Intermultiplet and angular momentum transfers of excited sodium atoms in collisionswith molecules. II. ModelsC. Desfrancois, J. P. Astruc, R. Barbe, and J. P. Schermann Citation: The Journal of Chemical Physics 88, 3037 (1988); doi: 10.1063/1.453947 View online: http://dx.doi.org/10.1063/1.453947 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/88/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Intermultiplet transfers of excited sodium atoms induced by nitrogen molecules J. Chem. Phys. 89, 251 (1988); 10.1063/1.455518 Intermultiplet and angular momentum transfers of excited sodium atoms in collisions with molecules. I.Experiment J. Chem. Phys. 88, 106 (1988); 10.1063/1.454642 Collisions of excited alkali atoms with O2. I. Intermultiplet transfer J. Chem. Phys. 87, 2084 (1987); 10.1063/1.453183 Transfer of electronic excitation in collisions of metastable argon atoms with nitrogen molecules. II J. Chem. Phys. 77, 5855 (1982); 10.1063/1.443750 Energy transfer in collisions of excited Na atoms with NO molecules J. Chem. Phys. 67, 839 (1977); 10.1063/1.434848

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Intermultiplet and angular momentum transfers of excited sodium atoms in collisions with molecules. II. Models

c. Desfrancois, J. P. Astruc, R. Barbe, and J. P. Schermann Laboratoire de Physique des Lasers, Universite Paris-Nord, Av. J. B.Clement, Villetaneuse 93430, France

(Received 23 September 1987; accepted 26 October 1987)

The use of ionic intermediate models for the description of collisions involving excited alkali atoms and molecules is discussed. It is shown that the multiple curve crossing and the quasifree electron models can be expressed with the same formalism. The results of these models are compared to experimental electronic to electronic transfer cross sections in the case of oxygen and nitrogen molecules for which detailed informations upon the resonant free electron-molecule scattering data are available.

I. INTRODUCTION

Collisions between moderately excited alkali atoms (with principal quantum number n in between 4 and 8) and molecules have received attention during recent years. Some potential energy surfaces of the most simple systems Na*­H/-3 and Na*-N2

4 have been calculated and electronic to electronic energy transfers5,6 as well as reactive processes 7,8

have been studied. For the quenching process

A(nJi) + MZ(VJi) + E-+A(nrif) + MZ(vf Jf ) + E +..:lE (1)

in the thermal range (ES I eV), pure quantal9 or classical trajectory, 10 dynamical calculations are only possible for the very first excited states since rather large numbers of poten­tial energy surfaces must be taken into account.

For more highly excited states, it is most often necessary to use models which are all derived from the Fermi model: process (I) is dominated by the scattering of the alkali va­lence electron by the molecule. Two cases must be distin­guished: if the electronic energy transfer AE = E(nflf ) - E(nJi) is negligible as compared to the vibrational fre­

quency of the molecule, the impulse approximation can be used either for very small energy transfers (AE S 10 cm - I )

as in I or n changing collisions, II or for resonant rotational energy transfers for which the momentum transfers ..:lE re­main small.6 The impulse approximation is valid when the electron-perturber collision time 7c is smaller than the char­acteristic time h / E (n ,I) of the electron motion in the atomic valence shell.

On the contrary, the temporary attachment of the atom­ic valence electron on the molecule favors the energy ex­change and leads to the vibrational excitation of the mole­cule. This is the well-known reason why the existence of ionic intermediates have been proposed to explain the large quenching cross sections corresponding to high AE values.

These ionic intermediate models will be discussed here in the case of collisions involving nitrogen and oxygen mole­cules for which recent experiments IZ

,13 have shown that the electronic to electronic transfer cross sections have the same order of magnitude and for which the vibrational excitation processes in free electron-molecule scattering are well estab­lished. 14,15 Section II will present two models which were originally proposed for the interpretation of the quenching

of the first resonant state of sodium: the multiple curve cross­ing model (MCC) of Bauer, Fisher, and Gilmore, 16 and the quasifree electron model (QFE) of Bottcher and Suku­mar.17 These two models will be described with the same formalism in order to emphasize their similarities and differ­ences. Section III will be devoted to the first quantitative comparison between these models and experimental results in the case of the two following processes:

Na(4P,4D) + Mz-+Na(5S,3D) + Mz (2)

and

Rb(7S) + M2 ~ Rb(5D) + M2. (3)

In this section we will also examine the use of the impact approximation and the necessity of an atomic core effect correction for the angular momentum transfer process:

Na(4D) + M 2 -+Na(4F) + M2 (2')

for which the electronic energy transfer is small (..:lE = 40 cm- I ).

II. MCC AND QFE MODELS

A. Principle and validity of ionic intermediate models

The atom-molecule collision problem can be formulat­ed in a vibronic network picture: the relative motion between the excited atom and the molecule, characterized by the co­ordinate R, is separated from the quantized molecular vibra­tional motion and the electronic motion. The total Hamilto­nian is taken as

I a2

H= ---+h=T +h 2"" aR 2 N'

where"" is the reduced mass. (4)

One thus considers two sets of potential energy curves corre­sponding to the diagonal terms of h in the diabatic elec­tronic-vibrational basis:

covalent system: A(n,l) + Mz(v) -+ V~ (R)

= (~cXvlh I ~cXv) = Vc(R) +Ev ' (5)

ionic system: A + + M 2- (v) -+ Vr (R)

= ( ~i X v·lh I~i X v,) = Vi (R) + Ev' . (5')

J. Chem. Phys. 88 (5), 1 March 1988 0021-9606/88/053037-08$02.10 © 1988 American Institute of Physics 3037

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3038 Desfrancois et al.: Sodium atoms in molecules. II

The nondiagonal coupling terms can be factorized as a prod­uct of an electronic term and a Frank-Condon factor:

h ~r = (¢i Xv' Ih I ¢e Xv) = hie (R) (Xv' IXv) . (5")

These expressions are valid if the vibrational quantiza­tion of the negative molecular ion is meaningful i.e., if its lifetime is at least equal to a vibrational period. This condi­tion is well verified for the O2- e1Tg) negative ion,14 but is questionable for the rather short-lived N2- (21Tg) shape reso­nance which is described by the boomerang model. 15

Another important validity condition is that the ionic­covalent curve crossing distances Re must be large enough to insure that the molecular negative ion M2- is not affected by the presence of the atomic ionic curve A *. From the work of Maessen and Cade,18 one can assume that this is verified in the case of molecules with positive electron affinities such as O2, in collision with excited alkalis,12.19 but it is clearly wrong for the first resonant state of sodium in collision with N24 and quite satisfactory for more highly excited states (Fig. 1). In fact, ab initio calculations would be necessary to ascertain the ionic character of the attractive curves in the crossing regions, and the absence oflarge distance humps in the entrance covalent curves which would have important effects in the here considered thermal range.

B. Cross section calculations

1. Multiple curve crossing model

At each ionic-covalent crossing distance Re such that

Vi(Re) +€v' = Ve(Re) +€v (6)

one computes the Landau-Zener transition probability20 (in atomic units)

P~~'(b) = l-exp[ -21Th~r'(Re)R~/V(b,Re)]' (7)

where V( b,Re ) is the radial velocity at Re and for the impact parameter b:

!J.lV 2(b,Re) =€(1-b2/R~). (8)

Each crossing is independently treated, ignoring inter-

eV V

2

-4

R o 50 A

FIG. 1. Internuclear crossing distances between covalent potential energy curves corresponding to the different excited states of sodium and ionic curves for different molecules. The negative molecular ions are, respective­ly, strongly or slightly perturbed in regions (I) or (2), and unperturbed in region (3).

ference effects, 2 I and one computes the total probability

P if( b) of transfer from the initial covalent curve correspond­ing to A(n;!i)+M2 (V i ) to the final curve A(nflf ) + M2 (vf)' If g is the degeneracy factor, the total cross sec­tion is obtain after summation over the impact parameters:

(9)

The calculation of the probability P if ( b) has often been performed with a Monte Carlo procedurel2 although a sim­ple algorithm permits us to evaluate it exactly by taking in account all the possible paths through the grid of crossings of the vibronic network.

2. Quasifree electron model

In free electron-molecule scattering, the discrete states of the negative molecular ion are coupled to the continuum (free electron + neutral molecule) states via the molecular internuclear motion. Bottcher and Sukumarl7 have ex­tended the formalism developed by Bardsley22 for the vibra­tional excitation of molecules by electron impact:

e- + M 2(vj> J i ) + M 2- (v', J') ..... e- + M 2(Vf , Jf ) (1')

to the electronic-vibration energy transfer in excited atom­molecule collisions [process (1)].

The radial wave functions S of the relative atom-mole­cule motion verify the following equation:

[TN + Veff(R)] S = ES, ( 10)

where the effective potential is given by

Veff(R) = VCR) +J(J+ 1)/2J.lR2

with V=Vr' or V~. (11)

The discrete ionic wave functions S r' are coupled to the con­tinuum covalent wave functions S ~ by the potential coupling term h ~r (R) [Eq. (5")]. The corresponding partial width is given by a Fermi Golden Rule expression

(12)

where S ~ is energy normalized and taken at the same total energy E as S r'. A JWKB calculation of the above matrix element leads to

h VV"(R )R 2 rvv' ( J) = 2£U el e e

el V( J,Re) (13)

where V( J,Re) is the radial velocity at R e, for the angular momentumJ,

(14)

and w is the classical oscillation frequency in the binding potential Vr (R), corresponding to the spacing between the energy levels of s r' at the total energy E. The semiclassical expression 14 is related to the classical expression8 by the rela-

tionship: bKo = ~ J( J + 1) where Ko is the initial relative momentum (€ = K ~/2M).

If one admits that the partial widths r~r are smaller than w, itself smaller than the experimental dispersion DE of the collision energy, one can calculate the cross sections with an energy averaged Breit-Wigner formula:

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Desfrancois sf at; Sodium atoms in molecules. II 3039

h r v' ~ r vv' were i=£.. ci' (15) c,v

rr is the inverse of the lifetime of the ionic intermediate A + + M2- (v').

3. Comparison between the two models

The MCC and QFE models have been originally de­scribed l6

,17 with two different formalisms, but we will here show that they are identical when the first order approxima­tion r~r <w is verified. The summation over J in Eq. (15) can be converted into an integral by means of the relation

1T/K~ ~ (2.1 + 1) = J 21Tb db. (16)

Equation (15) then becomes identical to Eq. (9) if we write the ionic-covalent transition probability as

(17)

This identity can be easily interpreted as follows: with the approximation r~r <w, Eq. (7) becomes, at first order

p~r = 21Th ~r2(Rc)R ~/V(b,Rc) = (1T/W )r~~'

and then Eq. (17) can be rewritten as

where Pc-v' = 2P~v; , ,

(18)

d P - rVfV'/rV' an V'-Cf - cfi i (19)

because the ratio r~~';n' between the probability per unit of time for switching from the ionic state (i,v') towards the covalent final state (cf,vf) and the inverse of ionic state life­time is, in fact, the probability Pv' -Cf for the system to be, after collision, in the final state/when incoming in an ionic state v'. Moreover, the switching probability PCi_V' from the

initial covalent state towards this ionic state is equal to 2P ~i~'. This is shown in Fig. 2 where the impact parameter b' is chosen in order that only three crossings are possible before the classical turning point T, with respective probabilities Po, PI' and P2. One gets at first order:

PCi_V' = 2 = (1 - PO)PI

+ (1 - Po)( 1 - P2)2(1 - PI )PI =2PI (20)

since the curve switching can occur before or after reaching T. Finally, from the identity p~r = (1T/W)r~r ofEq. (18), 1T/W appears as the time expansion of the scattered wave packet.

The QFE model thus appears as a first order treatment, for low values of the adiabatic probabilities, of the MCC model dynamic. This first order approximation is most prob­ably valid for O2 since the crossing radii Rc are large and, the couplings hic (Rc) are weak. This may not necessarily veri-

vY'IR) f Eb2

_-___ ~ R'

Re, Re1 Reo R

FIG. 2. Example ofionic~ovalent potential energy curve network (with centrifugal barrier) in the vicinity of the classical turning point T. Two dif­ferent possibilities are open for the switching of the system towards the ionic curves (here v' = 1).

fied for N 2, especially near the classical turning points where the velocity V(b,Rc) goes to zero.

C. lonic-covalent coupling term

1. Multiple curve crossing model

With the assumption that, at large crossing distances R c ' the vibrational states ofthe molecule M2 or the negative ion M2- are not modified, one can write20

(21 )

where hic (R) = ( ¢i Ihe/I'Pc; ¢c) where ¢i and ¢c are, re­spectively, the electronic wave functions ofM2- and M2 and 'Pc the atomic wave function of the valence electron. The angular dependence/(O), where 0 is the angle between the internuclear axis and the molecular axis, is determined by the symmetry selection rules in the three approach configu­rations20: 0=0 (Coo), 0=1T/2 (C2v ), and OE] 0;1T/2 [ (Cs ). The Franck-Condon factors (X v Ix v,) are calculated by resolution of the radial Schrodinger equation in the Morse potentials of M2 and M2- .

The main difficulty is thus the determination of hic (R) which reflects the overlap of the large atomic orbital 'Pc with the more localized molecular orbital ¢i' For the large values of Rc' 'Pc has an exponential dependence which justifies the semiempirical formula of Olson et al.23

:

h ~(R *) =AR * exp( -BR *), (22)

where

R * = R ~2IP(nl) and hic = h ~ ~IP(nl) 'IEAI . (22')

The experimental determinations of the ionic-covalent coupling elements of the ground-state alkali atom-halogen molecule systems are well fitted with A = 1.73 and B = 0.875. 24 With a slight adjustment, we have shown that the above expression is also valid for the Na(3P)-02 sys-

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3040 Desfrancois et al.: Sodium atoms in molecules. II

tem. 19 However, this expression is meaningless for mole­cules with negative electron affinities and thus, we will not give here any results for N2 with the MCC model.

2. Quasifree electron model

In this model, the coupling element h ~~' is determined from the partial widths of process (1'):

r~i" (k) = 217"1 ( <Pi Xv' IhM-lkLA; <Pc Xv) 12 , = r(k)l(xvlxv,)1 2, (23)

where hM - is the Hamiltonian of the e - M2 system and 2

IkLA) is the free spherical wave, energy normalized, which satisfies the resonance condition

!k 2=Ev' -Ev -EA. (24)

The shape resonances 2TIg of O2- and N2-, with respective parent states 3~g- and 1 ~g+ for O2 and N2, imply the symme­try conditions L = 2 and A = ± 1.

In the QFE approximation, one neglects, in the electron attachment process at large internuclear distances, the influ­ence of the atomic ionic core A +. The Hamiltonian h [Eq. ( 4 )] is thus reduced to h

M,- for the calculation of the cou-

piing element h ~r2(Rc)' The role of the ionic core A + is thus limited to the preparation of the angular momentum of the electron in the atomic reference frame [!It'. The atomic wave function <Pc can be approximated, for large distances R, by a free spherical wave in [!It' which will be transposed in the molecular reference frame [!It. At the crossing radius R c ' the momentum k of the electron must satisfy the resonance con­dition (24). If we take, for the potential curves, the simple expressions V; (R) = - 11 R - EA, and Vc (R)

- IP(nl), Eq. (6) becomes

- lIRc - EA + Ev, = - IP(nl) + Ev

or

(25)

in order that k appears as the classical momentum in the Coulomb potential of the atomic ionic core.

One then obtains the desired coupling element at the internuclear distance Rc:

h ~r2(Rc) = I~I (k) 1 ( <Pi Xv' IhM-lklm;<pc Xv) 9P' 12 , 2

where the form factor/nl (k) is defined as

Inl(k) = k 2(klml<pc)9P' .

(26)

(27)

The transposition from the atomic frame [!It' to the molecu­lar frame !7l is then performed by means of the transforma­tion formula25:

L,A Iklm)9P' = L (i)/'-I+L( _1)m

I',m'

x ../417"(21 + 1 )(21 ' + 1 )(2L + 1)

(I' L I) (II L

X 0 0 0 m' A

(28)

The symmetry conditions impose that, in this sum, the only

term different from zero, corresponds to L = 2 and A = 1 or A = - 1. When averaging over the angular orientations Rc of R c ' the crossed terms of the development disappear due to the orthogonality conditions

(29)

After averaging over the 21 + 1 values of m, we obtain

h ~~'2(Rc) = I~I (k) 1 ( <Pi Xv' IhM,-lkLA; <Pc Xv) 9P 12

(II L 1)2 Xf,: (2/'+ 1)(2L+ 1) 0 0 0 jT,(kRc)

(l' L

X~, m' A

The last sum is equal to 1I(2L + 1). Thus, finally, the ionic-covalent coupling elements will be calculated with the following expression:

r"v'(k) h ~r2(Rc) = e;17" I~I (k)C~ (Rc) (31)

with

(l' L 1)2

CURc) = L (21 ' + 1) 0 0 jT,(kRc)' (32) I' 0

The atomic termini (k) is obtained by means of wave functions <Pc computed from model potentials and the par­tial widths are given by Eq. (23), where r(k) is taken as a Wigner threshold lawl4 for O2:

r(k) = Ck 2L+ 1 = Ck 5 (33)

and a Blatt and Weisskopflaw26 for N2:

and

r(k) = C'kpVL (kp) with P = 2.66 a.u.

X4 v 2 (x) = --~-...,.

9+3x2+X4 (34)

This law is more realistic for N2 since the shape resonance corresponds to a rather high energy (~2 eV) as compared to O2,

In both cases, we can use the numerical values of the above constants C and C' which have been experimentally determined for process (1/).14.15 However, these values do not provide the experimental cross sections for the excited atom-molecule processes (2) and (3 ). This is not surprising since the lifetimes of the O2- and N2- are very sensitive to their environment and, thus, must depend upon the presence of the ionic core A *. This is not contradictory with the use of the QFE approximation to calculate the R dependence of the h ~r (Rc ) couplings, but one must keep the constants Cor C I

as the only adjustable parameters of the model.

3. Comparison between the two Ionic Intermediate models

In order to compare expressions (22) and (21), we have plotted, in Fig. 3, the logarithm of the reduced parameter h tJ R ~ as a function of R * defined by Eq. (22). This calcu­lation is performed for the 4D level of sodium. The straight line corresponds to the parameters A = 2.0 and B = 0.8 of

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Desfrancois et al.: Sodium atoms in molecules. II 3041

Na(4D) + M,

o

-5

-10

R' -15+-__ ~ __ ~ __ ~ __ ~ __ ~ ______ ~ __ ~~c 023 4 5 6 7 8

FIG. 3. Logarithm of the reduced ionic-covalent coupling element as func­tion of the reduced internuclear distance, for the Na( 4D)-02 and Na(4D)-N2 systems calculated with the quasifree electron (QFE) model. For comparison, a plot of the semiempirical Olson formula is given for the Na(3P)-02 system.

Eq. (22) determined l9 by means of the MCC model for the energy differential cross section of the process:

Na(3P) + 02el:g-) --Na(3S)

+02el:g-,IAg , or Il:g+) +AE (2")

studied by Hertel et al. 27

The curves corresponding to the QFE models for N2 and O2 are given for the absolute values of C and C' deter­mined from the experimental values discussed in the next section. Their comparison leads to three following com­ments:

(i) The R * dependence is strongly influenced by the form factorfn' (k), reflecting the nature of the atomic wave function fPc: both curves exhibit a minimum for R *::::::4 a.u. corresponding to a minimum of the atomic form factor and another one, equal to zero, for R = lIIP( 4D) which is the internuclear distance for which the classical momentum k goes to zero. As expected, the curves for the rubidium atom in the 7S level exhibit several similar minima. Nevertheless hie is, in all cases, a rapidly decreasing function of R.

(ii) For O2, one can observe that, for the here consid­ered 4D excited state and for the large values of R e , Eq. (22) overestimates the hie coupling. Thus, the total quenching cross sections calculated by means of the MCC model stead­ily increase when the initial excited state of the alkali atom becomes more and more excited. The Olson formula [Eq. (22) ] is thus satisfying for the first alkali resonant states, but the QFE model is more suitable for the study of higher excit­ed states.

(iii) In their respective crossing regions, the values of the hie couplings in the QFE model have the same order of magnitude for O2 and N2 and, thus, will lead to similar cross

TABLE I. Electronic to electronic energy transfer cross sections (in A 2 ) of excited sodium atoms in collision with molecular oxygen. The MCC model values are absolute. The QFE model values are normalized with respect to the Na( 4P- 3D) experimental value.

Electronic transfer Na(4P-3D) Na(4D-3D)

Experiment (Ref. 13;_ Ref. 12)

MCCmodei QFEmodel Relative collision energy (in me V)

41(7) 40 41 60

19(7) 100

14 60

Na(4D-5S)

4(3) 9(3) 11(3) 9 ""9.6 12.6 9 14 12

60 150 310

sections. Their R * dependence are identical, except for the small values of R * which correspond to large values of the momentum K for which Eq. (33) and (34) are different.

III. COMPARISON BETWEEN MODELS AND EXPERIMENT

We will first discuss the comparison between the predic­tions of the ionic intermediate models with the experimental determinations of the large electronic to electronic energy transfers (intermultiplet transfers) for collisions between excited sodium or rubidium atoms and molecules for which detailed free electron-molecule resonant scattering data are available: oxygen and nitrogen. We will further consider the weak Na( 4D--4F) angular momentum energy transfer within the frame of the impact approximation.

A. Intermultiplet energy transfers

1. Excited sodium atom-oxygen molecule

The experimental absolute values of the electronic to electronic cross sections, measured in crossed beam experi­ments l2

,13 are compared with the calculated values of the two preceding models in Table I. Although the three consid­ered processes correspond to different values of the energy transfer AE (1100,1350, and 5400 cm- I ), the agreement between models and experiments is quite satisfying, except for the value of the 4D--3D transfer in the MCC model.

2. Excited sodium and rubidium atoms nitrogen molecule

Our experimental results 13 and the values of the QFE model are compared in Table II, for a single collision energy of 66 meV. The agreement is also reasonable. In particular, the experiment confirms the weak transfer Na(4D--5S). Another test of the QFE model is given by the comparison

TABLE II. Electronic to electronic transfer cross sections (in A 2) of excit­ed sodium atoms in collision with molecular nitrogen. The QFE model val­ues are normalized with respect to the Na(4P-3D) experimental value. The relative collision energy is 66 meV.

Electronic transfer Na(4D-5S) Na(4D-3D) Na(4P_3D)

Experiment (Ref. 13) <4 28(8) 33(8) QFE model 0.5 29 33

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3042 Desfrancois et al.: Sodium atoms in molecules. II

80

70

60

50

40 .-

30 , ,r 20 JI 10 ! , 0 0.1 Q2 ~----~~------~------~~~----~O~A----~En~V~lo

FIG. 4. Comparison between the experimental cross sections of the Rb(7S .... 5D)-N2 and Rb(5D .... 7S)-N2 electronic to electronic energy transfers and the results of calculations from the quasifree electron model.

with the dependence of the cross sections of processes (3) as functions of the relative collision energy which have been measured by Paillard et al.28 (Fig. 4).

The exothermic process dependence is well verified by the QFE model, but the predicted values of the endothermic process must be divided by 2 in order to reproduce the ex­perimental measurements. The detailed balance principle, which is implicitly underlied in the QFE model is surprising­ly, not experimentally satisfied.

3. Excited rubidium atom-oxygen molecule

In Fig. 5 and Table III, the comparison is given for the Rb*-02 system. The detailed balance principle is here satis­fied and the QFE model predicts a correct collision energy dependence for both processes, while the MCC model gives a constant value for the exothermic process in the measure­mentrange.

B. Angular momentum transfer

In studies of highly excited atom-molecule collisions, the impulse approximation has been widely used and has proven to be quantitatively valid in several cases. 11.29 These collisions are considered as a three-body problem: the atom­ic or molecular perturber, the atomic optical electron, and

8

6

...... -----------~--

4

2

E/eV o 0.1 0.2 0.3 0.4

FIG. 5. Comparison between the experimental cross sections for the Na( 4D .... 5S)-02 electronic to electronic energy transfer (points with error bars, from Ref. 12) and the results of calculations from the multiple curve crossing model (solid line, from Ref. 12) and from the quasifree electron model (dotted line).

the positive ionic core. The validity conditions of the impulse approximation have been discussed into details by de Prune­le. 29 The core-perturber interaction can be neglected and, most generally, only the electron-perturber interaction is involved (first-order approximation). This is not wholly valid for the moderately excited states which are here con­sidered because the variation of the ionic core potential is too important. Then the valence electron-core interaction must be taken into account. The angular momentum transfer cross section can be expressed as a product of the elastic electron-perturber scattering cross section O'el and the squared form factor of the atomic transition Jl

:

21TL 2 ikf + kf

0'4D_4F = --2- F4D _ 4F (K) 'K'dK, V R kf-kf

(35)

where k; and kf are the initial and final momentum of the relative motion and L is the elastic scattering length related to O'el by

O'el = 41TL2 , (36)

where O'el must be taken at the mean energy of the electron in

TABLE III. Electronic to electronic transfer cross sections (in A 2 ) of excited rubidium atoms in collision with molecular oxygen.

Rb(7S .... 5D) Collision energymeV 96 132 150 175 200 310 Experiment' 56(25) 41(15) 39(12) .;;;32 MCCmodel 53 54 54 55 52 50 QFEmodel 52 46 44 42 39 30

Rb(5D .... 7S) Collision energymeV 100 165 206 223 245 360 Experiment' .;;;1 2.5(1) 4.2( 1.8) 4.9(2) 5.2( 1.8) 6.2(2.3) MCCmodel 1.6 5.5 7 7.5 8 8.7 QFE model 5.2 6 5.5 5.1 4.8 4.6

, Reference 28.

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Desfrancois et al.: Sodium atoms in molecules. II 3043

the 4D state. We calculate the form factor by means of the expression 11

F ,. (K) = _1_ " l(nlmleiK'r ln'I'm')1 2

nl-nl 21 + 1 ~,

or

(I I' L) Fn1_ nT (K) = (2/' + 1) L (2L + 1)

L 0 0 0

X [1"0 jL (Kr)Rnl (r)Rn'I' (r)r 2dr r (37)

where Inlm) and In'l 'm') are here the wave functions of the 4D and 4F states and Rnl and RnT their radial parts.

With the first-order approximation of the impulse ap­proximation, neglecting the atomic core effect, and with the values of the free electron elastic cross sections taken from the literature, we obtain the results shown in Table IV. This first calculation overestimates by a factor 45 the experimen­tal results and moreover the relative values do not fit them. It has been also observed in the resonant scattering process:

Rb(7S) +H2 (J= 1)-+Rb(5D) +H2 (J=3) (3')

that the predictions of the impulse approximation are much larger than the experimental values for moderately excited atoms.

As shown by de Prunele,29 the ratio of the core effect term to the first-order term of the impulse approximation is approximately equal to - L In, where n is the effective quantum number. For the low values of n here considered as well as in the above process (3'), a core effect correction must be employed. In order to obtain a tractable expression for this correction [which appears as a lengthly expression in its original formulation: Eq. (A 1 b) of Ref. 29], we introduce a simple cutoff function r(r) in the integral of the above expression of the form factor (37):

r(r) = 0 if r<AL and r(r) = 1 - AL Ir if r>AL.

The only adjustable constant is the parameter A and the com­parison between the experimental and the computed values of the angular momentum cross sections is shown in Table V for A equal to 10 which correspond to the best agreement, except for oxygen. The corresponding form factors [includ­ing the correction term r(r)] are displayed in Fig. 6.

A naive picture of this core effect correction of the im­pulse approximation is shown in Fig. 7. For transitions

TABLE IV. Angular momentum transfer Na(4D .... 4F) cross sections for different molecular perturbers. The experimental values of Ref. 13 are com­pared to the values (divided by a factor 45) calculated with the impulse approximation without atomic core effect correction.

Perturber H2 N2 O 2 N 20 SF6

Uel (A2) at 0.85 eV 10 9 5.5 6.5 26

Relative collision energy (meV) 57 66 59 63 64.5 if.~:'4F/45 (A2) 340 300 180 210 930

U::J'~4F (A2) 335(85) 96(19) 47(10) 89(34) 6(6)

TABLE V. Angular momentum transfer Na(4D .... 4F) cross sections for different molecular perturbers. The experimental values of Ref. 13 are com­pared to the values calculated with the atomic core effect correction of the impulse approximation.

Perturber H2 N2 O 2 N 20 SF6

U eJ (A2) atO.85eV 10 9 5.5 6.5 26 Relative collision energy (meV) 57 66 59 63 64.5 U;~J':.4F (A2) 310 94 96 87 1.5 U::J'~4F (A2) 335(85) 96(19) 47(10) 89(34) 6(6)

between atomic states (n;!i-+nif) where low values of the ratios I;lni or I;lnf are involved (because I~n or because 1= n - 2 with n small) the classical orbit have rather large excentricity. In the vicinity of the atomic ionic core, the va­lence electron-core interaction can no longer be neglected as in the first-order impulse approximation. The electron is only submitted to the perturber interaction (measured by L)

in the portion of its classical orbit which is far from the core.

IV. CONCLUSION

A quantitative comparison between two ionic interme­diate models has been performed after a discussion of their range of validity. It has been shown that although originally described with different formalisms, these models are in fact very close. Their main difference arises from the determina­tion of the ionic-covalent couplings hie responsible for the electron transfer. The multiple crossing model gives satisfy­ing results for low-lying excited states. For the more highly excited states, the couplings derived from the electron-mole­cule scattering experiments are more accurate in the QFE model. In particular, the collision energy dependence of the electronic transfer cross sections can only be reproduced by means of the QFE model. For angular momentum transfers of the moderately excited atoms, the impulse approximation can only be used if one takes into account the importance of

F.K

X10

H2 K

2 4 6

FIG. 6. Atomic form factorsF4D_4F (K) as a function of the momentum of the relative motion, including the atomic core effect correction, for several perturbers.

J. Chem. Phys., Vol. 88, No.5, 1 March 1988

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3044 Desfrancois et al.: Sodium atoms in molecules. II

FIG. 7. Schematic illustration of the atomic core effect correction. The va­lence electron is only sensitive to the perturber interaction outside a sphere of radius AL (L is the elastic scattering length).

the binding interaction between the optical electron and the atomic core.

ACKNOWLEDGMENTS

We thank Dr. Mestdagh and Dr. Berlande for providing us their experimental results before publication.

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