Some Techniques from Applied Mathematics useful
for Understanding the Dynamics of
Chemical Reactions and Molecular Collisions
School of ChemistryThe University of Manchester
Manchester M13 9PLEngland
J. N. L. Connor
Quantum Days in Bilbao VBilbao, Spain
July 13th-14th, 2015
Outline• Stationary phase approximation.• Uniform stationary phase approximation.• Hidden rainbows: the F + H2 reaction. • Pearcey, swallowtail,…, canonical integrals.• Bessoid canonical integral• Evenoid and oddoid canonical integrals.• Forward glories in the angular scattering of chemical
reactions.• Complex angular momentum techniques applied to chemical
reactions.
x
cos f(; x)
x
f(; x)Stationary Phase Method
f(; x)
x
x
cos f(; x)
cos f(; x)
cos f(; x)
x
x
Proceedings of Cambridge Philosophical Society. 1957, vol. 53, pp. 599‐611
“impact” for 1957 to 1967 of this paper was almost zero
Key step:Key step:
Uniform asymptotic approximation (bright side)
Transitional asymptotic approximation (bright and dark sides)
Recent comments by Nobel Laureates
(2010)
(2010)
2008
2011
Chengkui Xiahou Dong Hui Zhang
Nearside‐Farside analysis of the angular distribution for the F + H2 reaction (2008 expt)
2F F
2N N FN
2, ,f f f f
Phys. Chem. Chem. Phys., 20110 45 90 135 1803.5
2.8
2.1
1.4
0.7
0.0
_
_
_
_
_
θR / deg
log
σ (θ
R) /
(Å2
sr _ 1 )
θRr
θRr
PWS
PWS/F/r=3
PWS/N/r=3
000 300
(0,0,0) → (3,0,0)E = 0.3112 eVFXZ pes
bright dark
θ/deg
log DCS(θ)
Rainbows in Elastic Scattering of Atoms
From the book by
R.B. Bernstein,“Chemical Dynamics viaMolecular Beam and Laser Techniques”(1982)
θ/deg
bright dark
Journal of Chemical Physics, 1981
Hg - (H2)
θ/deg
bright dark
RF
Ai Ai
f
(generic formula)
Semiclassical theory of rainbow and glory scattering
cos~120
i21
JJJ
k PSJf
• Semiclassical rainbow and glory theory applies to Legendre series
• Applies to the chemical reaction C0,,AB0,,BCA fffiii mjvmjvusing exact quantum mechanics, or to many approximate theories, e.g., rotating linear model and its many extensions, CSH, IOS, DW, etc
R.,. bn
• Key quantity is the quantum deflection function:
JJS
Jd
~argd~
. .,n b S S
Quantum deflection function for F + H2 (2008 expt)
(0,0,0) → (3,0,0)E = 0.3112 eVFXZ pes
Phys. Chem. Chem. Phys., 2011
J
RF
Ai Ai
f0.0
0.00
σ (θ
R)
/ (Å
2 sr
_ 1 ) σ
(θR
)/ (
Å2
sr _ 1 )
θR / deg
θR / deg
θRr
θRr
PWS
PWSCSA
CSA
CSA
0 15 30 45
0.1
0.2
0.3
000 300
45 90 135 180
0.02
0.04
0.06
0.08
+SC/N/PSASC/N/PSA
+
(a)
(b)
SC/F/uAiry SC/F/tAiry
SC/F/uAiry
SC/N/PSA +
(0,0,0) → (3,0,0)E = 0.3112 eVFXZ pes
Phys. Chem. Chem. Phys., 2011
F + H2 (2008 expt)
θ/deg
bright dark (generic formula)
PWSθR
r
θR
SC/full
0 45 90 135 180
5
4
3
2
1_
_
_
_
_
log
σ (θ
R) /
(Å2
sr _ 1 )
θR / deg
r(0,0,0) → (3,3,0)Etrans = 0.119 eVE = 0.3872 eVSW pes
Black: PWSGreen: Semiclassical rainbow
theory
θ/deg
bright dark
F + H2 (1985 expt)
J. Phys. Chem., A, 2009
Classification of rainbows by Catastrophe (Singularity) Theory
Name Integral Unfolding
Fold Airy f(a;x) = x3 + ax
Cusp Pearcey f(a,b; x) = x4 + ax2 +bx
Swallowtail Swallowtail f(a,b,c; x) = x5 + ax3 + bx2 + cx
Butterfly Butterfly f(a,b,c,d; x) = x6 + ax4 + bx3 + cx2 + dx
ex, ,... , , ..p i ;. da b a bC f x x
Ai(a)
a a
b
, ;P a b x
3z 5 3 2exp i u zu yu xu
Bessoid integral
40
2
0, exp ix xJ J t t t t dy ty
•Diffraction theory of aberrations
•Design of optical instruments
•Design of highly directional microwave antennas
•Theory of image formation for high‐resolution electron microscopes
•Dry laser cleaning of surfaces
Bessoid integral: |J(x,y)|
Johannes Kofler, Institute for Applied Physics, J. Kepler University, Linz, Austria
Glory seen from an airplane
Uniform Semiclassical Approximation (USA, now called, UBA):
21
22121
20
221212
J
JI
21
where
Phys. Chem. Chem. Phys., 2004
(generic formula)
Derivation of USA[J.N.L.Connor, Molecular Physics, 103 (2005) 1715]
,;iexp~dd00
BSf
cos~arg,; SB
where
Make an exact local one‐to‐one change of variables:[J.N.L. Connor and H.R. Mayne, Mol. Phys., 37(1979)1]
Introduce new variables: ,;,; uu
221cos,; uAB
(0,0,0) → (3,0,0)E = 0.3112 eVFXZ pes
Phys. Chem. Chem. Phys., 2011
0.0
σ (θ
R)
/ (Å
2 sr
_ 1 )
θR / deg
PWS
CSA
USA
0 15 30 45
0.1
0.2
0.3
000 300
θ/deg
Glory analysis for the F + H2 reaction (2008 expt)
Phys. Chem. Chem. Phys., 2014
G. F. Carrier, J. Fluid Mech. 1966, tsunamis.W. H. Miller , J. Chem. Phys. 1968, elastic scattering.
Previous work:
Journal of Chemical Physics, volume 103, pages 5979-5998 (1995)
2 40
param1 exp exp polynomial up toJQ A A J i J
S matrix parameterization
3param param
0exp
quadratic phase in
nn
JJ Jnn
J
aS Q iJ J
J
Used to test the uniform CAM theory, etc
Then
0position of th Regge pole, e.g., 16.4 0.9nJ n J i
F + H2 (1985 expt)
θR/deg
DCSParameterized PWS
Numerical PWS
PWS DCSs
Linear plot
(0,0,0) → (3,3,0)Etrans = 0.119 eVE = 0.3872 eVSW pes
Physical meaning of Regge poles
• ReJn is related to the radius, R, of the interaction zone by ReJn ≈ k R.
• A Regge state is a short- or long lived “quasi-molecule” formed from the colliding partners. It corresponds to a pair of decaying surface
waves that propagate around the interaction region.
• 1/(2 ImJn) determines the life-angle of the system.
• rn is a measure of the probability of exciting the nth Regge state.
, 0,1, 2,...n
n
r nJ J
• The surface waves decay like exp(- ImJn θ ).
F + H2 (1985 expt)
(0,0,0) → (3,3,0)Etrans = 0.119 eVE = 0.3872 eVSW pes
Semiclassical DCSs
Thank you for listening!
UK Funding:
Overseas Research Students Awards Scheme
School of Chemistry, The University of Manchester
Leverhulme Emeritus Fellowship
