4480
J. Phys. Chem. 1985, 89, 4480-4483
An ab Initio Study of Isomerization O n the HONO System Almon G. Turner* Frank J . Seiler Research Laboratory, United States Air Force Academy, Colorado Springs, Colorado 80840-6528 (Received: April 12, 1985)
Ab initio calculations, Maller-Plesset perturbation theory carried to fourth order including triples, were used to study the cis-trans isomerization of HONO. The geometries were completely optimized, and the trans-cis energy differencewas calculated to be 3.5 kJ/mol. The activation energy for the process was found to be 53.5 kJ/mol. It was found necessary to employ a very large basis set, a doubly split valence level with complete polarization, and a set of diffuse functions on both heavy and hydrogen atoms in order to obtain a reliable estimate of the energy difference between the isomers. The calculations also indicate that a reexamination of the structural assignments for both isomers might be in order. A harmonic analysis of the rotational barrier is carried out and its origin discussed. The barrier is principally traceable to the distortion in electronic structure which occurs under rotation and can be ascribed to large contributions from the ionic structure H-O-N=O+.
Introduction Nitrous acid, HONO, is a system which offers an interesting area for ab initio study. Existing in both cis and trans f o r m , the structures of which are both known experimentally’ and characterized spectroscopically,2 H O N O provides a nontrivial ground for testing the accuracy of present-day ab initio quantum chemical calculation. The rotamers of HONO have been previously studied at the S C F level by Radom, Hehre, and P ~ p l eSchwartz, ,~ Hayes, and Rothenberg: Skaarup and B o g g ~ Farnell ,~ and Ogilvie,6 and others.’ Together these studies indicate the trans isomer to be more stable than the cis isomer by about 2 kcal/mol, although the size of the calculated barrier is strongly influenced by the choice of basis set. The only post-Hartree-Fock calculation to date has been that of Benioff, Das, and Wahl,8 wherein a 21configuration optimized valence configurations calculation attributed the origin of the trans-cis barrier to the presence of repulsions involving the oxygen atom lone-pair electrons. It is significant that Alleng has predicted on the basis of an energy component analysis that the cis-trans barrier will be attractive dominant for this system. This prediction will be found to be in accord with the calculations presented herein. It is fair to state that the origin of the rotational barrier is presently not clear, being unattributable to steric factors, hydrogen bonding, dipole-dipole interactions, or purely electrostatic effects. In this paper we shall reexamine the isomerization problem, and we shall attempt to determine the origin of this phenomenon by the analysis of post-Hartree-Fock ab initio calculations.
Previous study” in our laboratory has demonstrated the necessity for using a proper basis set in a study of an isomerization of this type. A basis set of at least split-level quality is required. Since we are interested in the relative energies of the three isomers involved-the trans “reactant”, the cis “product”, and the =transition state”-we can anticipate that we shall require a good description of the inner-shell orbitals and a fully polarized set of valence orbitals. Accordingly, calculations were carried out with the 3-21G,I2 6-31G(d),13 6-31G(d,p),I4 6-31 l G ( d , ~ ) , and ’ ~ ~631 1++G(d,p)15 basis sets. The calculations were done using the GAUSSIAN 82 computer program as obtained from Professor Pople. l6 Calculations were performed at the Hartree-Fock level with a gradient-based full-geometry optimization using the 631G(d) basis set. Calculations were then carried out through the post-Hartree-Fock fourth order of Mdler-Plesset perturbation theory. Using the HF/6-3 1G(d) geometries, the later calculations included all single, double, triple, and quadruple excitations and did not involve a frozen core approximation. The necessity for the inclusion of the triple substitutions in the calculation of the fourth-order perturbation correction to the energy has been demonstrated by previous workers.I7 These contributions must
Calculations Our approach is to perform a series of ab initio molecular orbital calculations using Mdler-Plesset perturbation theory,I0 employing a large enough basis set that we can confidently be certain that we have calculated the proper rotational barriers. This barrier will then be resolved by a harmonic Fourier analysis in terms of the rotational angle and its components examined. Previous studies have shown that the algebraic sign of the rotational energy change is not correctly predicted by S C F calculations with minimal basis sets.’ Calculations using small basis sets predict the cis form to be the conformation of lowest energy. As the size of the basis set is increased, the trans form becomes the conformer of lowest energy (see Table I). However, these conclusions are all based upon structures which were not fully optimized, and since the difference in energy between the two isomers is very small, the conclusions should be taken very cautiously. Indeed, we shall show below that when geometries are properly and fully optimized, the cis form is usually predicted to be the lowest energy conformer, the trans form becoming so only when a very large basis set is used together with higher order perturbation theory.
548. (3) Radom, L.; Hehre, W. J.; Pople, J. A. J . Am. Chem. SOC.1971, 93, 289. (4) Schwartz, M.; Hayes, E.; Rothenberg, S. Theor. Chim. Acta 1970, 19, 98. (5) Skaarup, S.; Boggs, J. J . Mol. Struct. 1976, 30, 389. (6) Farnell, L.; Ogilvie, J. F. Proc. R. SOC.London, A 1982, 381, 443. (7) Kleier, D. A.; Lipton, M. A. THEOCHEM 1984, 109, 39. (8) Benioff, P.; Das, G.; Wahl, A. C. J . Chem. Phys. 1976, 64, 710. (9) Allen, L. C. Chem. Phys. Lett. 1968, 2, 597. (IO) Merller, C.; Plesset, M. S. Phys. Rev. 1934, 4 4 , 618. ( 1 1 ) Turner, A. G. Inorg. Chim. Acta 1984, 84, 85. (12) Binkley, J. S.; Pople, J. A.; Hehre, W. J. J. Am. Chem. Sor. 1980, 102, 939. (13) Hariharan, P. C.; Pople, J. A. Theor. Chim. Acta 1973, 28, 211. (14) (a) Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A. J . Chem. Phys. 1980, 72, 650. (b) Hariharan, P. C.; Pople, J. A. Theor. Chim. Acta 1973, 28, 213. (15) Frisch, M. J.; Pople, J. A.; Binkley, J. S. J. Chem. Phys. 1984, 80,
*National Research Council Associate, 1984-1985. Present address: Department of Chemistry, University of Detroit, Detroit, MI 48221.
( I ) (a) Dor, L.; Tarte, P. Bull. Soc. R.Sci. Liege 1951,12,685. (b) Jones, L.; Badger, R.; Moore, G. J. Chem. Phys. 1951, 19, 1599. (2) (a) McGraw, G. E.; Bernitt, D. L.; Hisatsune, I. C. J. Chem. Soc. 1966, 45, 1392. (b) Hall, R.; Pimental J . Chem. Phys. 1963, 58, 1889. (c) Finnigan, D. J.; Cox, A. P.; Brittain, A. H. J . Chem. Sor., Faraday Trans. 1972, 68,
3265. (16) Pople, J. A.; Binkley, J. S.; Frisch, M. J.; DeFrees, D. J.; Raghavachari, K.; Whiteside, R. A.; Schlegel, H. B.; Fluder, E. M.; Seeger, R. GAUSSIAN 82 Computer Program, Release A, Carnegie-Mellon University, Pittsburgh, PA, 1983. (17) (a) Krishnan, R.; Frisch, M.; Pople, J. A. J . Chem. Phys. 1980, 72, 4244. (b) See also: Frisch, M. J.; Binkley, J. S.; Schaefer, H. F. Ibid. 1984, 81, 1882.
This article not subject to U S . Copyright. Published 1985 by the American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 21, 1985 4481
Isomerization in the H O N O System
TABLE I: HONO Optimized GeometriesC
NO ON HO ON0 HON
cis 1.161 1.327 0.910 11.39 107.6
this work” trans transition state 1.148 1.153 1.393 1.347 0.954 0.951 111.7 111.4 106.9 105.3
dihedral angle
0
180
88.5
ref 5 cis 1.179 1.425 0.972 113.1 108.5
trans 1.169 1.447 0.962 110.4 105.2
0
180
exptlb transition state 1.167 1.496 0.969 111.2 107.3 94.5
cis 1.186 1.399 0.989 113.6 103.9
trans 1.169 1.442 0.959 110.6 102.1
0
180
‘Calculated via the Hartree-Fock method using a 6-31G(d) basis set. bReference 2c. These are “averaged” structures determined from a quadratic potential function based upon determination of 11 of 16 possible force constants. cBond lengths in angstroms and angles in degrees.
be included to obtain a reliable activation barrier for a process of this type. The calculations were carried out serially in the sense that a basis set was selected and the Hartree-Fock calculation carried out followed by the MP4SDTQ calculation. A MP2 calculation was then performed with the next largest basis set and the results were combined, for example, MP4/6-31G(d,p) MP2/6-31 l++G(d,p) - MP2/6-31G(d,p). In this manner, we are able to avoid having to carry out the MP4 calculations for the largest basis set. The calculation of the trans-cis barrier was calculated in the following manner: MP4SDTQ/6-3 1lG(d,p) energies were calculated for five values of the dihedral angle, &the complement of the angle between the HO vector and a perpendicular to the O N 0 plane. The reactant, product, and transition states were calculated in their 6-3 1G(d) optimized geometries, i.e. MP4SDTQ/6-31 lG(d,p)//6-31G(d). The remaining points were calculated by using the geometry calculated for the transition state at the 6-31G(d) level but relaxing the rotational angle. These choices ensure that the reaction path calculated passes through the true transition state and at the same time passes close to the correct reactant and product states. The assumption made is that the stationary points calculated with the 6-31G(d) basis set are close in geometry to those calculated with the 6-31 lG(d,p) basis set. Primarily, on the basis of the magnitude of stretching type force constants, we estimate that the errors introduced by this factor should be a few tenths of a kcal/mol. For example, a representative stretching force constant in H O N O is about 0.01 hartreelbohr. Thus, an error of 0.05 A in a bond distance would introduce a change in energy of about 0.01 X 0.1 or 0.001 hartree, i.e. about 0.5 kcal/mol (see ref 18). The difference in geometries obtained by the use of the 6-31G(d) basis set in place of the 6-31 lG(d,p) is less than 0.05 A. The resulting energies were fitted to the equation
+
4
E ( @ = E a , cos (ne) n=O
(1)
in such a manner that the agreement between the equation and the numerical data was exact for the five data points. The five points corresponded to 0 values of 0, a / 4 , n/2, 3 ~ 1 4 and , no.
Results and Discussion Geometries. In Table I, we show the Hartree-Fock optimized geometries obtained for the cis or synperiplanar, trans or antiperiplanar, and transition state, respectively, together with the experimental spectroscopic average structures. A number of points can be made. The stationary-state calculations reported herein allowed for a full optimization of geometry. Nothing was assumed concerning the geometry, and care was taken not to start the calculation in the neighborhood of a saddle point on the potential energy surface. Thus, we have no reason to believe that our calculated optimized geometries do not correspond to true minima on the potential surface. There is some disagreement between our calculated structures and the experimental spectroscopic (18) This estimate is based upon the results obtained by Krishnan et al. (Krishnan, R.; et al. J. Am. Chem. SOC.1984,106, 5853) for a rather large number of 4-7-atom molecules. The work of Harding et al. (Harding, L. B.; Schlegel, H. B.; Krishnan, R.; Pople, J. A. J . Phys. Chem. 1980,84, 3394) also supports this estimate.
structures-differences which we believe to lie beyond our error. In accord with the work of Collins et al., who used similar basis sets, we estimate that our bond distances are reliable to *0.03 and that our bond angles are reliable to * 5 O . I 9 For both the cis and trans forms, it would seem as if the experimental single-bond distances are uniformly too long by about 0.05 8, while the N=O distance seems to be about correct. Since the experimental structures are not known very well and the best structures seem to be those of ref 2, the spectroscopist might consider a reexamination of the spectra and use this work as a guide for the development of the required force fields. In fairness one should note that the experimental system is a very complex one complicated by the presence of NO, NO2, N203,N 2 0 4 ,etc. The data of Table I also indicate some differences in our geometries and those determined by Skaarup and B ~ g g s .The ~ values calculated for the N - O single-bond length are about 0.1 8, shorter than those previously calculated. We can only attribute this difference to our use of a larger basis set. It is well-known that geometries based upon S C F calculations with minimal or small basis sets frequently differ from those determined experimentally. The problem seems to be more severe for distances associated with multiple bonds than for lengths associated with single bonds.19 The effect of the split valence orbital sets on the nitrogen and oxygen atoms seems to be to shorten the N-0 bond length. The calculated bond orders for the three structures are trans form:
N=O = 0.696; N-0
transition state: cis form:
= 0.189; H-0
N=O = 0.628; N - 4
N=O = 0.662; N-0
= 0.508
= 0.166; HO = 0.482
= 0.236; H-0
= 0.450
These values all indicate that the N - 0 bond, which involves the oxygen atom bonded to hydrogen, is somewhat less than an ordinary single bond. We remark in passing that as one passes from the trans form through the transition state and on into the cis form, the bond between the nitrogen atom and the oxygen atom bonded to the hydrogen atom weakens passing through a minimum in the transition state and becomes stronger again in the cis form. The same thing happens to the N=O bond except that it ends up weaker in the cis form than it was in the trans form. There is a progressive weakening of the 0-H bond as one passes from the trans form to the transition state and on into the cis form. The sum of bond orders is 1.393 for the trans form, 1.276 for the transition state, and 1.348 for the cis form. These are consistent with the trans isomer being the more stable conformer.20 Energetics. In Table I1 we display the results of various calculations of the total electronic energy for the cis and trans isomers of H O N O together with the energy of the transition state. The latter is the correct saddle point as evidenced by the presence of one and only one negative eigenvalue for the force constant matrix.21 Our calculations extend well beyond those previously From the Hartree-Fock calculations presented, we (19) These estimates are based upon the comparison of the results obtained from HF/6-3 1G(d) calculation with experimental geometries in molecules of known geometry as tabulated in the Carnegie-Mellon Quantum Chemical Archive, 2nd ed., July 1981, Carnegie-Mellon University, Department of Chemistry, Pittsburgh, PA. As such, they represent “averaged” values, and the deviation in any particular molecule could be larger or smaller.
4482
The Journal of Physical Chemistry, Vol. 89, No. 21, 1985
Turner
TABLE I 1 Total Energy Calculation for HONOP calculation HF/STO-3G HF/3-21G HF/4-31G HF/Gaussian lobe,b double-{ HF/Gaussian lobe: double-{ HF/6-3 1G(d) HF/6-3 1G(d,p)//6-3 1G(d) HF/6-31 lG(d,p)//6-31G(d) HF/6-3 11 ++G(d,p)//6-31G(d) OVCCI//4s2p/2s* MP206-31G(d) MP3/6-3 1G(d) MP4/6-31G(d) MP2/6-3 1G(d,p)//6-3 1G(d) MP3/6-3 1G(d,p)//6-3 1G(d) MP4/6-31G(d,p)//6-3 1G(d) MP2/6-31 l++G(d,p) MP4/6-3 lG(d,p)+ MP2/6-31 l++G(d,p) -MP2/6-3 1G(d,p)
cis -201.911 03 -203.468 50 -204.31089 -204.3 12 17 -204.4140 -204.639 94 -204.646 14 -204.701 21 -204.706 46 -204.765 53 -205.15955 -205.15553 -205.188 72 -205.17072 -205.16692 -205.200 44 -205.281 89
trans -201.90676 -203.46601 -204.3 1 1 9 1 -204.312 13 -204.417 3 -204.637 68 -204.643 99 -204.699 89 -204.706 65 -204.769 55 -205.157 14 -205.153 84 -205.186 25 -205.16841 -205.165 32 -205.19806 -205.283 30
transition state
-204.298 34 -204.402 9 -204.619 64 -204.626 30 -204.682 26 -204.688 72 -204.751 32 -205.136 12 -205.134 89 -205.16568 -205.14790 -205.14684 -205.17796 -205.262 67
Etram - Eci, +18.6 +6.5 +9.3 0.1 1 -9 +5.9 +5.6 3.5 -0.5 -10.5 +6.3 +4.4 +4.9 +6.1 +4.2 +6.3 -3.7
-205.311 61
-205.31295
-205.29273
-3.5
50
-2.5'
57.8'
-203.452 94
Etarrier
41 36 53 52 50 47 48 62 54 61 60 53 60 50
"Reference 3. These calculations did not include a full optimization of geometry but were carried out for an "idealized" structure. bReference 5 . This geometry employed herein was optimized for each H O N angle selected. The trans form had an optimized HON angle of 105.2O, while cis structure was optimized for an H O N angle of 108.5O. 'Reference 2a. "Reference 8. Utilized the geometry of ref 5 . C T ~ t aenergies l in hartrees; relative energies in kJ/mol. /Experimental value.
TABLE 111: Correlation Energies (in hartreed' isomer basis set cis 6-3 1 G(d) cis 6-31G(d,p) transition state 6-31G(d) transition state 6-3 lG(d,p) trans 6-3 1G(d) trans 6-3 lG(d,p)
MP2
MP4LSDO)
MP4SDTO
A
0.51961 0.524 58 0.51648 0.521 60 0.519 46 0.524 42
0.528 76 0.533 75 0.527 51 0.532 58 0.528 89 0.533 89
0.548 78 0.554 30 0.546 04 0.551 66 0.548 47 0.55407
0.553 75b 0.605 15' 0.551 13b 0.604 01' 0.553 53b 0.606 30'
4 T h e ~ eenergies are computed by performing the indicated calculation and subtracting the Hartree-Fock result using the same basis set. bMP4SDTQ/6-31G(d) MP2/6-31G(d,p) - MP2/6-31G(d). 'MP4SDTQ/6-3IG(d,p) MP2/6-31 l++G(d,p) - MP2/6-31G(d,p).
+
note that most of the S C F calculations indicate the cis isomer to be energetically lower than the trans isomer. It is only when we introduce diffuse functions into the basis that the energy of the trans form falls below that of the cis form at the Hartree-Fock level. We estimate the HartreeFock limit to be about -204.708 for both isomers. This in turn implies an energy difference of 2 kJ/mol (0.5 kcal/mol) or less between the two forms. This is in excellent agreement with the 0.506 f 0.25 kcal/mol difference previously suggested by Jones, Badger, and Moore.lb This estimate, of course, totally ignores the effects of electron correlation. Post-Hartree-Fock calculations have been carried out to the point where a large fraction of the correlation energy has been included. For example, at the MP4SDTQ/6-31G(d,p) level we estimate the fourth-order correction to the energy to represent all but about 8% or 14 mhartrees out of a total correction of 190 mhartrees. In Table 111, we present an analysis of the correlation energy for the three forms. At the highest level of calculation, the correlation energies of the three forms differ by about 1 mhartree, the largest correlation energy being associated with the more stable trans conformer and the smallest being associated with the looser bound transition state. Using the 6-31G(d) basis set, we calculated the zero-point energy of the two forms. The zero-point energy of the cis form is 14.53 kcal/mol, and that of the trans form is 14.48 kcal/mol. The zero-point energy difference is 0.2 kJ/mol. Thus, the zero-point energy difference contributes about 6% of the cis-trans energy difference. From Table I we can see that the best estimate of the energy difference between the cis and trans isomers is about 3.5 kJ/mol with the trans form being the more stable. This is in reasonable agreement with experiment, experimental estimates varying from 2.5 to 12 kJ/mol with the lower values probably being the more reliable.'** Other Properties. Table IV lists the rotational constants calculated for the three isomeric forms of HONO. Farnell and
+
TABLE I V Rotational Constants for HONO 6-31G(d) Basis Set (in GHz)" cis trans transition state calcd exptl calcd exptl calcd A 12.219 11.3643 11.937 11.1339 11.886 B 14.154 13.1690 13.519 12.6345 13.177 C 89.383 84.1018 101.95 93.6834 88.809 Experimental values taken from ref 2c.
TABLE V Dipole Moments for HF/6-3lG(d,p)//6-31G(d) Basis Set (in D)" cis trans transition state calcd exotl calcd exotl calcd Vx -0.297 0.306 0.724 1.383 0.512 1.579 1.394 2.513 1.347 1.771 VY 0 0 0 0 0 v, Ut,,,, 1.607 1.428 2.616 1.930 0.521 ~~
"Experimental values taken from: Allegrini, M.; Johns, J. W.; McKellar, A. R.; Pinson, P. J . Mol. Spectrosc. 1980, 79, 446.
Ogilvie6have previously calculated these quantities for the cis and trans isomers, obtaining 85.993, 13.316, and 11.530 for the cis isomer and 97.941, 12.783, and 11.307 for the trans isomer. These constants are related in the Born-Oppenheimer approximation to the reciprocals of the moments of inertia. The agreement between our calculated values, experimental values, and those values calculated previously is good. The largest error (cis, 6%; trans, 8%) is along the direction of the smallest principal axis of inertia, while the error along the larger principal axes is much smaller. Table V gives the calculated dipole moments for the three forms of HONO. The agreement between theory and experiment is very
Isomerization in the H O N O System TABLE VI: Ionization Potentials for HONO (in lOI9 kJ/molecule)a cis electron ionized 3-21G 6-31G(d,p) 1Oa’ 20.37 20.28 2a” 22.75 23.06 9a’ 25.76 26.31 1a” 29.32 30.80 8a‘ 31.96 32.93 7a’ 32.89 34.19
The Journal of Physical Chemistry. Vol. 89, No. 21, 1985 4483
trans 3-21G 19.38 22.29 26.38 27.35 30.50 36.62
transition stateb 6-31G(d,p) 19.56 22.76 26.76 30.38 31.93 34.94
3-21G 19.43 22.63 25.21 29.81 31.03 33.00
6-3 1G(d,p) 19.71 23.01 25.69 30.92 32.14 34.21
a 1 au = 2625.925 kJ/mol. bListed are the IPS for the six highest lying occupied MOs. The transition state belongs to the point group C,; hence, the a’-a” distinction is not applicable here.
TABLE VII: Fourier Coefficients for the Rotational Barrier i ai I ai 0 -0.517066 3 +0.241348 1 -0.246 561 4 -0.172 062 E(TS) -204.682 258 2 -0.000 687
good for the cis isomer, 1.428 vs. 1.607 (13% error). The agreement for the trans form is poorer, 2.616 vs. 1.930 (36% error). These results are consistent with the conclusion that a reexamination of the structural assignments is probably required. The discrepancy between the calculated and experimental results is beyond the error of the calculation. In Table VI we tabulate the calculated ionization potentials using the 6-31G(d,p) basis sets. These numbers should be of interest to the mass spectroscopist, photoelectron spectroscopists, and others interested in the ionization process. The Nature of the Rotational Barrier. In Table VI1 we list the Fourier expansion coefficients calculated for eq 1. We note the barrier has a large zero-, first-, third-, and fourth-order component and virtually no contribution from the second-order harmonic. The contributions from terms of n = 1 and n = 3 are very nearly equal but opposite in sign, indicating the symmetric character of the potential surface as it moves along the rotational coordinate in each direction. The barrier is to a large degree simply a sum of the zero- and fourth-order components. Various estimates have been given for the barrier. These estimates range from 12 kcal/mol (50 kJ/mol) by Jones, Badger, and MooreIb and 8.7 kcal/mol (36 kJ/mol) by D’or and Tartela to 9.7 kcal/mol(41 W/mol) by Hall and PimentalFb We calculate a bamer of 50 kJ/mol in agreement with the experiments of Jones, Badger, and Moore. The origin of the barrier has been attributed to the interaction of oxygen atom lone pairs: nitrogen atom nonbonded pairs,2 and the contribution of an ionic structure such aslb ti
\0-N=Ot
The structure calculated herein for the transition state gives some credence to the latter proposal. The isomerization process is accompanied by a decrease in the N=O distance and, at the same time, an increase in the N-0 distance. At the same time the 0-H distance is lengthened until finally in the transition state it attains a value of 0.954 A, a value closely resembling that of 0-H in the hydroxide ion (see Table I). In Figure 1, we diagram the change which occurs in the electronic structure of the system
1.327
0.
111
Figure 1. Diagrammatic representation of the trans-cis isomerization in HONO: geometries, 6-31G(d); energetics, MP4/6-31G(d,p) + MP2/ 6-31 l++G(d,p) - MP2/6-31G(d,p).
as the H O N O undergoes rotation from the trans to the cis form.
Summary Ab initio molecular orbital methods extended to the MP4SDTQ level when used with a sufficiently large basis set, 6-3 1l++G(d,p), are capable of describing the rotational isomerization of HONO, leading to values for the energy difference between isomers (cis-trans, 3.5 kJ/mol) and activation energy (53.6 kJ/mol) which are in excellent accord with experiment. The results can be utilized to evaluate experimental data and lead to an understanding of the effects of rotation upon the electronic structure of the system. Acknowledgment. The author recognizes the National Research Council, National Academy of Sciences, and the Office of Scientific Research, U S . Air Force, for the award of a NASA, National Research Council Associateship. He also thanks the staff of the Frank J. Seiler Research Laboratory, US.Air Force Academy, for the hospitality provided to him. He acknowledges the expert typing of Mrs. Deborah Landess. Registry No. HONO, 7782-77-6.
