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

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762 Part D Scattering Theory

adapted to cases where long range avoided crossings

are dominant [51.10]. If a semiclassical approximation

is introduced, the scattering equations become almost

identical to those obtained directly using translation

factors of the CTF (common translation factor) type

[51.11, 12].

Three representative systems are chosen to illustrate

the basic features of the problem: Al

/HandB

/He

which are typical of type I, and O

/H which is typical

of type II. In the Al

/H system, charge transfer takes

place via the reaction [51.13]





S +H → Al





P +H

(51.1)

involving a network of two Σ and one Π adiabatic states.

In the B

/He system, charge exchange occurs via a net-

work of three Σ and one Π states involving two avoided

crossings [51.14]. The two possible capture channels are





+He → B





+He

nl = 2s, 2p .

(51.2)

The O

/H system is a good example of a type II reac-

tion which plays an important role in astrophysical plas-

mas [51.15, 16]. The dominant reaction at low energies





P +H → O



2s2p



P +H

(51.3)

involves only quartet states.

51.1 Molecular Structure Calculations

The construction of the network of adiabatic states of

the molecular ion complex constitutes the ﬁrst step in

analyzing the dynamics of the ion–atom system. In prin-

ciple, standard techniques of quantum chemistry can be

employed. For doubly charged ions, where type II pro-

cesses dominate and the effective avoided crossings take

place at relatively short range, ab initio methods are re-

quired. On the other hand, for trebly and more highly

charged ions, where type I processes dominate, the ef-

fective avoided crossings occur at long range where ab

initio methods can be quite inaccurate. Model potential

methods are then often more satisfactory.

51.1.1 Ab Initio Methods

Both molecular orbital and valence-bond methods have

been extensively used and (provided a sufﬁcient number

of conﬁgurations are included) should be satisfactory

in principle. A simple test is the location of the long

range crossings, which is very sensitive to the en-

ergy difference of the initial and ﬁnal channels. But

this test is insufﬁcient. For example, in the C

system, for which both the molecular orbital and va-

lence bond methods yield comparable results in regions

well away from avoided crossings, there are signiﬁcant

discrepancies in the important region of the avoided

acrossings [51.17, 18]. In the present state of the art,

the absolute accuracy of theoretical calculations in the

crossing region is difﬁcult to ascertain. Of course, since

the computed electron capture cross sections depend

sensitively on the minimum energy separation, experi-

mental data at low collision energies makes some tests

possible. Unfortunately, there is not yet much reliable

data at low eV energies.

Figures 51.1 and 51.2 present the adiabatic network

of the

−

Π and

Π ,

∆ states of the O

system. These have been obtained by means of a conﬁgu-

ration interaction molecular orbital method with several

hundred conﬁgurations [51.15]. It is clear from these

calculations that the only effective avoided crossings are

those involving the

−

and

Π at 3.7and4.8 a

,for

R (a. u.)

0.6

0.5

0.4

0.3

0.2

0.1

–0.1

–0.2

3456789101112

E (arb. units)

Fig. 51.1 Adiabatic potential energy curves of the

−

and

Π states of the OH

molecular ion. The solid curves

designate

Π states, the dashed curves

−

states. The

dissociation limits A and B correspond respectively to [O

(

P) + H(

S)] and [O

(

P) + H

]

Part D 51.1

Ion–Atom Charge Transfer Reactions at Low Energies 51.1 Molecular Structure Calculations 763

R (a. u.)

0.6

0.5

0.4

0.3

0.2

0.1

–0.1

–0.2

3456789101112

E (arb. units)

Fig. 51.2 Adiabatic potential energy curves of the

and

∆ states of the OH

molecular ion. The solid curves

designate

states, the dashed curves

Π states and the

dotted curve a

∆ state. The dissociation limits A, B, C and

D correspond respectively to [O

(

S) + H(

S)], [O

(

D) + H(

S)], [O

(

D) + H+]and[O

(

P) + H(

S)]

which the energy separation is ≈ 0.1eV. The avoided

crossings involving the doublet states at 8.0, 8.5and

10 a

, for which the energy separation is ≈ 0.02 eV, are

diabatic. The avoided crossings for type II transitions oc-

cur at much shorter distances than for type I transitions.

51.1.2 Model Potential Methods

Model potential methods offer an attractive alternative to

ab initio methods in treating type I reactions. To illustrate

the method [51.19, 20], we consider an effective one-

electron system composed of a spherically symmetric

ion X

and a hydrogen atom. The model Hamiltonian

of the molecular ion XH

is written as

H = T +V

) −

(R), (51.4)

where T is the electronic kinetic energy, r

and r

are

respectively the position vectors of the Rydberg electron

with respect to nuclei A and B, and V

(r) is the effective

potential of the ion core. The latter is usually expressed

in the parametric form

(r) =−



q +(Z −q)



1 +αr +βr

+γr



−δr



(51.5)

where Z is the ionic nuclear charge and the parameters

α,β,γ,δ,... are optimized to the spectroscopic data so

that the asymptotic energies of those Rydberg states of

(q−1)+

which govern the charge transfer process are

essentially exact.

The eigenvalues of H for a given internuclear

distance R are determined by standard variational tech-

niques, using a basis set of Slater-type orbitals in

prolate spheroidal coordinates [51.3]. As an example,

we present in Fig. 51.3 the potential energy curves of

the Al

system which presents one Σ–Σ avoided

crossing around R

= 7.2 a

. Model potential methods

can also be used to treat multielectron systems such

as (XHe)

. In long range collisions, the transition

probabilities are primarily determined by the asymp-

totic form of the electron wave functions far from the

nuclei. Thus even when multielectron targets are in-

volved, the effect of dynamic correlation can be small.

Since the asymptotic form of the wave functions may

be easily generated by model potential techniques, the

effect of static correlation can be taken into account

in a rather simple way. The method proposed here is

quite similar to that used by Grice and Herschbach

[51.21] to treat the long range conﬁguration interac-

tion of ionic and covalent states in neutral atom–atom

collisions.

Internuclear distance

(

)

0.2

–0.2

–0.4

–0.6

46810

Adiabatic energies

(arb. units)

Fig. 51.3 Adiabatic potential energies of AlH

.Thefull

curve designates the

Σ state correlated to the [Al

(3p)+

H (1s)] entry channel. The dotted and dashed curves des-

ignate respectively the

Σ and

Π states correlated to the

[Al

(3p) + H

] electron capture channel

Part D 51.1

764 Part D Scattering Theory

For example, in the system X

He, where as previ-

ously X

is a closed shell ion, the energy separation at

a long range avoided crossing is given by [51.22]

∆ε(R

) ≡ 2

√



1σ





a

nlλ

+q/R|b , (51.6)

where the molecular orbitals a

nlλ



dissociating to

(q−1)+



and b

1σ

(dissociating to a 1s orbital of He)

are generated from a model Hamiltonian of the form

H = T +V

) −V

) +V

X,He

(R), (51.7)

where V

is an effective potential describing the 1s

orbital of He. The orbital b



is the 1s orbital of He

This expression is the optimal form of ∆ε(R

) which

can be achieved without the use of explicitly correlated

wave functions. Since both the a

nlλ

and b

1σ

orbitals

are generated from the same model Hamiltonian, the

calculation of the interaction matrix element in (51.6)is

simple. As an example, Fig. 51.4 presents the adiabatic

energies of the B

/He system which has two avoided

Σ −Σ crossings, one at 4.6 a

, the other at 7.4 a

A similar extension to treat the case of capture by an

ion with one electron in a p shell can be made along the

same lines. See [51.23] for an application to the reaction

(2p)

P +H → O

(2p3p)

P +H

. (51.8)

A combination of model potential methods to rep-

resent the core electrons and ab initio molecular orbital

methods for the valence electrons has been successfully

developed [51.24] to treat some complex systems such

as C

/H where both type I and II transitions take place.

The ﬂexibility of the model potential parameters makes

it possible to choose them so that the energies of the

initial and ﬁnal states are accurately reproduced.

51.1.3 Empirical Estimates

Electron capture cross sections at low energies are

largely controlled by two parameters: the crossing ra-

dius R

and the minimum energy separation ∆

.These

are the basic parameters required for the Landau–Zener

model, and it is useful to have a simple way of es-

timating them without having recourse to a complex

molecular structure calculation. Many empirical esti-

mates [51.25–27] have been proposed. However, it is

generally necessary to take into account the strong

-dependence of the electron capture channel states.

Deﬁning α =

√

/13.6, where I

is the ionization po-

tential of the target (in eV), then Taulbjerg’s formula

Internuclear distance (a

)

–0.5

46 810

Adiabatic energies (arb. units)

–1

Fig. 51.4 Adiabatic potential energies of BHe

.Thedot-

ted curve designates the

Σ state correlated to the [B

He(1s

)] entry channel. The short and long dashed curves

designate respectively the

Σ and

Π states correlated to

the [B

(2p) + He

(1s)] electron capture channel. The

full curve designates the

Σ state correlated to the [B

(2s) + He

(1s)] entry channel

is [51.27]

∆

) =

18.26

√

exp(−1.324αR

√

q ), (51.9)

where

= (−1)

n+l−1

√

2l+1 Γ(n)

[Γ(n+l +1)Γ(n −l)]

1/2

. (51.10)

For type I systems, (51.9) is generally quite sat-

isfactory when the energy levels of the capture states

are well separated [51.28]. In that case, it can be used

to predict the main electron capture reaction windows

and obtain a reliable ﬁrst estimate of the cross sec-

tions (in the energy range for which the Landau–Zener

model is valid). On the other hand, in cases where

there is a near degeneracy of the l states, serious er-

rors can occur [51.28] and the predictions are less

satisfactory.

For type II reactions, (51.9) must be modiﬁed by

a corrective factor to take account of the simultaneous

excitation of an electron from a 2s to 2p orbital. One

such modiﬁcation, proposed by Butler and Dalgarno

[51.29], may give some useful idea of the main electron

capture processes, but its precision is uncertain.

Part D 51.1

Ion–Atom Charge Transfer Reactions at Low Energies 51.2 Dynamics of the Collision 765

51.2 Dynamics of the Collision

The ﬁrst step is to introduce a suitable set of scattering

coordinates which can automatically describe both the

excitation and rearrangement channels. The particular

choice of coordinate system is conditioned by practical

considerations. We have found Eckart coordinates to be

convenient [51.10]. Their application is straightforward

since it involves nonadiabatic matrix elements which can

be calculated by the conventional techniques of quan-

tum chemistry. The practical implementation of Eckart

coordinates leads to the introduction of an adiabatic

variable ξ deﬁned as

ξ =

√



R+



where

s =

r · R



r −

r · R



, (51.11)

µ is the reduced mass of the colliding system, r the

coordinate of the active electron with respect to the c.m.

of the colliding system and Rthe relative position vector

of the nuclei.

The adiabatic Eckart and Born–Oppenheimer equa-

tions differ only by terms of the order of 1/µ.Itmay

be assumed that the Eckart states are given to sufﬁcient

accuracy by the Born–Oppenheimer adiabatic states des-

ignated by χ

(r; R). We expand the total wave function

of the system in the form

Ψ(r,ξ)=



(r;ξ)F

(ξ) . (51.12)

(For a many-electron system, r represents the ensemble

of electron coordinates.) Decomposing F

(ξ) on a basis

set of symmetric top functions according to

(ξ) =



K,M

(−1)



2K +1

4π



1/2

× D

Λ,M

(θ, φ)

(k)

(ξ)

(51.13)

where {θ, φ}are the spherical polar coordinates of ξ,then

the radial functions g

(k)

(ξ) are solutions of the equation

dξ

(K)

+2A

dξ

(K)

+Wg

(K)

= 0 , (51.14)

where

(ξ) =





∂

∂ξ

∂

∂z





δ(Λ

,Λ

(51.15)





∂

∂ξ





(51.16)



2µ(E −ε

) −



K(K +1) −Λ





+ B



K(K +1) L

, (51.17)

=∓





−2x

∂

∂z





δ(Λ

,Λ

±1),

(51.18)

z, x are the components of r parallel to and perpendicular

to the direction of ξ in the classical collision plane,

respectively.

Since the ratio 1/µ is small, it is legitimate to replace

the matrix element χ

|∂/∂ξ|χ

 by χ

|∂/∂R|χ

 in

(51.15)and(51.16). Furthermore, all the matrix elem-

ents A

and L

vanish asymptotically to ﬁrst order

in 1/µ. The modiﬁed radial and rotational matrix elem-

ents are identical to those obtained in the semiclassical

formalism with common translation factors.

The simplest way to solve (51.14) is to eliminate

the ﬁrst-order derivative by transforming to a diabatic

representation in which the radial matrix elements van-

ish [51.30]

(K)

= Ch

(K)

, (51.19)

where

dξ

C + AC = 0 , C(∞) = I .

(51.20)

Equation (51.14) then reduces to

dξ

(K)

−2µV

(K)



2µE −

K(K +1)



(K)

= 0 (51.21)

with

= V

−

√

K(K +1)

µR

, (51.22)

= C

−1

εC , V

= C

−1

LC . (51.23)

The solution of (51.21) and the subsequent extraction

of the scattering matrix S may be carried out us-

ing an extension of the log derivative method [51.31],

adapted to the case of a repulsive Coulomb poten-

tial in one or more of the scattering channels [51.32].

Part D 51.2

766 Part D Scattering Theory

This method is particularly stable and advantageous

to use at low energies for problems of the type con-

sidered here. Of course, at energies exceeding a few

hundred eV, rapid oscillations of the radial function ren-

der the method rather time consuming. But the method

is usable up to keV energies. However, for higher

energies exceeding 1 keV, semiclassical methods are

preferable.

51.3 Radial and Rotational Coupling Matrix Elements

Two independent methods are employed for the de-

termination of radial coupling matrix elements A

the ﬁrst based on a direct numerical differentiation of

the expansion coefﬁcients of the wave function, the

second on a variant of the Hellman–Feynman (HF)

theorem. In principle, the HF method is three times

faster than the direct numerical method, since the eigen-

vectors need to be calculated only at the value of R

concerned. However, the real gain in computing is not

always appreciable, the HF theorem being more sen-

Radial couplings

–0.5

Internuclear distance (

)

24 1068

Origin = Al

CTF

Origin = H

Fig. 51.5 Radial coupling matrix elements between the two

Σ states of AlH

.Thefull curve designates the ma-

trix element of the CTF (or Eckart) type. The short and

long dashed curves designate the radial derivative with the

origin of electron coordinates respectively on the Al and

H nuclei

Fig. 51.6 Rotational coupling matrix elements between

the

Σ and

Π states correlated to the electron capture

channel. The designation of the curves is the same as

for Fig. 51.5

sitive to errors in the wave function than the direct

method. To achieve comparable accuracy, a larger ba-

sis set would be required in the molecular calculations,

and this offsets much of the theoretical gain in com-

puting time. The rotational coupling matrix elements

are calculated numerically for each internuclear

distance.

Typical results for the radial and rotational matrix

elements are presented in Figs. 51.5 and 51.6 for the

/H system, illustrating the inﬂuence of translation

effects. The origin dependence of the matrix element

(without inclusion of translation) is weak in the vicin-

ity of an avoided crossing. Away from the crossing,

the origin dependence can be considerable. The corre-

sponding diabatic energies and couplings are given in

Figs. 51.7 and 51.8. For internuclear distances on the in-

ward side of the crossing, the diabatic states correspond

to a mixing of adiabatic states: it is clear that the mathe-

matical deﬁnition (51.20) of the diabatic representation

does not correspond to the empirical deﬁnition of the

Landau–Zener model (Sect. 51.5).

Internuclear distance (a

)

–0.6

–0.8

–1

–1.2

Rotational couplings

24 1068

CTF

Origin = H

Origin = Al

Part D 51.3

Ion–Atom Charge Transfer Reactions at Low Energies 51.4 Total Electron Capture Cross Sections 767

Diabatic energies (arb. units)

0.2

–0.2

–0.4

–0.6

Internuclear distance (

)

24 1068

Fig. 51.7 Diagonal elements of the diabatic matrix of

ALH

0.2

0.1

–0.1

–0.2

Internuclear distance (

)

24 1068

Origin = Al

CTF

Origin = H

Diabatic couplings

Fig. 51.8 Off-diagonal diabatic matrix element between the

two

Σ states of AlH

51.4 Total Electron Capture Cross Sections

Although the selection rules for electron capture at low

energies are primarily governed by the avoided cross-

ings between adiabatic states of the same symmetry,

the combined effects of radial and rotational coupling

can be quite complex. In general, rotational coupling

effects are weak when capture occurs via an S state,

but they can be strong for capture to P, D and higher

L states.

In the case of Al

/H, where capture to the (3p)

state of Al

takes place via a three-state network of

two Σ and one Π states, rotational coupling between the

Σ and Π states is strong in the vicinity of the avoided

crossing and enhances the capture cross section con-

siderably. This phenomenon is illustrated in Fig. 51.9,

which shows the calculated total cross section at differ-

ent energies as a function of ∆, the energy separation of

the quasidegenerate diabatic

Σ and

Π exit channels at

the

Σ –

Σ crossing radius. The total cross section has

amaximumfor∆ = 0.08 eV. The cause can be easily

seen from the corresponding adiabatic potential ener-

gies (Fig. 51.10). For ∆ =0.8 eV, the potential energies

of the adiabatic

Σ entry channel and the

Π exit chan-

nel become tangential to one another, thereby inducing

a resonant effect.

100

Cross sections (a

)

–2

–1

Energy separation σ

–π

(σ

)

E = 141.2 eV/amu

E = 84.7 eV/amu

E = 28.2 eV/amu

E = 8.5 eV/amu

Fig. 51.9 Results of three state (two

Σ and one

Π ) cal-

culations for electron capture in the Al

/H system. The

cross sections are plotted as a function of the parameter ∆

(see text). The extreme sensitivity to ∆ illustrates the ne-

cessity of knowing the

Σ –

Π energy separation to high

accuracy. The numerical values on the curves designate the

collision energies in units of eV/amu

Part D 51.4

768 Part D Scattering Theory

–0.4

–0.45

–0.5

–0.55

–0.6

56789

Internuclear distance (a

)

/H adiabatic energies (arb. units)

σ1s

σ3p

Fig. 51.10 Schematic diagram of the nondiagonal matrix

elements of AlH

in the vicinity of the curve cross-

ing. The three broken curves correspond to different

state potentials shifted from their calculated value by small

amounts

In the case of the B

/He system, four molecu-

lar states are implicated in the collision process: three

Σ states (Σ

correlated to the

S exit channel, Σ

cor-

related to the

P exit channel and Σ

correlated to the

entry channel) and one Π state (correlated to the

exit channel). In order to understand better the rota-

0.02

0.01

02468

Cross section (a

)

Impact parameter (a

)

/He

lab

= 6 keV

Fig. 51.11 Transition amplitudes for

Sand

P electron cap-

ture as a function of impact parameter in B

/He collisions.

The solid curve refers to the

P contribution, the dotted

curve to the

S contribution

02468

0.006

0.004

0.002

Cross section (a

)

Impact parameter (a

)

/He

lab

= 6 keV

with rot. coupling

without rot. coupling

Fig. 51.12 Inﬂuence of rotational coupling on the

P elec-

tron capture cross section in B

/He collisions. The

solid curve refers to the complete calculation, the dot-

ted curve to the calculation with only radial coupling

included

tional coupling mechanism, Figs. 51.11 and 51.12 show

the

Sand

P electron capture transition amplitudes as

a function of angular momentum (impact parameter)

for an energy of 6 keV, where rotational coupling is

of major importance [51.14, 33]. The

P electron cap-

ture transition amplitude exhibits 2 maxima, one for

an impact parameter of 1.6 a

, the other for an impact

parameterof5.7 a

, corresponding respectively to cap-

ture via Σ

–Π crossing around R = 2 a

and the outer

–Σ

crossing around R = 7.4 a

. It is clear from

Fig. 51.12 that at very short internuclear distances, ro-

tational coupling is much more important than radial

coupling. At the outer crossing (which is nearly dia-

batic), radial coupling is dominant. This result may be

generalized. When the avoided crossing has a largely

diabatic character, as for 3p capture in Si

/H [51.34]

or 3d capture in C

/H [51.10], the inclusion of rota-

tional coupling is fairly weak, affecting principally the

population of the sub-m levels and less appreciably the

total capture cross section into a given l state. In this

case, translation effects are of more importance than

rotational coupling.

In the case of O

/H (a particularly important sys-

tem in astrophysical plasmas), the existence of several

adiabatic states correlated to the entry channel leads to

many interesting features, typical of open p shell ions.

The favored reaction channel via the

−

and

Π states

involves simultaneous electron capture into a 2p orbital

Part D 51.4

Ion–Atom Charge Transfer Reactions at Low Energies 51.6 Differential Cross Sections 769

and excitation of a 2s orbital. The avoided crossing is due

to electron correlation. Rotational mixing of the

−

and

Π states leads to a large enhancement of the cross

section at energies exceeding a few tens of eV [51.16].

On the other hand, the avoided crossings involving the

and

Π states around 8 a

are too diabatic to con-

tribute to an electronic transition. As a consequence, the

metastable

D ions can only react via a curve crossing

at small distances (2.5 a

). The cross section is much

smaller than for ground

P state capture.

51.5 Landau–Zener Approximation

If a rapid estimation of the cross sections is all that

is needed, the Landau–Zener method can be used with

advantage, provided that the molecular structure par-

ameters are known accurately. This method, based on

an approximate solution of the dynamical equations in

a semiclassical formalism (Chapt. 49), is satisfactory

for the dominant channels. Aside from the transi-

tion being assumed to be localized at the crossing

point, the method can easily take account of trajec-

tory effects at low energies. On the other hand, it is

unreliable for the weaker channels and it makes al-

lowance neither for translation effects nor for rotational

coupling.

The cross section for capture into a state n via a curve

crossing located at R

is given by

= 2π

max



2p(1 − p)ρ dρ, (51.24)

where p, the probability for a single passage trajec-

tory (impact parameter) through the crossing, is given

p = exp



−2π



∆E



v∆F



(51.25)

∆F being the difference in slope of the covalent and

ionic diabatic curves at R

, ∆

the energy separation

of the diabatic curves at R

and v the radial vel-

ocity at R

. It should be recalled that there is no very

rigorous deﬁnition of the diabatic curves (which are re-

quired to obtain DF).For long distance crossings, the

simplest (and probably the most satisfactory) estimate

is that based on the asymptotic forms of the diabatic

states

∆F =

∂

∂R



(R) −

q −1



R=R

(R) =−

, (51.26)

where α

is the polarizability of the target. The radial

velocity is given by

v =





1 −

−

)



1/2

. (51.27)

The inclusion of the attractive polarization poten-

tial V

considerably increases the cross section at low

energies, since trajectories with impact parameters much

greater than R

contribute to the cross section. This

effect can introduce a negative energy dependence of

the cross section in the limit of low energies, of the

same kind as the Langevin model for ion–molecule

reactions.

51.6 Differential Cross Sections

Differential cross sections at large scattering angles (cor-

responding to small impact parameters) enable one to

probe details of the collision dynamics not readily ob-

tainable from total cross sections, which tend to be

dominated by the contribution from small angle scat-

tering (large impact parameters).

Their determination is staightforward once the S ma-

trix elements have been extracted from the asymptotic

solution of the coupled equations (51.21). The scattering

amplitude f

(ϑ) for scattering through the c.m. angle ϑ

is given by

(ϑ) =

2i(k

)

1/2

∞



l=0

(2l+1)



−δ



× P

(cos ϑ) exp[iα

(l)] , (51.28)

where k

and k

are the wave numbers of the initial

and ﬁnal channels, S

the S-matrix element for the i to

Part D 51.6

770 Part D Scattering Theory

–1

–2

0 0.5 1 1.5

0 0.2 0.4

0 0.1 0.2 0.3

–1

–2

1.5 2 3 42.5 3.5

–1

1 1.5 2 2.5 3

–1

0.5 1 1.5

0.6