Mathematical methods for understanding
reactive angular scattering
School of ChemistryThe University of Manchester
Manchester M13 9PLEngland
J. N. L. Connor
Quantum Days in Bilbao IVBasque Center for Applied mathematics, University of the Basque Country,
SpainJuly 15 -16th, 2014
Outline• Nearside-Farside (NF) theory of scattering.• Local Angular Momentum (LAM) Analysis.• Resummation of Partial Wave Series (PWS)• Glories in the angular scattering:
- Uniform semiclassical theory.• “Hidden” Rainbows in the angular scattering:
- Uniform semiclassical theory• Complex Angular Momentum (CAM) Theory:
- Regge poles, parametrized S matrix- Uniform semiclassical theory
Differential Cross Section (DCS)(or angular distribution)
( ) ( ) 2, ,i f i ffσ θ θ= R.,. θθ ≡bn
( ) ( ) ( ), ,0
12 1 cos
2 iJ
i f i f Ji J
f J S Pk
θ θ∞
== +∑ . .,n b S S≡
max 1J >> Localization Principle
differential cross section
PWS scattering amplitude
2008Experimental information:
Journal of Chemical Physics, 82, 3045-3066 (1985)
θ/deg
Experimental information:
Nearside-Farside picture of scattering
Nearside waves
Farside waves
Travelling angularwaves
( )12
ie
J θ− +
( )12
ie
J θ+ +
FULLER Nearside-Farside Decomposition
( ) ( ) ( )N Ff f fθ θ θ= +
where
( ) ( ) ( ) ( )2 i12 iN
0
12 1 cos cos2J J Jk
Jf J S P Qπθ θ θ
∞
=
⎡ ⎤= + ⎢ ⎥⎣ ⎦+∑
( ) ( ) ( ) ( )2i12iF
0
12 1 cos cos2J J Jk
Jf J S P Qπθ θ θ
∞
=
⎡ ⎤= + ⎢ ⎥⎣ ⎦−∑
( ) ( ) ( ){ }2i 1 12 41
cos cos exp iJ JJ
P Q Jπθ θ θ π>>
± ⎡ ⎤∝ + −⎣ ⎦∓n.b., Travelling angularwaves
Nearside-Farside analysis of the angular distribution for the F + H2 reaction (2008 expt)
( ) ( ) ( ) ( ) ( ) ( ) ( )2F F
2N N FN
2, ,f f f fσ θ θ θ σ θ θ σ θ θ= + = =
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
Nearside-Farside analysis of the angular distribution for the F + H2 reaction (1985 expt)
( ) ( ) ( ) ( ) ( ) ( ) ( )2F F
2N N FN
2, ,f f f fσ θ θ θ σ θ θ σ θ θ= + = =
0 45 90 135 180qR ê deg
-5
-4
-3
-2
-1
logsHqRLêHfi2sr
-1L
PWS
nearnear
farfar
F+H2 FH+H
(0,0,0) → (3,3,0)E = 0.3872 eVSW pes
Phys. Chem. Chem. Phys., 2004θ/deg
Nearside-Farside analysis of the angular distribution for H + D2 -> HD + D
(0,0,0) → (3,0,0)E= 2.00 eVBKMP2 pes
0 45 90 135 180qR ê deg
-7
-6
-5
-4
-3
-2
logsHqRLêHfi2sr
-1L
H+D2 HD+DPWS
farfar
nearnear
θ/deg
( ) ( ) ( ) ( ) ( ) ( ) ( )2F F
2N N FN
2, ,f f f fσ θ θ θ σ θ θ σ θ θ= + = =
Phys. Chem. Chem. Phys., 2004
Experimental information:
NEARSIDE
FARSIDEDef
lect
ion
angl
e / d
eg
b / angstrom
Nearside-Farside analysis at the University of Bordeaux
And more recent works
European Physical JJournal, 2006
Nearside-Farside theory for Local Angular Momentum (LAM)
( ) ( )θ
θθd
argdLAM f=
Full LAM:
Nearside and Farside LAMs:
( ) ( )NN
d argLAM
df θ
θθ
=
( ) ( )FF
d argLAM
df θ
θθ
=
Also ( ) ( ) ( ) ( )N FLIP LAM , LIP , LIPkθ θ θ θ=
(0,0,0) → (3,0,0)E = 0.3112 eVFXZ pes
Thiele rationalinterpolation
( )LAM θ
θ/deg
darkbright
Nearside-Farside LAM analysis of the angular distribution
for the F + H2 reaction (2008 expt)
Phys. Chem. Chem. Phys., 2011
Handout from Prof. Michael Polanyi’s1st year physical chemistry lectures at the
University of Manchester. About 1946.
( )cos 2b R θ=
( )( )
NLAM
cos 2kR
θθ=
−
θ
b
Partial wave series
( ) ( )0
1 cos2i J J
Jf a P
kθ θ
∞
== ∑
( ) ( )1 cos2i
m mJ J
J mf a P
kθ θ
∞
== ∑
( ) ( ) ( )( )
( ), ,
max ,
12i
f i f i
f i
m m m mJ J
J m mf a d
kθ θ
∞
== ∑
Jacobi function of the second kind
( ) ( ), cosnP α β θ ( ) ( ), cosnQ α β θ
( ) ( ) ( ),, cos
f iJ
nm md P α βθ θ∝ ( ) ( ) ( ),, cos
f iJ
nm me Q α βθ θ∝
f f i f in J m m m m mα β= − = − = +
Jet Wimp
This paper requires knowledge of:
( )2 1 ;, ,F a b c x
( )1 ,; ; ; ,F x yα β β γ′ ( )4 ,; ; ,;F x yα β γ γ ′
Casoratian identity Generating functions
θ/deg
Resummation of partial wave series (R=1)
( ) ( ) ( )
( )( ) ( ) ( )
0
11
1
0
1 cos 2 12i
1 1 cos2i 1 cos
J J J JJ
JJJ
f a P a J Sk
a Pk
θ θ
β θβ θ
∞
=
∞
=
= = +
=+
∑
∑
where( ) ( ) 1 11
1 1 11
2 1 2 3J J JJJ Ja a a a
J Jβ β β− +
+= + +− +
Choose
( ) ( ) 01 1
1
10
30JJ
J
aaa
β β =
== = ⇒ = − Raimondo Anni
(died 29/05/2003)
Resummation of partial wave series (R=2)
( ) ( ) ( )
( )( )( ) ( ) ( )
0
1 21 2
2
0
1 cos 2 12i
1 1 , cos2i 1 cos 1 cos
J J J JJ
JJJ
f a P a J Sk
a Pk
θ θ
β β θβ θ β θ
∞
=
∞
=
= = +
=+ +
∑
∑
where( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 2 2 1 1 2 11 12 1 1 11,
2 1 2 3J JJ JJ Ja a a a
J Jβ β β β β β β− +
+= + +− +
Choose( ) ( ) ( ) ( )2
210
21 21, 0 , 0J Ja aβ β β β= == =
Advantages of Nearside-Farside (NF) Theory
• It is exact (although approximate NF decompositions can be usedwhen convenient).
• The input is exact (or approximate) S matrix elements as calculated by standard (or non-standard) computer programs.
• It is easily incorporated into existing computer programs.
• Semiclassical techniques such as stationary phase or saddle point integration are not invoked, although the semiclassical picture isstill evident.
• Resummation can be applied to the partial series, followed by a NFdecomposition. This can improve the physical usefulness of theNF decomposition.
• NF and resummation can be incorporated into LAM-LIP analysis
The following techniques need to be used with caution
•Time delay for a single partial wave, e.g., J = 0.
• varying Jmax in PWS from Jmax = 0 to convergence:
• contribution from Jth partial wave to DCS by varying Jmax:
( ) ( )σ θ=
= ∑max
R m x0
a;J
J
J
( ) ( )σ θ σ θ− −R Rmax max; ; 1J J
Recent comments by Nobel Laureates
(2010)
(2010)
Glory seen from an airplane
Glory, or the “Spectre of the Brocken”, often seen in the Harz mountains, Germany
2011
Chengkui Xiahou Dong Hui Zhang
(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)
0 10 20 30 40qR ê deg
0.00
0.04
0.08
0.12
sHqRLêHfi2sr
-1L
CSACSA USAUSAPWS
F+H2 FH+H
Glory analysis for the F + H2 reaction (1985 expt)
(0,0,0) → (3,3,0)Etrans = 0.119 eVE = 0.3872 eVSW pesθ/deg
Phys. Chem. Chem. Phys., 2004
Uniform Semiclassical Approximation (USA):
( ) ( ) ( ) ( )[ ] ( )( )
( ) ( )[ ] ( )( )⎭⎬⎫−+
⎩⎨⎧ +=
+−
+−
21
22121
20
221212
θζθσθσ
θζθσθσθζπθ
J
JI
( ) ( ) ( )[ ]θβθβθζ −+ −= 21
where
Phys. Chem. Chem. Phys., 2004
Semiclassical glory analysis for the H + D2 reaction
(0,0,0) → (3,0,0)Etrans = 1.81 eVE = 2.00 eVBKMP2 pes
10 20 30 40qR ê deg
0.0000
0.0008
0.0016sHqRLêHfi2sr
-1L
0 2 4 6 8 10qR ê deg
0.000
0.004
0.008
0.012
sHqRLêHfi2sr
-1L
PWS
USAUSA
CSACSA
H+D2 HD+D(a)
(b)
PWS
USAUSA
CSACSA
Phys. Chem. Chem. Phys., 2004
θ/deg
I + HI -> IH + I
0,0,0->0,2,0 Etrans = 21.3 meV
Ashley Totenhofer
Black: PWSRed: USA
θ/deg θ/deg
Rainbows in Elastic Scattering
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
θ =′•• • + ••• •
f
Nearside-Farside analysis of the angular distribution for the F + H2 reaction (2008 expt)
( ) ( ) ( ) ( ) ( ) ( ) ( )2F F
2N N FN
2, ,f f f fσ θ θ θ σ θ θ σ θ θ= + = =
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
Semiclassical theory of glory and rainbow scattering
( ) ( ) ( )θθ cos~120
i21
JJJ
k PSJf ∑∞
=+=
• Semiclassical 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.,. θθ ≡bn
• Key quantity is the quantum deflection function:
( ) ( )JJS
Jd
~argd~ ≡Θ
. .,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
θ =′•• • + ••• •
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
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
Phys. Chem. Chem. Phys., 2014
Journal of Chemical Physics, volume 103, pages 5979-5998 (1995)
Complex Angular Momentum Theory of Scattering
• CAM theory is completely general.
• CAM theory describes both resonance and non-resonance scattering.
• CAM theory correctly describes scattering into angular regions that are classically allowed or classically forbidden.“Is a Regge rainbow all shadow? Answer: No!”
• The standard definition of a resonance in CAM theory is a pole in the first quadrant of the CAM plane as characterized by its position and residue at a fixed value of the total energy, E.
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 ).
Pade reconstruction of S(J) including Regge poles near the Re J axis
D. Sokolovski, E. Akhmatskaya and S. K. Sen, Comput. Phys. Commun. (2011)-a FORTRAN code.
( ) a polynomial ina (different) polynomial in
JS JJ
=
I have also used Thiele Rational Interpolation,implemented in Mathematica.
The QP decomposition
( )max
0exp
nn
J J Jnn
aS Q iJ J
φ=
⎛ ⎞⎜ ⎟= +⎜ ⎟−⎝ ⎠
∑
2J a J b J cφ = + +where
( ) ( ) ( ) ( ) ( )Q PR R Rf f fθ θ θ= +
Plot ( ) 2Rf θ ( ) ( )R
Q 2f θ ( ) ( )R
P 2f θ versus Rθ
DCSs for QmodPmod decomposition
θR/deg
DCS
F + H2 (1985 expt)
Linear plot
(0,0,0) → (3,3,0)Etrans = 0.119 eVE = 0.3872 eVSW pes
( ) ( )2 40
param1 exp exp polynomial up toJQ A A J i Jα⎡ ⎤ ⎡ ⎤= + −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
S matrix parameterization
( )maxparam ram
0
pa expn
nJJ J
nn
aS Q iJ J
φ=
⎛ ⎞⎜ ⎟= +⎜ ⎟−⎝ ⎠
∑
Used to test the uniform CAM theory
Then
F + H2 (1985 expt)
Parameterized S matrix
Numerical S matrix
Comparison of numerical and parameterized S matrix elements
J
modulus
phase
(0,0,0) → (3,3,0)Etrans = 0.119 eVE = 0.3872 eVSW pes
J
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
Watson transformation (1918) for PWS
12Jλ = +
Contour for background integral
Complex J plane
Re J
Im J
where
( ) ( ) ( ) ( )direct erfR R R R
reba sck cidue~f f f fθ θ θ θ+ +
and
Asymptotically: refc
( ) ( ) ( )pol a kR
b ceR Rf f fθ θ θ= +
( ) ( )
( )( ) ( )
max
max
R R0
12
pole pole
R10 2
i coscos n
n
nn
n n nJ
n n
f f
J rP
k J
θ θ
π θπ
=
=
=
+= − −
⎡ ⎤+⎣ ⎦
∑
∑
( ) ( ) ( ) ( ) ( )( )N1 1R R 2
back i cosJf k S J Q J d Jθ θΓ
−= +∫
Uniform CAM Theory
Linear plot
DCS
θR/deg
F + H2 (1985 expt)
Uniform CAM
PWS
(0,0,0) → (3,3,0)Etrans = 0.119 eVE = 0.3872 eVSW pes
Uniform CAM and PWS DCSs
(0,0,0) → (3,0,0)E = 0.3112 eVFXZ pes
F + H2 (2008 expt) Uniform CAM and PWS DCSs
Log plot
DCS
θR/deg
Uniform CAM
PWS
Xiao Shan
For the 0,0,0 -> 3,0,0 transition of the F + H2 reaction at E =
0.3112 eV on the FXZ pes, the S matrix has a pole in the
complex angular momentum J plane, given by
( ) 0
0
rS J
J J≈
−
where
J0 = 10.43 + 0.53i Implies a life-angle = 1/( 2 ImJ0) = 0.95 rad = 54o
r0 = – 0.013 – 0.029i
0J J≈for
Important Result F + H2 (2008 expt)
H + CD4 → HD + CD3
G. Nyman, Goteborg, Sweden
S matrix elements calculated by the rotating line umbrellamethod
Semiclassical CAM
PWS
θ/deg
Thank you for listening!
UK Funding:
Overseas Research Students Awards Scheme
School of Chemistry, The University of Manchester
Leverhulme Emeritus Fellowship
