4 theory of quantum scattering and chemical reactions · 4 theory of quantum scattering and...

4 Theory of quantum scattering andchemical reactions

Quantum scattering theory plays an essential role in describing chemical reactions andphotoionization. Although all of these phenomena are time dependent, scattering theoryis most accessible from a time-independent perspective. We will however introduce theconcept of scattering delays and then discuss two important applications of scatteringtheory, i.e. photoionization and reactive collisions.

4.1 Central-potential scattering from a time-independentperspective

In quantum mechanics, a collision between two particles A and B is described by ascattering problem. The kinetic energy of the relative motion is E = ~2k2

2µ with relative

momentum ~p = ~~k and the reduced mass µ = mAmBmA+mB

. The scattering wave function isgiven by

ψ~k(~R) = 4π

∞∑l=0

m=l∑m=−l

ilψkl(R)Y ml (R)Y m∗

l (k), (4.1)

where k and R represent the solid angles in the directions of vectors ~k and ~R, respectively.This ansatz gives equal weights to all l,m’s. Using the addition theorem of spherical

Figure 4.1: Definition of the coordinate system for the collision of two particles

48

4 Theory of quantum scattering and chemical reactions

harmonics gives

Pl(R · k) =4π

2l + 1

l∑m=−l

Y ml (R)Y m∗

l (k) (4.2)

which yields ψ~k(~R) =

∞∑l=0

(2l + 1)ilψkl(R)Pl(cos θ). (4.3)

In the absence of an interaction potential the solution is a set of plane waves. Thisresult can be obtained as follows. Since a plane wave can be written in a partial-waveexpansion as

ei~k·~R = eikR cos θ =∞∑l=0

(2l + 1)iljkl(R)Pl(cos θ), (4.4)

where jkl(R) = jl(kR) is the regular spherical Bessel function of order l with the asymp-totic behavior:

Gl(R) ∝ kRjl(kR)→

i2 e−i(kR−lπ/2)︸︷︷︸incoming from right

− ei(kR−lπ/2)︸︷︷︸outgoing

, forR→∞

0, forR→ 0 ,

we find that the potential-free scattering problem has the solution given by Eq. (4.3),

where ψkl(R) = jkl(R). j(kR)l has a sinusoidal R dependence until the point where the

first maximum of sin(kR − lπ2 ) occurs after which it declines to zero. The innermost

maximum occurs at the classical turning point (see Fig. 4.2).When an interaction potential is present the incoming and outgoing waves no

longer have the same amplitude. Instead:

Gl(R)R→∞−−−−→ i

2e−i(kR−lπ/2) − Slei(kR−lπ/2) (4.5)

where Sl is the scattering amplitude of the partial wave l.Conservation of angular momentum and energy implies that S is an orthogonal matrix.

In central-potential scattering, the angular momentum quantum number l is conservedand S is a diagonal matrix. Conservation of flux (incoming=outgoing) imposes |Sl| = 1

or Sl = e2iδl where the real number δl is the phase shift of the partial wave l. In thiscase

Gl(R)→

eiδl sin(kR− lπ

2 + δl) forR→∞0, forR→ 0

In elastic scattering, the potential only shifts the phase of the wavefunction. Thescattering event is therefore fully described by the phase shift δl. Scattering in non-centrally-symmetric potentials does not conserve the quantum number l, leading tooff-diagonal matrix elements of the scattering matrix S.

49


Figure 4.2: Scattering wavefunction (full line) in a repulsive potential and comparisonwith the corresponding plane wave, representing the solution in the absence ofpotential. The figure shows that a repulsive potential keeps the wavefunctionout of the region that is accessible to the classical motion without a potential.Outside the potential range, the wavefunction must describe a free motionand the scattering wave follows the oscillations of the plane wave with a phaseshift δl. Taken from R. D. Levine, Molecular Reaction Dynamics, CambridgeUniversity Press (2009)

4.1.1 Scattering amplitude

The scattering amplitude f(θ) is defined as the amplitude of the scattered wave underan angle θ from the incident wave defined by comparison with φ(~R) for free motion (aplane wave):

ψ~k(~R)

R→∞−−−−→ φ(~R) +ei~k·~R

Rf(θ). (4.6)

We find

f(θ) =1

2ik

∞∑l=0

(2l + 1)(e2iδl − 1)Pl(cos θ) (4.7)

The scattered intensity is I(θ) = |f(θ)|2, which is also known as the differential scatter-ing cross section (DCS).Description of collisions usually needs many partial waves because the de-Broglie wave-length of atoms and molecules (λ = ~

p with momentum p) is significantly shorter thanthe range of the potential. This leads to rapid oscillations in f(θ) that tend to cancel andyield classical scattering amplitudes. In practice, only a small range of partial waves

50


contribute significantly, at a given angle θ according to

θ = 2∂δl∂l. (4.8)

The cross section is obtained by integration

σ =

∫ ∫|f(θ)|2 sin θdθdφ =

4π

k2

∞∑l=0

(2l + 1) sin2 δl (4.9)

In many situations, the sum converges rapidly with l because the phase shifts δl decreasequickly with l. Since δl varies over many π as a function of energy, a useful and frequentapproximation consists in replacing δl with a random variable, yielding sin2 δl ≈ 1

2 . Thisis known as the random-phase approximation. With lc highest l that contributes to σ,we get

σ ≈ 4π

k2

lc∑l=0

(2l + 1)1

2≈ 2πl2c

k2≈ 2πb2c , (4.10)

where bc = lck is the maximal impact parameter contributing to the cross section. This

result is twice the value obtained in a classical treatment, an effect known as ”shadowscattering”, for details see R. D. Levine, Molecular Reaction Dynamics, CambridgeUniversity Press (2009).

4.1.2 Time delay in scattering

The scattering phase shift δl is a function of the angular-momentum quantum numberl and the energy E. The different weights of the partial waves describe an angulardeflection of the incoming particle. The energy dependence of δl similarly leads to a”deflection” or a shift in time. The time-dependent form of the outgoing wave of angularmomentum l is

ei(2δl+kR−

Et

~︸︷︷︸−lπ2

)

, (4.11)

where the term with underbrace represents the time-dependent phase of the stationaryscattering state.A wave packet can be formed as a linear superposition of scattering waves of differentenergies. Our goal however simplifies to ask at what time the wave packet will reach apoint R outside the range of the potential. The wave function becomes maximal whenthe rapidly varying part of the exponent a = (kR− Et

~ −lπ2 ) becomes nearly stationary.

With E = ~2k2

2µ we find

da

dk= R− ~k

µt = 0 (stationarity condition) whenR =

~kµt = vt. (4.12)

51


This is the classical result. If a potential is present the stationary phase of theoutgoing wave function is

R =~kµt− 2

∂δl∂k

= v(t− τ)

where τ =2

v

(∂δl∂k

)l

= 2~(∂δl∂E

)l

(4.13)

defines a time delay arising from scattering. What is the sign of this time delay? Toanswer this question, we distinguish the cases of repulsive and attractive potentials.

1. Repulsive potentials: The incoming wave cannot penetrate into the potentialwall, therefore the scattering wave comes out ahead of a plane wave of the sameasymptotic energy:

For low l′s∂δl∂k

= −d or δl ≈ −kd+lπ

2, (4.14)

where d denotes the range of potential.

2. Attractive potentials: The scattering wave also comes out ahead of a corre-sponding plane wave because the kinetic energy over the extent of the potentialwell is higher than without a potential.

These two examples show that scattering delays as defined here are usually negativefor central-potential scattering. Positive delays are, however, possible when the collidingparticles have internal structure. In this case the kinetic energy of the collision maybe partially converted into internal excitation, resulting in temporary trapping of theincoming particle. Examples include quantum tunneling through a centrifugal barrieras it occurs in shape resonances that will be discussed in the following Section.

4.2 Photoionization

Photoionization is one interesting application of scattering theory. It is of particularimportance in modern time-resolved spectroscopies (see chapter 5). The final stateis a continuum state described by a scattering wave function. The initial state is abound electronic state. The two states are connected by an electric dipole transition.We consider photoionization of an N -electron atom in LS coupling and neglect spin-orbitinteraction:

A(L, S,ML,MS , πA) + Υ(πΥ = −1, lΥ = 1,mΥ)→ A+(LSπA

+)εl(L′, S′,ML′ ,MS′),

(4.15)where mΥ = 0 for linearly polarized radiation and mΥ = ±1 represents circularly polar-ized radiation.

52


Selection rules:

1. L′ = L⊕ 1 = L⊕ l,ML′ = ML +mΥ = ML +ml

2. S′ = S = S ⊕ 12 ,MS′ = MS = MS +ms

3. πAπA+ = (−1)l+1

where A⊕B means A+B, A+B-1,......|A − B|, i.e. vector addition. The asymptoticboundary conditions are

ψ−αE(~r1s1, ......, ~rNsN )rN→∞−−−−→ θα(~r1s1, ......, ~rNsN )

1

i√

2πkα

1

rNei∆α

−∑α′

θα′(~r1s1, ......, ~rNsN )1

i√

2πkα

1

rNe−i∆αS+

α′α,(4.16)

where the phase ∆α ≡ kαrN − 12πlα + 1

kαlog 2kαrN + σlα︸︷︷︸

Coulomb phase shift

and σlα = arg Γ(lα + 1− ikα

).

ψ−αE has ”incoming wave” normalization, i.e. asymptotically, ψ−αE has outgoing sphericalCoulomb waves only in channel α and incoming spherical waves in all other channels.θα represents the wave function of the ion and the angular and spin parts of the pho-toelectron wave function. A key difference between scattering in short-range potentials,treated in section 4.1, and scattering in Coulomb potentials, is the logarithmic divergenceof the scattering phase which requires special attention in numerical treatments.

The dipole matrix element describing photoionization from an initial state Ψi writtenin the single-active-electron approximation as Ψi = RnliY

mli

is

d~k,~n(E) = 〈Ψi|~r · ~n|Ψ−k 〉 (4.17)

=1√k

∑lm

ile−i(σl+δl)〈Rnli |r|REl〉〈Ymili| cos θ|Y m

l 〉Y m∗l (Ωk), (4.18)

where ~k represents the momentum of the photoelectron and ~n the direction of the linearpolarization of the ionizing radiation. The differential photoionization cross section isgiven by

dσ

dΩ=

4π2Ek

c

∣∣∣d~k,~n(E)∣∣∣2 , (4.19)

which, in the case of single-photon-ionization of atoms and randomly-oriented moleculescan be written as

dσ

dΩ=

σ

4π(1 + βP2(cos θ)) , (4.20)

where β is called the asymmetry parameter, σ is the photoionization cross section andP2 is the Legendre polynomial of order 2.

53


4.2.1 Central-potential model

The exact electronic Hamiltonian of an atom Hexact is approximately given by a sum ofsingle-particle terms, the so-called central-potential Hamiltonian:

HCP =

N∑i=1

[p2i

2m+ V (ri)] (4.21)

with boundary conditions:

V (r)r→0−−−→ −Z

rand V (r)

r→∞−−−→ −1

r(4.22)

for an initially neutral atom.

HCP is separable in spherical coordinates. Its eigenstates are Slater determinants ofone-electron orbitals 1

rPεl(r)Yl,m(θ, φ) with Pεl(r) satisfying the equation

d2Pεl(r)

dr2+ 2

(ε− V (r)− ~2l(l + 1)

2µr2

)Pεl(r) = 0. (4.23)

High-energy behavior

• At high photon energies, inner shells have higher photoionization cross sectionsthan outer shells.

• Since Pnl(r) (bound state) is concentrated in a very small region of r, the maincontributions to the dipole matrix elements come from regions where Pnl(r) islargest.

• Therefore local approximations, such as the screened Coulomb potential, are oftensufficient.

Vnl(r) = −(Z − Snl

r) + V 0

nl (4.24)

For high but non-relativistic photon energies, the energy dependence of the cross sectionof the subshell nl is

σnl ∼ ω−l−72 without channel interactions, (4.25)

and σnl ∼ ω−92 for l > 0 with channel interactions.

Near-threshold behavior

• Shells with l > 1 often have non-hydrogenic behavior.

• Cross sections do not decrease monotonically; instead they display a delayed max-imum, followed by minimum.

54


Figure 4.3: Plot of the effective radial potential for the Xe atom for l = 3. Note thelocal maximum around 1.5 a.u., which leads to the appearance of a shaperesonance in the 4d → εf channel visible as a local maximum of the 4dphotoionization cross section shown in Fig. 4.4. The model potential istaken from A. Sarsa et al., J. Phys. B: At. Mol. Opt. Phys. 36, 4393(2003).

• Maximum: a ”shape resonance” originates from a local maximum in the effectiveradial potential for l > 2 (see figure 4.3)

Veff (r) = V (r) +~2l(l + 1)

2µr2. (4.26)

The photoionization cross section of the 4d shell of xenon displays such a maximumaround 100 eV (see figure. 4.4).

Double wells occur when an inner subshell with l = 2, 3 is being filled withincreasing Z. Effect on

– 3p of transition metal

– 4d of lanthanides metal

– 5d of actinides

• Another dynamical phenomenon frequently encountered in photoionization is theCooper minimum (see John W. Cooper, Phys. Rev. 128, 681 (1962)). Cooperminima arise when the sign of the dipole matrix element involving the bound-state and the continuum radial wave functions changes, which is only possiblewhen the bound-state radial function has a node. Figure. 4.5 (a) displays theenergy dependence of the ground-state wave-functions (Pnl(r)) and the dominantd-continuum radial wave functions (at zero scattering energy) for Ne, Ar and Kr.Qualitatively, one can infer that at low energies, the dipole matrix element will

55

http://stacks.iop.org/0953-4075/36/i=22/a=002

http://stacks.iop.org/0953-4075/36/i=22/a=002

http://dx.doi.org/10.1103/PhysRev.128.681


Figure 4.4: Partial cross sections σ and angular distribution parameters β as a functionof photon energy for xenon. [Taken from figure 20, chapter 5, VUV andsoft-X-ray photoionization, Uwe Becker, David Allen Shirley]

Figure 4.5: (a) Outer subshell radial wave functions Rnl(r) and d-continuum radial func-tions for Ne, Ar and Kr and ε = 0. (b) Radial matrix elements for p → d-transitions for Ne, Ar and Kr as a function of the scattering energy. [Takenfrom figures 2 and 3, John W. Cooper, Phys. Rev. 128, 681 (1962)).]

be positive for Ne and negative for Ar and Kr. As the energy increases, the d-wave will be ”pulled in” towards the nucleus, causing a decrease in the matrixelement amplitude for Ne and a sign reversal for Ar and Kr (cp. figure. 4.5 (b)).

56

http://dx.doi.org/10.1103/PhysRev.128.681


As already implied by the name, this situation leads to pronounced ”dips” inthe photoionization cross section σ and the photoelectron angular distributionasymmetry parameter β. The photoionization cross section of the 3p shell of argondisplays such a minimum around 50 eV (see figure. 4.6).

Figure 4.6: Partial cross sections σ and angular distribution parameters β as a functionof photon energy for argon. [Taken from figure 12, chapter 5, VUV and softX-ray photoionization, Uwe Becker, David Allen Shirley]

4.3 Molecular collisions

Molecular collisions can be classified as elastic, inelastic, or reactive collisions. Elasticcollisions are described by a formalism very similar to section 4.1, additionally includingthe non-spherical nature of molecules. Here, we only discuss reactive collisions. Wedistinguish between direct reactive collisions and compound collisions. In theformer case, the reactive complex formed from the association of the collision partnershas a very short lifetime whereas in the latter case, the lifetime can exceed the rotationalperiod of the complex. These two categories of reactive collisions yield very differentproduct angular distributions and are therefore best distinguished using crossed-beamexperiments.

The outcome of direct reactive collisions is mainly controlled by the impact pa-rameter and the orientation of the reacting molecular during the collision. Such reac-tions usually occur following close collisions during which the reactants experience the

57


Figure 4.7: Contour map: the flux (velocity-angle) distribution for the KI product ofthe K+ I2 reaction. The initial velocities are shown in the center-of-masssystem where the velocity of the relatively heavy I2 molecule is necessarilysmall compared to that of K. Taken from R. D. Levine, Molecular ReactionDynamics, Cambridge University Press (2009)

short-range repulsive potentials. The product angular distribution in such cases can beapproximated as

dσR

dω=d2

4P (b(θ)), (4.27)

where d is the distance at which the reaction takes place, b is the impact parameter and Pis the reaction probability. Backward scattering is observed in many cases because P (b)only contributes at low values of b. The higher the impact parameters b contributing tothe reaction probability are, the more forward the product scattering.

Within the class of direct reactive collisions, two reaction schemes can be distin-guished: Scheme 1: K + I2 → KI + I, Fig. 4.7: the KI product is mainly scatteredforward. The product angular distribution is forward-backward asymmetric which in-dicates that the process of reaction must be over quickly (compared to molecular rota-tion). K reacts with I2 according to the so-called ”harpoon” mechanism: an electronis transferred from K to I2, following which the K+ ion picks up an I− ion and carriesit forward. This is known as ”stripping mode” (spectator limit) which is associatedwith the characteristic angular distribution shown in Fig. 4.7. Reactions following theharpoon mechanism are usually associated with large cross sections, σR ≈ 125 Ain thecase of K + I2.

Scheme 2: CH3I + K → KI + CH3, Fig. 4.8: the KI product is mainly scattered tothe backward hemisphere. As in scheme 1, the product angular distribution is forward-backward asymmetric which indicates that the process of reaction must be over quickly(compared to molecular rotation). The reaction is dominated by low-impact-parameter(head-on) collisions which cause the diatomic product to rebound backwards. This is

58


Figure 4.8: Contour map: the flux (velocity-angle) distribution for the KI product of theK + CH3I reaction. Taken from R. D. Levine, Molecular Reaction Dynamics,Cambridge University Press (2009)

known as ”rebound mode” and leads to the angular distribution of products shown infigure 4.8.

In the case of compound collisions, the collision partners form a complex that lastsuntil the reaction has taken place. If the lifetime of the collision complex is much longerthan its rotational period τrot, a random distribution of products is observed. In this casethe product angular distribution is forward-backward symmetric but the differentialcross section is anisotropic because of the conservation of angular momentum

dσ

dω=

dσ

2π sin θdθ∝ 1

sin θ. (4.28)

This effect is known as ”sprinkler model”. One example is the F + C2H4→ C2H4F∗→ C2H3F+ H reaction illustrated in Fig. 4.9.

The clear differences in product angular distributions lead to a straightforward ex-perimental identification of direct reactive collisions as opposed to compound collisions.A backward-forward asymmetry is characteristic of a direct reactive collision where thelifetime of the complex is τrot. Backward-forward symmetry in product angular distri-bution with maxima in the direction perpendicular to the collision axis is characteristicof compound collisions, i.e. the lifetime of the complex is & τrot.

Finally, it is worth pointing out that the two cases of reactive collisions discussedabove have to be considered as limiting cases. This is illustrated by the example of thereaction RbCl + Cs → Rb + CsCl shown in Fig. 4.10. In this case, a nearly forward-backward-symmetric product angular distribution is found. This is characteristic of acollision complex that lives for approximately one rotational period.

59


maximal flux

lines of equal fluxvelocity axis

angle

Figure 4.9: Contour map: the flux (velocity-angle) distribution for the C2H3F productof the F + C2H4F reaction. See D. Herschbach, Nobel lecture 1986

maximal fluxvelocity axis

angle

Figure 4.10: Contour map: the flux (velocity-angle) distribution for the CsCl product ofthe RbCl + Cs reaction. See D. Herschbach, Nobel Lecture 1986

60

4 theory of quantum scattering and chemical reactions · 4 theory of quantum scattering and...

Documents