Volume 3. number 2 CHEMICAL PHYSICS LETTERS

a

February 1969

POTENTIAL ENERGYCURVES FOR THE HNO MOLECULE*

A. W. SALOTTO * * and L. BURNELLE New York University. De@vFment of Chemistry, New York. N. Y.. 10003. US4

Received 20 January 1969

Potential curves for the HNO molecule calculated by the unrestricted Hartree-Fock method are given. Com- parisons are made with experimental spectroscopic and kinetics results, and possible mechanisms explain- ing the observed ieatures are presented and discussed.

Although the spectrum of the FIN0 molecule

has been observed In both absorption and emis-

sion [I-31, little information is available on the potential surfaces for the system H + NO. Pre- dissociation has been observed in both absorption [2] and emission [3] but little is lmown about the

exact nature of the predissociation. Perturba- tiorls have been observed [I] in an excited state of the HNO molecule and here again little of the nature of the perturbing state has been estab- lished.

The kinetics of the reaction

H+NO+M-HNO+M

have been studied by Clyne and Thrush 141. Their interpretation of the results has led them to propose a set of potential curves, the validity of which has not yet been established. It is clear, therefore, that a theoretical approach to the problem of the potential surfaces might be help- ful in clarifying a number of the above-mentioned points.

Suitable methods have been lacking for theore- tical ah initio quantum mechanicai calculations of the potential surfaces for polyatomic systems. In particular, the application of the configuration interaction method in a rigorous manner is im- practical because it gives rise to prohibitive computations. We have, consequently-, looked into the use of the unrestricted Hartree-Fock (UHF) method for calculating potential curves. It is to be noted that the results of the UHF meth- od are not as good as those obtained by configu- ration interaction. This is especially true in the region of the potential minimum where the UHF

* Work supported by the U. S. A i-my Research Office - mrham.

** National Science Foundation Graduate Trainee.

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method obtains only a small fraction of the cor- relation energy.

The UHF method does have the advantage, as opposed to the restricted Hartree-Fock method, that for diatomic and those polyatomic molecules so far studied the correct dissociation products are obtained [S]. The method is characterized by the fact that calculated states need not be eigen- states of S8. The only case in which an eigen- state would have been expected to result from a calculation is at the equilibrium configuration for a closed shell system.

In attempting to correct for the above-men- tioned loss of spin eigenstate, projection proce- dures have been proposed. Amas and Hall [6] have proposed a procedure to calculate the ener- gy utilizing a single annihilation of the contami- nating state of next higher multiplicity. Harriman [7] and Hardisson and Harriman [8j have pre- sented a scheme for the complete projection of the spin eigenstate. Both methods of correcting do so after the optimization procedure has been completed and thus do not obtain the lower ener- gy possible by projecting prior to optimization.

In a preliminary study, the four methods men- tioned above (restricted, unrestricted, single annihilation, and complete projection) were each used to calculate disssociation curves for both the hydrogen molecule and the lithium hydride mole- cule. From an examicrltion of the results we have come to the conclusion that the UHF method gives the best general shape. Calculations of force constants and anharmonicity constants from the UHF curves confirm their adequacy.

For calculations on the HNO molecule a gaussian basis set of ?s, 3p on nitrogen and on oxygen, where the exponents for the atoms were optimized by Hornback [9] and 4s, lp on hydrogen,

Volume 3. number 2 CHEMICAL PHYSICS LETTERS February 1969

I I.5 2 RN-H (ii)

2s

Fig. 1. Computed potential energy diagram for the H + NO system.

as optimized for the atom by HLzinaga [lo], was the calculated molecular geometry and the corre- chosen. Integral evaluation over the basis func- sponding experimental values are presented in tions was by means of the Polyatom [ll] program. table 1.

Four states of the HNO molecule were calcu- lated. The ground state was found to be of 1~’ symmetry and to have the following electronic structure (if one uses the notation proper to the restricted method): (la’)2(2a’)2(3a’)2(4a’)2(5a’)2(6a’)2(law)2(7a’)2 ~~I”(8a’)0. The three excited states treated are

“, IA” and 3A’. The first two result from the single excitation 7a’ - 2a”, the last one from the excitation 7a’ - 8a’, both from the ground state.

Calculations show that the IA’ state at the equilibrium configuration has a value of (S2) equal to 0.32. This result was unexpected inas- much as we have found no other cases reported in the literature in which the unrestricted method has yielded a value other than zero at the equili- brium configuration for a molecule with an even number OF electrons and lacking degeneracy. For the HNO molecule the value of $2) decreases to- ward zero as the molecule is compressed along the NO axis and the energy rises.

Two adjustments made in the preparation of the potential curves presented in fig. 1 should be noted. As mentioned earlier, the IA’ ground state curve has a value of (Sz) equal to 0.32 at the minimum energy. In order to eliminate the ef- fect oi contaminants of higher multiplicity, a complete projection of the wavefunction [7,8] was carried out at the minimum of the curve. This resulted in a significant lowering of the energy; the UHF energy was equal to - 129.597 atomic units and the projection brought it down to - 129.623 a-u. This latter quantity provided a dissociation energy of 0.072 a.u. or 15800 cmmL_ This is to be compared with the value 17000 cm-l corresponding to the region where predissociation has been observed [2,3] and which is thought to be close to the dissociation energy. After pro- jection, the IA’ curve was adjusted at points be- low the dissociation energy by the same propor- tion as the minimum energy was changed.

The equilibrium configuration was determined by varying all three parameters: N-H bond length, N-O bond length and the bond angle. ‘Values for

A second adjustment was made in the IA” ex- cited state curve. Since this state is an open shell the average value of S2 calculated by the UHF method must of necessity be greater than

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Volume 3. number 2 CHEMICAL PHYSICS LETTERS February 1969

Table 1 I-IN0 molecular geometry

theoretical experimental [ll

RNH 1.038 ii 1.062 A

RNO 1.321 w 1.211 h

CHNO 110.4O 108.5O

one. ft is found in our calculation to be 1.02 at the energy minimum. This high value indicates that there is a large amount of mixing due to higher spin states, changing the energy signifi- cantly. Our projection program as presently written does not handle this case adequately. In order to fix better the position of this curve, the difference in energy between the 3An and the 1A” was determined using the virtual orbitals re- sulting from a restricted calcuiation on the 1A’ state at its energy minimum. As with the lAf curve, the iA” was then adjusted at points below the dissociation energy by the same proportion as the minimum energy was changed.

The resulting curves in fig. 1 show all four states converging to ground state products H(2S) and NO(gfl). We note mum appears in the i

however, that a slight maxi- A” curve. Whether a slight

maximum appears in the IA’ or 38” curve cannot be unambiguously determined from these calcu- lations. From theoretical considerations it is ex- pected that all the HNO states (LA’, IA-*, 3A’, SAN) formed from ground state products H(%) and NO(2ll) would be entirely repulsive. The theore- tically obtained 3A’ curve shown in fig. 1 is in agreement with this conclusion. The ground and low lying states at the HNO equilibrium configura- tion, IA’, 3A”, lA”, on the other hand, should correlate with H(2S)‘and excited N0(2A), (2X-) and (2A) states respectively. It is apparent, then, that the three curves, IA’, 3A”, IA”, shown in fig. 1 are each the result of an application of the non-crossing rule. Each of them actually results from the interaction of two states: a repulsive one going to the ground state NO and a bound state gcing to excited state NO. ln each case, our meth- od yields only the lower perturbed state. Our cal- culational methods yield no direct hint of the avoided crossing, with the exception of the aforementioned maximum In the IA” curve. A maximum or other irregularity would be observed in a curve when the interaction between the two unperturbed curves is very we&. Apparently this is the case with the 1A” state, while the interactions are greater for the 1A’ and 3AH states.

Dalby [l] has observed perturbations in some

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rotational levels of the 1A” state, the maximum magnitude of which was about 1 cm-I. Dalby sug- gests that the perturbations could arise from in- teractions with the 3A” state or with higher vi- brational levels of the IA’ state. The theoretical results presented here leave undetermined the exact state causing the observed perturbation. Interaction with either of the lower states is for- bidden according to electronic selection rules and any perturbation would be expected to be weak as indeed Dalby has observed.

Predissociation in the HNO spectrum was first noticed by Clement and Hamsay [3] in their emission study. Two bands of the I-IN0 molecule near 17000 cm-1 as well as three bands of the DNO molecule show a sharp breaking-off in the K-rotational structure. This is considered to be due to predissociation in the excited IA” state. Since the absorption spectrum as observed by Bancroft, Hollas and Bamsay [2] continues well above the assigned predissociation value, it is clear that the phenomenon is indeed predissocia- tion, not direct dissociation. Bancroft, Hollas and Bamsay [2] have found a “distinct though slight” diffuseness of rotational lines in one of the HNO bands. The observed predissociation is thus reported to be weak.

The theoretical curves indicate that the pre- dissociation is probably due to the small magni- tude of the interaction between the two closely approaching perturbed IA” states. That there is a maximum present at all in the lower IA” state helps confirm that the approach is G’CF~ and the interaction small. Herzberg [12] classifies this type of predissociation as type I, viz., where there is a finite probability for the vibrating molecule to pass from the lower curve to the upper one and back again. This could be thought equivalent to the molecule remaining as it vi- brates on the original unperturbed curve which has an HNO minimum and dissociates into ex- cited NO products.

In a kinetic study of the overall combination and light emission in the reaction of H and NO, Clyne and Thrush [4] have determined the rate constant for the formation of excited HNO (lA”) through a three body collision

They come to the conclusion that the latter re- action has a negative activation energy. They assume the presence of a large maximum in the potential energy curve of +he IA” state due to the avoided crossing; they explain the observed pre- dissociation and the ne tive activation energy by postulating that the P A* curve intersects the

. Volume 3. number 2 CHEMICAL PHYSICS LETTER8 February i969

IA’ curve at a point below the dissociation ener- gy, and presumably at an NH internuclear dis- tance smaller than equilibrium. As appears from fig. 1, we do not find an intersection between the 3A” and IA” curves in the cross section consi- dered here. Our calculations thus provide no support Ear Clyne and Thrush’s interpretation, although we carnot rule out the possibility of an intersection of the tx-? hypersurfaces elsewhere. As far as the cegitive activation energy is con- cerned, the sm;ll size of the maximum obtained in the computed IA” curves is in no way incon- sistent with this experimental fact.

from the New York University A.E.C. Computing Center is gratefully acknowledged.

REFERENCES [l] F.W. Daiby, Can. J. Phys. 36 (1958) 1336. 121 J. L. Bancroft. J. M. Hollas and D.A. Ramsay. Can. - - J. Phys.40 (1962) 322. [3] M. J. Y. Clement and D.A. Ramsay, Can. J. Phys. 39

(1961) 205. 141 M. A.A. Clyne &Id B. A-Thrush. Disc. Faraday Sot.

33 (1962) 139. [5] D. E. Ellis, Ph D.Thesis, bIaesaci-~~,-Xs mstitute

of Technology, 1966. [6] A. T.Amos and G. G. Hall. Proc. Roy.Soc. A263

(1961) 483.

ACKNOWLEDGEMENTS

We are thti. 1. to Dr. A.M. May, who has contributed signitmantly to the pro&rams used in this work. We also thank Dr. J. W. Moskowitz and the members of his group for providing us with the latest versions of the Polyatom system

[i’] j. E. Harriman, J. Chem. Phys.40 (1964) 281. 181 A. Hardisson and J. E. Harriman. 6. Chem. P&s. 46 . ,

(1966) 3639. IS] C. J. Hornback. Ph. D. Thesis, Case Institute of

Technology, 1967. [lo] S. Huzinaga. J. Chem. P&s. 42 (1965) 1293. (111 I. G. Csizmadia, BI. C. Harrison. J.W. Xoskotitz

and B. T. Sutcliffe. Theoret. Chim. Acta 6 (l966} 191. [12] G. Herzberg, Electronic Spectra of Polyatomic

of programs. Finally, a grant of computer time Molecules (Van Nostrand, Princeton, N. J. , 1966).

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