Well Defined and Accurate Semiclassical Surface Hopping

Propagators and Wave FunctionsMichael F. Herman

Department of Chemistry

Tulane University

New Orleans, LA USA

IMA Workshop January 16, 2009

OUTLINE

1. Background

2. Formal analysis of a surface hopping expansion

3. Improved efficiency, reduced statistical error

4. Forbidden transitions

Types of Problems of Interest

1. Photodissociation

2. Collisions involving change in electronic state

3. Nonadiabatic transitions in liquid phase

Initial wave function is a localized wave packet

H-K Propagator

Absorption spectra for the collinear photodissociation of carbon dioxide.The solid line shows the quantum result and the dashed line the HK-IVR

Comparison of adiabatic and diabatic potentials

R

Can cancel nonadiabatic terms in the Schrodinger equation by adding trajectories that hop between states.

Single hop terms:

An (inelegant) Analysis of a Semiclassical Surface HoppingExpansion for the Multi-Surface Wave Function (or Propagator).

Surface hopping: Trajectories can abruptly hop from one Vk( R) to another, where Vk( R) is the BO energy for electronic state φk.

Restrict analysis here to 1-dim, 2 state case for simplicity.

Same analysis has been performed for general case with any number of states and any number of degrees of freedom.

Conclusion: The surface hopping expansion formally satisfies the Schrodinger equation.

Because of singularities in semiclassical prefactor, generally notconvergent at all points.

ηij = < φi | 𝜵R φj >r = nonadiabatic coupling vector

< f | g >r = ∫ f * g dr

< φi | φj >r = 𝜹ij 𝜵R < φi | φj >r = < 𝜵R φi | φj >r + < φi | 𝜵R φj >r

= ηji + ηij = 0

Assume the φi are real.

Then, ηji = - ηij and ηii = 0.

In numerical calculations: Ignore momentum changes without hop. Use only T-type hops in allowed regions.

Comparison of adiabatic and diabatic potentials

R

Comparison of quantum and semiclassical transition probabilities for E > Ec

E PQ PS1(x > xt1) PS2(x > xt2)

0.38 0.618 0.440 0.576

0.40 0.951 0.819 0.918

0.45 0.142 0.179 0.143

0.50 0.835 0.761 0.838

0.60 0.543 0.508 0.544

0.75 0.356 0.348 0.356

0.90 0.118 0.120 0.118

1.20 1.87x10-2 1.77x10-2 1.86x10-2

1.40 0.184 0.182 0.184

CONCLUSIONS

• Surface Hopping Expansion Formally Satisfies SE. In General Not Convergent at All Points.

• The Surface Hopping Semiclassical IVR Methods are Capable of Providing Very Accurate Results for Many Surface Nonadiabatic Problems

• Recent Advances Show That It is Possible To Significantly Reduce The Statistical Errors in Monte Carlo Surface Hopping IVR Methods

Adventures in the Forbidden Zone

• Surface hopping expansion “formally exact” in classically forbidden region.

• Wave function fails at turning points and caustics

• Transition amplitudes have turning point singularities

Comparison of quantum and semiclassical transition probabilities of for E < Ec

E PQ PS2 PS2(FO) 0.36 0.275 0.261 0.2880.34 8.65x10-2 8.54x10-2 8.89x10-2

0.32 1.93x10-2 1.94x10-2 1.97x10-2

0.30 3.00x10-3 3.03x10-3 3.05x10-3

0.28 3.16x10-4 3.19x10-4 3.20x10-4 0.26 2.14x10-5 2.16x10-5 2.15x10-5

0.24 8.54x10-7 8.55x10-7 8.56x10-7

0.22 1.77x10-8 1.76x10-8 1.81x10-8

0.20 1.49x10-10 1.74x10-10 1.42x10-10

0.19 8.50x10-12 7.07x10-12 7.10x10-12

0.18 3.06x10-13 1.3x10-13 3.4x10-13 0.17 5.33x10-15 2.4x10-15 8.5x10-15

For transitions in forbidden zoneWave function on upper surface (Ψu) decays rapidly when moving into the forbidden zone from turning point.

Nonadiabatic coupling (η) sharply peaked around crossing point and is decaying when moving from crossing point toward turning point.

Product of Ψu and η is peaked in forbidden zone near turning point.

Suggests approximation based on behavior nearturning point may yield good results.

Probability of quantum state change in model collision system for forbidden transitions.. . . . exact quantum results, _____ semiclassical results, ----- results using “simple” approximation to semiclassical calculation.

For details see:

M. F. Herman, J. Phys. Chem. B 112, 15966 (2008).

and

P.-T. Dang and M. F. Herman, J. Chem. Phys., accepted for publication.

Why is this approximation for for-bidden transitions exciting (to me)?

• Curve crossing models use information from crossing point (where trajectory does not go).

• Local model (just uses information at turning point).• Momentum change due to hop occurs in direction of

nonadiabatic coupling vector, so hop is basically one dimensional in many dimensions.

• Since model is local and hop is one dimensional, should be possible to use model (or a generalization of it) for many dimensional problems.

Conclusions

• Surface Hopping Expansion provides very good transition probabilities even for strongly forbidden transitions.

• Cancellation between contributions from allowed and forbidden regions must be accurately accounted for.

• Good approximation obtained using only information evaluated at turning point.

Acknowledgements:

Funding: The National Science Foundation (USA)

Edward Kluk, Heidi Davis, J. Rudra, Julio Arce, Brianna Guerin, Guangcan Yang, Ouafae El Akramine, Michael Moody, Yinghua Wu, Xun Huang , Thanh Dang

Numerical Problems

• Different trajectories have different phases

• Leads to interference

• Add many terms, get result that is smaller than individual terms

• When integrations done by Monte Carlo, large relative statistical errors

• Need ways to reduce cancellation due to interference

Numerical Problems

• Different trajectories have different phases

• Leads to interference

• Add many terms, get result that is smaller than individual terms

• When integrations done by Monte Carlo, large relative statistical errors

• Need ways to reduce cancellation due to interference

2. Higher order (in size of hopping step) transition amplitudes for trajectory step

- Accounts for multiple hops in a single hopping step

- Accounts for phase difference between hopping trajectories

- Allows for use of much larger hopping steps

- Fewer hops along each trajectory

- Much of the phase cancellation is accounted for within hopping step

Numerical test of surface hopping using “optimal” representation and higher order transition amplitudes.Monte Carlo procedure for hop or no-hop choice for each hopping step.

M.F Herman and M. P. Moody, JCP 122, 094104 (2005).

The Mississippi River, 3,779 km (2,348 mi) long, is thesecond longest river, after the Missouri, in the United States.

Jackson Square is a historic park in the French Quarter andis in the heart of the French Quarter

The transition probability P12(t) is plotted. The solid line is the quantum-mechanical result. The dots are the semiclassical surf ace hopping results using the matrix multiplication method.

The transition probability P12(t) is plotted. The solid line is the quantum-mechanical result and thedashed line is the semiclassical result usingTully’s f ewest switchesmethod.

Semiclassical Results for Double Crossing Problems

E = 2.8 Quantum Transition Probability is 0.640PMM = matrix multiplication semiclassical transition probabilityPMC = Monte Carlo semiclassical transition probability, 10000 trajectories

Representation Amplitude Δx PMM PMC σ <hops>

Adiabatic Simple 0.005125 0.640 0.475 0.235 2.97 Adiabatic Simple 0.05125 0.645 0.745 0.213 2.65 Adiabatic Simple 0.1025 0.659 0.605 0.103 2.47

Adiabatic Phase Corrected 0.1025 0.641 0.569 0.086 2.36 Adiabatic Phase Corrected 0.205 0.651 0.619 0.045 1.95

App. Optimal Simple 0.1025 0.638 0.649 0.031 1.43

App. Optimal Phase Corrected 0.1025 0.636 0.644 0.029 1.36App. Optimal Phase Corrected 0.205 0.635 0.661 0.022 1.13