Download 4 Theory of quantum scattering and chemical reactions

4 Theory of quantum scattering and
chemical reactions
Quantum scattering theory plays an essential role in describing chemical reactions and
photoionization. Although all of these phenomena are time dependent, scattering theory
is most accessible from a time-independent perspective. We will however introduce the
concept of scattering delays and then discuss two important applications of scattering
theory, i.e. photoionization and reactive collisions.
4.1 Central-potential scattering from a time-independent
perspective
In quantum mechanics, a collision between two particles A and B is described by a
2 2
scattering problem. The kinetic energy of the relative motion is E = ~2µk with relative
momentum p~ = ~~k and the reduced mass µ = mA mB . The scattering wave function is
mA +mB
given by
~ = 4π
ψ~k (R)
∞ m=l
X
X
∗
il ψkl (R)Ylm (R̂)Ylm (k̂),
(4.1)
l=0 m=−l
~ respectively.
where k̂ and R̂ represent the solid angles in the directions of vectors ~k and R,
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
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4 Theory of quantum scattering and chemical reactions
harmonics gives
l
4π X m
∗
Pl (R̂ · k̂) =
Yl (R̂)Ylm (k̂)
2l + 1
(4.2)
m=−l
which yields
~ =
ψ~k (R)
∞
X
(2l + 1)il ψkl (R)Pl (cos θ).
(4.3)
l=0
In the absence of an interaction potential the solution is a set of plane waves. This
result can be obtained as follows. Since a plane wave can be written in a partial-wave
expansion as
e
~
i~k·R
=e
ikR cos θ
=
∞
X
(2l + 1)il jkl (R)Pl (cos θ),
(4.4)
l=0
where jkl (R) = jl (kR) is the regular spherical Bessel function of order l with the asymptotic behavior:

i(kR−lπ/2)
−i(kR−lπ/2)
i

 2 { e|
{z
} − |e {z }}, for R → ∞
Gl (R) ∝ kRjl (kR) →
outgoing
incoming f rom right

0,
for R → 0 ,
we find that the potential-free scattering problem has the solution given by Eq. (4.3),
(kR)
where ψkl (R) = jkl (R). jl
has a sinusoidal R dependence until the point where the
lπ
first maximum of sin(kR − 2 ) 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:
R→∞ i
Gl (R) −−−−→ {e−i(kR−lπ/2) − Sl ei(kR−lπ/2) }
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 conserved
and 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 this
case
(
eiδl sin(kR − lπ
2 + δl ) for R → ∞
Gl (R) →
0,
for R → 0
In elastic scattering, the potential only shifts the phase of the wavefunction. The
scattering event is therefore fully described by the phase shift δl . Scattering in noncentrally-symmetric potentials does not conserve the quantum number l, leading to
off-diagonal matrix elements of the scattering matrix S.
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4 Theory of quantum scattering and chemical reactions
Figure 4.2: Scattering wavefunction (full line) in a repulsive potential and comparison
with the corresponding plane wave, representing the solution in the absence of
potential. The figure shows that a repulsive potential keeps the wavefunction
out of the region that is accessible to the classical motion without a potential.
Outside the potential range, the wavefunction must describe a free motion
and the scattering wave follows the oscillations of the plane wave with a phase
shift δl . Taken from R. D. Levine, Molecular Reaction Dynamics, Cambridge
University Press (2009)
4.1.1 Scattering amplitude
The scattering amplitude f (θ) is defined as the amplitude of the scattered wave under
~ for free motion (a
an angle θ from the incident wave defined by comparison with φ(R)
plane wave):
~
i~k·R
R→∞
~ −
~ +e
ψ~k (R)
−−−→ φ(R)
f (θ).
(4.6)
R
We find
∞
1 X
f (θ) =
(2l + 1)(e2iδl − 1)Pl (cos θ)
(4.7)
2ik
l=0
The scattered intensity is I(θ) = |f (θ)|2 , which is also known as the differential scattering cross section (DCS).
Description of collisions usually needs many partial waves because the de-Broglie wavelength of atoms and molecules (λ = ~p with momentum p) is significantly shorter than
the range of the potential. This leads to rapid oscillations in f (θ) that tend to cancel and
yield classical scattering amplitudes. In practice, only a small range of partial waves
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4 Theory of quantum scattering and chemical reactions
contribute significantly, at a given angle θ according to
θ=2
∂δl
.
∂l
(4.8)
The cross section is obtained by integration
Z Z
σ=
|f (θ)|2 sin θdθdφ =
∞
4π X
(2l + 1) sin2 δl
k2
(4.9)
l=0
In many situations, the sum converges rapidly with l because the phase shifts δl decrease
quickly with l. Since δl varies over many π as a function of energy, a useful and frequent
approximation consists in replacing δl with a random variable, yielding sin2 δl ≈ 21 . This
is known as the random-phase approximation. With lc highest l that contributes to σ,
we get
lc
4π X
1
2πl2
σ≈ 2
(2l + 1) ≈ 2c ≈ 2πb2c ,
(4.10)
k
2
k
l=0
lc
k
where bc = 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 ”shadow
scattering”, for details see R. D. Levine, Molecular Reaction Dynamics, Cambridge
University Press (2009).
4.1.2 Time delay in scattering
The scattering phase shift δl is a function of the angular-momentum quantum number
l and the energy E. The different weights of the partial waves describe an angular
deflection 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 angular
momentum l is
Et
i(2δl +kR−
e
~
|{z}
− lπ
)
2
,
(4.11)
where the term with underbrace represents the time-dependent phase of the stationary
scattering state.
A wave packet can be formed as a linear superposition of scattering waves of different
energies. Our goal however simplifies to ask at what time the wave packet will reach a
point R outside the range of the potential. The wave function becomes maximal when
lπ
the rapidly varying part of the exponent a = (kR − Et
~ − 2 ) becomes nearly stationary.
2
2
With E = ~2µk we find
da
~k
=R−
t = 0 (stationarity condition)
dk
µ
51
when R =
~k
t = vt.
µ
(4.12)
4 Theory of quantum scattering and chemical reactions
This is the classical result. If a potential is present the stationary phase of the
outgoing wave function is
∂δl
~k
t−2
= v(t − τ )
µ
∂k
2 ∂δl
∂δl
where τ =
= 2~
v ∂k l
∂E l
R=
(4.13)
defines a time delay arising from scattering. What is the sign of this time delay? To
answer this question, we distinguish the cases of repulsive and attractive potentials.
1. Repulsive potentials: The incoming wave cannot penetrate into the potential
wall, therefore the scattering wave comes out ahead of a plane wave of the same
asymptotic energy:
For low l0 s
∂δl
lπ
= −d or δl ≈ −kd + ,
∂k
2
(4.14)
where d denotes the range of potential.
2. Attractive potentials: The scattering wave also comes out ahead of a corresponding plane wave because the kinetic energy over the extent of the potential
well is higher than without a potential.
These two examples show that scattering delays as defined here are usually negative
for central-potential scattering. Positive delays are, however, possible when the colliding
particles have internal structure. In this case the kinetic energy of the collision may
be partially converted into internal excitation, resulting in temporary trapping of the
incoming particle. Examples include quantum tunneling through a centrifugal barrier
as 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 particular
importance in modern time-resolved spectroscopies (see chapter 5). The final state
is a continuum state described by a scattering wave function. The initial state is a
bound 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-orbit
interaction:
A(L, S, ML , MS , πA ) + Υ(πΥ = −1, lΥ = 1, mΥ ) → A+ L̄S̄πA + l(L0 , S 0 , ML0 , MS 0 ),
(4.15)
where mΥ = 0 for linearly polarized radiation and mΥ = ±1 represents circularly polarized radiation.
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4 Theory of quantum scattering and chemical reactions
Selection rules:
1. L0 = L ⊕ 1 = L̄ ⊕ l, ML0 = ML + mΥ = ML̄ + ml
2. S 0 = S = S̄ ⊕ 12 , MS 0 = MS = MS̄ + ms
3. πA πA+ = (−1)l+1
where A⊕B means A+B, A+B-1,......|A − B|,
boundary conditions are
i.e. vector addition. The asymptotic
1 i∆α
1
rN →∞
−
e
ψαE
(r~1 s1 , ......, r~N sN ) −−
−−→ θα (r~1 s1 , ......, ~rN sN ) √
i 2πkα rN
X
1
1 −i∆α +
−
θα0 (~r1 s1 , ......, ~rN sN ) √
e
Sα0 α ,
r
i
2πk
α N
0
(4.16)
α
where the phase ∆α ≡ kα rN − 21 πlα +
1
kα
log 2kα rN + σlα
|{z}
Coulomb phase shift
and σlα = arg Γ(lα + 1 − kiα ).
−
−
has outgoing spherical
has ”incoming wave” normalization, i.e. asymptotically, ψαE
ψαE
Coulomb 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 photoelectron wave function. A key difference between scattering in short-range potentials,
treated in section 4.1, and scattering in Coulomb potentials, is the logarithmic divergence
of the scattering phase which requires special attention in numerical treatments.
The dipole matrix element describing photoionization from an initial state Ψi written
is
in the single-active-electron approximation as Ψi = Rnli Ylm
i
d~k,~n (E) = hΨi |~r · ~n|Ψ−
ki
X
1
i
= √
| cos θ|Ylm iYlm∗ (Ωk ),
il e−i(σl +δl ) hRnli |r|REl ihYlm
i
k lm
(4.17)
(4.18)
where ~k represents the momentum of the photoelectron and ~n the direction of the linear
polarization of the ionizing radiation. The differential photoionization cross section is
given by
2
dσ
4π 2 Ek =
(4.19)
d~k,~n (E) ,
dΩ
c
which, in the case of single-photon-ionization of atoms and randomly-oriented molecules
can be written as
σ
dσ
=
(1 + βP2 (cos θ)) ,
(4.20)
dΩ
4π
where β is called the asymmetry parameter, σ is the photoionization cross section and
P2 is the Legendre polynomial of order 2.
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4 Theory of quantum scattering and chemical reactions
4.2.1 Central-potential model
The exact electronic Hamiltonian of an atom Ĥexact is approximately given by a sum of
single-particle terms, the so-called central-potential Hamiltonian:
ĤCP =
N
X
p̂2
[ i + V̂ (ri )]
2m
(4.21)
1
Z
r→∞
and V (r) −−−→ −
r
r
(4.22)
i=1
with boundary conditions:
r→0
V (r) −−−→ −
for an initially neutral atom.
ĤCP is separable in spherical coordinates. Its eigenstates are Slater determinants of
one-electron orbitals 1r Pl (r)Yl,m (θ, φ) with Pl (r) satisfying the equation
d2 Pl (r)
~2 l(l + 1)
Pl (r) = 0.
+ 2 − V (r) −
dr2
2µr2
(4.23)
High-energy behavior
• At high photon energies, inner shells have higher photoionization cross sections
than outer shells.
• Since Pnl (r) (bound state) is concentrated in a very small region of r, the main
contributions to the dipole matrix elements come from regions where Pnl (r) is
largest.
• Therefore local approximations, such as the screened Coulomb potential, are often
sufficient.
Z − Snl
Vnl (r) = −(
(4.24)
) + Vnl0
r
For high but non-relativistic photon energies, the energy dependence of the cross section
of the subshell nl is
7
σnl ∼ ω −l− 2 without channel interactions,
(4.25)
9
and σnl ∼ ω − 2 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 maximum, followed by minimum.
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4 Theory of quantum scattering and chemical reactions
Figure 4.3: Plot of the effective radial potential for the Xe atom for l = 3. Note the
local maximum around 1.5 a.u., which leads to the appearance of a shape
resonance in the 4d → f channel visible as a local maximum of the 4d
photoionization cross section shown in Fig. 4.4. The model potential is
taken 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 effective
radial potential for l > 2 (see figure 4.3)
Vef f (r) = V (r) +
~2 l(l + 1)
.
2µr2
(4.26)
The photoionization cross section of the 4d shell of xenon displays such a maximum
around 100 eV (see figure. 4.4).
Double wells occur when an inner subshell with l = 2, 3 is being filled with
increasing Z. Effect on
– 3p of transition metal
– 4d of lanthanides metal
– 5d of actinides
• Another dynamical phenomenon frequently encountered in photoionization is the
Cooper minimum (see John W. Cooper, Phys. Rev. 128, 681 (1962)). Cooper
minima arise when the sign of the dipole matrix element involving the boundstate and the continuum radial wave functions changes, which is only possible
when the bound-state radial function has a node. Figure. 4.5 (a) displays the
energy dependence of the ground-state wave-functions (Pnl (r)) and the dominant
d-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
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4 Theory of quantum scattering and chemical reactions
Figure 4.4: Partial cross sections σ and angular distribution parameters β as a function
of photon energy for xenon. [Taken from figure 20, chapter 5, VUV and
soft-X-ray photoionization, Uwe Becker, David Allen Shirley]
Figure 4.5: (a) Outer subshell radial wave functions Rnl (r) and d-continuum radial functions for Ne, Ar and Kr and = 0. (b) Radial matrix elements for p → dtransitions for Ne, Ar and Kr as a function of the scattering energy. [Taken
from 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 dwave will be ”pulled in” towards the nucleus, causing a decrease in the matrix
element amplitude for Ne and a sign reversal for Ar and Kr (cp. figure. 4.5 (b)).
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4 Theory of quantum scattering and chemical reactions
As already implied by the name, this situation leads to pronounced ”dips” in
the photoionization cross section σ and the photoelectron angular distribution
asymmetry parameter β. The photoionization cross section of the 3p shell of argon
displays such a minimum around 50 eV (see figure. 4.6).
Figure 4.6: Partial cross sections σ and angular distribution parameters β as a function
of photon energy for argon. [Taken from figure 12, chapter 5, VUV and soft
X-ray photoionization, Uwe Becker, David Allen Shirley]
4.3 Molecular collisions
Molecular collisions can be classified as elastic, inelastic, or reactive collisions. Elastic
collisions are described by a formalism very similar to section 4.1, additionally including
the non-spherical nature of molecules. Here, we only discuss reactive collisions. We
distinguish between direct reactive collisions and compound collisions. In the
former case, the reactive complex formed from the association of the collision partners
has a very short lifetime whereas in the latter case, the lifetime can exceed the rotational
period of the complex. These two categories of reactive collisions yield very different
product angular distributions and are therefore best distinguished using crossed-beam
experiments.
The outcome of direct reactive collisions is mainly controlled by the impact parameter and the orientation of the reacting molecular during the collision. Such reactions usually occur following close collisions during which the reactants experience the
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4 Theory of quantum scattering and chemical reactions
Figure 4.7: Contour map: the flux (velocity-angle) distribution for the KI product of
the K+ I2 reaction. The initial velocities are shown in the center-of-mass
system where the velocity of the relatively heavy I2 molecule is necessarily
small compared to that of K. Taken from R. D. Levine, Molecular Reaction
Dynamics, Cambridge University Press (2009)
short-range repulsive potentials. The product angular distribution in such cases can be
approximated as
dσR
d2
= P (b(θ)),
(4.27)
dω
4
where d is the distance at which the reaction takes place, b is the impact parameter and P
is 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 to
the reaction probability are, the more forward the product scattering.
Within the class of direct reactive collisions, two reaction schemes can be distinguished: Scheme 1: K + I2 → KI + I, Fig. 4.7: the KI product is mainly scattered
forward. The product angular distribution is forward-backward asymmetric which indicates that the process of reaction must be over quickly (compared to molecular rotation). K reacts with I2 according to the so-called ”harpoon” mechanism: an electron
is transferred from K to I2 , following which the K+ ion picks up an I− ion and carries
it forward. This is known as ”stripping mode” (spectator limit) which is associated
with the characteristic angular distribution shown in Fig. 4.7. Reactions following the
harpoon mechanism are usually associated with large cross sections, σR ≈ 125 Åin the
case of K + I2 .
Scheme 2: CH3 I + K → KI + CH3 , Fig. 4.8: the KI product is mainly scattered to
the backward hemisphere. As in scheme 1, the product angular distribution is forwardbackward 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
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4 Theory of quantum scattering and chemical reactions
Figure 4.8: Contour map: the flux (velocity-angle) distribution for the KI product of the
K + CH3 I 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 in
figure 4.8.
In the case of compound collisions, the collision partners form a complex that lasts
until the reaction has taken place. If the lifetime of the collision complex is much longer
than its rotational period τrot , a random distribution of products is observed. In this case
the product angular distribution is forward-backward symmetric but the differential
cross section is anisotropic because of the conservation of angular momentum
dσ
dσ
1
=
∝
.
dω
2π sin θdθ
sin θ
(4.28)
This effect is known as ”sprinkler model”. One example is the F + C2 H4 → C2 H4 F∗ → C2 H3 F
+ H reaction illustrated in Fig. 4.9.
The clear differences in product angular distributions lead to a straightforward experimental identification of direct reactive collisions as opposed to compound collisions.
A backward-forward asymmetry is characteristic of a direct reactive collision where the
lifetime of the complex is τrot . Backward-forward symmetry in product angular distribution with maxima in the direction perpendicular to the collision axis is characteristic
of compound collisions, i.e. the lifetime of the complex is & τrot .
Finally, it is worth pointing out that the two cases of reactive collisions discussed
above have to be considered as limiting cases. This is illustrated by the example of the
reaction RbCl + Cs → Rb + CsCl shown in Fig. 4.10. In this case, a nearly forwardbackward-symmetric product angular distribution is found. This is characteristic of a
collision complex that lives for approximately one rotational period.
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4 Theory of quantum scattering and chemical reactions
angle
velocity axis
lines of equal flux
maximal flux
Figure 4.9: Contour map: the flux (velocity-angle) distribution for the C2 H3 F product
of the F + C2 H4 F reaction. See D. Herschbach, Nobel lecture 1986
angle
velocity axis
maximal flux
Figure 4.10: Contour map: the flux (velocity-angle) distribution for the CsCl product of
the RbCl + Cs reaction. See D. Herschbach, Nobel Lecture 1986
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