Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

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742 Part D Scattering Theory

izes the extent of the interaction region. For instance, for

the exchange interaction between two atoms, L

cor-

responds to the distance of closest approach of the

colliding particles, while a is the range of the exponen-

tial decrease of the interaction. Typically, L

noticeably

exceeds a .

49.1.2 Adiabatic

and Diabatic Electronic States

Let r refer to a set of electronic coordinates in

a body-ﬁxed frame related to the nuclear framework

of a colliding system, and let R refer to a set of

nuclear coordinates determining the relative position

of nuclei in this system. A conﬁguration of electrons

and nuclei in a frame ﬁxed in space is completely

determined by r, R, and the set of Euler angles Ω,

which relate the body-ﬁxed frame to the space-ﬁxed

frame. If the total Hamiltonian of the system is

H (r, R,Ω), the stationary state wave function satisﬁes

the equation

H (r, R,Ω)Ψ

(r, R,Ω)= EΨ

(r, R,Ω). (49.3)

The electronic adiabatic Hamiltonian H(r;R) is deﬁned

to be the part of H(r, R,Ω)in which the kinetic energy

of the nuclei is ignored. The adiabatic electronic func-

tions ψ

(r;R) are deﬁned as eigenfunctions of H(r;R)

at a ﬁxed nuclear conﬁguration R:

H(r;R)ψ

(r;R) =U

(R)ψ

(r;R). (49.4)

The eigenvalues U

(R) are called adiabatic potential

energy surfaces (adiabatic PES). In the case of a diatom,

the set R collapses into a single coordinate, the internu-

clear distance R,andthePES become potential energy

curves, U

(R). The functions ψ

(r;R) depend expli-

citly on R and implicitly on the Euler angles Ω.The

signiﬁcance of the adiabatic PES is related to the fact

that in the limit of very low velocities, a system of nuclei

will move across a single PES. In this approximation,

called the adiabatic approximation, the function U

(R)

plays the part of the potential energy which drives the

motion of the nuclei.

An electronic diabatic Hamiltonian is deﬁned for-

mally as a part of H,i.e.,H

= H +∆H.The

partitioning of H into H

and ∆ H is dictated by the

requirement that the eigenfunctions of H

, called dia-

batic electronic functions φ

, depend weakly on the

conﬁguration R. The physical meaning of this weak de-

pendence is different for different problems. A perfect

diabatic basis set φ

(r) is R-independent; for practical

purposes one can use a diabatic set which is consid-

ered as R-independent within a certain region of the

conﬁguration space R.

Two basis sets ψ

and φ

generate the matrices

φ

|H|φ

=H

φ

|H |φ

=H

+ D

ψ

|H |ψ

=U

(R)δ

. (49.5)

The eigenvalues of the matrix H

are U

. D

is the

matrix of dynamic coupling in the diabatic basis, and

is the matrix of dynamic coupling in the adiabatic

basis; the former matrix vanishes for a perfect diabatic

basis. All the above matrices are, in principle, of inﬁ-

nite order. For low-energy collisions, the use of ﬁnite

matrices of moderate dimension, will usually sufﬁce.

Diabatic PES are deﬁned as the diagonal elem-

ents H

. The signiﬁcance of the diabatic PES is that for

velocities which are high [but still satisfy (49.1)] the sys-

tem moves preferentially across diabatic PES, provided

that the additional conditions discussed in Sect. 49.3 are

fulﬁlled.

For a given ﬁnite adiabatic basis ψ

(r;R ), a perfect

diabatic basis φ



(r) can be constructed by diagonalizing

the matrix D

(R). The two basis sets are related by

a unitary transformation

(

r;R

)





(

)



(r). (49.6)

49.1.3 Nonadiabatic Transitions:

The Massey Parameter

Deviations from the adiabatic approximation manifest

themselves in transitions between different PES which

are induced by the dynamic coupling matrix D.Atlow

energies, the transitions usually occur in localized re-

gions of nonadiabatic coupling (NAR). In these regions,

the motion of nuclei in different electronic states is

coupled, and in general it cannot be interpreted as being

driven by a single potential.

An important simplifying feature of slow adiabatic

collisions is that typically the distance between differ-

ent NAR is substantially larger than the extents of each

NAR. This makes it possible to formulate simple models

for the coupling in isolated NAR, and subsequently to

incorporate the solution for nonadiabatic coupling into

the overall dynamics of the system.

For a system of s nuclear degrees of freedom, there

are the following possibilities for the behavior of PES

within NAR:

Part D 49.1

Adiabatic and Diabatic Collision Processes at Low Energies 49.2 Two-State Approximation 743

(i) If two s-dimensional PES correspond to electronic

states of different symmetry, they can cross along an

(s−1)-dimensional line. For a system of two atoms,

s = 1, and so two potential curves of different symmetry

can cross at a point.

(ii) If two s-dimensional PES correspond to electronic

states of the same symmetry, they can cross along an

(s−2)-dimensional line. For a system of two atoms,

s = 1, and so two potential curves of different symmetry

cannot cross. If they have a tendency to cross, they will

exhibit a pattern which is called an avoided crossing or

a pseudocrossing.

(iii) If two s-dimensional PES correspond to electronic

states of the same symmetry in the presence of spin–orbit

coupling, they can cross along an (s−3)-dimensional

line.

Statement (ii) applied to a two-atom system is known

as the Wigner–Witmer noncrossing rule.

The efﬁciency of the nonadiabatic coupling between

two adiabatic electronic states is determined, according

to the adiabatic principle of mechanics (both classical

and quantum), by the value of the Massey parameter ζ,

which represents the product of the electronic transition

frequency ω

and the time τ

nuc

that characterizes the rate

of change of electronic function due to nuclear motion.

Putting ω

≈ ∆U(R)/ ,(∆U is the spacing between

any two adiabatic PES), and τ

nuc

= ∆L/v(R),(∆L is

a certain range which depends on the type of coupling),

we get

ζ(R) = ω

nuc

= ∆U(R)∆L/ v(R). (49.7)

The nonadiabatic coupling is inefﬁcient at those con-

ﬁgurations R where ζ(R) 1. If ζ(R) is less than or of

the order of unity, the nonadiabatic coupling is efﬁcient,

and a change in adiabatic dynamics of nuclear motion is

very substantial.

The following relations usually hold for the param-

eters ∆L, a, L

for slow collisions:

∆L  a  L

. (49.8)

When the nonadiabatic coupling is taken into

account, the total (electronic and nuclear) wave func-

tion Ψ

can be represented as a series expansion in

or φ

(the Euler angles Ω are suppressed for brevity):

(r, R) =



(r;R )χ

(R)



(r)κ

(R). (49.9)

Here χ

(R) and κ

(R) are the functions which

have to be found as solutions to the coupled equa-

tions formulated in the adiabatic or diabatic electronic

basis, repectively [49.1, 2]. In general, different contri-

butions to the ﬁrst sum in (49.9) can be associated with

nonadiabatic transition probabilities between different

electronic states.

A practical means of calculating functions χ

(R)

[or κ

(R)] consists of expanding them over certain

basis functions Ξ

nν



),whereR



denotes all coordi-

nates R except for the interparticle distance R. Writing

(R) =



nν



)ξ

nνE

(R), (49.10)

one arrives at a set of coupled second-order equations

for the unknown functions ξ

nνE

(R) (the scattering equa-

tions) [49.1]. In the semiclassical approximation, these

equations become a set of ﬁrst-order equations for the

amplitudes of the WKB counterparts of ξ

nνE

(R).Atthe

next step of simpliﬁcation, in the common trajectory ap-

proximation, the variable R is changed into the time

variable t, the latter being related to R via the classical

trajectory R = R(t) [49.2]. In the adiabatic approxima-

tion, the total wave function is represented by a single

term in the ﬁrst sum of (49.9):

(r, R) = ψ

(r;R)χ

(R). (49.11)

49.2 Two-State Approximation

49.2.1 Relation Between Adiabatic

and Diabatic Basis Functions

In the two-state approximation, the basis of electronic

functions consists of two states. In this case, the elements

of the matrix C in (49.6) are expressed through a single

parameter only, a mixing or rotation angle θ:

(r;R) = cos θ(R)φ

(r) +sin θ(R)φ

(r),

(r;R) =−sin θ(R)φ

(r) +cos θ(R)φ

(r).

(49.12)

The rotation angle, θ(R), is expressed via the diag-

onal and off-diagonal matrix elements of the adiabatic

Hamiltonian H in the diabatic basis φ

,φ

tan 2θ(R) =

(R)

(R) − H

(R)

(49.13)

Part D 49.2

744 Part D Scattering Theory

The eigenvalues of H in terms of H

are

1,2

(R) =

[

(R) + H

(R)

]

/2 ±∆U(R)/2 ,

(49.14)

where

∆U(R) =



[

(R) − H

(R)

]

+4H

(R)



1/2

(49.15)

The matrix elements H

are expressed via the adia-

batic potentials and the rotation angle by

(R) + H

(R) = U

(R) +U

(R),

(R) − H

(R) = ∆U(R) cos 2θ(R),

(R) = (1/2)∆U(R) sin 2θ(R).

(49.16)

49.2.2 Coupled Equations

and Transition Probabilities

in the Common Trajectory

Approximation

A two-state nonadiabatic wave function Ψ(r , R) can be

written as an expansion into either adiabatic or diabatic

electronic wave functions:

Ψ(r;R) = ψ

(r;R)α

(R) +ψ

(r;R)α

(R),

Ψ(r;R) = φ

(r)β

(R) +φ

(r)β

(R), (49.17)

in which the nuclear wave functions satisfy two coupled

s-dimensional Schrödinger equations [49.1].

In the common trajectory approximation, the motion

of the nuclei is described by the classical trajectory, i. e.,

by a one-dimensional manifold Q(t) embedded in the

s-dimensional manifold R. A section of PES along this

one-dimensional manifold determines a set of effective

potential energy curves (PEC). In the case of atomic

collisions, Q coincides with the interatomic distance R,

and the effective PEC are just ordinary PEC.

A common trajectory counterpart of (49.17)is

Ψ(r, t) = ψ

[

r;Q(t)

]

(t) +ψ

[

r;Q(t)

]

(t),

Ψ(r;t) = φ

(r)b

(t) +φ

(r)b

(t). (49.18)

The adiabatic expansion coefﬁcients a

(t) satisfy the set

of equations

(Q)a

Qg(Q)a

=−i

Qg(Q)a

(Q)a

, (49.19)

where g(Q) =ψ

|∂/∂Q|ψ

=dθ/ dQ,andQ =Q(t).

The diabatic expansion coefﬁcients b

(t) satisfy the set

of equations

= H

(Q)b

+ H

(Q)b

= H

(Q)b

+ H

(Q)b

. (49.20)

Clearly, for a system of two atoms, Q ≡ R.

Solutions to (49.19)and(49.20) are equivalent, pro-

vided that the initial conditions are matched, and the

transition probability is properly deﬁned.

For a given trajectory, it is customary to identify

the center of the NAR with a value of Q = Q

which

corresponds to the real part of the complex-valued co-

ordinate Q

at which two adiabatic PES cross. The

crossing conditions in the adiabatic and diabatic rep-

resentations are

) −U

) = 0 , (49.21)



) − H

)



+4H

) = 0 . (49.22)

Since Q represents a one-dimensional manifold, the

crossing condition (49.22) is satisﬁed for a complex

value of Q =Q

unless H

=0. Then, by deﬁnition, the

location of the NAR centeris identiﬁed with Q

through

= Re(Q

), (49.23)

where Q

is that value of Q

which possesses the

smallest imaginary part, and Re denotes the real part.

For the case when the regions of nonadiabatic coup-

ling are well localized, the function g(Q) possesses

a pronounced maximum at (or close to) Q

, the width

∆Q

of which determines the range of the NAR; nor-

mally ∆Q

is about Im (Q

), with Im denoting the

imaginary part. The two Eqs. (49.19) decouple on both

sides of this maximum. A solution of the equations for

the nonadiabatic coupling across an isolated maximum

of g(Q) yields the so-called single-passage (or one-way)

transition amplitude and transition probability. For this

problem, the time t = 0 can be assigned to the maxi-

mum point of g[Q(t)]. Assuming that away from t = 0

the decoupling occurs rapidly enough, the nonadiabatic

transition probability

=|a

(∞)|

, (49.24)

provided that a solution to (49.19) corresponds to the

initial conditions,

(−∞) = 1 , a

(−∞) = 0 . (49.25)

Part D 49.2

Adiabatic and Diabatic Collision Processes at Low Energies 49.2 Two-State Approximation 745

In the limit of almost adiabatic conditions where P

very small, the following equation holds [49.3]:

= exp



−









[Q(t)]−U

[Q(t)]









(49.26)

where t

is a root of

Q(t

) = Q

. (49.27)

Here t

is any real-valued time. Equation (49.26) is valid

when the exponent is large, so that P

is exponentially

small.

The property of the function g(Q) to pass through

a single narrow maximum ensures that the rotation angle

away from the maximum tends to constant values, and

adiabatic functions in these regions are expressed by

certain linear combinations of diabatic functions with

constant mixing coefﬁcients. These linear combinations

should serve as the initial condition on one side of the

coupling region, and as the proper ﬁnal state on the

other, when the problem of a nonadiabatic transition

between adiabatic states is treated in the diabatic rep-

resentation. The same property of the function g(Q)

implies that a common trajectory needs to be deﬁned

only locally, within a given NAR, and not globally, in

the full conﬁguration space.

49.2.3 Selection Rules

for Nonadiabatic Coupling

In the general case, the coupling between adiabatic states

or diabatic states is controlled by certain selection rules.

The most detailed selection rules exist for a system of

two colliding atoms, since this system possesses a high

Table 49.1 Selection rules for the coupling between diabatic and adiabatic states of a diatomic quasimolecule (w = g, u;

σ =+, −)

Interaction

2S+1

(σ)

nomenclature Ω

(σ)

nomenclature

Conﬁguration interaction ∆Λ = 0, ∆S = 0 ∆Ω =0

(electrostatic) g  u, +  − g  u, + −

Spin–orbit interaction ∆Λ =0, ±1, ∆S = 0, ±1 ∆Ω = 0

g  u, + − g  u, +  −

Radial motion ∆Λ = 0, ∆S = 0 ∆Ω = 0

g  u, +  − g  u, +  −

Rotational motion ∆Λ =±1, ∆S = 0 ∆Ω =±1

g  u, +  − g  u, +  −

Hyperﬁne interaction ∆Λ = 0, ±1, ∆S = 0, ±1 ∆Ω =0, ±1

g u, + − g u, + −

symmetry (C

∞v

or, for identical atoms, D

∞h

point sym-

metry in the adiabatic approximation). In the adiabatic

representation, the coupling is due to the elements of

the matrix D . They fall into two different categories:

those proportional to the radial nuclear velocity (coup-

ling by radial motion or radial coupling), and those

proportional to the angular velocity of rotation of the mo-

lecular axis (coupling by rotational motion or Coriolis

coupling).

In a diabatic representation, provided that the effect

of the D matrix is neglected, the coupling is due to the

parts of the interaction potential neglected in the deﬁ-

nition of the diabatic Hamiltonian H

. In typical cases,

these parts are the electrostatic interaction between dif-

ferent electronic states constructed as certain electronic

conﬁgurations (H

corresponds to a self-consistent ﬁeld

Hamiltonian); spin–orbit interaction (H

corresponds

to a nonrelativistic Hamiltonian); hyperﬁne interaction

ignores the magnetic interaction of electronic and

nuclear spins as well as the electrostatic interaction be-

tween electrons and nuclear quadrupole moments). The

selection rules for the above interactions in the case

of two atoms are listed in Table 49.1 for two conven-

tional nomenclatures for molecular states: Hund’s case

(a),

2S+1

(σ)

and Hund’s case (c), Ω

(σ)

[49.3].

For molecular systems with more than two nuclei,

the selection rules cannot be put in a detailed form

since, in general, the symmetry of the system is quite

low. For the important case of three atoms, a general

conﬁguration is planar (C

symmetry); particular con-

ﬁgurations correspond to an isosceles triangle if two

atoms are identical (C

symmetry), to an equilateral

triangle for three identical atoms (D

symmetry) or

to a linear conﬁguration. For the last case, the selec-

tion rules are the same as for a system of two atoms.

Part D 49.2

746 Part D Scattering Theory

The selection rules for the dynamic coupling between

adiabatic states classiﬁed according to the irreducible

representations of the C

and C

groups are listed in

Table 49.2.Inthistable,z and y refer to two modes of

the relative nuclear motion in the system plane, R

and

refer to two rotations about principal axes of iner-

tia lying in the system plane, and R

refers to a rotation

about the principal axis of inertia perpendicular to the

system plane.

Table 49.2 Selection rules for dynamic coupling between

adiabatic states of a system of three atoms







z y, R

y, R

z R



z y, R

y, R

49.3 Single-Passage Transition Probabilities: Analytical Models

49.3.1 Crossing

and Narrow Avoided Crossing

of Potential Energy Curves:

The Landau–Zener Model

in the Common Trajectory

Approximation

The Landau–Zener model applies to a situation when

the effective adiabatic PEC cross or show a narrow

avoided crossing. The latter is deﬁned by the condition

that the spacing between adiabatic PEC within a NAR is

much smaller than the spacing between adiabatic PEC

aw ay from the NAR. The cases of crossing and narrow

avoided crossing of adiabatic PEC can be considered

within a uniﬁed model since a narrow avoided crossing

of adiabatic PEC corresponds to a crossing of diabatic

PEC. Therefore, in both cases, one considers the cross-

ing of zero-order PES (adiabatic or diabatic) and the

interaction between them (dynamic or static). Ho wever,

the deﬁnition of transition probability is different for

crossing and avoided crossing.

In the framework of the Landau–Zener model [49.4–

6], the two zero-order PEC which cross at

a point Q

along a trajectory Q(t) are approx-

imated by functions linear in ∆Q = Q −Q

,and

the off-diagonal matrix element is assumed to be

a constant.

For the avoided crossing adiabatic PEC (crossing

diabatic PEC), the matrix H

within a NAR is approxi-

mated as

(Q) = E

−F

(Q −Q

(Q) = E

−F

(Q −Q

(Q) = V = constant , (49.28)

from which the spacing between adiabatic PEC is

∆U =



(∆F)

(Q −Q

)

+4|V|



1/2

, (49.29)

with ∆ F =|F

−F

The common trajectory is assumed to be a linear

function of t,

Q = Q

t , (49.30)

where v

is the velocity of Q-motion at point Q

For this model, adiabatic wave functions on both

sides of the nonadiabaticity region (in the limits

−∞ < t < +∞) coincide with diabatic functions, but

their ordering is reversed. Explicitly,

= φ

; ψ

= φ

for t →−∞,

= φ

; ψ

=−φ

for t →+∞. (49.31)

The transition probability between pseudocrossing adi-

abatic curves for the Hamiltonian (49.28)andthe

trajectory (49.30) is given by the Landau–Zener formula

psc

= P

= exp



−2πζ



;

= V

/( ∆Fv

), (49.32)

where ζ

is the appropriate Massey parameter at the

pseudocrossing point. Note that for the LZ model, the

single-passage transition probability depends on one di-

mensionless parameter ζ

. A remarkable property of

the Landau–Zener model is that the probability P

givenby(49.26) for an arbitrary value of the exponent,

and not only for large ones when the probability is very

low.

With a change of the velocity from very low to very

high values, the transition probability varies from zero

to unity. In the near-adiabatic limit, ζ

 1, the nuclei

preferentially move across single adiabatic PEC, while

in the sudden limit, ζ

 1, they move across single

diabatic PEC. It is the latter property of the LZ transition

probability that allows one to interpret diabatic energies

and H

as the potentials which drive the nuclear

motion at high velocities.

Part D 49.3

Adiabatic and Diabatic Collision Processes at Low Energies 49.3 Single-Passage Transition Probabilities: Analytical Models 747

The region of applicability of the Landau–Zener

formula is determined by the condition that the e x-

tension of the region of nonadiabatic interaction ∆Q

should be small compared with the range a over which

potential curves deviate substantially from linear func-

tions. The condition ∆Q a actually implies the two

conditions [49.2]

2V/∆F  a ,

(49.33)

and



/∆F



1/2

 a . (49.34)

Clearly, the range parameter a does not enter the LZ

formula since it controls the behavior of adiabatic curves

away from the crossing point.

The constant velocity approximation (49.30)im-

poses yet another condition:

V  µv

/2 . (49.35)

The actual application of (49.32) requires the speciﬁ-

cation of V and v

for each particular trajectory Q(t).

For the case of avoided crossing between two poten-

tial curves of a diatom, V does not depend on the

trajectory and represents, according to Table 49.2,the

matrix element of the electrostatic interaction, spin–orbit

interaction or hyperﬁne interaction.

For crossing adiabatic PEC, the Landau–Zener

model assumes the following approximation for adia-

batic potentials and the dynamic coupling:

(Q) = E

−F

(Q −Q

(Q) = E

−F

(Q −Q

(Q) = D = constant , (49.36)

with the trajectory parametrization given by (49.30)

where v

is replaced by v

. Since the ordering of adia-

batic PEC for crossing and pseudocrossing is reversed

on one side of a NAR, the following relation exists be-

tween transition probabilities for the crossing case P

and the survival probability for the pseudocrossing case

1 − P

psc

= 1 − P

psc

, (49.37)

provided that D and v

in the crossing situation are

replaced by V and v

in the pseudocrossing situation.

Conditions (49.33)and(49.34)appliedtothecaseof

the dynamic coupling often imply that this coupling is

weak [49.2]. Therefore, (49.37) yields

2πD

∆Fv

. (49.38)

Usually, the matrix element D is related to the Coriolis

coupling, and it is proportional to the angular velocity of

rotation of the molecular frame at the crossing point Q

49.3.2 Arbitrary Avoided Crossing

and Diverging Potential

Energy Curves: The Nikitin Model

in the Common Trajectory

Approximation

The restrictions of narrow avoided crossing [(49.33)and

(49.34)fortheLZ model] are relaxed in a more gen-

eral model suggested by Nikitin [49.7]. This model uses

a more ﬂexible exponential parametrization, instead of

the linear parametrization for diabatic matrix elements

(49.36).

In a diabatic basis, the model is formulated with the

Hamiltonian

(Q) =U

(Q)−∆ E/2+(A/2)cos 2ϑ exp(−αQ),

(Q) =U

(Q)+∆ E/2−(A/2)cos 2ϑ exp(−αQ),

(Q) =(A/2) sin 2ϑ exp(−αQ). (49.39)

The spacing between adiabatic PEC is

∆U = ∆E



1 −2cos2ϑ exp



−α



Q −Q



+exp



−2α



Q −Q



1/2

, (49.40)

where Q

is introduced instead of A via (49.21)and

(49.23). At the center of an NAR,whereQ = Q

,the

spacing between adiabatic PEC, ∆U

=∆U(Q

),is

∆U

= 2∆E sin ϑ. (49.41)

The common trajectory within the NAR is taken in

a form identical to (49.30)inwhichv

is now the

velocity of Q motion at the center of the coupling

region Q

For this model, adiabatic wave functions coincide

with diabatic functions before entering the coupling re-

gion (in the limit α(Q −Q

)  1), but after exiting the

coupling region [in the limit α(Q −Q

) −1] they are

linear combinations of the diabatic functions

(r;Q) = φ

(r), ψ

(r;Q) = φ

(r),

for α(Q −Q

)  1 ;

= φ

cos ϑ +φ

sin ϑ,

=−φ

sin ϑ +φ

cos ϑ,

for α(Q −Q

) −1 . (49.42)

The latter equation identiﬁes the parameter ϑ that enters

into the deﬁnition of the diabatic Hamiltonian in (49.39)

Part D 49.3

748 Part D Scattering Theory

with the asymptotic value of the mixing angle θ (49.12)

for α(Q −Q

) −1.

The transition probability P

between adiabatic

PEC for the Hamiltonian (49.39) and the trajectory

(49.30)is

= exp(−πζ

)

sinh(πζ −πζ

)

sinh(πζ)

(49.43)

where ζ = ∆ E/( αv

) and ζ

= ζ sin

ϑ. With the

change in velocity from very low to very high values,

the transition probability varies from zero to cos

ϑ.As

ϑ changes from very small values to π, the pattern of

adiabatic potential curves changes from narrow to wide

pseudocrossing and ultimately to strong divergence.

Since the single-passage transition probability

(49.43) depends on two parameters, the Nikitin model

is more versatile than the Landau–Zener one.

In three limiting cases, ϑ 1,ζ1, ϑ =π/4, ζ ar-

bitrary, and ϑ arbitrary, ζ =0, (49.43) may be simpliﬁed.

In the ﬁrst case, the diabatic Hamiltonian (49.39)be-

comes the Landau–Zener Hamiltonian (49.28). Also,

(49.43) reduces to a single exponential which gives the

LZ transition probability,

= P

= exp(−2πζ

), (49.44)

with ζ

identical to ζ

. The two conditions ϑ  1

and ζ  1 are equivalent to the two conditions (49.33)

and (49.34). In the second case (ϑ = π/4), the diabatic

Hamitonian reads

(R) = E

−∆E/2 ,

(R) = E

+∆E/2 ,

(R) = ( A /2) exp(−αQ). (49.45)

The transition probability in this case is given by the

Rosen–Zener–Demkov formula [49.8, 9]

RZD

exp(−πζ)

1 +exp(−πζ)

,ζ=∆ E/(

α) . (49.46)

In the third case (ζ = 0), also called the resonance

case since ∆E = 0, the transition probability reads

Res

= cos

ϑ. (49.47)

Equation (49.47) is a particular example of transitions

between initially degenerate states. This kind of tran-

sition occurs in the recoupling of angular momenta

in collisions of atoms possessing nonzero electronic

angular momentum [49.10].

For the general case, the nuclei preferentially mo ve

across single adiabatic PEC in the near-adiabatic limit,

ζ  1, while in the sudden limit, ζ  1, they move

across both diabatic PEC, unless the condition of narrow

avoided crossing, ϑ  1, is fulﬁlled.

49.3.3 Beyond the Common Trajectory

Approximation

The common trajectory approximation is valid when the

spacing between adiabatic PEC within an NAR is small

compared to the local kinetic energy of the nuclei. The

relaxation of this restriction is not unambiguous since

one should pass from a one-dimensional manifold (time

as a progress variable) to a multi-dimensional coordi-

nate (conﬁguration space manifold). Only if the latter is

one-dimensional (a single coordinate as a progress vari-

able, as is the case for atom-atom collisions), one can

suggest a generalization of the common trajectory tran-

sition probability. We consider this case, taking R to be

such a single coordinate, and assume that the quantum

motion across the adiabatic PEC satisﬁes standard qua-

siclassical conditions [49.3]. For a two-state problem

with adiabatic potentials U

(R) and U

(R), the gen-

eral condition of the common trajectory apporximation

reads

E −



) +U

)





) −U

)



(49.48)

where E is the total (conserved) energy and R

is the

coordinate of the NAR center.

The quantum generalization of the expression for

the transition probability in the near-adiabatic condition,

(49.26), is given by the original Landau formula [49.4, 5]

= exp







−











2µ



E −U

(R)



−





2µ



E −U

(R)















(49.49)

where µ is the reduced mass of the colliding atoms,

is the complex-valued coordinate of the crossing

of U

(R) and U

(R) and R is any value of the co-

ordinate in the classically accessible region of motion

of the nuclei. The nonadiabatic transition is local-

ized in the region of width ∆R = Im(R

) centered

at R

= Re(R

). Equation (49.49) becomes the com-

mon trajectory equation (49.26) under the condition

(49.48).

The quantum generalization of the LZ transition

probability can not be represented by an exact analyti-

cal expression though it is known that it depends on two

Part D 49.3

Adiabatic and Diabatic Collision Processes at Low Energies 49.4 Double-Passage Transition Probabilities and Cross Sections 749

parameters (and not on one, as in the case for the com-

mon trajectory approximation) [49.2]. A recommended

approximate expression for E > E

reads [49.11]

= exp







−2πζ







1+

−2



160



(ζ

)

+0.7







1/2







(49.50)

where



µv

∆F

√

Equation (49.50) becomes the common trajectory

(49.32) under the condition 

 1; it turns out that the

latter condition may be less restrictive than the general

condition (49.48).

The quantum generalization of the Nikitin

transition probability is possible provided U

(R)

in (49.39) is given by an exponential func-

tion, U

(R) ≈ exp(−αR). The transition probability

reads [49.11]:

= exp(−πδ

)

sinh(πδ −πδ

)

sinh(πδ)

(49.51)

and depends on three parameters of the model (and not

two as is the case for the common trajectory approxi-

mation). These parameters enter into δ

and δ through

complicated contour integrals. If the general condition

of the common trajectory approximation, (49.48) , is ful-

ﬁlled, δ

and δ reduce to ζ

and ζ so that (49.51) becomes

(49.43).

More discussions of two-state models within and

beyond the common trajectory approximation can be

found elsewhere [49.2, 11–14].

49.4 Double-Passage Transition Probabilities and Cross Sections

49.4.1 Mean Transition Probability

and the Stückelberg Phase

In the case of an atomic collision, the set R shrinks into

a single coordinate R. If there is only one NAR over

the whole range of R, the colliding system traverses it

twice, as the atoms approach and then recede. In this

case, there are two different paths between the center

of the NAR, R

, and the turning points R

and R

on the adiabatic potential curves U

(R) and U

(R).

The double-passage transition probability P

is ex-

pressed via the single-passage transition probability P

the single-passage survival probability 1 − P

,andthe

Stückelberg interference term cos ∆Φ

[49.15],

= 2P

(1 − P

)(1 −cos ∆Φ

)

= 4P

(1 − P

) sin

(∆Φ

/2). (49.52)

The Stückelberg phase ∆Φ

/2 is expressed as the

phase difference ∆Φ

(0)

/2 which is accumulated during

the motion of a diatom from the center of the NAR to

the turning points, together with an additional phase φ

∆Φ

/2 = ∆Φ

(0)

/2 +φ

. (49.53)

Generally, (49.53) is valid provided ∆Φ

/2  1. For

transitions between electronic states of the same ax-

ial symmetry, the relative angular momentum  of the

colliding atoms is conserved, and we have

∆Φ

(0)

/2 =





2µE −

( +1)/R

−2µU

(r)



1/2

dR/

−





2µE −

( +1)/R

−2µU

(r)



1/2

dR/ ,

(49.54)

where R

and R

are the turning points for adiabatic

motion on potential curves U

and U

, E is the total

energy. Once R

is chosen, ∆Φ

(0)

/2 is well-deﬁned and

is independent of the dynamic details of a nonadiabatic

transition. On the other hand, P

and φ

do depend

on these details. In particular, for the models discussed

in Sect. 49.3, the velocity v

that enters into the Massey

parameter, is





1/2

E −U

−

( +1)

2µR

1/2

(49.55)

where U

≈U

) ≈U

). As a function of E and ,

the double-passage transition probability is symmetric

with respect to the initial and ﬁnal states.

Part D 49.4

750 Part D Scattering Theory

In many applications, one can use the mean transi-

tion probability P

, which is obtained from P

averaging over sev eral oscillations:

P

=2P

(1 − P

). (49.56)

The important limiting cases of the double-passage

Nikitin model are:

(i) Double-passage Landau–Zener–Stückelberg equa-

tion,

LZS

= 4exp



−2πζ



1 −exp



−2πζ



×sin

∆Φ

(0)

/2 +φ

LZS

(49.57)

where ζ

is given by (49.32), ∆Φ

(0)

/2by(49.54)

and the expression for φ

LZS

is available [49.2]. When

changes from zero to inﬁnity,

LZS

passes through

amaximum,P



max

= 1/2, and φ

LZS

decreases

from π/4tozero.

(ii) Double-passage Rosen–Zener–Demkov equation:

RZD

sin



∆Φ

(0)

/2 +φ

RZD



cosh

(πζ/2)

(49.58)

where ζ is given by (49.46) and the expression for φ

RZD

is available [49.2]. Under certain conditions [49.2, 9],

the Stückelberg phase in (49.58) can be identiﬁed with

the phase accumulated during the motion of a diatom

from inﬁnitely large distance to the turning points.

(iii) Double-passage equation for a resonance process

(∆E =0):

res

= sin



2ϑ



sin



∆Φ



. (49.59)

The general resonance case (zero ener gy change,

∆E = 0) is also called an accidental resonance. For

the accidental resonance, the diagonal diabatic matrix

elements are not equal to each other. A particular case

of an accidental resonance is a symmetric resonance,

for which the diabatic matrix elements are the same. For

the Nikitin model, symmetric resonance corresponds to

ϑ = π/4, and (49.59) reads

symm

= sin



∆Φ



. (49.60)

Equation (49.60) also follows from (49.58) in the limit

ζ → 0. Actually, (49.60) is valid for any symmetric

resonance case [not necessarily for the model Hamil-

tonians (49.39)and(49.45)] and for the arbitrary values

of the phase ∆Φ

/2. This phase can be identiﬁed

with ∆Φ

(0)

/2 from (49.54) provided R

is taken to be

inﬁnitely large.

49.4.2 Approximate Formulae

for the Transition Probabilities

Several approximate formulae are available for P

 in

the case where H

depends on time in a bell-shaped

manner, and ∆H = H

− H

can be represented as

ω +∆V, with ∆V also having a bell-shaped form.

Deﬁne

v =





+∞



−∞

(t) dt ;





+∞



−∞

(t) exp(iωt) dt ,

w =





+∞



−∞

(t) exp



(i/ )



∆H(t)dt



dt ,

u =





+∞



−∞

∆ V

(t) dt , (49.61)

and

S(t) = (1/ )





∆H

(t) +4H

(t)



1/2

dt . (49.62)

Then the various approximate formulae, as suggested by

different authors [49.2], read

∼

/v)

sin

v, (49.63)

∼

sin

w, (49.64)

∼

sin



), (49.65)

∼



+∞



−∞

(t) exp[iS(t)]dt/



, (49.66)

∼

sin



cosh



. (49.67)

In (49.67), S



and S



are the real and imaginary

parts of the complex quantity S

= S(t

) from (49.62).

The complex-valued time t

is found from

∆H

) +4H

) = 0 , (49.68)

under the condition that t

possesses the smallest im-

aginary part of all roots of this equation.

Part D 49.4

Adiabatic and Diabatic Collision Processes at Low Energies 49.5 Multiple-Passage Transition Probabilities 751

49.4.3 Integral Cross Sections

for a Double-Passage

Transition Probability

The quasiclassical inelastic integral cross section σ

for

the transition i → f is related to P

µE

∞



d, (49.69)

where E

is the initial collision energy, E

= E−U

(∞).

Thecrosssectiondeﬁnedby(49.69) typically shows the

following qualitative dependence on the collision ve-

locity: σ

increases rapidly with E

at low energies,

reaches a maximum and then slowly falls off at high

energies. The position of the maximum roughly cor-

responds to the energy E

= E

∗

at which the relevant

Massey parameter at the NAR center, ζ(R

),isofthe

order of unity. The conditions E

< E

∗

and E

> E

∗

cor-

respond to the near-adiabatic and strongly nonadiabatic

(also called diabatic) regimes, respectively.

In calculating σ

, one usually neglects the Stück-

elberg oscillating term and sets the upper limit in the

integral in (49.69)toavalue = 

beyond which the

integrand begins to fall off quickly. Yet another sim-

pliﬁcation is possible, in the framework of the impact

parameter approximation, when the relative motion of

atoms is described by a rectilinear trajectory R(t) with

constant velocity v and impact parameter b =

/µv:

R(t) =





1/2

. (49.70)

For instance, for the Landau–Zener model 

µvR

/ , and the cross section depends on one dimen-

sionless parameter γ = 2πV

/(∆F v) according to

(γ) =2πR



exp(−γ/

√

x )



1−exp(−γ/

√

x )



= 4πR



(γ) −E

(2γ)



, (49.71)

where E

(z) is the exponential integral.

For the symmetric resonance, the cross section reads

(v) = 2π

∞



sin

∆Φ

Res

(b,v)

bdb ≈

(v) ,

(49.72)

where b

is found from the Firsov criterion [49.2]:

∆Φ

Res





∞



∗



(R) −U

(R)



−b

dR =

(49.73)

The cross section in (49.71) ﬁrst increases and then de-

creases with the collision velocity v, while that in (49.72)

slowly decreases with v.

49.5 Multiple-Passage Transition Probabilities

49.5.1 Multiple Passage in Atomic Collisions

In the case of atomic collisions, there is only one

nuclear coordinate R. If there exist several NAR on

the R-axis, those which are classically accessible (for

given total energy E and total angular momentum J)

can be traversed several times. In the semiclassical

approximation [49.16], the multiple-passage transition

amplitude A

for a given transition between inital state i

and ﬁnal state f can be calculated as a sum of transi-

tion amplitudes A

, over all possible classical ways

L which connect these states, and which run along

a one-dimensional manifold R:



, (49.74)

where each A

can be expressed through the probabil-

ity P

and the phase Φ

by [49.13]





1/2

exp



iΦ



(49.75)

The net transition probability is then

=|A

(49.76)



L,L









1/2

cos



−Φ





The ﬁrst sum runs over all different paths, and the sec-

ond (primed) over all different pairs of paths. The primed

sum usually yields a contribution to the transition prob-

ability which oscillates rapidly with a change of the

parameters entering into P

(i. e., E and J) and repre-

sents a multiple-passage counterpart to the Stückelberg

oscillations.

If the Stückelberg oscillations are neglected, P

equivalent to a mean transition probability P

:

P

=



. (49.77)

For one NAR, there are two equivalent paths, and P

(1)

(2)

= P

(1 − P

).Then(49.77) yields (49.56).

Part D 49.5