5
25. D. R. Rosseisky, Chem. Rev., 65, 467 (1965). 26. H. F. Hall• and S. C. Nyburg, Trans. Faraday Soc., 59, 1126 (1963). 27. N. A. Izmailov, Dokl. Akad. Nauk SSSR, 149, 1364 (1963). 28. O. A. Osipov and V. I. Mink• Handbook of Dipole Moments [in Russian], Vysshaya Shkola, Moscow (1965). 29. G. H. F. Diercksen, W. von Niessen, and W. P. Kraemer, Theor. Chim. Acta, 31, 205 (1973). 30. S. L. Chong, R. A. Myers, and J. L. Franklin, J. Chem. Phys., 56, 2427 (1972). 31. G. H. F. Diercksen and W. P. Kraemer, Chem. Phys. Lett., ~, 419 (1970). 32. M. A. Haney and J. L. Franklin, J. Chem. Phys., 50, 2028 (1969). 33. C. Salez and A. Veillard, Theor. Chim. Acta, ii, 441 (1968). 34. L. Pauling, General Chemistry, W. H. Freeman (1970). USE OF ELECTRON SCATTERING DATA IN CALCULATING THE EXCHANGE INTERACTION OF EXCITED ATOMS WITH MOLECULES G. K. Ivanov UDC 539.186 The interaction of an excited atom A* (or negative ion A-) with a molecule is determined at large distances by the two most characteristic ranges of motion of a weakly bound elec- tron. The electron spends most of its time close to the particle binding it, where the long- range potential U l of the A* -- M (or A- -- M) system is formed under the action of the weak molecular field. On the other hand, the electron can penetrate close to the molecule and in- teract strongly with it. This motion is responsible for the exchange potential of the sys- tem Uexch = -- 2~B (E, n) I (R)1~, (i) where r is the unperturbed electron Wave function, ~ is its radius-vector measured from its atom, R is the center of mass coordinate of the molecule (having in mind the state of zero projection of electron angular momentum on the R axis), B(E, n) is a quantity measuring the electron scattering amplitude at a negative energy E = -- (a2/2) < 0, and n is a unit vec- tor characterizing the molecular orientation (e = ~ = m =i). Theproportionality of Uexch to the square of the wave function I~(R) I2 was first estab- lished [I] by a method, equivalent to modeling the force field of the perturbing particle by a potential of vanishing range, assuming the existence of a simple analytic relation between the quantity B(E) and the s-scattering amplitude A s of a free electron by this particle B(E) = As_(--E). In an electron--molecule interaction, however, there exist always long-range effects characterized by a potential V(r) decreasing according to a power law at large dis- tances D p /rn\ Q /rn\ 2r4 where D and Q are the dipole and quadrupole moments of the molecule, 8o = (1/3) (~n ~ 2~• 2 ~] =-3(~;I--~-); ~g~• are the principal values of its polarizability tensor, r is the elec- tron coordinate measured from the molecule, and Pl(x) are Legendre polynomials. Such be- havior of the potential at infinity strongly affects the scattering properties of the mole- Institute of Chemical Physics, Academy of Sciences of the USSR, Moscow. Translated from Teoreticheskaya i Eksperimental'naya Khimiya, Vol. 12, No. 2, pp. 163-168, March-April, 1976. Original article submitted April ii, 1975. This material is protected by copyright registered in the name of Plenum Publishing Corporation, 227 West 1 7th Street, New York, N. Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $7.50. 122

Use of electron scattering data in calculating the exchange interaction of excited atoms with molecules

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25. D. R. Rosseisky, Chem. Rev., 65, 467 (1965). 26. H. F. Hall• and S. C. Nyburg, Trans. Faraday Soc., 59, 1126 (1963). 27. N. A. Izmailov, Dokl. Akad. Nauk SSSR, 149, 1364 (1963). 28. O. A. Osipov and V. I. Mink• Handbook of Dipole Moments [in Russian], Vysshaya Shkola,

Moscow (1965). 29. G. H. F. Diercksen, W. von Niessen, and W. P. Kraemer, Theor. Chim. Acta, 31, 205

(1973). 30. S. L. Chong, R. A. Myers, and J. L. Franklin, J. Chem. Phys., 56, 2427 (1972). 31. G. H. F. Diercksen and W. P. Kraemer, Chem. Phys. Lett., ~, 419 (1970). 32. M. A. Haney and J. L. Franklin, J. Chem. Phys., 50, 2028 (1969). 33. C. Salez and A. Veillard, Theor. Chim. Acta, ii, 441 (1968). 34. L. Pauling, General Chemistry, W. H. Freeman (1970).

USE OF ELECTRON SCATTERING DATA IN CALCULATING THE EXCHANGE

INTERACTION OF EXCITED ATOMS WITH MOLECULES

G. K. Ivanov UDC 539.186

The interaction of an excited atom A* (or negative ion A-) with a molecule is determined at large distances by the two most characteristic ranges of motion of a weakly bound elec- tron. The electron spends most of its time close to the particle binding it, where the long- range potential U l of the A* -- M (or A- -- M) system is formed under the action of the weak molecular field. On the other hand, the electron can penetrate close to the molecule and in- teract strongly with it. This motion is responsible for the exchange potential of the sys- tem

Uexch = -- 2~B (E, n) I �9 (R)1~, (i)

where r is the unperturbed electron Wave function, ~ is its radius-vector measured from its atom, R is the center of mass coordinate of the molecule (having in mind the state of zero projection of electron angular momentum on the R axis), B(E, n) is a quantity measuring the electron scattering amplitude at a negative energy E = -- (a2/2) < 0, and n is a unit vec- tor characterizing the molecular orientation (e = ~ = m =i).

Theproportionality of Uexch to the square of the wave function I~(R) I 2 was first estab- lished [I] by a method, equivalent to modeling the force field of the perturbing particle by a potential of vanishing range, assuming the existence of a simple analytic relation between the quantity B(E) and the s-scattering amplitude A s of a free electron by this particle B(E) = As_(--E). In an electron--molecule interaction, however, there exist always long-range effects characterized by a potential V(r) decreasing according to a power law at large dis- tances

D p /rn\ Q /rn\ 2r4

where D and Q are the dipole and quadrupole moments of the molecule, 8o = (1/3) (~n ~ 2~•

2 ~] =-3(~;I--~-); ~g~• are the principal values of its polarizability tensor, r is the elec-

tron coordinate measured from the molecule, and Pl(x) are Legendre polynomials. Such be- havior of the potential at infinity strongly affects the scattering properties of the mole-

Institute of Chemical Physics, Academy of Sciences of the USSR, Moscow. Translated from Teoreticheskaya i Eksperimental'naya Khimiya, Vol. 12, No. 2, pp. 163-168, March-April, 1976. Original article submitted April ii, 1975.

This material is protected by copyright registered in the name o f Plenum Publishing Corporation, 227 West 1 7th Street, New York, N. Y. 10011. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, wi thout written permission o f the publisher. A copy o f this article is available f rom the publisher for $7.50.

122

cule, destroying at the same time the simple procedure noted above of continuing the scatter- ing amplitude into the negative energy region (due to the appearance of divergences). A simi- lar feature of long-range potentials was pointed out in [2] in considering the scattering properties of an atom (D = Q= 81 = 0) in the negative energy region.

To overcome this difficulty the exchange interaction is treated in the present paper as a component of the total interaction of the particle, containing a weakly bound electron (for the sake of simplicity a negative ion is considered in what follows). The asymptotically ex- act equation for the energy level shift of a weakly bound electron [3]

S �9 (3)

is valid when the level displacement is much smaller than its distance to the boundary of the continuous spectrum. In Eq. (3) t is an integral operator satisfying the equation

t = v + VGt, (4)

where G = 2n ] :---- r ' I e-air-el is the Green function of a free electron at E= --(a2/2) < 0.

The result of the calculation is subsequently compared to the expression for the scattering amplitude A(k', k) (k and k' are the initial and final electron moments):

I ~ e._ik,r teikrdr ' A (k' , k) = - - 2--~- 0

(5)

while the operator t in (5) is defined for positive energies

E = -~- = 2 > 0 G = 2 h i r - - r ' I e~ ' r - rq "

The f u n c t i o n r a p p e a r i n g i n (3) i n c r e a s e s e x p o n e n t i a l l y w i t h r = P - - R ; i t s i n c r e a s e , how- e v e r , i s n o t unbounded and i t s smooth maximum i s s h i f t e d by p ~ l / ~ . Thus , i n s t e a d o f t h e d i v e r g e n c e s wh ich would o c c u r by d i r e c t u s e o f Eq. (5) i n t h e n e g a t i v e e n e r g y r e g i o n , Eq. (3) g i v e s f i n i t e e x p r e s s i o n s , c o r r e s p o n d i n g to t h e c o n t r i b u t i o n o f t h e l o n g - r a n g e p o t e n t i a l o f the system.

We specify the following procedure of the calculation by restricting it to a homonuclear diatomic molecule X2 (D = 0). We represent the electron --molecule interaction as a superposi- tion of two fields, a short-range Vo (Vo = 0 at r > ro) and a long-range V~ [coinciding at r>ro with (2), while for r<ro VI =0], so that the electron wavelength ~= 12EI-I/2 exceeds the radius ro, and the potential VI satisfies the requirement Vx << i/r~. These conditions together with

IEl'~0<<l, 1s ]EI~I<<I (6)

allow one to treat the short-range part of the potential in the longwave approximation, and the long-range part by perturbation theory.

It is easily shown that by first- and second-order perturbation theory in VI the opera- tor t transforms to the form

t = t o q- (1 q- Gto) (V 1 + V1GV1) (1 +Gto), (7)

where to =Vo+VoGto, while (l+Gto)~ is the electron wave function, distorted by the short- range potential Vo. For the scattering amplitude by this part of the potential one can use the well-known longwave expansion [4] (valid also for a nonspherical scatterer [5])

A0 (k', k) = - - L o q- ~L~ q- b0 (Q', ~) ~ q- . . . , (8)

where Lo is a parameter, bo(~', ~) is some angular function, and ~, ~' are the angles char- acterizing the directions of the vectors k and k' relative to the molecular axis. The ex- pansion (8) is, naturally, also valid for negative energies (in this case ik is replaced by --a) �9

123

The use of the scattering amplitude (8) in the calculations together with the potential Vo allows not to specify the form of the electron --molecule potential at distances of the order of atomic distances, and include in the treatment effects of electron exchange. This is the basic difference between the approach used here and the usual methods of the theory of electron scattering bymolecules, based on simulating by a model potential at r~---i (see, for example, [6] and references therein).

f-- The calculation by Eqs. (3) and (7) with the functions ~(p) = V~-~e-aP/p, correspond-

ing to a weakly bound state of the negative ion A- [,the action of the operator to on ~(~) re- duces to multiplying 2~[Lo + aL~ +bo(~R, ~R)a2]bY r verifies the possibility noted above of dividing the total interaction of the A,--X2 system into a long-range part

Uz ---- -- --~ Pz 2R' ~ Pz (9)

and an exchange potential of type (i), while in this case

B ( E . n , = - - L - - ~ L 2 - - 3 L Q P = ( - ~ - ) + 4 L~ocZ2Incz+g(QR)a*4- . . . (io)

The quantity L in (i0) is the following combination of the parameters Lo and ro earlier in- troduced:

(li)

and g(~R) is an unknown function, depending on the molecular orientation with respect to the R axis. The contribution of V: to Uexch consists of separating the features at r§ 0 (lead- ing to renormalization of the scattering length L = Lo -- Bo/ro) and of taking into account the interference between V, and Vo.

The exchange interaction in an excited atom--molecule system is also of such a form. To treat this case it is necessary to improve in (3) the potential in the neighborhood of atom A (but not close to the molecule), using the method described in [7], and to include in the

COU lomb f i e l d e f f e c t ~ -- ~ --2 I.~.~-=l, The l o n g - r a n g e po t en t i a l U ~. i s now quantity ~ the

by R = smaller than for the A---X2 system (9), so that there exists a range of distances and angular orientations of the molecule, in which all expansion terms in (I0) are important in comparing U~ and Uexch.

We compare the result obtained (I0) with the corresponding expansion of the scattering amplitude of an electron by a molecule, which could be obtained by the same method on the basis of Eq. (5):

2 QP2 (cos Qq) - - ~ A (k', k) = - - L - - --~ --4- ~oq - - " -~ ~lqP~ (cos Oq)

ik - - kQZ~ (fg, ~2) + ikL z -~ ...ff LQ [-P2 (cos Ok) + P2 (cos Ok, )] - - 4 L~ok ~ In k - - b (fl', Q) k z + . . . . ( 1 2 )

where

[P2 (cos %)12 ] (~, ~) = 2 [p~ (~o~ %,_~) ~(co~ %-0 ~ p~ dp ' ~ l ~ - p ~ + m

§ k' (for ~ + 0), e v is the angle between the vector ~ and the molecular axis, and q = k -- . The quantity L appearing in (12) is the same constant as in expression (I0) for the exchange po- tential. Equation (12) coincides with expansion (3.21) of [8], found by a different method accurately up to terms linear in k. The use of our method allows one to find the following

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smaller expansion terms, as well as to establish a relation between the angular function b(~', ~) and the function g(~) in Eq. (I0) for the exchange potential

g (~) = Re b (~, ~), (13)

and w h i l e f o r a s p h e r i c a l s c a t t e r e r Re b ( ~ ' , ~) = Yo + Y t ( k k ' / k 2 ) , where 7o, Y1 a r e c o n s t a n t s , for a molecule the similar representation of the function Re b (~', ~) contains three unknown constants.

Comparing Eqs. (i0) and (12) solves the problem of the relation between the exchange in- teraction in the excited atom --molecule system and electron scattering data. Due to the ab- sence of a direct contribution of the quadrupole moment the exchange potential is generally more spherically symmetric than the angular distribution in electron scattering by a molecule.

According to Eqs. (i0) and (12), the problem of determining the coefficient B(E, n) in Eq. (i) for the exchange potential reduces to finding the quantity L. At the same time it is possible to use experimental results on total cross sections of electron scattering in the low-energy region using the equation

16 nQa, (14) Oo----4uL2+-~-ff .

or more accurate spectroscopic data on the energy level shift of strongly excited alkali-met- al atoms in the presence of an impurity molecular gas, interpreting them by the Fermi equa- tion [9]

A = 2 ~ L N . (15)

where N is the gas density.*

Such spectroscopic data, giving the value of L as well as its sign, are for hydrogen and nitrogen [i0]: LH2= +1.05, LN2 = +0.4. At the same time, using experimental values of to- tal cross sections OH== 7.5 .~I0 -I~ cm = [ii] and ON2 = 1"4"10-16 cm2 [12], we find from Eq. (14) (QH2 = 0.49, QN2 = --i'i) LH== 1.4, LN2 = 0.46. The total cross sections reported in [ii, 12] for small, but finite, energies give for L > 0 somewhat higher L values. This fact, the increase (for L > 0) or decrease (for L < 0) in cross section with increasing electron energy, can also be used to determine the sign of the scattering length L.

The problem of electron scattering (free or weakly bound) by a molecule possessing a small dipole moment D ~ i (in atomic units) is similarly solved. In this case perturbation theory can also be applied, including long-range effects by the prescription formulated above.t

In the range of negative electron energies, however, terms linear in a do not survive if the polarizability of the molecule is sufficiently high. To establish the structure of the coefficients for terms proportional to ~ a more rigorous condition is required, since an indefinite term ~uDr~ appears in the calculations, which can be small compared to other terms, proportional to ~, only when D ~ 8~I. A similar uncertainty also arises in the imaginary part of the scattering amplitude at E > 0. The linear part in k of ReA remains, however, the same as in (12). Therefore, for D ~ 1 the total scattering cross section is established ac- curately up to terms linear in k, inclusive.

The previous expansion terms of the amplitude A(k', k) and the quantity B(E, n) are now of the form

�9 D 2 2 A (k', k) = 2//)q Pl (cos 0q) q- T ~F ( f s ~ ) - - L - - - ~ QP2 (cos 0q) - - L D [P, (cos Ok) q- P, (cos 0_k.)l.

2k S P, (cos 0k._. ) P, (cos 0p_ k) ~ F ( f l ' , Q ) = - ~ i k , __ p l (k2 __ p2 + i.l) l k __ p l dp: B (E) = - - L .

(16)

*The Fermi equation contains an average over positions and orientations of many gas mole- cules; therefore, due to (12), the molecular quadrupole moment does not explicitly affect the shift of the optic level. #Rigorous numerical calculations performed in [13] for scattering by a two-center dipole show the high accuracy of the Born approximation for values up to D < 0.5.

125

According to (16), to determine L it is convenlent to use the transport cross section at D2/k << i, kSo << 1

8~ D 2 16 ~Q~. (17) %r = T k "--T + 4aL2 +

For example, treating the experimental curves Otr(E) for the dipolar molecules CO (D = 0.038; Q=1.55) and N20 (D=0.063; Q=3.1)by this equation, [14] gives ILcoI = 0.9 and ILN2OI = 2.8.

The requirement that the electron wavelength exceed the dimension of the molecule, re- flected in the inequality IEI8o ~ I, is the basic restriction on the range of electron ener- gies considered. For molecules such as H=, N2, 02, C08o--~ i0 (~H2 =5"5; 8N== 11.8; BO~ = 10.6; 8CO = 13.1 [7]) this condition starts being satisfied for energies of the order of sev- eral tenths of an electron-volt, that is, practically for IEi < 1 eV. Concerning electrons in this range, the negative ions of many elements coincide, as well as highly excited atomic states.

Highly excited atoms, despite their comparatively short lifetime, play an important role in several ~ physicochemical effects, and are presently investigated experimentally [15]. They are effectively formed in a plasma and participate in a number of plasma-chemical pro- cesses, such as strong vibrational excitation of molecules (in the lower ionosphere a similar buildup mechanism of vihrationally excited molecules gives rise to ionization and an enhanced concentration of free electrons [16]). In calculating such processes, andalso in investi- gating the possibility of formation of long-lived A*M complexes (leading to a more profound reorganization of particles) knowledge of the asymptotic behavior of terms of the excited atom--molecule system is of fundamental value.

LITERATURE CITED

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1__3, 363 (1970). 4. L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Pergamon Press (1965). 5. Yu. N. Demkov and V. S. Rudakov, Zh. Eksp. Teor. Fiz., 5-9, 2035 (1970). 6. K. Takayanagi and S. Geltman, Phys. Rev., A138, 1003 (1965). 7. B.M. Smirnov, Asymptotic Methods in the Theory of Atomic Collisions [in Russian],

Atomizdat, Moscow (1973). 8. T. F. O'Malley, Phys. Rev., A134, 1188 (1964). 9 E. Fermi, Nuovo Cimento, I~I, 157 (1934).

i0 T.Z. Ny and C. Y. Chen, Phys. Rev., 5__4, 1045 (1938). ii J.L. Pack and A. V. Phelps, Phys. Rev., 121, 798 (1961). 12 A.G. Engelhardt, A. V. Phelps, and C. G. Risk, Phys. Rev., 135, 1566 (1964). 13 K. Takayanagi and Y. Itikawa, $. Chem. Soc. Japan, 2__4, 160 (1968). 14 J.L. Pack, R. E. Voshall, and A. V. Phelps, Phys. Rev., 127, 2084 (1962). 15. S. E. Kupriyanov, Zh. Eksp. Teor. Fiz., 5_! , i011 (1966). 16. A. D. Danilov and M. N. Blasov, Photochemistry of Ionized and Excited Particles in the

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