THEORETICAL STUDIES OF IMPORTANT PROCESSES IN 

PLANETARY AND COMET ATMOSPHERES 

Renewal Request 

April 21, 1989 

NASA Grant NAGW 1404 

Steven L. Guberman 

Principal Investigator 

Institute for Scientific Research 

(Employer Identification B 22-248-7207) 

33 Bedford St., Suite 19a 

Lexington, MA 02173 

The NASA Technical Officer for this Grant is 

Dr. J. T. Bergstralh 

Solar System Exploration Division, Code EL 

NASA Headquarters, Washington, D. C. 20546 

Principal Investigator Institute Official 

/ &72i5L-m& 

Steven L. Guberman 

Physic is t 

61 7-861 -7900 

p, 3" 

Susan L. Greenblatt 

Grants Administrator 

61 7-861 -7900 

[NASA-CR-l84951) THEORETICAL STUDIES CF 189 -25099 IHPOBTANT PROCESSES IN PLBIETAtZY AND CONET 

BTUOSPHERES. RENEREL BEQUEST Proqress 

Report, 1 Sep. 1989 - 31 Oct. 1990 Unclas 

[Institute for Scientific Research) 30 p G3/91 0201504 
I h 

This report is submitted to request the renewal of NASA Grant NAGW 1404 

c 

for the second year of funding, September 1, 1989-October 31, 1990. 

I. Progress Report 

Our current efforts have focused on the dissociative recombination (DR) 

Of O:, a process of great importance in planetary atmospheres. This process 

is difficult to study experimentally because of the need to determine the . 

dependence of the product electronic states and kinetic energies upon the 

vibrational distribution of the ion and the electron temperature. The 

knowledge of these characteristics of DR is needed to accurately model 

planetary ionospheres. Using a theoretical quantum chemical approach, we 

have studied in detail the generation of O(*S) from DR, 

0: + e- + O('S) + O(rD), 

where e' is an electron, both 0 atom products are excited, and the 0; 

molecular ion may be vibrationally excited. Building upon earlier 

which showed that (1) occurs along a '1: molecular state, we 

have now included the indirect dissociative recombination mechanism. The 

latter process is similar to (1) except that instead of direct electron 

capture into a repulsive molecular state leading to dissociation to atoms, 

electron capture first occurs into an intermediate vibrationally excited 

Rydberg state followed by predissociation into atoms along the repulsive 

state used in the direct mechanism, 
where O,(n,v) is a neutral Rydberg state with principal quantum number n and 

vibrational level, v. These vibrationally excited Rydberg levels lie above 

the v-0 level of 0: and are resonances in the electron-ion continuum. When 

the energy of the incoming electron coincides with the position of the 

Rydberg vibrational level a marked perturbation occurs in the DR cross 

section. We have found that the DR rate can also be altered by these Rydberg 

states. 

The calculation of the full DR cross section including both the direct 

and indirect processes has required the calculation of the quantum defect Of 

the '.Zt Rydberg series and the autoionization width, both as a function of 

the internuclear distance. The cross sections have been calculated using 

molecular quantum defect theory.4 Further details on these calculations are 

given in the first semi-annual progress report. 

Figure 1 shows the cross sections for both the direct and indirect DR 

processes. The lower curve at low electron energies is relatively smooth 

across the range of electron energies plotted and corresponds to the direct 

DR cross section, i.e. there are no Rydberg excited vibrational levels 

included in the calculation. However, the direct DR curve has several abrupt 

decreases at 0.23eV, 0.46eV, 0.68eV, and 0.90eV. These energies correspond 

to the positions of the v=1,2,3 and 4 vibrational ion levels above the v=O 

ion level. At each of these vibrational levels, a new channel for emitting 

the electron opens up and the DR cross section consequently drops. Some of 

these sudden decreases are partially obscured by the infinite series Of 

resonances below each ion vibrational threshold which contribute to the 
. 3 

indirect DR process which is also plotted in the Figure. When the electron 

is emitted, the ion can be left in an excited vibrational level. The net 

result is electron impact vibrational excitation. Cross sections for the 

latter process have also been calculated and are discussed below. 

The full cross section curve describing both the direct and indirect 

process is also shown in Figure 1. This curve has a very rich resonance 

structure giving rise to rapid variations in the cross section over narrow 

energy intervals. The full cross section curve is above the direct cross 

section curve even at low energies, where it is far from the first 

discernable resonance at 0.36eV. The source of the higher full cross section 

curve is the Rydberg level with principal quantum number n=6 and v=2. This 

level actually falls 0.0147eV below the v=O level of the ion. However it is 

broadened by predissociation by the dissociating 'Cc state. The broadening 

allows a wing of this resonance to appear above threshold. This resonance 

plays a large role in increasing the DR cross section at typical ionospheric 

temperatures (see below). The first resonance appears at 0.036eV and 

corresponds to the n=9 Rydberg 'C: state in the v=1 vibrational level. The 

resonance has a typical Beutler-Fano profile corresponding to a positive 

Fano q, i.e. the cross section first decreases and then increases as the 

energy is increased through the resonance position. The next resonance for 

n=lO,v=l appears at 0.076eV with a similar shape to that for n=9. As the 

electron energy approaches the v=l ion level at 0.23eV, the infinite series 

of v=l resonances interferes with resonances due to n=7,v=2 and n=4,v=6 at 

0.12eV leading to a complex cross section shape in this region. 

Figure 2 shows the DR rate constant for electron temperatures up to 

3000K. The upper curve is from the full DR calculation and the lower curve 
.I 4 

is for only direct DR in which the resonance states have been omitted. The 

full DR rate is 3.0 times larger that the direct DR rate at 300K and 2.6 

times greater near 800K. The increase is primarily due to the n=6, v=2 

resonance discussed above. 

Figure 3 shows the calculated vibrational excitation cross section 

corresponding to the following process: 

The cross section for process (3) is calculated simultaneously with the 

calculation of the DR cross sections. Direct vibrational excitation by 

electron impact is often thought of as a slow process. However, its rate can 

be substantially increased if electronic resonance states can act as 

intermediate states. This is well known to be the case in the analogous 

excitation of neutral molecules where negative ion states play the 

intermediate role. There is very little in the literature on the excitation 

of molecular ions using neutral resonance states to assist the process. In 

the case of O:, the cross section for excitation of the v=1 ion level has a 

sudden onset at the energy of the v-1 level, O.23eV. Both the direct cross 

section, shown as the lower curve in Figure 3, and the full cross section, 

including both the direct and indirect processes, are shown. The direct 

cross section has abrupt breaks at the onset of higher excited vibrational 

levels. At the breaks, the cross section for v=l excitation drops abruptly 

since a higher vibrational level can also be excited at these energies. At 

energies just below threshold, the cross section for the full DR process is 

considerably higher than that for direct DR and shows structure due to many 
interfering resonance states. 

The vibrational excitation rates are shown in Figure 4. The lower curve 

does not include the Rydberg resonances while the upper curve includes both 

direct and indirect DR. Clearly, the resonances play an important role in 

determining the magnitude of the excitation rate. The rate has a value Of 

about 1.7 x 10-1 

important to remember that the repulsive '1: state, which is studied here, 

does not cross the ion within the turning points of the v-1 vibrational 

level. However, there are other states which are to cross the ion 

within these turning points, e.g. 'Z, and 'Au. These states could lead to 

high vibrational excitation for v=l. Studies of role of these states in 

vibrational excitation are currently in progress. 

cm3/sec at 800K. While this rate is not very large, it is 

11. Plan of Work for 9/1/89-10/31/90 

The plan of work for the next funding period will proceed in accord with 

the research discussed in the original proposal. The plan includes the 

completion of the ab initio calculation of the Rydberg and valence potential 

curves needed for the study of both direct and indirect DR leading to 'P 

atoms. Only states having significant electron capture widths will be 

determined. These states will be used to calculate cross sections and rates 

for 0: DR leading to 3P atoms, using techniques developed during the first 

year of funding. The electronic widths of states leading to 'D atoms will be 

calculated. For those states with large widths, the potential curves will 

also be determined. The important Rydberg states contributing to indirect 

recombination in the generation of 'D atoms will be determined. DR cross 
6 

sections and rates for generating rD atoms will be calculated for both the 

direct and indirect processes. The results will yield a detailed picture of 

the DR of 0; and will be invaluable for the modelling of the chemistry of 

planetary ionospheres. The kinetic energies and quantum yields of excited 

and ground state atoms will be determined as a function of electron 

temperature and ion vibrational excitation. 
7 

I I I. References 

1. S. L. Guberman, Nature 327, 408 (1987). 

2. S. L. Guberman, Planet. Space Sci. - 36, 47 (1988). 

3. S. L. Guberman, Ab Initio Studies of Dissociative Recombination, in 

Dissociative Recombination: Theory, Experiment, and Applications, ed. by J. 

B. A. Mitchell and S. L. Guberman (World Scientific, Singapore, 19891, p. 

45. 

4. A. Giusti, J. Phys. B - 13, 3867 (1980). 

5. S. L. Guberman, Potential Energy Curves for Dissociative Recombination, 

in Physics of Ion-Ion and Electron-Ion Collisions, ed. by F. Brouillard and 

J. W. McGowan (Plenum, New York, 1983) ,p.167 
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12 

IV. Budget for Year 2 

September 1, 1989 - October 31, 1990 

Steven L. Guberman,P.I. 

Fringe benefits 

Phone 

Supplies, postage, copying 

Page charges 

Trave 1 

Secretarial Services 

Total direct charges 

Indirect cost @47.02% 

Total 

Per son 

months 

6 

Amount 

29,493 

10,399 

165 

260 

500 

1,320 

550 

42,687 

20,071 

$62,758 
13 

~ V. Publications I 

1. S. L. Guberman, Ab Initio Studies of Dissociative Recombination, in 

Dissociative Recombination: Theory, Experiment, and Applications, ed. by J. 

B. A. Mitchell and S. L. Guberman (World Scientific, Singapore, 19891,~. 45. 

2. S. L. Guberman and J. B. A. Mitchell, Introduction to Dissociative 

Recombination, in Dissociative Recombination: Theory, Experiment, and 

Applications, ed. by J. B. A. Mitchell and S. L. Guberman (World Scientific, 

Singapore, 19891,p.l. 

3. K. Yoshino, A. S.-C. Cheung, J. R. Esmond, W. H. Parkinson, D. E. 

Freeman, S. L. Guberman, A. Jenouvrier, B. Coquart, and M. F. Merriemme, 

Improved Absorption Cross Section of Oxygen in the Wavelength Region 205- 

240nm of the Herzberg Continuum, Planet. Space Sci. 36, 1469 (1988). 

4. J.-H. Yee, S. L. Guberman, and A. Dalgarno, Collisional Quenching of 

O(’D) by O(’P), to be submitted. 

- 
AB INITIO STUDIES OF DISSOCIATIVE RECOMBINATION 

Steven L. Guberman 

Institute for Scientific Research 

33 Bedford St., Suite 19A 

Lexington, MA 02173 

ABSTRACT 

Quantum chemical calculations of the dissociative 

recombination of 0; and N; are reported. An approach for 

calculating autoionization widths from high principal 

quantum number Rydberg states is summarized and an 

example is presented for the lowest 311g dissociative 

state of 0,. For O:, the 'E: state is the sole source of 

O('S) from the lowest 10 vibrational levels of the ion. 

For O( 'D), the B3Ci, 'Et, and 1 'Au states are the sources 

of O('D) from the lowest three vibrational levels. Rate 

coefficients for generating O('S) and O('D) at ionospheric temperatures are reported. For N:, states of 'nu, 'E: 

and 3Ag symmetries intersect the ion at the v=O level. 

The configurational structure of these states indicates 

that the 311u states are likely to be the most important 

routes for dissociative recombination from v=O. 

1. INTRODUCTION 

Interest in dissociative recombination (DR) first arose out of the 

need to understand the properties of the Earth's ionosphere. 1 I,21 

Initial attention focused on the DR of O:, N: and NO'. While there has 

been much progress in the measurement of the total DR rate coefficients 

of these ions, the dependence of the quantum yield of excited and 

ground state atoms upon the electron temperature and ion vibrational 

level is only currently being unraveled. Impediments to experimental 

progress have included the difficulty of generating ions in specific 

vibrational levels and simultaneously identifying the atomic products. 

. 
I 31,Ql ,51 Nevertheless there has been recent progress toward this goal. 

An alternative theoretical approach based on spectroscopic identification of the potential curves which provide routes for DR is difficult 

since these states often do not have dipole allowed transitions from 

the ground state. Furthermore, electron capture occurs at high energies 

on the repulsive wall, well above the dissociation asymptote and out Of 

range of the RKR approach. Additional difficulties include the paucity 

of information on electron capture widths although progress has been 

made in spectroscopic analysis of Rydberg-valence 

interactions 6],7],81,9] from which widths can be derived. 

1- 

Here, I review recent theoretical ab i n i t i o r e s u l t s obtained in this 

laboratory for the DR of 0; and describe some preliminary results on 

the DR of N:. For O:, potential curves, electron capture widths and 

rate constants have been obtained for generating 'S and 'D atoms, the 

upper states of the green and red lines respectively. For N:, potential curves for DR of the v=O ion level have been computed. 

2. WAVE FUNCTIONS 

The wave functions calculated here are based on orbitals determined 

in either multiconf iguration or complete active space self consistent 

field (MCSCF or CASSCF) calculations. Each of the orbitals, 4j, is 

expanded in a large set of Gaussian basis functions, xi, 

Some of the xi are individual Gaussian primitiveslO] while others are 

fixed combinations (contractions) of Gaussian basis functions determined from atomic calculations.l1] The orbitals are used to construct 

large scale configuration interaction (CI) wave functions containing 

many configurations, each of which describes a possible distribution of 

the electrons in the molecular orbitals for a particular state symmetry. For example, for the *Ci ground state of N,, the most important 

configuration near the equilibrium separation is: 
The la and 2a orbitals are mostly N atom 1s and 2s orbitals, respectively. The 3ag orbital is a bonding combination consisting mostly of 

the N atom 2p orbitals lying along the bond axis. The 1ru orbitals are 

out of plane bond orbitals consisting mostly of the N 2p orbitals that 

are perpendicular to the bond axis. Each configuration can have one or 

more spin couplings (referred to below as terms) for a given total Spin 

symmetry. A wave function is constructed consisting of a superposition 

of many terms, @i, each of which has the same total spin and spatial 

symmetry , 

N 

In the CASSCF12] approach a set of active orbitals is defined and 

N(CASSCF) terms are generated in (3) by taking all possible excitations 

of the electrons within this set while restricting each term to have 

the total spin and spatial symmetry of interest. Both the cj,i and the 

Cj,i are determined by variationally optimizing the total energy of $j. 

The MCSCF approach is identical except that usually a subset of the 

full active space is used. Since the orbitals are expanded in large 

basis sets, xi, the number of orbitals used in the CASSCF is only a I 

subset of the orbitals that can be formed from the basis set. The CI 

wave functions are constructed by exciting electrons out of a small set 

of reference configurations into the remaining (virtual) orbitals. The 

reference configurations often consist of all the configurations needed 

to properly dissociate that particular state to the proper separated 

atoms limit. This approach leads to wave functions that are not biased 

to favor particular internuclear distances. Some of the wave functions 

reported here for N2 include the 20 orbitals in the active space and 

use the entire CASSCF wave function as a reference set. Taking at most 

one or two electrons from each of the reference configurations and 

promoting them to the virtual and other valence orbitals leads to a 

singles and doubles CI denoted CISD. The CI wave functions are expanded 

over the CASSCF orbitals and have the form of (3) with 
N(CI)>>N(CASSCF). Some of calculations used here employ large scale 

wave functions having mre than 100,000 terms with the CI coefficients 

determined by direct CI techniques. l33 Calculations have recently been 

completed on several states of N, and 0, which provide examples of the 

sort of accuracy which can be expected with these techniques, For the 

ground state of N,, 14 reference configurations (needed to properly 

dissociate the ground state to “S atoms) are used in the CI with a 

basis set consisting of 6s,3p,2d and If type contracted Gaussians 

leading to a CI wave function having 176,536 terms. The calculated 

(experimentall lr3 1 spectroscopic constants are 2340.57(2356.56cm-l) for 

Wen 14.27(14.3244~m-~) for and 2.0856(2.074347a0) for the 

equilibrium distance, Re. Similar calculations on the ground state of 

02 using a basis set of the same size leads to a 228,036 term wave 

function. The calculated (experimental’ 53 results are 1565.47( 1580.19 ) 

for we, 10.93(11.98~m-~) for @exes and 2.2999(2.2819ao) for Re. For O,, 

a series of vertical excitation energies have been calculated. The 

calculated (experimentally derived) results are 5.8982(5.90*0.05eV) for 

Xu; 6.1 3 1 8 ( 6.1 1 fO .05 eV ) for ’ A u, 6.2766( 6.19+_0.05eV) for Cs, and 

8.66(8.61+0.05eV1 61) for ’Iu: All the experimental energies are derived 

from Ref.15 (except for ’1; by extending the RKR results with an ab 

in it io repulsive wall. 

Recent results have been obtained for N, using a C4s,3p12d,lfl 

contractecj Gaussian basis set and a CISD reference set consisting of 

the CASSCF configurations with 2a in the active space. For the C311u 

state the calculated (expcrimental)14] results are 11.01 (11.05eV) for 

Te, 1986(2047cm-’) for we, 25.05(28.44cm-’) for WeXe, and 

2.1932(2.1707a0) for Re. For the wlAu state the results are 

1 

8.83(8.84eV) for Te, 1532(1559cm-’) for we, 11.67(12.01~1~-~) for WeXe, 

and 2.4192(2.3977a0) for Re. The highly accurate values of the excitation energies and the other spectroscopic constants give us confidence 

that these repulsive curves will be located quite accurately relative 

to the ion potential curve. 

3. POTENTIAL CURVES FOR 0, 

Potential curves describing DR in 0, and based on a C3s,2p,ldI 
contracted Gaussian basis set and first order CI wave functions have 

been described previously. 73 These calculations showed that the ' Zi 

state is the only state that can generate O('S) from the low vibrational levels. It crosses the ion between the outer turning points of the 

v=l and v=2 vibrational levels and dissociates to the 'D + 'S asymptote. The next accessible state is 5'IIg which crosses the outer turning 

point of 1-10. Among the states dissociating to O('S) there are three 

routes which cross the a411u metastable state between the turning points 

for PO. These routes with their dissociation limits shown in parentheses are 5'IIg('P + 'SI, 5'll,('P + 'SI, and 4'IIg('D + 'SI. Because of its 

spin symmetry, the state will have only a small DR rate coefficient from the metastable state. The remaining two states can lead to 

hot 0 atoms, each with 3.4eV kinetic energy. Future experiments 

designed to measure the 'S quantum yield must be careful to be certain 

that no a4nU 0; is generated which would interfere with the determination of the 'S generated from the ion ground state. 

For generating O('D), in addition to the 'Cs state, the B'Z; state 

crosses the ion near the inner turning point of v=O and leads to 'P + 

'D atoms. A 'Au state leading to two 'D atoms crosses the ion near the 

outer turning point of v-0. The lowest valence state of 'IIg symmetry 

crosses the ion near the inner turning point of v=O and can lead to 'D 

atoms via a curve crossing with the 2'IIg state. At higher energies, the 

lowest 'ifg state crosses the ion near the inner turning point of the 

v=3 ion level and can lead to 'D atoms via an avoided crossing with the 

2'11g state. The next accessible states for 'D production are 2'Cp + 

'PI, 2'IIg('D + 'PI, and 2 'llU('D + 'PI near the outer turning point of 

v=3, 3'IIg('D + 'P) and 2'IIg('D + 'D) near the outer turning point of 

v=4. As has been pointed out earlierl71, several of these routes are 

expected to have small electron capture widths. 

The reader is referred to Ref. 17 for a discussion of additional 

routes that lead to only ground state atoms. 

The important B'ZI;, 'Au, and 'Zt states identified above have been 

the subject of larger scale calculations involving [6s,3p,2d,lfI 

contracted Gaussian basis sets and CI wave functions that include all 

single and double excitations to the virtual ~pace.l~1,~~1 Second order 
Fig. 1. Dissociative first order (dashed) and second order 

(solid) potentials for 0, plotted with the ion potential 

from Ref. 15. 

curves for the llg states have not been obtained because of their small 

widths (see below). Both the earlier curves and the large scale curves 

are plotted in Figure 1. The ion curve is the RKR15] curve plotted at 

the experimental ionization potential above the large scale calculated 

ground state. The ion curve has been shifted to larger R to compensate 

for the difference between the calculated and experimental ground state 

, 

Re. In order to plot the first order curves on the same figure they 

have betn shifted to larger R to correct for the difference between the 

first order and second order calculated Re's. Also, the first order 

curves are shifted in energy so as to be plotted relative to the first 

order g ound state. Relative to the ion, after the shift in the energy 

and distance coordinates, the curves are qualitatively similar. The 

largest difference is for 'Au for which the second order curve is 

shifted by about 0.02a0 to smaller R near v=O relative to the first 

order rI?sult. 

4. ELECTRON CAPTURE WIDTHS FOR 0, 
electron capture or autoionization width, r, a matrix element of the 

Hamiltonian operator, H, given by Fermi's golden rule: 

Yp is the wave function of the properly antisymmetrized product of a 

Itfree" electron in a coulomb like orbital and the wave function for the 

ion. Yf is the electronic wave function for the neutral state which 

describes the DR products and p is the density of states. Rydberg 

orbitals with high principal quantum numbers, n, have been used to 

describe the coulomb like orbital. Except for the Rydberg orbital, all 

the orbitals in (4) are bound and have large amplitude only near the 

molecule. As a result, it is only important to know the amplitude of 

the Rydberg orbital near the molecule. The inner part of coulomb 

orbitals are well known to be very similar to the inner part of high 

Rydberg orbitals.20] With the coulomb orbital represented as a high 

Rydberg orbital all the orbitals in (4) are bound and are represented 

in terms of Gaussian basis functions. 

For n*>>R, where R is the angular momentum quantum number, the 

matrix element in (4) can be closely represented by 

I <Yf I HI Yp> I - (1 /n*3) kexp(-c/n*) (5) 

where k and c are constants determined from the calculation of the 

matrix elements on the left side of (5) for the highest Rydberg states. 

These Rydberg states are nearly hydrogenic and the energies can be 

represented as E=-(1/2(n*I2) where n*=n-6 and 6 is the quantum defect. 

The density of states can be written as 

p = l/(E(n*-1/2) - E(n*+1/2)) 

which can be closely represented as 
r - 21rkexp(-(c/n*)-(.5/n*~)). 

Taking the limit of' n* -+ gives the threshold capture width, 

r = 21rk. (9 1 

The Rydberg wave functions have been refined further by optimizing them 

in the field of an optical potential due to the Yf states. Feshbach 

projection operators21] have been defined which partition the total 

space such that Y = PI + QY where PY contains the Rydberg states to be 

optimized and QY are the Yf states described above. In the usual 

Feshbach projection operator formalism, an eigenvalue equation is 

derived for PI that is diffic'ult to solve because it contains an 

energy-dependent optical potential. However, using partitioning 

PY can easily be determined by solving the usual CI 

problem. Writing the total wave function as Y = PY + QY, we can write 

the Schroedinger equation in matrix form as 

("u ::) I:) = E [I) 

where Hpp = PHP, HQP = QHP, etc. 

Multiplying the matrices in (10) leads to: 

HppPY + H~QQY = EPY 

and 

HQPPY + HQQQY = EQY. 

(10) 

From Eq.(12) we have, 

QY - HgpPY/(E - HQQ). (13) 

Substituting Eq. (1 3) into Eq. (1 1 ) le.ads to a matrix optical potential 

for PY, 
, It is difficult to solve (14) directly for PY since E is on both sides 

of the equation. Nevertheless, PY can be determined by retaining the 

coefficients of the PI terms resulting from the diagonalization of the 

H matrix in Eq.(lO). Before diagonalizing H we must project out the low 

energy QY roots in order to prevent them from mixing into PY. There- 

- 

~ 

fore, it is necessary to first solve for the QY roots by diagonalizing 

HQQ. The HQQ portion of the H matrix in Eq.(lO) is transformed to 

project out the physically meaningful QY roots. PI is then determined 

by diagonalizing the new H matrix with the transformed HQQ. The net 

effect is to provide additional correlation to the PY space from the 

nonphysical roots in the Q space, PY's for successive principal quantum 

numbers are used to represent YE in Eq.(4). The widths obtained by this 

procedure are then extrapolated to the continuum as described above. 

The importance of the correlation added to the P space from the higher 

terms in Q space has been emphasized by Hazi who has described an 

approach similar to that used here.23] 

An application of this approach to the calculation of the widths of 

the B and L 'II states of NO has already been An 

additional example of this approach is given in Table 1. for the 0, 3JIg 

width. This is the only 0, width for which there are experimentally 

derived matrix elements for comparison. An analysis of the released 

kinetic energy line width for the v=l level of the C3ng state of 0, 

indicates that the electronic matrix element between the C Rydberg 

state and the lowest valence 311g state is 0.079eV.8] An additional 

determination based on the experimental line widths of the lowest 4 

vibrational levels of the C state obtains a matrix element of 

0.0625(+0.006, -0.004) eV .93 

, 

All calculations in Table 1. have been done at R=2.2819a0. A 

[3s,2p,ld] basis set on each center was supplemented with a set a 18 

diffuse s Gaussians at the midpoint and the Rydberg orbitals were 

determined by the Improved Virtual Orbital meth0d.~5] The 'lIg valence 

state was described with a full valence CI consisting of 46 terms. For 

the C state, a valence CI on the ion ground state was coupled with the 

Rydberg orbital and each of the three valence virtual ag orbitals 

leading to a 636 term CI. Using the procedure outlined above leads to 
the widths shown in Table 1. Using the widths for n=8,9 gives the 

expression, rp 0.00217exp~1.39/n*-(0.5/n*z)). The extrapolated width 

for n*- is 0.00217eV. This width is about two orders of magnitude 

smaller than the widths for the three important dissociative routes 

identified above. As a result, the Illg state was not included in the DR 

rate coefficient calculations discussed belowTable 1. Electron-Ion Capture Widths (r) from Rydberg States 

n n* p(eV--') rW> 

9 7.8897 17.903 0.00257 

8 6.8896 11.892 0.00263 

7 5.8902 7.4022 0.00272 

6 4.891 3 4.21 10 0.00285 

5 3 A930 2.0972 0.0031 1 

4 2.8959 0.84005 0.00367 

3 1.8984 0.21774 0.00522 

The matrix element calculated here for na3 is 0.0617eV. Since the 

electronic matrix element varies with R, the experimentally derived 

matrix element is an averaged value. However, since the variation of 

the matrix element with R over the relevant range of nuclear vibrations 

is estimated to be less than about lo%, the results show that relatively small wave functions can yield interaction matrix elements that 

are in good agreement with experiment. 

5. DR RATE COEFFICIENTS FOR 0, 

The expression for the direct DR cross section derived by Giusti261 

has been used to calculate cross sections and rate coefficients for 

generation of O(lS) and O('D) along the 'Z;, BIZ;, and 'Au diabatic 

potential curves described above. For DR leading to O('S), 181 the v=2 

level of the ion has the highest rate coefficient for the lowest eight 

vibrational levels and is a factor of 78 greater than the coefficient 

from v=O at an electron temperature (Te) of 300K. A comparison of the 

calculated rate coefficients to those deduced from atmospheric models . 

shows that the atmospheric models apply to molecular ions which are not 
% in a Boltzmann distribution but which are instead significantly 

populated in the v>O levels. In contrast to the situation for O(’S) 

generation, the total rate coefficient19] for producing O(’D) is 

greatest from v=O and is a factor of 1.9 greater than the coefficient 

from v=2 at Tez300K. The rate coefficient for v=O is in good agreement 

with those deduced from atmospheric models. The calculations show that 

the dominant route for producing O(’D) from v=O is ’Au and ’Z; is the 

Table 2. DR Rate Coefficients for Production of O(’S) for 6006TeS1000K. 

Ion Vibrational Level Rate Coefficient (crn’/se~)~ 

0 0.24(+. 12,-.08) C-0.141 

1 5.0(+1.2,-1 .1 )C-0.361 

2 15.(+2. ,-3. )C-0.501 

3 4.8(+3.3,-2.3>C-0.771 

4 4.9 (+0.6, -1 .2) [-0.321 

5 1.2(+1.9,-0.9)c-0.901 

6 5.1 (+0.6,-1 .1)[-0.56] 

7 1.4(+0.9,-0.9)[+0.06] 

a. All rate coefficients have been multiplied by TO9. The exponent of 

the temperature dependence is in square brackets. The rate coefficient 

for v=O is 2.4(+1.2,-0.8)x10-’0x((Te/800)-0~’~)~ms/se~. 

dominant route from v=l and v=2. 

Since the electron temperatures in ionospheric models are near 800K, 

we list in Table 2. the rate coefficients for producing O(’S) for 

600STeSlOOOK for each of the lowest eight ion vibrational levels. 

Table 3. DR Rate Coefficients for Production of O(’D) for 60OSTeS1000K. 

Ion Vibrational Level Rate Coeff icient(cm’ /secIa 

0 1.37(+.14,-.16)C-0.511 

1 1.09 (+ .12 ,-. 14) C-0.551 

2 0.78 (+ . 1 3 ,-. 1 6) [ -0.4 1 1 

a. All rate coefficients have been multiplied by 10’. The exponent of 

the temperature dependence is in square brackets. The rate coefficient 

for v=O is 1 .37 (+ .I 4 ,-. 1 6)xi O-’X( (T,/800)” 

Because of the position of the intersection of the dissociative curve 

with the v=5 level, the coefficient from v55 is highly sensitive to an 

energetic displacement of the dissociative route within its expected 

uncertainty (0.leV). Table 3 has the O(’D) rate coefficients in the 

)cm’/sec. 
same temperature range from each of the lowest three ion vibrational 

levels. 

6. DISSOCIATIVE ROUTES FOR N, 

Preliminary results are reported here which identify the important 

dissociative routes for the vi0 level of the ground state of N:. These 

results are based on large scale second order CI calculations Using 

atomic natural orbital basis ~ets.~7] The details will be reported 

s e par a t e 1 y . 

2P), 3lIIU('D + 'D), and C'Ag('S + 'D), are shown in Fig. 2. 

The dominant dissociative routes from v=O, 2lL';('D + 'D), C''Ilu('S + 

The dominant configuration of the 2'L$ state is 

where . . . . denotes (lag)2(lau)2(2ag)2(2u~)2. This configuration differs 

by a triple excitation from the dominant configuration of the ion plus 

a "free" or Rydberg electron, 

The matrix element of the Hamiltonian between these two configurations 

is zero and these dominant terms will not contribute to the electron 

capture width. The magnitude of the width will be determined by matrix 

elements between secondary terms in both states and may be small. As a 

result, 2'Z; may make only a small contribution to the total DR rate 

coefficient from v=O. 

The C state also passes through the v-0 level of the ion and 

dissociates to 'S + 'D giving atoms with 1.72eV kinetic energy from 

v=O. The dominant configurations of this state, 

c 
differ from the'ion plus a continuum electron by a triple excitation. 

Therefore as for Z2ZiJ the dominant configurations will not contribute 

__ - -- _. - 

0 

1.7 2.1 2.5 

INTERNUCUAR DISTANCE 

Fig. 2 Dissociative routes for the v=O level of Nb. 

The ion potential is from Ref. 14. 

to the width and this state is likely to have a small width and a small 

DR rate coefficient. This is an important conclusion for the escape of 

N atoms from Mars since this state and possibly the C' state (through 

an avoided crossing with the C state) are the only states that can 

supply enough kinetic energy for escape from v=O. 

The primary configuration of the C' and 3'nU states, 

differs by a double excitation from the ion ground state plus a 

continuum electron. The capture width for this state may be larger than 

that for 2lI; and G3Ag and it appears that these 'nu states may be the 

dominant states for DR from 1-0. The importance of the 'I, states is in 
c 

good agreement with the results of Michels.28] The C' state has an 

avoided crossing with the lower C'nU state which dissociates to 'S + 'D 

atoms. The determination of the states of the product atoms due to 

initial dissociation along the C' state is currently under study. 

7. THE ROLE OF RYDBERG STATES IN DIRECT DR OF Ni 

Rydberg state potential curves having the ground state of the ion as 

core cannot cross the ground state of the ion and in addition have very 

small electron capture widths. These states are usually not important 

for direct DR electron capture. However, because the first excited 

state of Ni is only 1.1 eV above the ground state, Rydberg states having 

the excited state core can cross the ground state of the ion and 

provide routes for DR. 

For 'E; symmetry, an N, Rydberg state can be formed by binding a mu 

Rydberg orbital to the excited A2nU state of Ni. This state will have 

an avoided crossing with the 2'E; state described above and can provide 

a channel for DR. A comparison to analogous states in 0, indicates 

that the 4pn 'E: state having the A core will pass through the ground 

state of N: near the equilibrium separation. If the 4pn state 

should prove to have a large capture width, this route could be an 

important route for DR in Ni. The calculation of the potential curve 

for this state and the electron capture width are in progress. While 

the Rydberg states are diffuse, their primary configurations differ 

from the ion by a double excitation and they may have larger electronic 

widths than the valence state which has primary terms that differ 

from the ion plus a continuum electron by a triple excitation. It 

appears that a multistate treatment involving both the valence states 

and the excited core Rydberg states may be needed to calculate the 

cross sections for DR of NZ. 

8. ACKNOWLEDGEMENT 

This research was supported by the National Science Foundation under 

grant ATM-8616776 and by the National Center for Atmospheric Research 
which is sponsored by the National Science Foundation. Support was also 

provided by NASA Grant NACW-1404, by the Air Force Office of Scientific 

R'esearch under grant AFOSR-84-0109 and by NASA Cooperative Agreement 

NCC 2-308. 

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I 

b