Theoretical problems in the interpretation of associative attachment experiments

Theoretical problems in the interpretation dissociative attachment experiments Florence

Fiquet- Feyard. Laboratoire de Collisions Electroniques (asso&

au CNRS).

of

Universitt! Paris-Sud. 91405

Orsay, France

The various problems arising in the quantitative application of resonance theory to dissociative attachment are reviewed, with particular attention to the problem of a unique definition for the resonant state, and to the assumptions involved in the use of Born-Oppenheimer approximation. Classical approximations are discussed. Recent results for O2 and HCI are presented.

1. Introduction Dissociative attachment has been recognized for a long time as a process involving a breakdown of the Born-Oppenheimer approximation.’ But,, although breakdown of the BornOppenheimer approximation in other processes, such as predissociation, has received considerable attention, dissociative attachment-which can be called a predetachment problemhas been totally neglected. In 1962 Schulz2 observed a resonant structure in the vibrational excitation of Nz and the interpretation was given by Herzenberg and MandL3 The concept of resonance turned out to be the key to dissociative attachment theory. In 1963 Bardsley et c~l.“-~ used the KapurPeierls definition of a resonance to formulate the problem of dissociative attachment. O’Malley’ developed a similar theory, using the formalism of projection operators. These authors defined a complex potential curve V” (R)-i r (R)/2 for the negative ion, v’ (R) being the potential curve of the neutral. Then the process is understood as an electronic transition from AB + e to AB-, which then dissociates. The cross-section is proportional to an electronic transition moment, dependent on 1‘ (H), ;nd to a Franck-Condon factor. The upper state nuclear wa?.e function which enters the Franck-Condon factor is calculated for the complex potential, and is damped because of the negative imaginary term -Z/2. A classical interpretation of r can be given: the extra electron can autodetach, with a unimolecular rate constant I’lfi and this process can compete efficiently with dissociation. Hence a survival probability can be defined for the molecular negative ion:

potential of 0,-(211.) and was able to fit ,the temperature dependance of the DA data. I3 Birtwistle and Herzenberg’” parametrized the complex potential of Nz- (‘II,) and fitted the VE data.15 But, in fact, as we shall see in the course of this paper, the assumptions involved in the complex local potential theory are such, that in many cases the theory cannot be quantitative. The formation of Cl- in HCl is an example we shall discuss later. Let us develop here the simple example of H2-(ZCU+). Numerous data have been obtained in the range O-10 eV and a few of them are summarized in Figure 1. We notice that

I

3

I,\ 4

-z

survival

probability

(

24/number

11 /12.

0%

OH-/H,

>

where F is a mean value of the width I’ (R) and rd is the necessary time to reach the crossing point of v’ (R) and V” (R). Vibrational excitation occurs because the detached electron can “fall back” into various vibrational channels of the neutral. Vibrational excitation (VE) and dissociative attachment (DA) are two complementary aspects of the same phenomenon. This theory allowed many qualitative explanations of experimental effects. In HZ, Bardsley et aZ6 found the right order of magnitude for the DA and VE cross-sections. Isotope effectss*9 were explzined by the dependance of 7d upon the reduced mass, and temperature effect in NzO by its dependance upon the bending angle.lO*ll Quantitative calculations were possible in a few cases: O’MalIeylZ parametrized the complex Vacuum/volume

5

a 03

F - xrd

= exp

,

Pergamon Press &d/Printed

1 =o, Exp. Date

6

Electron Energy

I

I

)

7

8

eV

10 I .12_20cm’ LI .4_46 cmZ

Schulz and Asundi 1967 i Ehrhardt et al.

19S!l

Figure 1. Experimental data for VE and DA in Hz for the low energy

resonance. o,, IS very small, and that the energy dependance varies remarkably from DA to VE. Now we use the local potential theory and adjust V” (R) and I? (R) in order to reproduce the DA data, including the isotope effect and the energy dependante. We find that this is possible for a family of curves, one in Great Britain

533

Florence Fiquet-fayard:

Theoretical problems in the interpretation of dissociative attachment experiments

2. Short-range and long-range interaction between an electror and a neutral molecule or atom 2.1 Screened coulomb field and multiplet

structure. Slater? rules*’ allow us to evaluate the effective Z seen by the extrz electron of a negative ion. In the case of C- (Is)* (2~)~ (2~)~ for example, 2 (effective) = 6 - 2(0.85) - 4(0.35) = 2.9. Clearly, the binding energy of the extra electron must increast with Zerf, but the third column of Table 1 is in contradictior with this simple prediction. This apparent disagreement is dua

4 > z5 3

Table 1. Effective Z for the extra electron in the first row negativt

ions Stabilization from multiplet structure (eV)

2

Ion

Zeff

F-

4.85 4.2 3.55 2.9 1.2

0NCNe-

1

%,

0 0 0.93 2.26 0

3.40 1.465 to 1.25 t0

structure of C- and N- : the experimenta’ electron affinities of the third column are given for the lowesr multiplet of the configuration, and not for the average energq of the configuration. The other multiplets of C- and N- art not known (and in fact they are unstable), but we can calculate weighted average energies for the isoelectronic atoms N and 0 They are 2.26 and 0.93 eV higher than the lowest multipleh respectively. If we now subtract these corrections from the electron affinities, we find the order predicted in the firsi column. This screened coulomb field falls off exponentially with r! and it is a typical short range potential, It creates a potentia! well which may or may not be able to trap an electron. The situation in molecules is similar: the screened coulomb field is larger in 02- (T# than in NO- (n9)’ or Nz- (r,), as confirmed by the electron affinities, 0.43,” 0.0223 and -l.914 respectively.

to the multiplet 5

’

R(A)

1.5

Figure 2. Potential curves of H,. The Hz- curve is hypothetical,

of them being shown in Figure 2. But for any of these curves the calculated cross section for VE shows a rapid decrease as a function of energy, similar to the decrease in aDA,and totally different from the experimental behaviour shown in Figure 1. Yet the magnitude of the calculated ool at 3.9 eV is very close to the experimental value, and thus we cannot suggest that the resonant contribution is buried in a large direct contribution. The failure of the local potential approximation is explained as follows. The calculated uye are shown on the left side of Figure 2, for the energy E. We see that cro_lo-though it is much lower than aol---is still larger than uDA=0.12 x lo- 2o cm*; the channel Hz v’ = 10 is thus in effective competition with the dissociative attachment channel, and so will be the channels v’ = 11 and 12 when the energy E is increased. Then the rate constant of autodetachment, r/h, will increase with the total energy, but this effect cannot be taken into account in a local potential r(R). An important advance in the direction of a more general theory is due to Bardsley, I* but numerical applications of this theory are still scarce, and its range of validity unclear. Difficulties arise from the fact that the requirements of resonance theory and of Born-Oppenheimer approximation have not yet been connected to each other, and we do not know if they are compatible. Our study will be limited to the lowest state ofnegatiue ions. In Section 2 we analyse schematically the interaction between the electron and the neutral molecule, and use this analysis to classify the negative ions (Section 3). We then present the resonance theory in the formalism of Lippmann and O’Malleylg (Section 4) and the concept of diabatic states introduced by SmithZo (Section 5). In Section 6 we discuss Bardsley’s equations for DA and VE, and in Section 7 we use them to discuss the validity of the local potential and classical approximations. 534

2.2 Polarization potential and centrifugal barrier. Table 2 enables us to find which partial waves will be coupled to the negative ion state under study, knowing that the irreducibk representations of the s, p and d-waves are S, P” and D res. pectively. For example, in 02- (T,)~, which belongs to the Dmh group, the incoming electron must find a place in a 4 Table 2. Resolution of the irreducible representations of the group 01 the atom into those of groups of lower symmetry C 3” C Z” 0, Dmll S”

A lg A

P P” (P) D 6-I)

Az + E T:: n”, + &AI-I-E n. + IL+ Tl" Eg + Tz, A,+ n,+ Ccl' Al+2E

D”

E. +

S (s)

Al A2

gc

T2u

A.

+

n,

+

z.-

A2

Al AL

+

2E

Az + & + B: Al + BI i-B: 2Ai+A2+ BL +& AI

+ BI

Example

SF6

02

NH,

24 +

f &

H@

orbital, and we see that the s and p waves are uncoupled but the d wave is coupled. In S02-(8a#(3b1) the group is Czv and wt see that the p and d waves are coupled to the negative ion. Ir

F/orence Fiquet-Fayard: Theoretical problems in the interpretation of dissociative attachment experiments

NO~-(4b~)‘(6a1)* the s, p and d waves are coupled to a1 but the next electronic state has a configuration (4b#(6a1)(2b1) and is coupled to the p and d waves only. Table 2 has been limited to the three lowest partial waves because higher partial waves give a negligible contribution in the case of electron impact, as opposed to heavy particle collisions: if E = 1 eV the classical distance of closest approach is 1.95 I A and for large I the incoming electron does not penetrate the dense part of the electron cloud. For the same reason the lowest allowed partial wave is usually the most important. If the angular momentum is I, the radial Schrodinger equation will contain a repulsive centrifugal term.

3.

Classification of the lowest electronic state of negative ions

Figure 4 shows the three types of negative ions. OH is an example of type 1: in the fixed nuclei approximation the negative ion is stable at the equilibrium distance R.’ of the neutral. O2 is an example of type 2: the negative ion is unstable in R,‘, but it is trapped below the high centrifugal barrier of I= 2, and thus it is a narrow shape resonance. HCl is an example of type 3 : the negative ion is unstable, and there is no centrifugal barrier to trap the electron, since s-wave coupling is allowed between HCl IX+ and HCI- %+. Another example of type 3 is HZ- YZU+(Figure 2), in which the negative ion state is located above the centrifugal barrier, in the fixed nuclei approximation (upper part of Figure 3).

h2 1(1+ 1) (1)

2m,r2 On the other hand the potential 1

due to polarization

is:

ue2 (2)

-=2r4

where C(is the polarizability of the neutral molecule. When I f 0, these two potentials combine to form a barrier, known as the centrifugal barrier. Typical potentials are shown in Figure 3; the data used for the upper and lower curves correspond to Hz and OZ respectively.

. I’s’

10-a'

.$DA in Ha

5zi \

-:;. 0

‘.

-I k

I i

I 1 1

2

3

4

0

_

t

‘L

1 %I

E k

a..858cm3

4

1.4

‘&g

c.1

z

1.2

)

5& >

:-

3

‘I+,,

-f~~;\_.+~~~

I 1.2

/’

__--I R(S)

==!I c 1.4

r (A)

J-2 a.1.7cm’

lOma’

A

I

1

I

1.2

I

I

I

I

1.4 R (61) 1.6

e

Figure 4. Potential curves of OH, O2 and HCl (full line), and their negative ions (dashed line). HCl- potential curve from ref 56.

Figure 3. Typical centrifugal barriers for I = 1 (upper curve) and I = 2 (lower curve). The square well, in broken line, symbolizes the short range attractive well. Let us now consider a molecule, as 02, in which this short range attractive well weakens when the internuclear distance R decreases: the negative ion level, which is bound for large R, moves upward, becomes a narrow resonance, then a broad resonance, and eventually disappears above the barrier.

3.1 Type 1: stable state. The potential curve of OH- in Figure 4 has been drawn from experimental data of Branscomb,24 which have been confirmed by more recent photodetachment experiments.25*26 Other examples of type 1 negative ions are SOz-(‘B1) and NO,-(‘AI). The potential curves of Figure 5 correspond to the simultaneous stretching of both bonds. They have been drawn with the following data: EA(S02)

= 1.123 eV26

EA(N02)

= 2.38 eV2’.

The equilibrium distances, angles, and vibrational frequencies are solid state data for NO;z8-33 and SO-.3* Herzberg’s data35 were used for the neutral molecules. The 535

Florence Fiquet-Feyerd: Theoretical problems in the interpretation of dissociative attachment experiments I

I

I

I

SYMMETRIC

!in LETCH

I

l-

I

I

?I,1090

I MODE

1

IllI w h

SRNCHE &SEHUlZ

N02 4 ELECTRON

ENERGY,

(eV)

Figure 6. Resonances in NO1. (From ref 37.)

Figure 5. Potential curves of SO2 and NO* (full line) and their negative ions (dashed line) for simultaneous stretching of both bonds. angles shown for the negative ions are indicative only. The dotted line shows the potential curve of SOz-(‘Bl), when the angle is the same as in SO#A1), calculated from the bending frequency of SOz- . Autodetachment is energetically allowed for the upper vibrational levels of the negative ions, but it is probably very slow, as we can infer from the following experimental data. (a) Doverspike et al36 observed OD- with internal energy as large as 4 eV, approximately 2.2 eV above the threshold for autodetachment. In their experiment it took approximately lo-* for the OD- to reach the detector after formation. (b) Low lying resonances were not seen in SOZ, in the transmission experiment of Sanche and Schu1z3’ (c) NOz-(‘AI) is probably not seen in the same transmission experiment. The experimental curve of Sanche and Schulz is shown in Figure 6. The large peaks are likely to be due to symmetric stretch structure, and the small peaks are an indication of a small change of angle, as in 2A1 (134”) + ‘B1(131”). The bending mode would be more excited if the transition was *A, (134”) 4 ‘A, (116”). Then the resonant structure observed by Sanche and Schulze at low energy is probably due to the 3B1 state interacting with p-wave electrons. Nevertheless, we must point out that three-body attachment of thermal electrons does occur in N02js and S02.39 The mechanism of this process is unknown.38,39 Type 1 negative ions always exist when the molecule possesses a large dipole moment, as shown by the work of Crawford4’ and many other authors. If the nuclei were fixed, the critical dipole moment would be 1.625 Debye and then HZ0 (EL= 1.84 Debye) would be able to bind an electron, We must point out that the dipole moment of a molecule goes to zero when 536

R -+ co or R-+ 0. Then, these negative ions can possibly disappear into the continum when R is increased; in this case they would never dissociate into a stable negative ion. Nevertheless they may be stabilized by other forces than the dipole field, namely short range forces. Garrett4’ and Bottcher4* have shown that the critical dipole moment is increased to 2-2.2 Debye when rotation of the molecule is taken into account. The binding energy of the added electron in the fixed nuclei approximation, calculated by Turner et a143 is shown in Table 3 for various dipole moments. Alkali halides have large dipole moments, 7.4, 10.5 and 10.2 Debye for CsF, and CsCl CsI respectively, and table 3 predicts fairly large electron affinities for these molecules. Type 1 negative ions have not been studied theoretically. Table 3. Binding energies for an electron in an electric-dipole field. (rl> is the average value of the distance of the electron from the positive dipole charge (from ref 43) Binding Dipole energy moment ;z; (eV) (Debye) 1.625 1.83 2.54 4.17 5.08 7.63

0l (-7) 2.7 (-2) 1.2 2.4 5.2

:8 (+3) 8.4 1.5 1.2 0.9

3.2 Type 2: below-a-barrier resonance. 02-, N2-, CO-, NO-, N02-(3B1) etc are, examples of this type, which will be studied in further detail later in this paper. 3.3 Type 3: above-a-barrier resonance. For many negative ions --HZ%, SFs-, HCI-, HBr-, C12- etc coupling with s-wave electrons is allowed. Nevertheless large cross-sections have been measured for DA and VE in HaS, SFs, HCl and HBr. On the contrary, H,O- *A1 has never been observed either in DA experiments

or in scattering

experiments.

Florence Fiquet-Feyerd: Theoretical

Various assumptions

problems

in the interpretation of dissociative attachment experiments

can be thought of:

li’r

(a) The matrix element of s-wave coupling happens to be small, for reason unknown, and p-wave coupling is dominant.

w, E = 6eV (resonance )

(b) The negative ion does exist as long as V” (R) < v’(R) but disappears at the crossing point R,. When R, happens to be in the Franck-Condon region, the negative ion can be formed by electron impact. (c) The negative ion is trapped below a short range barrier, and is in fact a type 2 negative ion. In SF6 for example the barrier could be due to the electronegative fluorine atoms.44 (d) Quantum reflection, on the steep wall of the short range potential well, builds up a broad resonance. These resonances are described for square potential wells in various textbooks, such as Messiah’s. Type 3 resonances are not understood theoretically but they are of major importance since they seem to occur for a very large number of electronegative molecules.

.f m I

23

E 5

/

E=GeV

Xr

/’ :

: \ \ \\ ‘1

7r

7P

.

1

4. Theoretical description of shape resonances

H=

-$ +V(r).

The exact solution 4 of the SchrGdinger equation (H-E)#=O

(3)

is easily calculated for this one-dimensional case; 4 is shown in Figure 7 for two different energies. We normalize # to a delta function of energy and thus its asymptotic form is, in the chosen units:

3

\ ’

r(ll)

potential

4.1. The scattering wave function. We consider a single electron

moving in the potential shown in the lowest part of Figure 3; this potential V(r) is a square well at small internuclear distances, and at large distances it is the sum of the repulsive centrifugal term corresponding to I = 2 and the attractive polarization term. The square well is chosen in such a way that the resonance will occur around 6 eV and will be rather broad. If we choose R,, and a0 as units, the radial Hamiltonian is:

2

well

Figure 7. Total wave function # for the potential of Figure 3b.

x,: Arbitrary resonant state; dr: potential scattering wave function; +?: resonant part of 4. given by Fano44*45 but is more general and better adapted to the problem of shape resonances. Both treatments use standing wave functions. We choose a bounded wave function x,, which is thought of as being an approximation for the small r part of $. We shall discuss in Section 4.3 the influence of the choice of x,. We define the projection operators: p=

IXXXJ,

e = 1 -p.

We now calculate the eigenfunctions Q. They satisfy:

Q(H - GQ4,

4,, of H in the subspace

(5)

= 0

and Ic/ -4-

I-a m

zl,l~l,4

sin(kr - In/2 + S)

(4)

where k = E1lz is the momentum of the electron. In the vicinity of the resonance the phase shift 6 and the amplitude of the wave function inside the well vary rapidly, as visible in Figure 7. This property makes the wave function unsuitable for Born-Oppenheimer approximation; we shall see in Section 5 that Born-Oppenheimer approximation is valid when the electronic wave function varies slowly with the internuclear distance R; but, when R varies, the location of the resonance varies and thus 4, calculated at constant energy E, varies rapidly when the resonance energy is close to E. The purpose of resonance theory is to express $ as a linear combination of a potential scattering wave function +, and a resonant wave function d,, both of them varying slowly as a function of E. 4.2 Decomposition of the wave function into a potential scattering part aad a resonant part. We shall use the procedure of Lippmann

and O’Malleylg

which is similar to the treatment

(6)

= 0. Equation (5) means that (H-E) (H - EM,

$,, is in subspace P:

= -CL

where c is a constant, to be determined later. Since 4 is solution of equation (3), the general regular solution of this equation is: &, = c’l// + cGlx,> where G is the Green’s function of H and the constant c’ wilr be determined by the normalization condition. Equation (6) can now be written: c’ + c<~rlGl~,)

= 0.

Then :

In order to normalize 4, we must calculate its asymptotic form. By using either the eigen function expansion of the 537

Florence Figuet-Feyerd;

Theoretical problems in the interpretation of dissociative attachment experiments

Green’s function or its closed form, the asymptotic Glx,> is found to be

form of

V,(E)= <&MIX,> E, =

GM

----f - *d r-m

($1x,> cos@

>

- 1x12 + 6).

(8)



(16)

A(E) = (~rlffG,Hl~,) we obtain :

We combine (8) with (4) and we find that 9, is normalized to a delta function of energy if: c’ = cos 6,

(9)

!

with : tans,=

V,(E)

(

* = cos 6, (bp+

_~i(iixr>l2

then the asymptotic form of 4, is:

k;z

(17)

&i

(11)

sin(kr - In/2 + 6 - 6,).

If our choice of xI was good, a,, given by equation (lo), is such that 8 - 6, does not vary much in the vicinity of the resonance energy, though 6 does vary; 4, is the potential scattering wave function. Let us now proceed to the calculation of the resonant part of $. By definition, the component of 4 in subspace P is . Then: II/ = C&)x,

+

(12)

Qlcl.

If we substitute (12) in the Schriidinger multiply on the left by Q we obtain:

equation

(3) and

E-&,-A(E)

(18)

4%

4.3 Influence of the choice of the resonant state x,. It is a puzzling characteristic of resonance theory that the choice xI is somewhat arbitrary. In this paragraph we shall consider a family of functions xr; they are eigen functions for square wells of different depth and same width, and we shall identify them by their energies Ed= . For each X, we can compute (x,1 JI) and
(19)

In Figure 8, Es,V, (Es) and (dAldE)E, are shown for this family of xr. The most striking feature of this figure is that Es is rather insensitive to the choice of x,. A first order change in xI does not make a first order change in Es but only a second order change; the curvature of the curve Es suggests that Es is stationary.

QW - WQ$ = -(x,l$>Q%.

c

The genera1 regular solution of this equation is:

Qti = ““4p + <~,l$)G,~x, in which Go is the Green’s function of QHQ. Then, $ can be written : ‘4 = ~“4~ +

Cd,W,

+ G,ffxJ.

Using the properties of Green’s functions we find the asymptotic form of GoHx,:

G&X, - --) ---$p 1-m Since the asymptotic conditions :

<4,IHlx,> Wkr

- W2 + 6 - &>.

form of 4 must be (4), we find two t

c” = cos 6, and : sin 6, = -

~<~~llCl><4~lHlx~>.

(13)

5.61

The function 4 is now expressed as the linear combination of the potential scattering function +,, and the resonant part +,:

-

$ = cos s&P, + 6 = xr + G&%r.


(14) (I 5)

If we multiply (14) on the left by
/

Er(eV) -different

X,

c

Figure 8. Influence of the choice of the resonant state x,. In order to prove theoretically this result we should prove that it is possible to find a particular function j,, associated to a particular value & with the following property: when

Florence F&et-Fayard:

Theoretical problems in the interpretation of dissociative attachment experiments

x”?is changed into % + 6x, the change in ES is second order. We know from equations (19) and (10) that & is the zero of the function g(E) = <%lGlX).

(20)

If ES is stationary, then g(E) is stationary in ES. Let us change j$ into 1, + axI, 6x, being in subspace 0 ((jr/ 8x,> = 0). We keep ES constant and thus the Green’s function is constant. Sg = 2<&,IGl%).

(21)

We see that the function G;?, must be orthogonal to any function of subspace 0, in order to ensure 6g = 0, and orthogonal to 2, in order that g(E,) = 0. This is impossible. The apparent stability of ES in Figure 8 is due to the choice of 6~~; our 6xr are limited to the small r range and 6g, given by equation (21), is small because Gxl. is small in this range: the function Gxl can be seen in Figure 7 since it is proportional to & at resonance energy (equation 7). We choose a reasonable xr empirically. The wave function labelled by rr = 9.5 eV in Figure 8 is satisfactory because the residual phase shift, 6 - 6,, is small within a large range of energy O-8 eV. This function was used for Figure 7 and will be used throughout the next paragraph. We must point out that this wave function is by no means an approximation to the unique resonant function 2, defined by Lippmann and O’Malley.” These authors require that A(&, jr) = 0, but for our wave function A (ES) = 3.5 eV. 4.4 Study of the coupling matrix element as a function of the electron energy and the internuclear distance. In this paragraph we keep xI constant and we vary E. Since the potential V(r) corresponds to I = 2, the Wigner threshold law predicts lhE5/2

The middle curve of Figure 9 shows ve4j5 for a different potential V(r), in which the square well has been modified. The well is now deeper and the resonance energy is 0.6 eV. This modified potential V(r) corresponds to a larger internuclear distance of the diatomic molecule. We see in Figure 9 that the threshold law is valid in the same range as before but the curve is about 30% lower. The third curve corresponds to a larger internuclear distance, the potential well is still deeper, the resonance disappears and is replaced by a bound state. If we look at the potential curves of 02, in Figure 4, we notice that the resonance energy, v”(O,-) - v’(O,), varies from -0.6 to 0.6 eV only, in the Franck-Condon region. Thus the variation of V, with R will be negligible; a variation of 6% is evaluated by interpolation of the data of Figure 9. This is a very convenient property of the coupling matrix element: it is independant of the internuclear distancein first approximation.

Another simplification arises from the fact that the Wigner threshold law is valid within a large energy range. This is illustrated in Figure 10 where the cross-section for vibrational excitation is shown as a function of the energy E of departing electrons, the other parameters being roughly kept constant; this cross-section is expected to be proportional to r (see Section 6.2) and thus to E’+ l/2 in which I = 1 for CO and 1 = 2 for Nz. BONESS

8 SCHULZ

r v--

IO t

OPRESENT

(23”)

PRESENT

(90”)

l

t

xt;t

EHRHAROT a et 01 (1968)

or VeccE514.

The ;pper curve of Figure 9 shows Ve415 for the potential V(r) we have studied in the preceding paragraphs. This curve is labelled 6 because the resonant state is located around 6 eV. We see that the deviation to the threshold law is very small from 0 to 1.1 eV.

8 SLOPE

.3/z

ENERGY

6

4-v

SLO’PE -5/Z

OF DEPARTING

ELECTRONS

c (eV)

Figure 10. Experimental energy dependence for VE cross-sections. (From M J Boness and G J Schulz.) 5. Born-Oppenheimer

approximation

A fundamental problem of dissociative attachment theory, and it is yet unsolved, is to decide whether we must use the diabatic representation or the adiabatic representation. We define them in the next paragraph for the case of two bound electronic states. 2

Figure 9. Energy

E (eV)

dependence of the coupling matrix element for three values of the potential well.

5.1 Smith’s deEnition of diabatic states. The concept of diabatic

states has been rigorously introduced by Smith” and a complete treatment can be found in his paper. In this section we

Florence Fiquet-Fayard: Theoretical problems in the interpretation of dissociative attachment experiments

shall study the simple case of (1) two electronic states 4i and 42, which are assumed to form an orthonormal and complete set (2) a constant orientation of the internuclear axis of a diatomic molecule. Then the nuclear kinetic energy operator is : TN=

M aR

=-

in which the unit of energy is R,, the unit of length is a0 and M is the reduced mass of the nuclei. The total Shrodinger equation is, for the total energy Et: (T, + V(R,r) + TN - E,)$(R,r) = 0

= &(Kr)pi(R)

+ +2(RPr)p2(R).

If we apply TN to a function

T&p = +T,p + h 2

(24)

4 (R,r) p (R) we obtain:

+h’p

(25)

where

&?A?g m. 824

h’s

___.

M

dR2

We shall need the matrix elements of h, h’ and U = T, + V. We take the functions 41 and 42 as real and we use the bracket notation for integration over r alone. We remark first that u1, =

(27)

u2,

u,2

~2@)

+h12;R+;-&h12.

+

(304

>

U22

+$-

h122 0

(

-

P,(R)

4 1

(23)

where T, is the electronic kinetic energy, V is the entire interaction potential including internuclear repulsion and r represents the coordinates of all electrons. Since 4r and 4z form a complete set, the total wave function can be written:

h

>

(

4

for p1

u,, + $$,,2- Et P,(R)

(22)

---2

$(Kr)

AN+

0

a2

m,

relations (27)-(29). We obtain two coupled equations and p2:

because U is hermitian and real, and:

=-

K2-h12$R-$;Rh12.

>

P,(R)

If h12 and U12 were zero, we could choose pZ = 0 and pi solution of the homogeneous equation (3Oa), or p1 = 0 and p2 solution of (30b). Then the total wave function would be 41 p1 or 42 pz, and the Born-Oppenheimer approximation would be perfect. If the second members of 30 are not zero, we can try to minimize them by a suitable change of the basis set 41 and 4Z ; we build new functions, which are linear combinations of 41 and 42; this transformation involves four parameters but 3 of them are determined by the two normalization conditions and the orthogonality condition. The fourth parameter can be adjusted by the condition iJt2 = 0 or by the condition h12 = 0, but both cannot be satisfied. In the former case we obtain the adiabatic electronic wave functions 4i” and 424, and the diagonal matrix elements U,,” (R) and UZ2* (R); they are the familiar non-crossing potential curves shown in Figure 1la. The coupling is due to h12 and F. Smith shows that the term of interest is the product of the coupling and the velocity of the nucleus; this product is shown qualitatively in Figure lla. In the other case we adjust h12 = 0 and we obtain the diabatic wave functions 4id and 42d; the potential curves are now Ulld and UzId, shown in Figure llb together with the coupling matrix element U, 2d.

hll =h,,=O hi2 =

-h21.

(28)

The proof involves only the orthogonality and normalization of 41 and 42. If we differentiate, for example, <41 I42) = 0 we obtain h12 + hzl = 0. Furthermore we can express the matrix elements of h’ in term of those of h and their derivatives. Using the completeness of C1 and CZ we can write

-.zrno 5j&

h’ 12

h’

1 d =TjdR

22 ==o

Mh

42

=

h2242

h 12

+

h24,

=

t&24,

(29)

2 l2 .

Let us now substitute (24) in the Schrodinger equation (23), use (25), multiply on the left by < 4rI or < 421 and use the 540

(b)

Figure 11. Adiabatic (a) and diabatic (b) potential curves. (From ref 20.)

Florence Fiquet-FayardzTheoretical

problems in the interpretation of dissociative attachment experiments

We remark that if l_Jlzd is very small in the diabatic representation, then the potential curves Ulla and Uzza of the adiabatic representation will approach each other very closely at the pseudo-crossing point, and they will have a large curvature to avoid each other; then the derivative 6 &” I6R will be very large, hi2 will be large, and the adiabatic representation will be unusable. Similarly, in the negative ion case, we expect that the diabatic approach will be adapted to the case of weak coupling (narrow resonances) and the adiabatic approach will be adapted to the case of broad resonances. But the only theoretical work using the adiabatic representation so far, is the work of Crawford,46 and no definite conclusions can be drawn yet. 5.2 Equations for the negative ion case. 5.2.1 Generalization of the two states equations. The complete basis set #1 and 42 is now replaced by a set x1 and &, where x1 is a bound electronic wave function representing the negative ion and & is a continuous set of real wave functions representing the free electron of energy E interacting with the target. Each of these states is nondegenerate, all degeneracy having been removed by specification of 1 and m, or any adequate set of quantum numbers. For all possible basis sets bE the target wave function is the same: this is reasonable if Born-Oppenheimer approximation is valid for the neutral molecule. Then :

5.2.2 Application to type 2 resonances (below-a-harrier). In order for hEE, to be small the wave function & must be weakly dependent on R. From the results of Section 4 we see that a reasonable choice is x1 = xr and & = &,. Then, hopefully, the matrices h and h’ will be small, U,,, will be small, and the coupling will occur through UIE only; UIE is the coupling matrix element V,(E,R) we have studied in Section 4.4 for a simple model of one-electron one-partial wave shape resonance. Equation (32) can now be written as: (AN + V”(R) - E&,(R) (AN + JV)

= -~dE’V,(E’,R)p,.(R)

+ E - E&,(R)

(W

= - ~e@,R)p,(~).

(33b)

5.2.3 Application to type 3 resonances (above-a-harrier or no harrier). It is not clear which set x1 and & we must take in order to minimize h,,, and U,,,. The implicit assumption of Crawford46 is that the adiabatic states are adapted to this problem. The coupled equations for p1 and pE will be similar to (33), but V”(R) will be modified by the diagonal correction Wll(R) and V.(E’,R) will be replaced by the matrix elements of h and h’. An important feature of Crawford’s calculations is that Wll(R) is infinite at the crossing point (Figure 12); then the nuclear wave function of the negative ion, pi(R), is entirely located on the right of the crossing-point. 3

where V’(R) is the potential curve of the neutral molecule. Uil will be called Y”(R), potential curve for the negative ion, and we notice that it is not the same for different basis sets. It is easily shown that the diagonal matrix element of h is still zero, and that the matrix of h’ is diagonal when h is diagonal. The total wave function is now, instead of (24): $ = x~J)P#)

+ !dE’&@Q9&@

where JdE’ stands for C,,,jdE’ and the functions are solutions of the coupled equations (A, + WO = -jdE’

(31) p1 and

pE

+ &I’ - E&,(R) UIE, + hl\E, $

+ h’,+,.(R)

(324

(4 + v’(R) + E + h’m - Eh(R) I I.0

= - JdE’

U,,. + hEE, $

I

I I.2

,

INTERNUCLEAR

d - UlE + h,, dR + h’lE p,(R). Wb) > These equations will be simplified if it is possible to find a set x1, & such that UEE, and hEE, be negligible as compared to hlE. We saw in Section 5.1 that, in the general case, it is not possible to satisfy simultaneously iJ,,t = 0 and h,,, = 0, but we may hope that these conditions are approximately satisfied for some particular cases. Then the coupling is entirely due to x1 and we shall call this case the pure resonance case.

I I.4

I

I I.6

+ h’,+,.(R) DISTANCE

I

td,

Figure 12. Adiabatic potential curves and diagonal correction for a negative ion. (From ref 46.) 6. Dissociative attachment pure resonance case

and vibrational

excitation

for the

6.1 Bardsley’s equation. We should now solve equations

and the and But

(33a) (33b). If we took real wave functions, we would obtain standing wave solutions # of the Schrijdinger equation their asymptotic forms would give us the reactance matrix. in fact we shall now use complex wave functions, because 541

Flcrence Fiquet-Fayard: Theoretical problems in the interpretation of dissociative attachment experiments they were used by BardsIey18 whose results we want to present. The functions & are now the &+ solutions of the LippmannSchwinger equation for the potential scattering problem. They are specified by E and /; and not by E, I, m, as were the real functions in the preceding section. The function p,(R) is a one dimensional outgoing wave, since we assumed a constant direction for the internuclear axis. BardsIey18 obtains the final equation :

(AN + v(R)

- Et)p,(R) = - Ve(kiJJp,‘(R)

- C”,P”,P’“,(R)(P’“,lP1)

IK(kE’)12 E, - E,, - E’ + ie

(&-+ 0)

(35)

In these formula it has been assumed that V, is independent of R. This assumption, which was checked in Section 4.4, allows considerable simplifications in the equations. Equation (35) shows that /3”, is complex, and is a function of E,-E,, alone; Et-E,, is the energy of a scattered electron when the target undergoes the transition 0 + v’. Then we can write :

j?,, = -i

KS

(40) where u” is the vibrational level of the negative to E, and I”” is the partial width for the transition

ion closest v”-u’ (41)

(34)

in which E, is the total energy, p’,,(R) are the vibrational wave functions of the neutral target, Et and kI are the energy and direction of the incident electron, and the initial vibrational state of the target is taken as v’ = 0. The coefficients /3,, are

/I,, = SdE’dii’

coupled negative ion. It can easily be shown that there is only one term in expansion (39), when 1V,i’ is much smaller than the vibrational spacing of the negative ion, and that the crosssection is:

- E,,> 2 + A(E, - Et,,>.

We have found a Breit-Wigner formula for the cross-section when V, + 0. We remark that the width I‘f includes a Franck-Condon factor, which is always smaller than I. Therefore, shape resonances will be narrower in molecules than in atoms and they will be easier to observe. It is interesting to see how the Lorentzian shape of u in equation (40) is modified when V, is increased, the other parameters being kept constant. In Figure 13, V, is twice as large for curve 2 than for curve 1, and thus the width is 4 times as large; ool has been calculated with equation (34), assuming A (E) = 0. The calculation shows interferences between adjacent resonance peaks, as long as there are only two open exit channels: these channels are v’ = 0 and v’ = I in Figure 13. This effect was predicted by Mies and Krauss4’ in a different context. These interferences disappear when many exit channels are open.

(36) A

v’; 1 P//

In conclusion, all information concerning dissociative attachment and vibrational excitation is contained in the functions V”(R), r(E) and A(E), as long as the process is purely resonant, and V, is independent of R. If we calculate the flux in the various outgoing channels we find the cross sections for dissociative attachment and vibrational excitation :

V”

v”+l 1 AB-

I

(37)

6.2 Narrow resonances. Breit-Wigner formulae. Let us study the limiting case V, + 0 in equation (34). The first term in the right hand side is proportional to V,, and the second term is proportional to 1VJ2 and pl. If p1 remains finite when V, + 0, the second term will be negligible. If the total energy Et is larger than the dissociation limit of the negative ion, dissociation is possible and it provides an exit channel independent of V,. Then pi(R) remains finite and the second term is negligible. If the total energy Et is such that dissociation is impossible, the amplitude of the wave function pi(R) is limited by the second term only, and this term then cannot be neglected. We expand p1 in the form p,(R)

= c “Uc”..p;,,(R)

(39)

where the functions p”“” are the eigen functions of T,v + V”-E: they are the vibrational wave functions of the un542

I .5 electron

i\ \

‘. .6

.

energy (eV)

Figure 13. Interference between two adjacent vibrational negative ion. (From H Abgrall, unpublished results.)

states of a

6.3 Application to 02- 211s. Photodetachment studies 22have shown that the electron affinity of O2 is 0.440 i 0.008 eV and that the equilibrium inter-nuclear distance of 02- is R,” = 1.341 f 0.01 A. On the other hand resonant vibrational excitation of O2 has been studied by Linder and Schmidt.48 Their results are shown in Figure 14. The vertical lines are the successive levels of 02-, and they correspond to we” = 0.135 eV and wenx,” = 0.001 eV. The resonances are narrower than the electron beam distribution and thus the amplitudes of the

Florence Figuet-Fayard; Theoretical

problems

in the interpretation

of dissociative

attachment f 0

normalization LINDER

A

and

experiments

SCHMIDT

r; = 1.340

Q,

1 I

Linder and Schmidt ,

theory

ILL

Xl 3

1

_

o--1

6

9

7

10 I,

v(o;)

11 13 14

x2

I I I i i ii ii i iii I

,

I

I

I

.5

I

.,

I

excitation

D-2

iaLL 8 9 10 1, ,a 13 14 ,I

vyo; )

,I,.,

t

I

I

/

1.

collision energy

Figure 14. Vibrational

H

I

I I 1.5

I

I

I

(eV) c

xl0

of 0,. (From ref 48.)

0 m-3 0 F

:

peaks

in Figure 14 are proportional to the energy-integrated cross-section. When formula (40) is integrated, (42) is obtained: cr;;‘:,(integrated)

2nZ I-“” ru” = g2 0 ki C”J-::

10

PARLANT

,a

11

13

14

IS

v-(0;)

17

16

and

FIQUET-FAYARO

(42)

where g is a statistical factor. The electron energy in Figure 14 is low enough for the Wigner law to be valid (see Section 4.4), and then:

‘2iv 13

14

,I

17

16

”

18

(ol) -

Figure 15. Comparison between experiment (vertical bars) and theory

(open circles) for VE in Oz. T(E) = yE5”. The four known parameters we”, o~“x,“, R,” and AE (0,) determine the potential curve of 02-. Koike and Watanabe49 have previously calculated the order of magnitude of the cross-sections, using a Landau-Zener formula for the FranckCondon factors. In the present work we use the Numerov method and Ridley’s matching procedure to compute numerically p’ and p”. Adjusting R,” within the error limits given by Celotta et a1,22 we found the best results for R,” = 1.340 A (Figure 15). The open circles are theoretical values and the vertical bars experimental values. We point out that the experimental data of Figure 14 provide us with at least 14 accurate relative amplitudes, and that there is only one adjustable parameter in the calculation: R,“. Thus the excellent agreement of Figure 15 proves that for narrow resonances theory is fairly quantitative, and conversely it proves the reliability of the experimental set up of Linder and Schmidt. Figure 16 shows how the cross sections are sensitive to R,“: the change from the open circles to the triangles is due to a 0.003 A change in R,“, and it is larger than the deviation from experiment in Figure 15. If we include in the calculation a R dependence, interpolated from the data of Figure 9, the open circles are changed into the squares, and the effect is seen to be insignificant.

0

02

“:

f

d : Xl

n Y

s

D r’ = 1.340

1

A

ZJ.343

lI

.

with

o-

R dependance

1

i 6

7

I

P

10

1,

3

A

:

,a

a

(

13

14

s 3

x2 “,

PARLANT

O--a

“a

I

B P

V”(Oi)

10

I,

12

a

13

14

15

13

14

IS

V” (0;)

and

FIQUET-FAYARO

6.4 The problem of NO-. As the next step we would like to study NO- because the resonances are broader, yet still well defined. Experimental results have been obtained recently by Tronc and ReinhardPO, and the elastic cross section is shown as an example in Figure 17 for two scattering angles. Since the

16

17

II

vyo;)

Figure 16. Influence of various parameters on the theoretical results for VE in 0,.

543

Florence

Fiquet-Fayard:

Theoretical

problems

in the interpretation

of dissociative

attachment

experiments

in order to be transferred into the left hand side of equation (34). We expand (46) in the form:

fc“,C”9”JW

NO

with c,, = SdRp’,,(R)r,(R)p,(R). Since p’“, and p1 are oscillating functions of R and rl is not, the integral builds up essentially in the point RUTfor which the two functions have the same period:

Y”

\

i

c,, = I-,(~“~)
(48)

Substituting (48) into (47) we obtain a term which is identical to (45) under the condition:

r,(R,.)= I-(E,- E,,). TADNC

&

p1 corresponds to a classical motion with total energy E, in the potential V” and pfv, to a motion with total energy E, in the potential V’. In R,, the kinetic energies are equal:

AElNHARllT

Figure 17. Elastic scattering in NO. (From ref 50.)

Et -

electron affinity of NO is 0.024 eV, the vibrational level u” = 1 of NO-(3C-) is visible in the elastic channel. The series U” = 1,2,3,4 is approximately regular, but the fifth peak shows an aberrant angular dependance, and is almost split in two at 90”. The authors suggest that NO-(3X:-) v” = 5 and NO(‘A) a” = 0 overlap and interfere. Burrows1 observed the same anomalous intensity for v” = 5 in a back scattering experiment. The dissymmetry of the a” = 3 and a” = 4 peaks at 90” is particularly interesting because one does not expect to find NO-(‘IA) at such a low energy (respectively 0.44 and 0.60 eV). The dissymmetry may be due to interferences between adjacent vibrational peaks of the same electronic state, as we saw in Figure 13. 7. Local potential approximation and classical approximation 7.1 The local potential approximation as a classical limit. Bardsleyl’ has shown that, if I’, (E,R) is independent of energy, then the local potential approximation is valid with a local I’#) given by: T,(R) = 2nJdkl Ve(R)j2

(9

This property shows that, in these particular circumstances, T,(R) does exist and has the same R dependence as II’&?)1’, but it does not preclude the existence of I’,(R) in more general cases, with another R dependence. We shall now prove that the local potential approximation is valid when a classical approximation can be used for the wave functions P’~. and p1 ; this approximation is similar to the one used by Landau and Lifshitz5* to derive the LandauZener formula for predissociation. We neglect the shift A.(E). The last term of equation (34) is: + f C”J-(J% - E”,>P’,,(R)(P’“,lPl>.

(45)

This term must be shown to be equivalent to: (46) 544

(49)

V”(&)

= E,, - I”(&)

and (49) will be satisfied if r, is the implicit function of R

r,(R)= r[v(R) - v’(R)]. The calculation brings about the important correspondance between I’(E) in Bardsley’s equation and l?,(R) in the local potential theory: they are the same mathematical function of the ejected electron energy, but this energy is Et-E,. in the former case and V”(R)-V’(R) in the latter case. V”(R)-V’(R) is the energy of the ejected electron in the fixed nuclei approximation. When r is large, the function p1 may be very small in R,. because of the large damping, and equation (48) may be not valid. But for the case of small I? we can safely state that the local potential approximation is a classical limit of the general case, when V, is independant of R. Figure 18 illustrates the procedure for the case r-+0; then pi(R) is proportional to p,-(R); R,. is shown by arrows, and we see it is located on the left or the right of the crossing point for open or closed channels respectively.

I

classical transition

open channel

*I

closed channel

Figure 18. Classical transitions for the same resonant level v” and two different levels of u’ of the neutral.

7.2 Classical approximation. Fiquet-Fayard et a/,54 have shown that good results are obtained in a pure classical calculation, if a proper representation is taken for the target initial wave function pa’(R). In classical mechanics the target has definite position R and momentum p = Mu at a given instant, and Wigner53

Florence Fiquet-Fayard:

Theoretical problems in the interpretation of dissociative attachment experiments

gave a general formula for the probability P(R,p)dRdp of observing them. This formula is given in Figure 19 for the particular case of the O-level of a harmonic oscillator. The Franck-Condon principle, in Fran&s formulation, states that R and p are both unchanged during an electronic transition: in the final state the potential energy is V”(R), the kinetic energy is p2/2M and the total energy is the sum. For half of the transitions p points to the right, and dissociation will occur rapidly; for the other half, p points to the left and the particle will be reflected at the turning point. If l? was zero these two beams of particles would interfere, but, when I? is large enough, the second beam is damped because: (1) I? is large at small R, and the rate constant for electron ejection is I’/& (2) its path is longer, (3) its velocity is zero at the turning point. h r, V

rm=.la~

laA ; ‘1

.

---

In II h ,

rm,.3d

\

\

!L p,,

r,.l.l \\ ‘\

2.

\

‘1

I

d

0

>

I’

E

6

rm=1.1eV

\ A. ,.

:

4’

E

-

quantum

--___

classical

E

@

Figure 20. Comparison between quantum calculations (for a local potential) and classical calculations. 1. Type 3 resonances: HCI 1.5

.5

0 P(R,~)dRdp.~exp(-~-~)dRdp a

Figure 19. Pure classical model for DA. P(R,p) is the Wigner function for the target. Figure 20 compares cross sections for a quantum calculation with a local potential, and for a classical calculation. The left part is for the attractive curve of Figure 19 and some interference effect is still visible when the average width is 0.1 eV. The right part is for the repulsive curve of Figure 19. In the course of the calculation it is easy to find in which range R the transition does occur. Because of the damping of the wave function, the transition does not occur at the turning point-or rather, when it occurs there the negative ion does not survive-but slightly on the right of R’,, equilibrium distance of AB. Then, if we want to calculate some average survival probability, we must not compute it for a motion starting at the turning point with zero velocity, if the curve V” is attractive and I? is large. The good results obtained for diatomics with a pure classical approximation, show that classical trajectories computations should be able to describe DA in triatomics. Calculations are in progress for COZ.

8.1 Adlalmtlc approach. Crawford46 uses this approach, but comparison with experiments5 is difficult because his computational procedure is oversimplified. Firstly, he assumes that the extra electron is bounded in a square well; the potential V(r) changes suddenly from Vo(R) to zero in r = ro; the wave function #(R,r) of the electron and its derivative a#/& are continuous in ro, but the derivative a#laR is not. Then the computation of the matrix elements of h and h’ (see Section 5) is rather uncertain. Nevertheless I feel this is not a major drawback. Of more importance is the assumption made by Crawford that in first approximation, instead of (31),

A thorough examination of the equations, and the experimental cross-sections, shows that equation (31) should be used without any further simplification; Crawford’s approximation would be valid in the case of much smaller cross sections. In hydrogen halides, the system of coupled equations should be solved by the Green’s functions method, as Bardsley did for the diabatic approach. Nevertheless Crawford’s results show that his approach is worth more extensive calculations. 8.2 Dial&c approach. Fiquet-Fayards6 assumes that a diabatic resonant state can be defined. Equation (34) is used with V, = constant or zero for open or closed channels respectively. The shift A(E) is neglected. The purpose of the calculation is to show that all experimental results can be explained simultaneously: the absolute cross-section, the isotope effect,57 and the shape of the ionization curve.55 Agreement is good.56 A comparison between the experimental and calculated ionization curves is shown in Figure 21. Similar calculations5* in HBr give excellent agreement with experiment. Let us remark that the structure in Figure 21 is due to the opening of new exit vibrational channels, HCl a’ = 3,4,5 545

Florence

Fiquet-Fayard:

Theoretical

problems

in the interpretation

of dissociative

Ziesel

and

experiments

and not upon R, and a shift A(E); then, a local potential is obtained as a classical limit, with a local width rl (R) depending upon R only.

f HCI ____talc. . . . . exp. unfolded

attachment

Schulz

I References

’ H S W Massey, Negative Ions. Cambridge University Press, London (1950). z G J Schulz, Phys Rev, 125, 1962, 229. 3 A Herzenberg and F Mandl, Proc Roy Sot, A270, 1962, 48. 5 J N Bardsley, A Herzenberg and F Mandl, in Atomic Collision Processes, London Conference 1963 (edited by M R C McDowell) (1964). 5 J N Bardsley, A Herzenberg and F Mandl, Proc Phys Sot, 89, 1966, 305. 6 J N Bardsley, A Herzenberg and F Mandl, Proc Phys Sot, 89, 1966, 321. ’ T F O’Malley, Phys Rev, 150, 1966, 14. 8 T F O’Mallev. J them Phvs. 47. 1967. 5457. Electron

Figure 21. Ionization curve in HCl. Comparison and resonant state theory. (From ref 56.)

energy (eV)

between experiment

successively; in the case of H2 discussed in Section 1, we already invoked the opening of new exit channels to explain the sharp decrease of the cross-section with energy. But, as we discussed in Section 3.3, the physical nature of the hypothetical resonant state is not known.

9. Conclusion In this paper we intended mainly to compare the familiar local complex potential theory with the more general theory given by Bardsley in 1968. Many problems were left aside and a more extensive survey can be found in Bardsley and Mandl’s review,59 Chen’s review,6o and Taylor’s review?l considerable progress has been made recently in the theory of angular distribution of the ejected electrons.62-66 Our first conclusion is that the connection of the BornOppenheimer approximation with resonance theory is not yet fully understood. As we pointed out in Section 4, the choice of the resonant state is somewhat arbitrary, and we can state that the best resonant state must allow at most a simple treatment for the breakdown of Born-Oppenheimer approximation. We saw in section 5 that considerable simplification occurs when he,, and U,,, can be made small as compared to h,, and lJIE; for some processes it will be impossible to fulfil this condition, whatever resonant state is chosen, and then both resonant and direct contributions will have to be taken into account. But, in a few cases, it will be possible to describe the process as purely resonant, with a proper choice of the resonant state. For a narrow shape resonance, as 02- ‘&, this resonant state is probably a diabatic state, but more work is necessary to give a precise definition of the best resonant state, and this definition should be valid for broad resonances as well. Our second conclusion is that the local complex potential approximation is a classical limit of Bardsley’s equation. In this equation it is reasonable to assume that all information concerning the resonant state is contained in a potential curve V”(R), a width I’(E), depending upon the electron energy E 546

9 J C Y Chen and J L Peach&, Phys Ret?, 163, 1967, 103. lo P J Chantry, J them Phys, 51, 1969, 3369. ‘I J N Bardslev. J them Phvs. 51, 1969. 3384. I2 T F O’Malley, Phys Rev; lk5, ‘1967, ‘59. I3 W R Henderson, W L Fite and R T Brackmann, Phys Rev, 183, 1969, 157. Is D T Birtwistle and A Herzenberg, J Phys B, 4, 1971, 53. I* H Ehrhardt and K Willman, 2 Phys, 204, 1967,462. I6 G J Schulz and R K Asundi, Phys Rev, 158, 1967, 25.

I7 H Ehrhardt L Langhaus, F Linder and H S Taylor, Phys Reo, 173, 1968, 222.’ I8 J N Bardsley, J Phys B, 1, 1968, 349. I9 B A Lippmann and T F O’Malley, Phys Rev, A2, 1970, 2115. *OF T Smith, Phys Rev, 179, 1969, 111. *’ J C Slater, Phys Rev, 36, 1930, 57. ** R H Celotta R A Bennett, J L Hall, M W Siegel and J Levine, Phys Rev, A6, ;972, 631. *3 M W Siegel, R J Celotta, J L Hall, J Levine and R A Bennett, Phys Rev, A6, 1972, 607. I4 L M Branscomb, Phys Rev, 148, 1966, 11. 25H Hotop, T A Patterson and W C Lineberger, J them Phys, 60,

1974, 1806. 26 R J Celotta, R A Bennett and J L Hall, J them Phys, 60, 1974, 1740. 27 D B Dunkin, F C Fehsenfeld and E E Ferguson, Chem Phys Left, 15, 1972, 257. 28J W Sidman. J Am Chem Sot. 76. 1957. 2675. 29 J W Sidman; J Am Chem Sot; 79; 1957; 2669. 3o R Kato and J Rolfe, .I them Phys, 47, 1967, 1901. 3’ R M Hochstrasser and A P Marchetti, J them Phys, 50, 1969, 1727. 32 D E Milligan, M E Jacox and W A Guillory, J them Phys, 52, 1970, 3864. 33 D E Milligan and M E Jacox, J them Phys, 55, 1971, 3404. 34 D E Milligan and M E Jacox, J them Phys, 55, 1971, 1003. 35 G Herzberg, Electronic Spectra of Polyatomics Molecules.

Van Nostrand, New York (1966). 36 L D Doverspike, R L Champion and S K Lam, J them Phys, 58. 1973. 1248. 37’L Sanche and G J Schulz, J them Phys, 58, 1973, 479. 38 B H Mahan and I C Walker. J them Phvs. 47. 1967. 3780. 39 L Bouby F Fiquet-Fayard and C Bodeie,‘Int’Mass’Spectrom ion Phys, 7, 19j1, 415. do 0 H Crawford, Proc Phys Sot, 91, 1967, 279; and subsequent papers. J’ W R Garrett, Chem Phys Lett, 5, 1970, 393. ” C Bottcher, MO/ Phys, 19, 1970, 193. -13J E Turner, V E Anderson and K Fox, Phys Rev, 174, 1968, 81. 44 J L Dehmer, J them Phys, 56, 1972, 4496; B Cadioli, U Pincelli, E Tosatti, U Fano and J L Dehmer, Chem Phys Lett, 17, 1972, 15. 45 U Fano, Nuoao Cimento, 12, 1935, 156; Phys Rev, 124, 1961, 1866. 46 0 H Crawford, J them Phys, 60, 1974, 4512. 47 F H Mies and M Krauss, J them Phys, 45, 1966, 4455. 48 F Linder and H Schmidt, Z Naturforsch, 26a, 1971, 1617. a9 F Koike and T Watanabe, J Phys Sot Japan, 34, 1973, 1022; F Koike, J Phys Sot Japan, 35, 1973, 1166.

Florence

Fiquet-Fayard:

Theoretical problems in the interpretation of dissociative attachment experiments

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