quantum scattering calculationson chemical reaction

1st Day

schedule

day 1.1.Introduction

day 2.2.three-atom reactions

day 3.3. four-atom reations4. five and six-atom reactions5. conclusion

* This article covers the period 1995-2003Up to 1995, Bowman & Schatz, Ann. Rev. Phys. Chem. 46:169-195

scattering

wave function

-incoming plane wave :

-outgoing spherical wave :

-scattering amplitude :

-S function :

cross section

differential cross section : the probability to observe a scattered particle in a given quantum state per solid angle unit

cross section: integral of the differential cross section on the whole sphere of observation

coordinate system

Jacobi coordinate(ri,Ri)

reaction path coordinate(u,s)

hypersphericalcoordinate(θ,ρ)

coordinate system

Jacobi coordinate (3 different ways)

state-to-state reaction dynamics

state-to-state reaction probability (P)

Pαtα’t’ = |Sαtα’t’|2 (α: reagents or products, t: electronic structure)

A schematic diagram of state-to-state reaction dynamics in the vibrationally adiabatic basis

reaction probabilities for H+HD(v=0,j=0) → D+H2(v ’, j ’)

1.introduction

quantum scattering calculation

To get informations from dynamics of chemical reactions

simplist reationA + BC(v, j) -> AB(v', j') + C

objectivesrigorous PES for chemical reaction

It is possible to calculate kinetic quantities, such as rate constants, from the dynamics calculationsand these can have useful applications in models of reacting systems(such as atmospheres, interstellar clouds, and combustion processes).

scope of this articleapplication of quantum molecular collision theory to bimolecular chemical reactions in the gas phase

time-independent Schrődinger eqn.

variational methodsinvolve expansion of wavefunction in basis function for vibrational-rotational statesof reactants and products

hyperspherical methodJacobi coordinates transformed to set of polar coordinatesgeneral code for differential cross section for atom-diatom rxn. (using hyperspherical method)

time-independent Schrődinger eqn.

4-atom reactionsreduced dimensionally(RD) theoryex.rotating bond approximation(RBA)

3-atom reactions(atom-diatiom)accurate state-to-state reaction probablities for J = 0(J > 0, at least twice as many angular basis functions are required)

approximation for J > 0 (J-shifting approximation): PJ(E)(J > 0) from PJ=0(E)(J = 0)

treat explicitly : s, R, r, and θaveraging over : γ,φ(φ: out-of-plane torsional motion of AB w.r.t. CD)

“not rotationally selectivein CD”

time-dependent Schrődinger eqn.

solve TD eqn. -> reaction dynamics on the potential surfaceThe solutions are time-evolving quantum wave packets

Ψ(t) = exp(-iHt/ħ) • Ψ(t=0) (time-evolution operator exp(-iHt/ħ))

exp(-iHt/ħ) : computed by propagation algorithm

Ψ(t) : represented by grid

time-dependent Schrődinger eqn.

split-operator method :

Chebychev series :

Ψ(t) : represented by grid

exp(-iHt/ħ) : computed by propagation algorithm

Lanczos method (iteration method)

In some case, different methods are used by TD Hamiltonian

time-dependent Schrődinger eqn.

advantage physical picture of dynamicsvector-matrix multiplication method (scale : TD→less than N2, TI→N3)available for six-dimensional four-atom, three-atom with large basis set calculation

disadvantageunable to the dynamics involves long-lived resonance (efficient for TI)artificial absorbing potentials in TD typically reflect small part of wave packet(thus, TI often gives more accurate results than TD)if state-to-state probabilities or cross sections are required,

* efficiency of a TD method depends on the basis set and propagation method

time-dependent Schrődinger eqn.

reactant product decoupling(RPD) approachavoid the coordinate problem by splitting the (exact) Schrodinger eqn.

quantum scattering calculationson chemical reaction

2nd Day

2. Three-atom reactions

H+H2 → H2+H

quantum scattering calculation (1975)

Liu-Siegbahn-Truhlar-Horowitz(LSTH) PES (late 1980s)converged integral and differential cross sections

geometric phase(GP) associated with conical intersection(1990)

geometric phase(Berry phase) associated with conical intersectionex. Jahn-Teller effect conical intersection

H+H2 → H2+H

geometric phase(Berry phase) associated with conical intersection

cancelation of geometric phase

H+H2 → H2+Hcancelation of geometric phaserepresentative classical trajectories for H+H2(v=1,j=0) reaction

H+D2 / H+HD → HD + D/H

qunatum scattering calculation (since 1995)

BKMP2 PES

methods used in BKMP2 calculationsKohn variational methodhyperspherical coupled channel methodTD wave packet mathod

overall behavior of the cross sectionsusing quantum and quasiclassical trajectory(QCT)backward scattering dominates for low value of j’shifting to sideway scattering at higher j’

forward scatteringtime-delay in forward scattering

H+D2 / H+HD → HD + D/H

quasiclassical trajectory(QCT)“In the quasiclassical trajectory method, molecules are prepared in discrete internal energy states corresponding to the quantum state of the molecule.Once the trajectory is begun, this quantum restriction is relaxed so that the time evolution of the system is governed solely by classical mechanics. Asimilar “quantization” is often employed on the analysis of product moleculeinternal energy state” Truhlar and Muckerman, ‘Reactive scattering cross section’

*exampleO(3P) + HCl → OH + Cl

H+D2 / H+HD → HD + D/H

forward scatteringschematic illustration of the time-delayed reaction mechanism

F+H2 → HF+H

importancebenchmark exothermic reaction

Stark-Werner(SW) surface first accurate ab initio potential energy surfaceStark & Werner, 1996correctly predict a bent TSrealistic barier hightagreement with exp. of Neumark group

QCT/quantum scattering calculation on SW surface

Feschbach resonance

spin-orbit coupling

F+H2 → HF+H

resonance in F+HD reactionexperimenatal data

F+H2 → HF+H

resonance in F+HD reactionexperimenatal data / calculation

F+H2 → HF+H

spin-orbit coupling

Cl+H2

importanceimpotant model for transition state theoryatmospherically important reaction

PES G3 (Allison et al.)Bian-Werner(BW)

Bian-Werner(BW) surface well describes experiment (compare to others)-van der Waals well

insertion-type reations

insertion-type reactions1. O(1D)+H2 → OH(2Π)+H2. N(2D)+H2 → NH+H 3. C(1D)+H2 → CH+H 4. H+O2 → OH+O 5. O(1D)+HCl → OH+Cl / ClO+H6. N(4S)+O2 → NO+O

featuredeep wellsabsence or near absence of reaction barriers

heavy-light-heavy reactions

reactionsO(3P) + HCl → OH + ClCl + HCl → HCl + ClCl + HBr → HCl + BrF + HCl → HF + Cl

featurebarriersvery low skew angles

O-H-Cl angle(a) 10°, (b) 80.4°, (c) 131.4°, (d) 180°

O(3P) + HCl → OH + Cl

metal included reactions

reactionsLi+HF → LiF+HH+LiH → Li+H2 / HLi+H

featurelate, noncollinear TSwell in both the reagent and product channel

PES for Li + HF

Li-F-H angle(a) 180°, (b) 106°, (c) 74°, (d) 45°

ion-molecule reactionsinvolving three atoms

reactionsD++H2 → D+H2

+ / HD+H+ / HD++HHe+H2

+ → HeH++H

featuredeep wellslong-range potetialspossibility of strong nonadiabatic effect associated with charge transfer

reactive collisions at ultracold temp.

aimBose-Einstein condensation between molecules

reactionsNa + Na2 (Soldan et al.) F + H2F + D2

featureultracold temperature (<10-3K)

calculation (Na+Na2)hyperspherical close-coupling methodobtained J=0 cross sections down to 10-9Kaccordance with Wigner threshold laws

quantum scattering calculationson chemical reaction

3rd Day

3. Four-atom reactions

OH+H2 → H2O+H

Potential Energy SurfaceSchatz & Elgersma PESCollins PES Ochoa de Aspuru & Clary PES

calculation (1995)TD wave packet method

reduced dimensionrotating bond approximation(RBA)semi-rigid vibrating rotor target(SVRT)

OH+H2 → H2O+H

rotating bond approximation(RBA)

OH+H2 → H2O+H

rotating bond approximation(RBA)

exact rovibrational Hamiltionian for the isolated ABC molecule

Hamiltonian(J=0,Ω=0)

OH+H2 → H2O+H

rotating bond approximation(RBA)

R-matrix propagator methoddiagonalize the internal Hamiltionian Hl

fix,

Gaussian basis function

expanding V1 in Legendre series nL

Hθ is diagonalized

oprimized basis set,

OH+H2 → H2O+H

rotating bond approximation(RBA)

configuration intereraction(CI)

initial state k is expanded

close-coupling equation

OH+H2 → H2O+H

rotating bond approximation(RBA)

reaction cross sections obtained form S matrixinitial state

initial translational energy

S matrix

hyperspherical basis function

OH+H2 → H2O+H

rotating bond approximation(RBA)

real calculation

convergence w.r.t. number of basis functions

calculated cross sections σ(0,0→n,m)

OH+H2 → H2O+H

Ochoa de Aspuru & Clary PESlargest angle generalization of rotating bond order(LAGROBO)

OH+H2 PES in Ochoa de Aspuru & Clary PES

OH+D2 → HOD+D

mode specific behavor

energy level diagram HOD product are labeled (m,n)

m : quantum number for bending moden : OD local stretching mode

D atom product time-of-flight spectrasolid line : simulation based on best-fit

translation energy and angulardistribution

OH+D2 → HOD+D

mode specific behavor

comparision of three different theoretical predictionenergy level diagram HOD product are labeled (m,n)

m : quantum number for bending moden : OD local stretching mode

H2+CN → HCN+H

calculationfirst quantum-dynamical calculation (Clary et al.)temperature dependent J-shifting procedure(Zhang)wave packet calculation(Zhu)SVRT model(Ma et al.)

reduced dimensionalityL2 eigenstate method(Skokov & Bowman)RBA calculationextended RBA(Takayanagi & Schatz)-include CN stretching motion (CN bond have some effect)

OH+CO → CO2+H

importancemain reaction for producing CO2 in flame and in the Earth’s atmosphere

calculationfirst 6D wave packet calculation-initially state-selected reaction probabilities-similar to 5D result

resonancelong-lived “HOCO” intermediates.complex potential method was developed to characterize resonances

OH+HCl → Cl+H2O

importanceimportant source of Cl in the Earth’s atmosphere

calculation3D theory-Born-Oppenheimer type separation(for light and heavy nuclear motion)-comparison to RBA-extended to planar treatment of the Cl+H2Ovibrationally adiabatic model

other four-atom reaction

calculationTI method

..

4. five and six atom reactions

featuremore than four atoms are very hard to carry outapproximations are normally needed

H+CH4→ H2+CH3

reducing dimensionalitySVRT, RBA, modified-RBA, RLU

calculationTD reduced dimensionality method-consider rotational motion of CH4

-agree with the rate constants at higher temperature

O(3P)+CH4→ CH3+OH

importancekey reaction of CH4 in flame

reducing dimensionalityRBA calculation-large mode-selective effect-very low vibrational excitation of CH3

-cross sections about twice as large (Fig.7)RLU calculationSVRT calculation

Cl+CH4→ CH3+HCl

importancemajor source of the HCl in the atmosphere

reducing dimensionalityRLU calculationwave packet calculation

Cl-+CH3Cl→ CH3Cl+Cl-

Cl-+CH3Br→ ClCH3+Br-

importanceimportant prototype reaction in physical organic chemistry

PESdeep “ion-dipole” wells in reaction and product region

reducing dimensionalityRBA calculationnew TI quantum scattering theory

C(3P)+C2H2→ C3H+H

other polyatomic reactions

conclusion

in this paperbimolecular chemical reactionspast seven yearsover 40 different reactions

time-dependent methodwave packet method

free radicals.

potential energy surface.

quantum scattering calculations on chemical reactions

1.Introduction

2.three-atom reactions

2-1. H+H2,

2-2. F+H2,

2-3. Cl+H2,

2-4. O+H2, N+H2, C+H2, H+O2, O+HCl, N+O2

2-5. O+HCl, Cl+HCl, Cl+HBr, F+HCl

2-6. Li+HF, Na+HF, Mg+HF, Li+H2

2-7. ion-molecule reactions involving three atoms

2-8. reactive collisions at ultracold temperatures

3. four-atom reations

3-1. OH+H2

3-2. H2+CN

3-3. OH+CO

3-4. OH+HCl

3-5. other four-atom reactions

4. five and six-atom reactions

4-1. H+CH4

4-2. O+CH4

4-3. Cl+CH4

4-4. Cl-+CH3Cl, Br-+CH3Cl

4-5. C+C2H2

4-6. other polyatomic reactions

5. conclusion

0. preliminary

scattering

* wave function

-incoming plane wave :

-outgoing spherical wave :

-scattering amplitude :

-S function :

Cross section

* differential cross section

: the probability to observe a scattered particle in a given quantum state

per solid angle unit

* cross section

: integral of the differential cross section on the whole sphere of observation

Jacobi coordinate(ri,Ri)

reaction path coordinate(u,s)

hypersphericalcoordinate(θ,ρ)

Jacobi coordinate (3 different ways)

state-to-state reaction probability (P)

Pαtα’t’ = |Sαtα’t’|2 (α: reagents or products, t: electronic structure)

A schematic diagram of state-to-state reaction dynamics in the vibrationally adiabatic basis

Coordinate system

state-to-state reaction dynamics

1. introduction

* This article is for the period 1995-2003

Up to 1995, Bowman & Schatz, Ann. Rev. Phys. Chem. 46:169-195

quantum scattering calculation

* To get informations from dynamics of chemical reactions

* simplist reation

A + BC(v, j) -> AB(v', j') + C

* objectives

-rigorous PES for chemical reaction

-It is possible to calculate kinetic quantities, such as rate constants,

from the dynamics calculations and these can have useful applications

in models of reacting systems

(such as atmospheres, interstellar clouds, and combustion processes).

*scope of this article

-application of quantum molecular collision theory to bimolecular chemical

reactions in the gas phase

time-independent Schrődinger eqn. (TI)

* variational method

-involve expansion of wavefunction in basis function for vibrational-rotational

states of reactants and products

* hyperspherical method

: Jacobi coordinates transformed to set of polar coordinates

-general code for differential cross section for atom-diatom rxn.

(using hyperspherical method)

* 4-atom reactions

-reduced dimensionally(RD) theory

rotating bond approximation(RBA)

treat explicitly : s, R, r, and θaveraging over : γ,φ(φ: out-of-plane torsional motion of AB w.r.t. CD)

“not rotationally selectivein CD”

* 3-atom reactions(atom-diatiom)

-accurate state-to-state reaction probablities for J = 0

(J > 0, at least twice as many angular basis functions are required)

approximation for J > 0 (J-shifting approximation)

: PJ(E)(J > 0) from PJ=0(E)(J = 0)

time-dependent Schrődinger eqn. (TD)

* solve TD eqn. -> reaction dynamics on the potential surface

-solutions are time-evolving quantum wave packets

Ψ(t) = exp(-iHt/ħ) * Ψ(t=0) (time-evolution operator exp(-iHt/ħ))

exp(-iHt/ħ) : computed by propagation algorithm

1.split-operator method :

2.Chebychev series :

3.Lanczos method (iteration method)

4.In some case, different methods are used by TD Hamiltonian

Ψ : represented by grid

* advantage

-physical picture of dynamics

-vector-matrix multiplication method (scale : TD→less than N2, TI→N3)

available for six-dimensional four-atom, three-atom with large basis set

calculation

* disadvantage

-unable to the dynamics involves long-lived resonance (efficient for TI)

-artificial absorbing potentials in TD typically reflect small part of wave packet

(thus, TI often gives more accurate results than TD)

-if state-to-state probabilities or cross sections are required,

* reactant product decoupling(RPD) approach

-avoid the coordinate problem by splitting the (exact) Schrodinger eqn.

* efficiency of a TD method depends on the basis set and propagation method

-various propagation methods

1. TI wave packet

-Lippmann-Schwinger eqn.

2. damped Chebyshev

-Chebyshev Operator : transform time/energy to order/angle formulation

-numerical advantage

3. real wave packet

4. filter diagonalization

5. symplectic integrator

6. L2 methods

* reagent to product coordinate transformation

-applying it at just one time-step

often difficult, owing to spreading of the wave packet

2.three-atom reactions

H+H2 → H2+H

quantum scattering calculation (1975)

Liu-Siegbahn-Truhlar-Horowitz(LSTH) PES (late 1980s)

-converged integral and differential cross sections

geometric phase(GP) associated with conical intersection(1990)

ex. Jahn-Teller effect conical intersection

-cancelation of geometric phase

-representative classical trajectories for H+H2(v=1,j=0) reaction

-cancelation of geometric phase

H+D2 / H+HD → HD + D/H

qunatum scattering calculation (since 1995)

BKMP2 PES

methods used in BKMP2 calculations

-Kohn variational method

-hyperspherical coupled channel method

-TD wave packet mathod

overall behavior of the cross sections

-using quantum and quasiclassical trajectory(QCT)

-quasiclassical trajectory(QCT)

'In the quasiclassical trajectory method, molecules are prepared in discrete

internal energy states corresponding to the quantum state of the molecule.

Once the trajectory is begun, this quantum restriction is relaxed so that the

time evolution of the system is governed solely by classical mechanics. A

similar "quantization" is often employed on the analysis of product molecule

internal energy state' Truhlar and Muckerman, 'Reactive scattering cross section'

-backward scattering dominates for low value of j'

-shifting to sideway scattering at higher j'

forward scattering

-time-delay in forward scattering

F+H2 → HF+H

importance

-benchmark exothermic reaction

Stark-Werner(SW) surface

-first accurate ab initio potential energy surface

-Stark & Werner, 1996

-correctly predict a bent TS

-realistic barier hight

-agreement with exp. of Neumark group

-QCT/quantum scattering calculation on SW surface

Feschbach resonance

-resonance in F+HD reaction

* experimenatal data

* experimenatal data / calculation

spin-orbit coupling

Cl+H2

importance

-impotant model for transition state theory

-atmospherically important reaction

PES

-G3 (Allison et al.)

-Bian-Werner(BW)

Bian-Werner(BW) surface

-well describes experiment (compare to others)

*van der Waals well

insertion-type reations

insertion-type reactions

1. O(1D)+H2 → OH(2Π)+H

2. N(2D)+H2 → NH+H

3. C(1D)+H2 → CH+H

4. H+O2 → OH+O

5. O(1D)+HCl → OH+Cl / ClO+H

6. N(4S)+O2 → NO+O

feature

-deep wells

-absence or near absence of reaction barriers

Li-F-H angle(a) 180°, (b) 106°, (c) 74°, (d) 45°

heavy-light-heavy reactions

reactions

1. O(3P) + HCl → OH + Cl

2. Cl + HCl → HCl + Cl

3. Cl + HBr → HCl + Br

4. F + HCl → HF + Cl

feature

-barriers

-very low skew angles

metal included reactions

reactions

Li+HF → LiF+H

H+LiH → Li+H2 / HLi+H

PES for Li + HF

ion-molecule reactions involving three atoms

reactions

1. D++H2 → D+H2+ / HD+H+ / HD++H

2. He+H2+ → HeH++H

feature

-deep wells

-long-range potetials

-possibility of strong nonadiabatic effect associated with charge transfer

O-H-Cl angle(a) 10°, (b) 80.4°, (c) 131.4°, (d) 180°

O(3P) + HCl → OH + Cl

reactive collisions at ultracold temp.

aim

Bose-Einstein condensation between molecules

reactions

1. Na + Na2

2. F + H2

3. F + D2

feature

ultracold temperature (<10-3K)

calculation (Na+Na2)

hyperspherical close-coupling method

obtained J=0 cross sections down to 10-9K

accordance with Wigner threshold lows

3. Four-atom reactions

OH+H2 → H2O+H

Potential Energy Surface

-Schatz & Elgersma PES

-Collins PES

-Ochoa de Aspuru & Clary PES

calculation (1995)

-TD wave packet method

reduced dimension

-rotating bond approximation(RBA)

-semi-rigid vibrating rotor target(SVRT)

*rotating bond approximation(RBA)

exact rovibrational Hamiltionian for the isolated ABC molecule

Hamiltonian(J=0,Ω=0)

R-matrix propagator methoddiagonalize the internal Hamiltionian Hl

fix,

Gaussian basis function

expanding V1 in Legendre series nL

Hθ is diagonalized

oprimized basis set,

configuration intereraction(CI)

initial state k is expanded

close-coupling equation

reaction cross sections obtained form S matrixinitial state

initial translational energy

S matrix

hyperspherical basis function

real calculation

convergence w.r.t. number of basis functions

calculated cross sections σ(0,0→n,m)

Ochoa de Aspuru & Clary PESlargest angle generalization of rotating bond order(LAGROBO)

OH+H2 PES in Ochoa de Aspuru & Clary PES

OH+D2 → HOD+D

mode specific behavor

energy level diagram HOD product are labeled (m,n)

m : quantum number for bending moden : OD local stretching mode

D atom product time-of-flight spectrasolid line : simulation based on best-fit

translation energy and angulardistribution

comparision of three different theoretical predictionenergy level diagram HOD product are labeled (m,n)

m : quantum number for bending moden : OD local stretching mode

H2+CN → HCN+H

calculation

-first quantum-dynamical calculation (Clary et al.)

-temperature dependent J-shifting procedure(Zhang)

-wave packet calculation(Zhu)

-SVRT model(Ma et al.)

reduced dimensionality

-L2 eigenstate method(Skokov & Bowman)

-RBA calculation

-extended RBA(Takayanagi & Schatz)

include CN stretching motion (CN bond have some effect)

OH+CO → CO2+H

importance

-main reaction for producing CO2 in flame and in the Earth's atmosphere

calculation

-first 6D wave packet calculation

initially state-selected reaction probabilities

similar to 5D result

resonance

-long-lived "HOCO" intermediates.

-complex potential method was developed to characterize resonances

OH+HCl → Cl+H2O

importance

-important source of Cl in the Earth‘s atmosphere

calculation

-Born-Oppenheimer type separation(for light and heavy nuclear motion)

4. five and six atom reactions

feature

-more than four atoms are very hard to carry out

-approximations are normally needed

H+CH4→ H2+CH3

reducing dimensionality

-SVRT, RBA, modified-RBA, RLU

calculation

-TD reduced dimensionality method

consider rotational motion of CH4

agree with the rate constants at higher temperature

O(3P)+CH4→ CH3+OH

importance

-key reaction of CH4 in flame

reducing dimensionality

-RBA calculation

large mode-selective effect

very low vibrational excitation of CH3

-RLU calculation

-SVRT calculation

Cl+CH4→ CH3+HCl

importance

-major source of the HCl in the atmosphere

reducing dimensionality

-RLU calculation

-wave packet calculation

Cl-+CH3Cl→ CH3Cl+Cl- / Cl-+CH3Br→ ClCH3+Br-

importance

-important prototype reaction in physical organic chemistry

PES

-deep ‘ion-dipole’ wells in reaction and product region

reducing dimensionality

-RBA calculation

-time-independent calculation

conclusion

in this paper

-bimolecular chemical reactions

-past seven years

-over 40 different reactions

time-dependent method

-wave packet method as a major technique

free radicals

-open shell calculation

-nonadiabatic reaction calculation

potential energy surface

-functional representation of such surface for polyatomic molecules

remains a major problem