Ab InitioPotential Energy Surface and Internal Torsional–Wagging States of Hydroxylamine

JOURNAL OF MOLECULAR SPECTROSCOPY 186, 162–170 (1997)ARTICLE NO. MS977425

Ab Initio Potential Energy Surface and InternalTorsional–Wagging States of Hydroxylamine

Jan Makarewicz,* Marek Kre– glewski,* and Maria Luisa Senent†

*Faculty of Chemistry, A. Mickiewicz University, Grunwaldzka 6, 60-780 Poznan, Poland; and†Instituto de Estructura de la Materia, C. S. I. C., Calle Serrano 123, 28006 Madrid, Spain

Received May 1, 1997; in revised form August 7, 1997

The two-dimensional potential energy surface describing the interaction of the large-amplitude torsional and waggingmotions in hydroxylamine has been determined from ab initio calculations. This surface has been sampled by a largeset of grid points from a two-dimensional configuration space spanned by the torsional and wagging coordinates. Ateach grid point, the geometry optimization has been performed using the second-order Møller–Plesset perturbationtheory with the basis set 6-311 / G(2d , p) . At the optimized geometry, the single-point calculation of the electronicenergy has been carried out using a larger basis set 6-311 / G(3df , 2p) . This method was verified to yield the resultscomparable to those obtained by a direct optimization of the geometry with the basis set 6-311 / G(3df , 2p) whichhad been used by A. Chung-Phillips and K. A. Jebber (1995. J. Chem. Phys. 102, 7080–7087) to calculate the energiesof only three points in the potential energy surface of hydroxylamine. The trans and cis local minima have been foundon the determined potential energy surface. The localization features of the torsional–wagging states have been studiedby solving the two-dimensional Schrodinger equation for the coupled torsional and wagging motions. q 1997 Academic Press

INTRODUCTION theory (MP2) with the 6-311G(d , p) basis set, yielded thetorsional potential with only one minimum at the trans con-formation of NH2OH. However, such a basis set appearedHydroxylamine (NH2OH) is a favorite model molecule

for experimental and theoretical studies of the interaction to be insufficient to provide a correct shape of the potentialenergy surface (PES). This fact has been recently revealedbetween the internal motions. This is one of the simplest

molecules exhibiting two strongly anharmonic modes, the in the most accurate to date calculations with the basis set6-311 / G(3df , 2p) , where an additional local minimumlow-frequency torsion and the wagging mode of the amino

group NH2. Such modes were intensively investigated in at a cis conformation (10) had been found.The goal of this paper is to calculate the reliable torsionalmethyl amine and hydrazine which have higher symmetry

than NH2OH. From spectroscopic investigations of NH2OH, potential energy function for the hydroxyl torsion of NH2OHby using a high level of the ab initio theory. Furthermore, tovery little is known about its internal dynamics. The equilib-

rium structure of NH2OH was determined long ago from the describe correctly the tunneling motions between the trans andthe cis local minima, the wagging mode will be considered.microwave spectra (1) . The fundamental frequencies of the

vibrational modes of NH2OH, except for the not observed Thus, the two-dimensional PES for the torsional and the wag-ging motions will be calculated and then used for determinationtwisting mode, were assigned by using the infrared spectros-

copy (2–4) . Taubmann and Jones observed the diode laser and analysis of the torsional–wagging internal states.spectrum in the region of the stretching, wagging, and tor-sional bands (5) . The spectrum in the region of the OH AB INITIO CALCULATIONS OF THE POTENTIALbending mode was reported by Birk and Jones (6) . Recently, ENERGY SURFACEthe lowest torsional bands have been studied in detail withthe high-resolution infrared spectroscopy (7) . However, the To construct the PES, the electronic energy of NH2OH

was calculated on a discrete set of grid points lying in thepotential energy along the torsional mode has not been de-rived from these data. two-dimensional space spanned by the torsional and the wag-

ging coordinates. The definition of these coordinates and ofThe theoretical ab initio calculations of the equilibriumgeometry and vibrational frequencies of the normal modes the remaining internal coordinates, which specify the config-

uration of the NH2OH molecule, is illustrated in Fig. 1. Eachhelped in a correct vibrational analysis for NH2OH (8, 9) .Significantly less work has been devoted to determine the configuration of NH2OH is specified by four bond lengths,

as the stretching coordinates, and five angles. The anglestorsional potential. Recent calculations of this potential (9) ,employing the second-order Møller–Plesset perturbation a and b are defined as the bond angles NOH and HNH,

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TORSIONAL–WAGGING STATES OF HYDROXYLAMINE 163

the MP2 method (full treatment) was used with the basis set6-311 / G(2d , p) , further called basis I. At the optimizedgeometry, the electronic energy was recalculated at the MP2level (full treatment) with the basis set 6-311 / G(3df ,2p) , hereafter basis II. All computations were carried outusing the GAUSSIAN 92 package programs (11) .

Such a method was chosen after a good number of testcalculations. Two sets of d polarization functions and one

FIG. 1. Definition of the angular internal coordinates of NH2OH. set of diffuse functions on the heavy atoms were necessaryto obtain good values of the optimized angular coordinates.The equilibrium geometry for the cis and trans conformers

respectively. r is the angle between the bisector of the HNH optimized with the bases I and II is compared in Table 1.angle and the O–N bond and is used to describe the NH2 For a given conformer, the difference between the two setswagging. t is defined as the dihedral angle HONX, where of parameters optimized by using two bases, I and II, isX is the point lying on the HNH bisector. Naturally, this lower than 1%, i.e., not larger than the difference betweenangle is zero for the cis configuration of NH2OH. t describes the geometry calculated with the larger basis II and the exper-the internal rotation (torsion) of OH relative to NH2. The imental geometry. It is interesting to notice that the geometryangle g describes the NH2 twisting and is defined as p– d, optimized with the smaller basis I is closer to the experimen-where d is the dihedral angle ONXH. tal geometry than that optimized with the larger basis II.

Since we constructed the two-dimensional PES, the two The length of the N–O bond was especially sensitive tocoordinates t and r were treated as dynamical variables and high angular momentum orbitals of heavy atoms. It seemsthe remaining ones were considered as the flexible geometry that inclusion of more d and f orbitals or eventually addingparameters. This means that for each fixed point (t, r) the g orbitals would be essential in improving the results, but

they are not available in the GAUSSIAN 92 program.geometry parameters were optimized. In the optimization,

TABLE 1Comparison of the cis and the trans Structures of Hydroxylamine, Optimized Using

the Treatment MP2(full ) /6-311 / G(2d , p ) and MP2(full ) /6-311 / G(3df , 2p )

Copyright q 1997 by Academic Press

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MAKAREWICZ, KREGLE– WSKI, AND SENENT164

TABLE 2Molecular PI Symmetry Group G4 , Isomorphic to C2h , and Its Action

on the Internal Coordinates of NH2OH

Note. The symmetry types of the trigonometric functions for n Å 0, 1, . . . are given in the last column.

We checked at several points (t, r ) that the energy for actions of the PI operations on the large amplitude motioncoordinates, t and r, are summarized in Table 2. It is im-the geometry optimized with basis II is systematically lower

by only 20–30 cm01 than the energy calculated with the portant to notice that the remaining internal coordinates alsotransform under the action of the operations from G4 . Thebasis II using geometry optimized with basis I. The specific

numerical results are available from the authors on request. transformation rules for g and r{ Å r(NH(1) { r (NH(2))are given in Table 2. The new variables r/ and r0 are moreThus, the simpler treatment MP2(full) /6-311 / G(3df ,

2p) / /MP2(full) /6-311 / G(2d , p) gives for NH2OH the convenient than the original coordinates r(NH(1)) andr (NH(2)) since they transform under the symmetry opera-results comparable to the most accurate results obtained up

to date (10) by the optimization with basis II. tions in a simple way, similar to the familiar symmetricand antisymmetric stretching normal modes. The remaininginternal coordinates are fully symmetric, so they are notANALYTIC REPRESENTATION OF THE PESincluded in this table. The symmetry properties of standardtrigonometric functions, employed in the fitting of the PES,The study of the torsional–wagging states of NH2OHare also presented in the table.requires the representation of the PES in the form of an

The PES calculated on a discrete set of 130 grid pointsanalytic function of two variables t and r. Such a function(tn , rn) in the range 07 £ tn £ 1807 and 07 £ rn £ 1107must possess correct symmetry features reflecting the dy-was fitted to the fully symmetric analytic expressionnamical symmetry of the molecule. Because the torsional

and the wagging modes are large amplitude motions, thesymmetry theory for nonrigid molecules has to be used. The

V (t, r) Å ∑4

nÅ0

R2n(r)cos 2ntmolecular symmetry group of NH2OH can be described inthe framework of the permutation-inversion (PI) formalism(12, 13) . A very rigorous approach of Gunthard et al. (14 ) / ∑

3

nÅ0

R2n/1(r)cos(2n / 1)t, [1]or the physical operations description of Smeyers (15 ) couldalso be applied to NH2OH. However, for such a simplemolecule the standard PI formalism seems to be the most whereconvenient.

In our problem only two feasible symmetry PI operationsR2n(r) Å ∑

4

kÅ0

a(2n , k)cos krgenerate the molecular PI group, G4 , namely the permuta-tion P of the hydrogen nuclei of NH2 and the inversion E*of all particle Cartesian coordinates. Here, the group G4 is / ∑

8

kÅ5

a(2n , k)cos kr exp(0ckr2) [2]

considered to be isomorphic to the point group C2h . The


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FIG. 2. Schematic contour plot of the ab initio torsional–wagging PESFIG. 4. The optimized scissoring angle b as function of the torsionalof NH2OH.

and wagging dynamical variables t and r.

minima, along the t coordinate, exhibit an energy of 2530andcm01 . The critical points (t, r) Å (07, 07) and (07, 1807)have an energy of 3930 cm01 .

R2n/1(r) Å ∑5

kÅ1

a(2n / 1, k)sin kr It is interesting to point out that the PES is constant alongthe line r Å 07. This property follows from the fact thatV (t, r) is calculated for the optimized angle g(t, r) , which/ ∑

8

kÅ6

a(2n / 1, k)sin kr exp(0ckr2) . [3]

adjusts to given r and t. For r Å 07, the optimum configura-tion of NH2OH is singular, with the OH bond perpendicularto the ONH2 plane. For such a configuration g takes theThe numerical values of the coefficients a(n , k) and ck arevalue g Å t if t £ 907 or g Å t 0 1807 if t ú 907. Thus,available from the authors on request. This analytical func-g is discontinuous at t Å 907. Near the discontinuity, it wastion reproduced the ab initio PES at the grid points with thenecessary to optimize g on a quite dense set of the griderror of about 1 cm01 , except for only a few points lyingpoints (tn , rn) , so the initial set was extended. The unusualfar from regions of the local minima. Similar expressions,behavior of g as function of t and r is depicted in Fig. 3.adapted to the proper symmetries, were used to fit the opti-It should be stressed that despite of the discontinuity of g,mized geometry parameters as functions of t and r.the Cartesian coordinates of all nuclei change continuouslyThe shape of V (t, r) is illustrated in Fig. 2. Two localwith variations of t and r. Thus, the pathologic behavior ofminima trans on the PES, at t Å 07, r õ 07 and t Å 1807,g does not produce singularities in the torsional–waggingr ú 07, are equivalent due to the symmetry. For the sameHamiltonian.reason, the two cis minima are equivalent too. The cis min-

Figure 4 shows the strong relaxation of the scissoringima lie 1510 cm01 above the trans minima. The equivalentsaddle points located on the paths from the trans to the cis

FIG. 5. The potential function V (t Å 907, r) of NH2OH calculatedwith the optimized twisting angle g ( the solid line) and with g fixed at 07FIG. 3. The optimized twisting angle g as function of the torsional and

wagging dynamical variables t and r. ( the dashed line) .


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TABLE 3 ging model. Thus, the effective one-dimensional torsionalHarmonic Frequencies (cm01 ) of Normal Modes Calcu- potential was calculated by treating the wagging coordinate,

lated for the cis and trans Conformers of NH2OH by Em- r, as the geometry flexible parameter which was optimizedploying the Treatment MP2(full ) /6-311 / G(2d , p ) together with all others. Using the results of this optimiza-

tion, the torsional potential and the geometry parameterswere fitted by using trigonometric functions of a propersymmetry. In this way, the flexible model for the torsionalmotion was determined. This means that all components ofthe nuclear position vectors, rna , were defined as functionsRna(t) of the torsional coordinate t. These functions andtheir derivatives are necessary to calculate the kinetic energytensor G(t) which specifies the kinetic energy operator.The quantum mechanical Hamiltonian for the torsional mo-tion can be expressed by the tensor G in the form (17)

HO (t) Å 0 12

\ 2 d

dtGtt(t)

d

dt/ U0(t)

[4]

/ dU1(t)dt

/ V (t) ,

where V (t) is the torsional potential energy function andU0 and U1 are the pseudo-potential terms given by

U1(t) Å 12

\ 2Gtt(t)gt(t) [5]angle b with variations of t and r, as described in methylamine (16) . The relaxation of the remaining geometry pa-rameters is much weaker, so it is not presented graphically.

andThe geometry relaxation is very essential for NH2OH.Neglecting it produces too high potential energy in the regionnear r Å 07. This fact can be seen in Fig. 5, where two U0(t) Å U1(t)gt(t) [6]potential functions calculated for t Å 907 with and withoutrelaxation of g are compared. The difference between both

withpotentials for r Å 07 is about 3000 cm01 .In the calculated PES the contribution from the zero-point

vibrational energy of small-amplitude normal modes, E0(t,r) , is not taken into account. To estimate this energy, the gt(t) Å 0 1

4d

dtln g(t) , [7]

harmonic frequencies of the normal modes were calculatedfor the trans and cis minima. The results are collected inTable 3. Comparison of the harmonic frequencies for both where g(t) is the determinant of G(t) . The method of aminima indicates that E0(t, r) lowers the cis minimum by fully numerical derivation of the tensor G(t) and of theabout 70 cm01 relative to the trans minimum. Thus, the pseudo-potential terms directly from the functions Rna(t)modification of the whole torsional–wagging PES by E0(t, was described in Ref. (17) .r) , should be of the same order of magnitude. The eigenvalues and eigenfunctions of the torsional Ham-

iltonian H(t) were calculated by diagonalizing the Hamilto-ONE-DIMENSIONAL MODELS FOR THE TORSIONAL nian written in the Meyer’s basis functions (18) . In that

AND WAGGING MOTIONS basis, all matrix elements of an arbitrary one-dimensionalHamiltonian can be evaluated very easily. We applied thecomputer program, described in detail in Ref. (17) , to solveOne-dimensional models for the torsional and wagging

motions in NH2OH are very helpful for understanding the eigenvalue problem for the Hamiltonian H(t) and obtainedconvergent eigenvalues with about 40 basis functions.results obtained from the two-dimensional torsional–wag-


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The first lowest torsional states are depicted in Fig. 6,together with the torsional potential. Since the local minimaof the torsional potential are relatively deep, the torsionalwavefunctions resemble the harmonic oscillator wavefunc-tions localized around the trans or cis minimum. Thus, thequantum numbers nt ( trans) or nt (cis) were assigned tothem. The index nt counts the nodes of torsional wavefunc-tion inside of the trans or of the cis local minimum.

An analogous one-dimensional model for the waggingmotion was defined by fixing the torsional coordinate at thevalue t Å 07. The wagging potential function with the firstlowest wagging wavefunctions is illustrated in Fig. 7. Thelowest wagging states lying below the potential barrier are FIG. 7. The first wagging states localized on the trans minimum (solidalso localized around the trans or cis minimum. They can lines) or on the cis minimum (dashed lines) of the one-dimensional wag-

ging potential for NH2OH. The first state above the potential barrier isbe assigned by quantum numbers, analogous to those useddelocalized.for the torsional states.

It is important to notice that the one-dimensional torsionaland wagging states have a broken symmetry, because theydo not transform according to irreducible representations of The basis functions fn(t) were determined from the one-the symmetry group C2h . However, they properly describe dimensional torsional eigenequation with the model torsionalthe localization of the states. So, they help us to understand potentialthe character of two-dimensional wavefunctions.

V0(t) Å (Vb /2)(1 0 cos 2t) [9]TWO-DIMENSIONAL MODEL FOR THE COUPLED

TORSIONAL AND WAGGING MOTIONSwhich is invariant under the operations of the symmetrygroup C2h . This potential with the barrier Vb Å 2500 cm01

To solve the two-dimensional problem for the coupledis similar to the torsional potential obtained from ab initiotorsional and wagging motions, the Hamiltonian for the flex-calculations only near the trans local minima.ible model was constructed numerically on a discrete set of

The basis functions cm(r) were determined from the one-the grid points which were already determined for the one-dimensional eigenequation with the model wagging potentialdimensional models. The eigenvalues and eigenfunctions of

this Hamiltonian were calculated by applying the variationalW0(r) Å Wb(1 0 (r /re ) 2) 2 . [10]method with the basis functions of the form

Fn ,m(t, r) Å fn(t)cm(r) . [8] This potential with the barrier Wb Å 3900 cm01 resemblesthe ab initio wagging potential near the trans local minimum,but is symmetric with respect to the operation s:r r 0r.

The choice of the fully symmetric model potentials forgenerating the one-dimensional basis functions allowed fora simple selection of the two-dimensional basis functions ofcorrect symmetry. The basis functions in a simple productform of Eq. [8] have already good symmetry types de-pending on combinations of the numbers n and m .

Let us write n as n Å 4i / gt , where i is 0 or a naturalnumber. Then, for gt Å 0, 1, 2, and 3 the torsional basisfunctions fn(t) can be classified according to the irreduciblerepresentation of C2h :Ag , Bu , Au , and Bg , respectively.

The wagging basis functions should be classified withrespect to the symmetry group Cs Å {E , s}. Let m Å 2i /gr , then for gr Å 0 and 1 the functions cm(r) are symmetricand antisymmetric with respect to s, respectively.FIG. 6. The first torsional states localized on the trans minimum (solid

Now, the symmetry types of the two-dimensional func-lines) or on the cis minimum (dashed lines) of the one-dimensional tor-sional potential for NH2OH. tions Fn ,m(t, r) can easily be determined by taking into


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TABLE 4 blets arise due to a localization of the wavefunctions inSymmetry Types of the One-Dimensional (gt , gr) two symmetrically equivalent minima trans or cis. Theand of the Two-Dimensional (G ) Basis Functions splitting is caused by the tunneling motions between the

minima. There are two types of doublets depending on thesymmetry of the states, namely (Ag , Bu ) and (Au , Bg ) . Thewavefunctions of the first doublets are shown in Figs. 8a–8d. The strong localization of the states is clearly visible.As a consequence of this localization, the approximatequantum numbers, considered in the case of the one-dimen-sional torsional and wagging models, can be ascribed totwo-dimensional wavefunctions. These quantum numbers,nt and nw , are indicated for torsional–wagging states col-lected in Table 5. Only one component of each doublet, Ag

or Au , is included into the table, because the doublet split-ting for these states is on the order of magnitude from 1006

to 1003 cm01 .account the symmetry types gt and gr of the one-dimen- The one-dimensional torsional (Et ) and wagging (Ew) en-sional basis functions. They are summarized in Table 4. ergies, calculated from the models depicted in Fig. 6 and

For each symmetry type, the Hamiltonian matrix was Fig. 7, are also included Table 4. The interaction of theformed and diagonalized to obtain the torsional–wagging torsional and wagging modes is noticeable when comparingenergy levels and wavefunctions. It was necessary to take Et and Ew with the exact energy E of the torsional–waggingabout 250 functions of each symmetry type to obtain the state. As an example, let us consider the state localized onenergies with an accuracy of 1007 cm01 for the lowest the cis minimum with nt (cis)Å 0 and energy EtÅ 1512 cm01

states. Such an accuracy was necessary because the states and the close lying state localized on the trans minimum withnt ( trans) Å 4 and energy Et Å 1523 cm01 ; see Fig. 6. Theform doublets with extremely small splitting. These dou-

FIG. 8. The torsional–wagging wavefunctions for the first lowest states calculated from ab initio PES of NH2OH: (a) the state Ag(nt , nw) Å (0, 0)trans, (b) the state Bu(0, 0) trans, (c) the state Au(1, 0) trans, and (d) the state Bg(0, 0) trans.


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TABLE 5 sults comparable to those obtained by a direct optimizationEnergies (E in cm01 ) of the Torsional–Wagging States of of the geometry with 6-311 / G(3df , 2p) basis set.

NH2OH Calculated from the ab initio PES and Labeled by The results of the geometry optimizations revealed unusu-the Symmetry Types and the Quantum Numbers for the One- ally strong relaxation of the angle g ( twist of NH2) and bDimensional Torsional (nt ) and Wagging (nw) Motion (scissoring of NH2). Thus, we expect that a four-dimen-

sional model including the g and b coordinates as dynamicalvariables would be useful in more accurate description ofthe interaction between the NH2OH internal modes.

To study the global features of the torsional–waggingPES, the obtained values of the potential energy and ofthe geometry parameters were fitted to symmetry adaptedanalytical functions. The trans and cis local minima werefound on the PES. To check whether these minima are deepenough to bind the localized states, the torsional–waggingHamiltonian was derived numerically for the flexible geome-try and, then, its eigenfunctions were calculated using thevariational method.

The analysis of the feasible PI symmetry group of NH2OHallowed us to simplify the solution of the torsional–waggingproblem and to classify the states. The lowest states, local-ized in the trans or cis minima, appeared to be arranged intodoublets with extremely small splittings. This finding is inagreement with the experiment because any tunneling split-tings were not observed in the spectra of NH2OH.

The torsional excited states, determined from the ab initioPES, appeared to be systematically higher in energy thanthe corresponding observed states. The one-dimensionalmodel for the torsion described the lowest states quite well.For example, the energy of the first exited torsional stateobtained from the one- and two-dimensional models wasdetermined to be 411 and 419 cm01 , correspondingly. Thesevalues are lower than the fundamental torsional frequencyenergies of these states, calculated from the two-dimensionalof 432 m01 , calculated from the harmonic model. Naturally,model, are 1488 and 1543 cm01 . Thus, it is clear that thethis fact is caused by the anharmonicity of the torsionaltorsional–wagging interaction in these states is strong sincemode.the separation of these states is much larger than that ob-

The ratio of the calculated to the observed torsional energ-tained from the one-dimensional model. This interaction alsoies is about 0.92 for the first three excited torsional states.enhances the delocalization of some states. For example,This result can be extrapolated to improve higher calculatedtwo closely lying states nt ( trans) Å 5 and nt (cis) Å 1 aretorsional states. For example, the fourth torsional statemore delocalized than the corresponding states calculatednt ( trans) Å 4 should be located at 1420 cm01 .from the one-dimensional model.

The first excited wagging state calculated from the abinitio PES was lower in energy (1033 cm01) than the corre-

CONCLUSIONS sponding observed state (1115 cm01) . The calculated har-monic frequency of 1155 m01 for the wagging mode deviatesconsiderably from the values determined from the PES. ItThe two-dimensional torsional–wagging PES of NH2OH

was determined for the first time from ab initio calculations, is clear that this deviation cannot be attributed only to theanharmonicity, but also to the dynamical interaction of theby taking into account the electron correlations at MP2 level.

This PES has been calculated on a discrete set of the torsional wagging mode with the remaining internal modes, whichwere not included explicitly in the two-dimensional model.and wagging coordinates (t, r) . For each point (t, r) con-

sidered, the geometry parameters were optimized with the The effects discussed in this work are analogous to thosefound recently for methyl amine (16) .basis set 6-311 / G(2d , p) . At the optimized geometry,

single-point calculations were performed using the larger The present findings may help in a search for excitedtorsional–wagging states localized in the cis potential energybasis set 6-311 / G(3df , 2p) . This procedure yielded re-


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5. G. Taubmann and H. Jones, J. Mol. Spectrosc. 123, 366–381 (1987).wells and in the explanation of some perturbations observed6. H. Birk and H. Jones, J. Mol. Spectrosc. 129, 333 (1988).in the spectra of hydroxylamine.7. B. Hanoue, I. Morino, and K. Kawaguchi, J. Mol. Spectrosc. 174, 172

(1995).8. N. Tanaka, Y. Hamada, and M. Tsuboi, Chem. Phys. 94, 65–75 (1985).ACKNOWLEDGMENTS9. J. Tyrrell, W. Lewis-Bevan, and D. Kristiansen, J. Phys. Chem. 97,

12768–12771 (1993).The authors acknowledge financial support from the European Commu- 10. A. Chung-Phillips and K. A. Jebber, J. Chem. Phys. 102, 7080–7087

nity in the form of a Human Capital and Mobility Network SCAMP (Con- (1995). [See references cited therein]tract CHRXCT93-0157 and its supplementary agreement CIPDCT94- 11. M. J. Frish, G. W. Trucks, M. Head-Gordon, P. M. Gill, M. W. Wong,0614). The authors thank Professor Y. G. Smeyers for reading and com- J. B. Foresman, B. G. Johnson, H. B. Schlegel, M. A. Robb, E. S.menting on the manuscript as well as for his hospitality during the visit of Replogle, R. Gomperts, J. A. Andres, K. Raghavachari, J. S. Binkley,one of the authors (J.M.) in the Instituto de Estructura de la Materia in C. Gonzalez, R. L. Martin, D. J. Fox, D. J. Defrees, J. Baker, J. J. P.Madrid. Stewart, and J. A. Pople, ‘‘GAUSSIAN 92,’’ Gaussian, Pittsburg, 1992.

12. H. C. Longuet-Higgins, Mol. Phys. 6, 445–469 (1963).13. P. R. Bunker, ‘‘Molecular Symmetry and Spectroscopy,’’ Academic

REFERENCES Press, New York, 1979.14. H. Frei and Hs. H. Gunthard, ‘‘Lecture Notes in Physics,’’ Vol. 79,

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Ab InitioPotential Energy Surface and Internal Torsional–Wagging States of Hydroxylamine