15473
J. Phys. Chem. 1995,99, 15473-15478
Static Characteristics of Penning Ionization in the He(23S)-D2 Collision System? Jan Vojtik” J. Heyrovsb Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, 182 23 Prague 8, Czech Republic
Richard Kohl Department of Physical and Macromolecular Chemistry, Charles University, I28 40 Prague 2, Czech Republic Received: March 7, 1995; In Final Form: June 27, 1995@
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Vibrational population factors for the nascent Penning ions Dz+(v’)(. *He)and energy of the Penning electrons are calculated for the ionization event He(23S)-D~(v”=0) [He***D2+(v’)] e- taking place at a range of the He*-DZ separations and orientations accessible by the system during low-energy collisions. The vibrational populations are obtained from the local widths of the He(23S)-D~(v”=0) state with respect to autoionization to Dz+(“*He) in its uth vibrational level. The resulting initial overall picture of the autoionization event is consistent with the very accurate Penning electron spectra of the He(23S)-H2(v”=0) system. On the other hand, the population factors are different from the approximate average populations used in model considerations about the Penning processes in the system. Sample dynamical calculations on the system are presented, yielding a similar picture of the autoionizing event. The calculated static quantities vary quite smoothly with changes in the He*-D2 separations and orientations, thus indicating that the analytical representations of the present 2D surfaces might form a sound basis for future detailed study of the dynamics of the collision system.
I. Introduction In 1993, one of us’ proposed and tested a two-dimensional (2D) classical trajectory model of the dynamics of the ionization event in thermal energy collisions in autoionizing A* BC systems with repulsive interactions. At these energies, the autoionization process is best viewed as proceeding in two microscopic In the first step,
+
A*
+ BC - [A.**BC+]+ e-
(la)
the system leaks into a continuum and, under the framework of the Millei‘ and N a k a m ~ r a ~ theory , ~ of Penning ionization, undergoes a vertical transition from the Born-Oppenheimer resonant surface Vgo(R) to the ionic one, Go(R). In the second step, the collision continues on the ionic surface: [A. -BC+]
-
BC+
+A
-AB++C (ABC)+
-+A+B++c
(1b)
Here the products represent, respectively, the Penning, rearrangement, associative, and dissociative channels. In thermal energy collisions in repulsive A* -BC collision systems, the BC+ molecule-ion formed in step (la) is only slightly perturbed by the A collision partner. Hence, the most prominent quantum features of the ionization event are connected with the vibrational motion of the perturbed BC+(v’)o *A) and exhibit themselves in the vibrational structure of the corresponding Penning electron spectra. The main motivation behind the approach of ref 1 was to adequately describe the above quantum ‘Dedicated to Professor Zdenek Herman on the occasion of his 60th birthday. Abstract published in Advance ACS Abstracts, October 1, 1995. @
0022-365419512099-15473$09.0010
+
features of the ionization event, without losing the possibility of following the dynamics of the rather complicated step (lb) of the collision process. The approach makes full use of the fact that before the ionization event A* BC(v”) [A...BC+(v‘)] e- takes place in repulsive systems the BC coordinate is only very weakly coupled to the remaining coordinates. For this stage of the process, a separate quantum mechanical treatment of the perturbed BC(v”) [BC+(d)] vibrational motion is adopted and included into the calculation of static characteristics of the system. These characteristics thus depend on the distance R from A* to the center of mass I of the BC collision partner, the (AI, BC) angle y , and the pertaining vibrational quantum numbers of the perturbed BC and BC+. In the dynamical part of the description of this step of the process, the BC collision partner is treated as a rigid rotor. The basic idea of the model is to single out and properly (quantum mechanically) treat that degree of freedom of the repulsive atom-diatom autoionizing collision system which determines most prominent features of the ionization event and thereby influences the course of the collision process after ionization of the system. In this sense, the approach is similar in spirit to the model which Herman and Cermik7 used to discuss the mechanism of Penning and associative ionization in noble gas metastable-metal collisions. In their discussion of the mechanism of reactions in attractive atom-atom autoionizing systems, they, of course, were concerned with the degree of freedom corresponding to the internuclear separation of the collision partners. On the other hand, they considered a general shape and mutual position of the relevant potential energy curves and thereby the position, shape, and range of the vibrational wave functions the potentials support. These are just the features of the (perturbed) Hz(“.He*) and H*+(“.He) vibrational wave functions, the analysis of which prompted one of us8 to propose the 2D classical trajectory model for the atomdiatom autoionizing collision systems mentioned above. It should be also noted that the approach of Herman and Cermik’
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0 1995 American Chemical Society
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15474 J. Phys. Chem., Vol. 99, No. 42, 1995
Vojtik and Kotal
proved to be very useful in general considerations and, more importantly, tumed out to be close to the conceptual basis of the Mille# and Nakamura5q6theory of Penning ionization. The method of ref 1 was applied to the He(23S)-H2(v”=0) system with collision energy ranging from 10 to 150 meV (the average 51 meV) and led to a picture of the
-
He(23S)-H2(v”=O)
-
[He***H;(v’)]
+ e-
(2a)
ionization event, which is very similar to that provided by very accurate Penning electron spectra measurement^.^ The necessary analytical representations of the 2D surfaces for the resonant energy H2(d’=O)(‘ **He*),the H2+(d)(‘ *-He)vibrational population factors, and the energy of the corresponding Penning electrons for the ionization event were obtained from the values of the 2D static characteristics of the system calculated in our previous work.8 As mentioned above, a geometrical configuration of the repulsive A*-BC system is in ref 1 described by a distance R from A* to the center of mass I of the BC collision partner and the (AI, BC) angle denoted y . The static characteristics (see later) in the approach are taken to be functions of these variables, are dependent on the vibrational quantum numbers of BC(BC+) and, in addition, were allowed in refs 1 and 8 to be dependent on the end-over-end rotational quantum number N of the perturbed BC(BC+) collision partner. The effect of changes in N on the 2D static characteristics was found to be very smalL8 A nonzero choice of the value for N , however, is inconsistent with the radial parts of the rovibrational wave functions being calculated for a given value of the (AI, BC) angle y . Therefore, in what follows, we do not consider dependence on N of the static characteristics. This means, for example, that the motion of the system (with the BC collision partner treated as a rigid rotor) will be governed by the potential Vt,(R,y), where u” stands for the vibrational quantum number of the perturbed BC(. *A*)diatomic. Information about the value of this potential for R, y can be obtained (cf. ref 8) from the one-dimensional Schrodinger equation:
the 2D static characteristics of the ionization event for He(23S) D2(~”=0).
+
11. Method and Calculations
The electronic Bom-Oppenheimer surfaces for the resonant energy, ionic potential, and the width of the resonant state with respect to ionization are denoted,8 respectively, by VEo(R,y;r), co(R,y;r), and TBo(R,y;r). Here R stands for the distance between the He*(He) atom and the center of mass I of Dz(D2+), y is the He*I-D2 (HeI-D2+) angle, and r is the Dz(D2+) internuclear separation. The necessary vibrational wave functions and corresponding energy levels for the perturbed D2(v”=O)(. **He*)and D2+(v’)(‘ -.He) are obtained from the onedimensional Schrodinger equation (3), where p is the reduced mass of the Dz(D2+) pair and the superscript J stands for (resonant surface) or (ionic surface). Some comment on the role of the eigensystems of the Schrodinger equation (3) in the present approach (cf. refs 1 and 8) appears to be warranted. The vibrational wave function X$,(R,y;r) forms a 1D part of the approximate picture of the initial state of the autoionization event
*
+
He(23S)-Dz(v”=0)
-
[He***D,+(v’)]
+ e-
(2b)
taking place at R, y . Analogously, the X:(R,y;r) wave function is related to the final state of the event (2b). The probabilities of formation of the nascent Penning D2+(v’)(‘ -He) ions at the configuration R, y are obtained from the corresponding local widths* of the He(23S)-D2(v”=0) state with respect to autoionization to Dz+(‘* .He) in its v’th vibrational level, r(v”,V’;R,y) = I&$,(R,y;r)l [rBo(R,y;r)l
+
112lkv@,l’;r))r12
(4)
which were found8 to provide a physically more relevant picture of the autoionization event in the He(23S) Hz(v’=O) system than the simpler Franck-Condon factors,
+
The energy of the Penning electrons corresponding to the ionization event is taken as the difference between the pertinent eigenenergies of the Schrodinger equation (3), Here p is the reduced mass of the DZpair and the superscript j stands for (resonant surface).
*
It should be remarked that unlike the (3D) electronic BomOppenheimer surfaces for the resonant energy, the width of the resonant state with respect to autoionization, and the surface for the ionic BC+(. *A) potential, the 2D static characteristics discussed in our previous papers’,*are dependent on the isotopic variant of the BC collision partner. An analysis of these 2D surfaces for He(23S)-H~(v”=0) and the corresponding dynamical calculation on the system revealed that the main features of the theoretical description of the ionization event (2a) could have been deduced from the values of the 2D static characteristics, prior to performing rather demanding dynamical calculations. Hence, it appears tempting to obtain a static picture of the ionization event in other autoionizing collision systems with repulsive interactions. As far as we know, the only existing sufficiently accurate and complete information about the pertinent 3D Bom-Oppenheimer surfaces for systems of this class is that for the triplet He*-H2. In view of this, one of us has recently calculated the static characteristics for He(23S) HD(v”=O).’O In this work we will concem ourselves with the remaining isotopic variant of this autoionizing collision system and report calculations of
.
+
Otherwise, the role of the eigenenergies V$,(R,y) and (R,y) in the present model of the dynamics of the autoionizing systems’ is less symmetrical. As already noted, V,,(R,y) is taken as an approximate (resonant) potential which govems the 2D motion of the system before the autoionization event takes place. More specifically, a trajectory starts on the pt,(R,y) surface and is determined by numerical integration of Hamilton’s equations given in ref 1. At each integration time step i in the trajectory, the probability P(Ri,yi) of leaking of the system into the continuum is computed from the expression
where Ati is the length of the numerical integration time step. A pseudorandom number & is then computed and compared with this probability. If P(Ri,yi) 5 &, the trajectory continues on the resonant surface. If P(Ri,yi) > Ci, however, an electron is ejected and the system leaks into the continuum. The
J. Phys. Chem., Vol. 99, No. 42, 1995 15475
Penning Ionization in the He(23S)-D2 Collision System vibrational state of the nascent Penning ion formed in the transition is based on a comparison of an extra pseudorandom number ttrwith the set of (appropriately normalized) local widths for the BC(v”)(. *A*) BC+(v’)t *A) e- transition taking place at the configuration of the system corresponding to the integration time step of the ionization. Once the quantum number v‘ is determined, one can calculate the energy of the Penning electron released in the transition from eq 6. Further, the actual value of C ( R , y ) , together with the corresponding 1D cut through the 3D Bom-Oppenheimer ionic potential Go(R,y;r),the actual values of the corresponding 2D general coordinates, and their conjugate momenta’ can be used to specify the starting conditions for the rest of the trajectory. Of course, the starting value of the phase of the BC+(v’)(. ‘A) vibrational motion has to be randomized with the help of an additional pseudorandom number. This second part of the trajectory describes a 3D, ionic part of the motion of the system which is governed by the (3D) Born-Oppenheimer potential (R,y;r). Tuming back to the calculation of the 2D static characteristics for the He(23S)-D~ system, we first note that the information about the cuts through the 3D Born-Oppenheimer surfaces Go(R,y;r), G0(R,y;r), and rBo(R,y;r) for the He(23S)-H2 system employed in the present calculations is the same as that used in the previous work of one of US.^.'^ The analytical surfaces for the resonant state are related to the DIM model for the ~ y s t e m , ” - ’which ~ was f o ~ n d ’ to ~ ~give ’ ~ @,(R,y;r) and rBo(R,y;r) values comparable in accuracy with the sample ab initio calculations.’6~’7The ionic surface G o ( ~ , y ; rwe ) use is a Sorbie-Murre11 type analytical representationI8 of the CI ab initio potential of McLaughlin and T h o m ~ o n . ’The ~ Schriidinger equation (3) was solved by the Cooley-Numerov technique.20-2’ +
+
Go
111. Results and Discussion
TABLE 1: Vibrational Population Factors P,,(Normalized to 100 at v’ = 3) and Penning Electron Energies E,, (eV) for [He Dz+(v’)] the Ionization Event He(Z3S) D*(v’’=O) e- Taking Place at RH~-D? = 6.5 au He*-D2 angle OD 30” 60” 90” P,, E,, P,, E,, P,, EL/ P,, EL: V‘ -
+
- ...
+
0 28.73 4.416 27.82 4.406 26.15 4.386 25.09 4.377 1 67.53 4.217 66.16 4.207 63.63 4.188 61.98 4.179 2 93.41 4.027 92.50 4.017 90.81 3.998 89.68 3.990 3 100.00 3.845 100.00 3.835 100.00 3.817 100.00 3.808 4 92.10 3.670 92.92 3.661 94.50 3.643 95.58 3.635 5 77.05 3.504 78.36 3.494 80.93 3.477 82.70 3.469 6 60.47 3.345 61.94 3.335 64.88 3.318 66.93 3.311 7 45.45 3.194 46.86 3.184 49.72 3.167 51.74 3.159 8 33.18 3.050 34.41 3.039 36.94 3.022 38.75 3.015 9 23.76 2.914 24.77 2.902 26.88 2.885 28.41 2.878 10 16.82 2.785 17.61 2.772 19.30 2.754 20.54 2.748 14.72 2.624 13.75 2.631 11 11.82 2.663 12.44 2.649 12 8.76 2.533 9.76 2.514 10.52 2.508 8.28 2.550 7.51 2.398 6.93 2.405 13 5.80 2.445 6.17 2.425 14 4.93 2.302 5.37 2.296 4.36 2.324 4.05 2.349 15 2.79 2.264 3.09 2.231 3.52 2.207 3.85 2.201 2.53 2.119 2.78 2.113 2.19 2.147 16 1.66 2.192 17 1.56 2.071 1.82 2.040 2.01 2.033 0.73 2.142 1.46 1.961 0.76 2.116 1.10 2.005 1.32 1.968 18 1.07 1.898 19 0.77 2.079 0.74 1.951 0.96 1.904 0.77 1.842 20 0.74 2.041 0.23 1.909 0.69 1.849 21 0.69 1.999 0.16 1.884 0.49 1.802 0.55 1.796 0.33 1.766 0.37 1.760 0.16 1.871 22 0.64 1.955 0.22 1.734 0.20 1.740 23 0.58 1.909 0.16 1.857 0.07 1.720 24 0.52 1.860 0.15 1.843 0.08 1.725
selecting the relevant ones, we have adopted here the approach of ref 8. As a consequence, the data in Table 1 for the linear configuration are related either to solutions with vibrational wave functions concentrated in the inner region of the D2+C .He) internuclear separations (v’ i 16) or to those of the intermediate or one-minimum-like character (cf. refs 8 and 22) which lead to largest population factors. The angular dependence of the static characteristics for individual vibrational levels at a given R can be described through the corresponding Legendre moments. The average Penning electron energy E;(R, y), for example, can be expanded as follows:
-
In Table 1 we show the vibrational population factors and energy of the corresponding Penning electrons for the ionization event in the He(23S)-D2(v”=0) system taking place at R = 6.5 au and the He*-D2 angles y = 0”, 30”, 60”, and 90”. The distance R = 6.5 au is singled out because the interaction energy of the system for the corresponding configurations, as obtained from the eigenvalues of the Schrodinger equation (3) f o r j = *, u = v“ = 0, was found to range from 49.23 meV ( y = 90”) to 59.90 meV ( y = 0”). Each of these interaction energies is not too far from the average collision e n e r g i e ~typical ~ . ~ of Penning ionization electron spectroscopy (PIES) measurements. In this respect, the situation here is similar to that found for the R = where E,.,r(R), 1 = 0, 2, 4, 6, are the Legendre moments. As far as the angular dependence of the individual vibrational 6.5 au configurations in the He(23S)-H~(v”=0) system.8 population factors and the corresponding Penning electron However, because of the H2 D2 change in the reduced mass energies of Table 1 are concerned, we here only note that due of the diatomic collision partner, 25 vibrational levels instead of 17 ones reported8 for the ***HZ case (and 20 for the *.*IDto the change in the reduced mass, the degree of anisotropy of these quantities is lower than that found for He(23S)-H2(v”=0),8 isotopic variant)I0 were found here to be supported by the 1D the vibrational population factors being more anisotropic than cuts through the 3D ionic surface. It should be remarked in the corresponding Penning electron energies. These findings this connection that for the linear configurations the 1D cuts are also illustrated by Figure 1, in which we display the through the 3D Born-Oppenheimer ionic surface have a doubleminimum character and lead to more complicated eigensystems Legendre moments of the vibrational population factors of Table 1, and by Figure 2, where the Legendre moments of the of the Schrodinger equation (3) than those found for the other corresponding Penning electron energies are shown. In Table R = 6.5 au configurations considered here. The problems 2 we give the 1 = 0 Legendre components of the vibrational connected with the treatment of the situations such as this were population factors for He(23S)-D2 at the distance R = 6.5 au, discussed in some detail in our previous communication.8 The approach employed in that work led to a physically relevant together with those for the *H2 and *HD isotopic variants, static description of the ionization event (2a) and, in addition, and corresponding previous information on the autoionization formed a basis for a picture of the dynamics of the ionization event in the systems. In addition to the vibrational populations obtained from eq 4, the values calculated for the systems with event in the He(23S)-H2(v”=0) system’ which is very close to the help of eq 5 are also included in Table 2. It can be seen that provided by precise PIES measuremenkg In view of this, from this table that the spherically symmetric components of in classifying the vibrational levels in linear configurations and
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Vojtik and Kotal
15416 J. Phys. Chem., Vol. 99, No. 42, 1995 100 80
0
1=2 -20
*
0
l
5
‘
l
s
10
“
l
1s
-
‘
20
VI
Figure 1. Legendre moments of the vibrational population factors (normalized to 100 at u’ = 3 ) for the ionization event He(2%) D2(u”=O) [He*-*D2+(v’)] e- taking place at RH^-^^ = 6.5 au. Notation: 1 = 0 (+); 1 = 2 (A); 1 = 4 (x); 1 = 6 (0).
-
+
+
2 4
d
t 0I 0
5
10
15
20
IV. Concluding Remarks
V‘
-
Figure 2. Legendre moments of the Penning electron energies E,,(eV)
+
+
[He...D2+(u’)] efor the ionization event He(23S) Dz(u”=O) taking place at R H ~ - D=* 6.5 au. Notation: 1 = 0 (+); l = 2 (A); 1 = 4 (x); and 1 = 6 (0).
the He(23S)-H2, R = 6.5 au, vibrational population factors are quite close to the populations obtained in very accurate PIES measurement^.^ Further, this table shows that the He(23S)H2 population factors based on eq 4 are closer to the accurate experimental values than those obtained from the simpler Franck-Condon factors, eq 5 . On the whole, the He(23S)-H2 data of Table 2 illustrate our finding’,8 that for this type of collision system the spherically symmetric components of the vibrational population factors and the Penning electron energies, when calculated for the distance R at which the interaction energy of the collision partners is close to the average experimental collision energy, give a physically reasonble initial picture of the ionization event. There is a connection between this fact and the fact that, in the region of the configuration space of the system corresponding to thermal energy collisions, the anisotropy of both the resonant and ionic 3D BomOppenheimer potentials is rather small. The main features of the difference between the spherically symmetric components of the [He-*.H~+(V‘)],~ the [He.. -HD+( v’)],’O and the present [He- -D2+(v’)] vibrational population factors of Table 2 can be understood in terms of the changes in the position, shape, and range of the corresponding vibrational wave functions Xi(R,y;r) (cf. eqs 3 and 4) caused by the H2 HD D2 change in the reduced mass. Also shown in Table 2 are the approximate average populations for [He***Dz+(v’)] and
-
He- +ID+(v’)] used in the model interpretati~n~~ of the product branching in the He(23S)-D2/HD collision systems. These sets of approximate average populations are substantially different from our 1 = 0 Legendre moments for R = 6.5 au. To understand this difference, we f i s t note that the approximate average population^^^ were derived from the experimental populations for the ***HZ system reported in ref 3. These latter populations must be considered to be less accurate than those of ref 9. The method of obtaining the ***D2+and HD+ population^^^ from the *H2+ ones was based on the relation between the sets of the vibrational energies of the (unperturbed) collision partners. One is therefore not surprised by the fact that their approximate sets of vibrational population factors end up with v’ = 13 (. *D2+)and v’ = 11 (. *HD+). The remaining differences in the * *D2+populations of Table 2, including the overestimated initial approximate value of the population for the ground vibrational level and the reversed order of the populations for v’ = 2 and v’ = 4, can be ascribed to the fact that their approach, being connected with the relations between the corresponding sets of vibrational energies, does not account for the changes in the position, shape, and range of the pertinent vibrational wave functions &(R,y;r). In Table 3 we give spherically symmetric components of the vibrational population factors for a set of the He*-D2 separations representing the whole range of the He*-D2 configurations which are classically accessible by the collision system under conditions typical of PIES measurements. What is important in the present context is that even for the strongest interactions considered (-150 meV), the main features of the present view of the ionization event are the same as for R = 6.5 au. It should be also added that with increasing He*-D2 separation the vibrational population factors corresponding to individual vibrational levels of [He* *D~+(V‘)](and the related Penning electron energies) were found to become more isotropic.
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The vibrational population factors for the nascent Penning ions [He* *D~+(v’)] and energies of the corresponding Penning electrons were calculated for quite a large number of the He*D2 distances R and He*-D2 angles y . The set of the configurations considered here forms a fairly dense grid in the 2D configuration space which is accessible by the system in low-energy collisions. For each of the vibrational quantum numbers v’ of the nascent Penning ion, both the quantities were found to vary quite smoothly with R and y. This means that for each v’ the data reported here can form a sound basis for a construction of well-behaved analytical representations of the corresponding 2D surfaces. To substantiate our feelings in this respect, we have constructed analytical representations for all the 2D static characteristics of the He(23S)-D2 system, corresponding to quantum numbers v” = 0; v’ = 0, ..., 25. In the analytical representation of all the surfaces, the angular dependence of the characteristics was accounted for by a truncated (1 = 0 , 2 , 4 , 6 ) Legendre expansion, and the R dependence of the individual Legendre components was described by cubic tautspline fits.24 Having obtained the analytical representations we have started performing classical trajectory calculations of the dynamics of the autoionization event in the He(23S)-D2 system. The choice of the starting conditions for the trajectories is exactly the same as in the He(23S)-H2 case,’ except that the BC distance for the D2 isotopic variant is kept fixed at 1.414 08 au. At present, we have calculated 1 000 000 trajectories. Of these 21 780 (2.18%) trajectories led to ionization. The corresponding total cross section is (5.54 f 0.04)m2, which is consistent with the values reported’ for the He(23S)-H2 system.
Penning Ionization in the He(23S)-D2 Collision System
J. Phys. Chem., Vol. 99, No. 42, 1995 15477
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TABLE 2: Spherically Symmetric Component of Vibrational Population Factors (Normalized to 100 at the Most Populated Level) for the Ionization Events He(23S) Dm/I-Iz(d’=O) [He Dz+/HD+/H2+(v‘)I e- Taking Place at RH~-D”* = 6.5 au and Previous Data
+
He(23S) 1“
theory 26
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
26.37 63.96 91.02 100.00 94.3 1 80.63 64.54 49.39 36.66 26.65 19.12 13.62 9.66 6.85 4.87 3.48 2.48 1.77 1.28 0.92 0.59 0.43 0.3 1 0.20 0.1 1
25.52 62.66 90.16 100.00 95.11 81.94 66.04 50.86 37.96 27.74 19.99 14.30 10.18 7.25 5.17 3.70 2.65 1.89 1.37 0.99 0.63 0.45 0.33 0.22 0.13
+
+
+ Dz(u”=O)
U’
...
He(23S) HD(u”=O)
He(23S)
theory
theory, ref 8
10.‘
2b.c
ref 23
1“
26
ref 9
ref 3
68.70 87.02 96.95 100.00 87.02 76.34 68.70 58.02 42.75 28.24 17.56 12.98 10.69 8.40
42.28 83.49 100.00 95.15 79.57 61.51 45.26 32.31 22.65 15.73 10.86 7.52 5.26 3.70 2.61 1.85 1.29 0.88 0.56 0.25
41.03 82.29 100.00 96.28 81.31 63.41 47.05 33.83 23.88 16.68 11.61 8.08 5.63 3.93 2.73 1.91 1.24 0.79 0.54 0.32
71.15 94.23 100.00 91.67 77.56 66.67 53.21 32.69 20.5 1 13.46 11.54 8.33
54.95 92.98 100.00 86.5 1 67.02 48.70 34.09 23.37 15.88 10.77 7.32 4.89 3.43 2.28 1.41 0.90 0.54
53.21 92.43 100.00 87.60 68.52 50.21 35.43 24.48 16.75 11.42 7.79 5.31 3.61 2.43 1.48 0.95 0.58
55.0 1.5 94.7 f 1.0 100.0 85.6 % 1.0 65.5 i 1.0 47.4 f 1.0 32.4 i 1.0 21.8 f 0.8 14.5 f 0.8 9.8 f 0.8 6.5 0.8 4.4 f 0.8 2.9 f 0.6 1.9 f 0.5 1.3 0.4 0.9 f 0.3
72.78 95.00 100.00 81.11 70.00 55.56 32.22 18.33 12.78 10.56 7.22
r included in the matrix elements, eq 4.
TABLE 3: Spherically Symmetric Component of the Vibrational Population Factors (Normalized to 100 at d = 3) for the Ionization Event He(2%) Dz(d‘=O) [He Dz+(v’)l e- Taking Place at Different R H ~ - Separations D~
-
+
...
5.5
6.0
6.5
7.0
7.5
*
* *
The Franck-Condon factors, eq 5. From ref 10.
TABLE 4: Vibrational Population Factors P,,(Normalized to 100, t/ = 3) and Penning Electron Energies E,, (eV) for the Ionization Event He(23S)+D~(v”=0) [He Dz+(v’)l e-
-
Ru.-ma U‘
experiment
ref 23
The Born-Oppenheimer width function
+
+ H*(u’’=O)
9.0
10.0
0 26.40 26.69 26.37 26.12 25.87 25.40 25.31 25.39 1 64.14 64.50 63.96 63.57 63.18 62.44 62.48 62.41 2 91.23 91.42 91.02 90.76 90.48 89.97 90.11 89.96 3 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 4 93.97 93.88 94.31 94.56 94.83 95.30 95.06 95.26 5 79.99 79.90 80.63 81.04 81.49 82.25 81.83 82.04 6 63.72 63.67 64.54 65.03 65.54 66.40 65.94 65.87 7 48.53 48.52 49.39 49.87 50.38 51.20 50.80 50.28 8 35.84 35.86 36.66 37.09 37.53 38.27 37.97 36.97 9 25.94 25.98 26.65 27.02 27.38 28.00 27.80 26.43 10 18.53 18.57 19.12 19.42 19.71 20.21 20.09 18.51 11 13.14 13.18 13.62 13.85 14.07 14.48 14.40 12.78 12 9.20 9.33 9.66 9.84 10.01 10.33 10.28 8.75 13 6.51 6.60 6.85 6.99 7.11 7.37 7.33 5.98 14 4.64 4.66 4.87 4.98 5.07 5.27 5.24 4.10 15 3.30 3.30 3.48 3.56 3.62 3.78 3.75 2.83 16 2.32 2.37 2.48 2.55 2.60 2.73 2.70 1.97 17 1.52 1.70 1.77 1.84 1.88 1.97 1.96 1.39 18 1.12 1.20 1.28 1.32 1.36 1.43 1.42 0.99 19 . 0.85 0.80 0.92 0.95 0.98 1.04 1.03 0.72 20 0.66 0.59 0.59 0.68 0.70 0.75 0.52 0.75 21 0.51 0.44 0.43 0.44 0.50 0.37 0.53 0.53 22 0.39 0.32 0.31 0.30 0.30 0.33 0.25 0.36 23 0.29 0.22 0.20 0.19 0.19 0.15 0.19 0.22 24 0.20 0.13 0.11 0.10 0.09 0.09 0.06 0.07
The distances are in atomic units.
The ratio of the present He(23S)-D2 total cross section and that obtained in ref 1 for He(23S)-H2 is 1.23. This value is consistent with the experimental values 1.15, 1.43, and 1.31 reported in ref 25 and is very close to the theoretical value of 1.22 that we obtain using the formula given by Miller.4 The resulting vibrational population factors for the nascent Penning ions P,, and the corresponding average Penning electron
U’
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1“ 26.37 63.96 91.02 100.00 94.3 1 80.63 64.54 49.39 36.66 26.65 19.12 13.62 9.66 6.85 4.87 3.48 2.48 1.77 1.28 0.92 0.59 0.43 0.3 1 0.20 0.11
+
E”,
Pd
8.0
...
26
1“
26
25.0 62.0 88.6 100.0 93.7 81.3 62.7 50.3 35.6 26.6 18.1 13.7 9.1 7.3 5.1 3.5 2.5 1.7 1.2 1.2 0.6 0.7 0.3 0.2 0.1
4.390 4.191 4.002 3.820 3.646 3.480 3.321 3.170 3.026 2.888 2.758 2.634 2.518 2.409 2.307 2.213 2.127 2.049 1.980 1.919 1.868 1.827 1.796 1.773 1.758
4.38 4.18 3.99 3.81 3.63 3.47 3.31 3.16 3.01 2.87 2.74 2.62 2.50 2.39 2.29 2.19 2.11 2.03 1.95 1.90 1.83 1.78 1.76 1.73 1.74
a
Spherically symmetric components of P,,and E,, for the ionization event taking place at R H ~ - D = ~6.5 au. bValues based on sample dynamical calculations.
energies E,, are shown in Table 4. Also given in this table are the spherically symmetric components of their static counterparts obtained for the ionization event taking place at R = 6.5 au. It should be stressed, however, that despite the fact that the number of trajectories run here is greater than that in the dynamical
15478 J. Phys. Chem., Vol. 99, No. 42, 1995
Vojtik and Kotal
study of the He(23S)-H2 system,’ the “dynamical” values given in this table should still be taken with some caution and the present trajectory calculations should be classed as sample ones. The main reason lies in the fact that, due to the H2 D2 change in the reduced mass, the number of vibrational levels of the nascent Penning ion [He.*-D2+(d)] is now greater and so is the number of those vibrational states for which the population factors (and the numbers of the trajectories leading to the nascent Penning ions with corresponding vibrational levels) are very small (cf. Tables 2 and 4). There is no doubt, however, that the present 2D static characteristics for He(23S) D2(v”=O) can form a sound basis for a detailed study of the dynamics of the ionization even in the system by the approach’ mentioned above. The data given in Tables 1-3, and those illustrated by Figures 1 and 2, give an idea about what changes in the “dynamical” vibrational population factors of the nascent Penning ion [He* *D~+(v’)] and average Penning electron energies of Table 4 can be expected when the present dynamical calculations are extended to become fully converged. It is clear from these tables and figures that for quite a wide range of collision conditions the dynamical calculations’ will fumish a picture of the ionization event (2b) which will be consistent with that for the He(23S) H2(~’=0) system.'^^ As a consequence, the resulting vibrational population factors will differ from the approximate average population^^^ discussed above and reproduced in Table 2.
-
+
+
Acknowledgment. This work was sponsored by the Grant Agency of the Czech Republic (Grant No. 203/95/1055). The authors would like to thank the referees for helpful comments and suggestions.
References and Notes (1) Vojtik, J. Chem. Phys. Lett. 1993, 214, 425. (2) Hotop, H.; Niehaus, A. Z. Phys. 1968, 215, 395. (3) Miinzer, A,; Niehaus, A. J . Electron Spectrosc. Relat. Phenom. 1981, 23, 367. (4) Miller, W. H. J . Chem. Phys. 1970, 52, 3562. (5) Nakamura, H. J . Phys. SOC.Jpn. 1969, 26, 1473. (6) Nakamura, H. J . Phys. SOC.Jpn. 1971, 31, 574. (7) Herman, Z.; term&, V. Collect. Czech. Chem. Commun. 1966, 31, 649. (8) Vojtik, J. Chem. Phys. 1993, 170, 209. (9) Bregel, T.; Yencha, A. J.; Ruf, M.-W.; Waibel, H.; Hotop, H. 2. Phys. 1989, 013, 51. (10) Vojtik, J. Int. J . Quantum Chem., in press. (11) Vojtik, J.; Paidarovi, I. Chem. Phys. Lett. 1983, 99, 93. (12) Vojtik, J.; PaidarovB, I. Chem. Phys. Lett. 1984, 103, 305. (13) Vojtik, J.; Paidarovh, I. Chem. Phys. 1986, 108, 115. (14) Vojtik, J.; Paidarovh, I.; Schneider, F. Chem. Phys. 1987, 114, 369. (15) Vojtik, J.; Paidarovh, I. Chem. Phys. 1991, 157, 67. (16) Cohen, J. S.; Lane, N. F. J . Chem. Phys. 1977, 66, 586. (17) Hickman, A. P.; Isaacson, A. D.; Miller, W. H. J . Chem. Phys. 1977, 66, 1483. (18) Jones, T.; Sathyamurthy, N. J. Chem. Phys. 1987, 86, 704. (19) McLaughlin, D. R.; Thomson, D. L. J. Chem. Phys. 1979,70,2748. (20) Numerov, B. Publ. Obs. Cent. Astrophys. Russ. 1933, 2, 188. (21) Cooley, J. W. Math. Comput. 1961, 15, 363. (22) Savrda, J.; Vojtfk, J.; Paidarovi, I. Chem. Phys. Lett. 1991, 182, 524. (23) Martin, D. W.; Weiser, C.; Sperlein, R. F.; Bemfeld, D. L.; Siska, P. E. J. Chem. Phys. 1989, 90, 1564. (24) de Boor, C. A practical guide to splines; Springer: Berlin, 1978. (25) Penton, J. R.; Muschlitz, E. E. J . Chem. Phys. 1968, 49, 5083. JP950667L
