified by removing the vdW part, leaving only the covalent well.Results: Expected symmetry effect was not observed... Agreement with classical trajectory simulations was surprisingly good (small effect in wrong direction).

Quantum Symmetry Effect in a Model System (Pack)

This simple problem allows to carry out very clean VRIOSA calculations:- masses are slightly modified in order to have exactly the same reduced masses, same energies and lifetimes of all states. - there is no DZPE here, any difference

seen is due to symmetry in the Ne2* + M collision.

- number of states is small and easy to treat.

16Ne + 18Ne 1618Ne2* (+ M) 1618Ne2 17Ne + 17Ne 1717Ne2* (+ M) 1717Ne2

Pack and Walker, JCP 121, 806 (2011).

State-to-state ( v, j ) rate coefficients in 1618:

Results:- weak (strict) selection rules for state to state

transitions in 1618 (1717). Quantum effect, classical trajectories would not reproduce.

- this opens additional pathways for the energy transfer and increases the recombination rates.

11% isotope effect !

Scaling Problems in the Quantum Dynamics Calculations

Scaling with J- Size of the Hamiltonian matrix scales

linearly with J : 1296 x 1296 for J=0

41248 x 41248 for J=31- Cost of calculations

scales as ~ J3, J2.

Kendrick, JCP 114, 8796 (2001).

)63(

computed seigenvalue #

sizematrix NCe

- exponential scaling problem.

B. Poirier and co., J. Chem. Phys. 124, 144107 (2006).

Scaling with Number of Atoms

- Number of vibrational degrees of freedom is 3N–6. - Most QM methods use Direct

Product Basis sets (DPB) to express the wave function. As result:

Parallelization Problem ( Poirier )

It is found that when the quantum dynamics calculations are parallelized using standard math libraries the efficiency significantly drops after p ~10 or so (communication).

Chen and Poirier, J. Comp. Phys. 219 (2006) 185.

B. Poirier has shown that using the sparsity pattern of the Hamiltonian matrix, a speedup close to linear and efficiency close to one can be achieved with large number of processors for large systems.

~ 2x107

- Block Jacoby diagonalization;

- Domain decomposition;- Data distribution;- Load balancing.

• The bending angle is “relaxed” to convert 3D PES into a 2D PES, V(r1,r2).

• The overall topology of the surface is preserved:

De, w, “reef”, vdW wells, channels.

• Two channels allow studying the DZPE effect.

• The effect of the bending enters through the bending energy correction and the partition function.

Adiabatic Bending Model (Babikov)

2 3 4 5 6 7 8 9 10 11

2

3

4

5

6

7

8

9

10

11

r1 (a.u.)

r2 (a.u.)

vdW wells

16O16O + 18O

16O

+ 1

6O

18O

Approach based on fast vs. slow degrees of freedom (Born-Oppenheimer like):

Ivanov and Babikov, JCP 134, 174308 (2011).

Mixed Quantum-Classical Theory of Energy Transfer (Babikov)

Ivanov and Babikov, JCP 134, 144107 (2011).

The internal vibrational motion is treated with QM using the TDSE (wave packet): resonances, DZPE and permutation symmetry.

The collisional motion O3* + M and rotation of O3* are treated using classical trajectories: computational advantage.

Energy is exchanged between translation, rotation, vibration.Total energy is conserved.

16O18O16OJ=19, Ka=4, Kb=12

b(a0)

Classical degrees of freedom allow intrinsic massive parallelization.

Can Quantum Isotope Effects Contribute to S-MIF ?

Several gas-phase reactions involve S-atoms and may exhibit the DZPE and symmetry isotope effects: Gao and Marcus, JCP 127, 244316 (2007).

S + S2 (+ M) → S3 Sm + Sn (+ M) → Sn+m

SO + O (+ M) → SO2

SO2 + O (+ M) → SO3

SO + H (+ M) → HSOSO2 + OH (+ M) → HSO3

CI Recombination by ET:

Farquhar et al., J. Geophys. Res. 106, 32829 (2001).

Pavlov and Kasting, Astrobiology 2, 27 (2002).

Several other:

S + SH → S2 + HS + OH → SO + HHS + O → H + SOS2 + O → S + SO S + O2 → SO + O

Relative Rates (Exp.) (Mauersberger and co., PCCP 3, 4718, 2001)

Although not really a one-day job, the classical trajectory simulations can be carried out for variety of chemical reactions relatively easily.

Size of the molecule - the number of degrees of freedom, is not really a problem. (Well, given the potential energy surface…)

Can Dynamics Methods be Applied to S-MIF ?

Ivanov and Schinke, JCP 126, 54304 (2007).

2181614

141816 NOin 2~

3Oin 6.1~

Note: It is relatively straightforward to set up such calculations for SO2

Example: Isotope effect in the O + NO (+ M) → NO2 recombination.

Classical + ZPE method of Schinke was applied and predicted the isotope effect (larger than that in ozone):

Construction of the Potential Energy Surfaces

Before the nuclear dynamics is studied, the electronic structure problem is solved for many nuclear configurations (independently). Dependence of electronic energy on nuclear configurations gives the potential energy surface. Motion of the nuclei on this surface (dynamics) is studied next.

Two major methods for building a continuous surface from the descrete ab initio data points:

- Spline (highly accurate, but practical only for small molecules);

- Analytic fit (the only way to go in the case of larger polyatomics).

Permutational invariance is important in the context of the isotope effects,which affects the choice of

- Coordinates;

- Functional form of the fitting function.

Ground State PES of Ozone

Ab initio electronic structure: MOLPRO: icMRCI+Q/cc-pVQZ, CASSCF(12,9).Spline on a 3D grid: 11 x 28 x 20 = 6160 points

(Schinke and co., JCP 116, 9749, 2002)

• Spectroscopically accurate at low energies; wrong behavior in the barrier region.

• Dissociation energy:

DVQZ = 1.027 eV,

DEXP = 1.132 eV.

O

O

O117°

2.4 a0

“ Reef ” Along the Minimum Energy Path

• Correction is smooth in 3D;• Only upper part of PES;• Corrects barrier and Dexp.

• Van-der-Waals tail.

(Babikov and co., JCP 118, 6298, 2003)

E = – 1.0 eVE = – 0.9 eVE = – 0.8 eVE = – 0.7 eVE = – 0.6 eVE = – 0.5 eVE = – 0.4 eVE = – 0.3 eVE = – 0.2 eVE = – 0.1 eVE = – 0.03 eVE = – 0.02 eVE = – 0.012 eVE = 0E = – 0.027 eV

r

, q f

E = DZPEO

O

O

O

O

O

O

O

O

1.

2.

3.

OO

O

OO

O

OO

O

I.

II.

III.

OO

Oi.

ii. O

OO

( r, , ; , , )

Euler

ShapeSize

Shadow

1D Slice along the MEP:3D surface: 3D surface

Potential Energy Surface of S3

There is no global potential energy surface available for S3 at this time. Exploratory work showed many similarities to O3:

Francisco and co., JCP 123, 54302 (2005).

Francisco and co., JCP 125, 84314 (2006).

- Isoelectronic, structural similarities;

- Two isomers, cyclic-S3 is at much lower energies, 4.39 kcal/mol.- Calculations at MRCI+Q/CBS are needed to reproduce spec. const.

- Covalent well is much deeper, 2.7 eV, vibrational frequencies are smaller (translates into density of states and number of coupled channels).

Isotopic shifts predicted for 32S3/34S3 mixture:

Analog of the Hartley band in ozone:

l~272 nmTheir hypothesis:

S + S2 → *S3

*S3 → S2 + *S(1D) +hv

+M

*S(1D) → OC*S, *SO2 +R

Potential Energy Surfaces of SO2

- accurate empirical PES for the ground X 1A1 state.

H. Guo, Chem. Phys. Lett., 329, 503 (2000).

Ab initio PES for the C 1B2 state:~

~

Photo-absorption Spectra of SO2 Isotopomers (Gua)

H. Guo, Chem. Phys. Lett. 439, 280 (2007).

bending mode progressions

Calculated absorption spectra:

- Intensities are quite similar among the isotopomers.

- Frequency shifts are regular.

Vibrational states of the excited PES(adiabatic calculations)

… … … … …

Conclusions Very neat quantum mechanical effects lead to MIF in O3: - DZPE effect; - symmetry effect; - scattering resonances.

Challenges for theory and experiment on ozone: - spectroscopically accurate PES near O + O2 threshold; - collisional stabilization of O3* and the symmetry effect; - dynamics and spectroscopy near threshold.

Work done on the S-containing species: - Predictions of statistical theory for SO2; - PESs of SO2; preliminary work on S3; - Photo-absorption spectra of SO2 .

In the near future: - Accurate PESs for S3 ( MRCI-CBS level ); - Classical trajectory studies for S3 and SO2 ( aka Schinke ); - Energy transfer in the S3 + M collisions ( mixed Q-C ).

Acknowledgments: NSF Atmospheric Chemistry Program ($$$)

j =0.

j =117 deg.

j =180 deg.

j ~ 80 deg.