Threedimensional quantum calculation of the vibrational energy levels of ozoneO. Atabek, S. MiretArtes, and M. Jacon Citation: The Journal of Chemical Physics 83, 1769 (1985); doi: 10.1063/1.449365 View online: http://dx.doi.org/10.1063/1.449365 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/83/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Exact three-dimensional quantum mechanical calculation of ozone photodissociation in the Hartley band J. Chem. Phys. 114, 10651 (2001); 10.1063/1.1374580 Quantum threedimensional calculation of endohedral vibrational levels of atoms inside strongly nonsphericalfullerenes: Ne@C70 J. Chem. Phys. 101, 2126 (1994); 10.1063/1.467719 Quantum freeenergy calculations: A threedimensional test case J. Chem. Phys. 97, 3668 (1992); 10.1063/1.462949 Quantum exact threedimensional study of the photodissociation of the ozone molecule J. Chem. Phys. 92, 247 (1990); 10.1063/1.458471 ThreeDimensional MorsePotential Calculation of Vibrational Energy Transfer: Application to DiatomicMolecules J. Chem. Phys. 45, 4710 (1966); 10.1063/1.1727560

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Three-dimensional quantum calculation of the vibrational energy levels of ozonea)

o. Atabek and S. Miret-Artes Laboratoire de Photophysique Mo!eculaire du CNRS, hi Universite de Paris-Sud, 91405 Orsay, France

M. Jacon Laboratoire de Recherches Optiques et ERA au CNRS no. 541, Faculte des Sciences, 51062 Reims Cedex, France

(Received 2 October 1984; accepted 8 March 1985)

The Fox-Goodwin propagator associated with an iterative matching procedure is used for the exact quantum mechanical calculation of the vibrational-rotational energy levels of a triatomic molecule. The method starts with the specification of a potential energy surface monitoring the relative motion of the atoms and utilizes the well-known close coupled equations technique of molecular scattering theory formulated in a body-fixed reference frame. The number of equations i~ optimized by the choice of some judicious local basis. Accurate values for the lowest energy eIgenvalues obtained for ozone molecule in its fundamental electronic state and corresponding to zero total angular momentum are presented and compared with results arising from variational and spectral methods. The method seems to be an accurate tool not only for bound states calculations but also for resonances occurring in photodissociation processes.

I. INTRODUCTION

The photochemistry of ozone as a tool for understand­ing the dynamical processes taking place in this atmospheric layer which filters the energetic UV part of the solar spec­trum is now developing very fast both experimentally and theoretically.I-5 In order to get a complete theoretical de­scription of the UV photolysis of 160 3 , one needs all together precise information concerning the potential energy surfaces of the various electronic states that monitor the dissociation process and methods that relate these data to the observed cross sections.

A primary step in such a full quantum mechanical cal­culation is the accurate evaluation of the vibrational-rota­tional energy levels in the fundamental electronic state using a method which could be applied not only to molecular bound states but also to molecular resonances. There are two main routes not involving the traditional use of perturbation theory that are available in this context. 6 One way is to ex­pand the wave function in terms of a complete set of basis functions with weighting coefficients determined by a linear variational calculation. This is in fact the widely used tech­nique for the determination of molecular bound states.7 Per­turbations in molecular spectra,8 vibronic levels in excited dimers or polymers,9 the Jahn-Teller,1O and Renner-Tell­er ll effects are some additional examples treated by such linear expansions. It is to be noted that the recently devel­oped rotated complex coordinate method by building square integrable functions for the resonances have made it possible to treat them using expansions in a basis of integrable func­tions. 12 Another possible way is to separate the coordinates describing the various degrees of freedom into two classes, one group containing all of them except one of particular interest which forms, in tum, the second class. The total

alWork supported in part by UER 52, Universite Pierre et Marie Curie. hi Laboratoire associe a I'Universite Paris-Sud.

wave function can be developed on a complete basis of known functions depending on the variable of the second class as a parameter. A set of coupled equations in this coor­dinate is generated by premultiplication of the Schrodinger equation by the functions of the basis and integration over the variables of the first class. These equations are identical to those encountered in scattering theory problems either nonreactive or reactive, the main difference lying in the boundary conditions. Several methods have been developed dealing with bound states as solutions of the coupled equa­tions. Among them, the piecewise analytic method of Gor­don, 13 the renormalized N umerov method of Johnson, 14 and the finite elements method of Duff et al. 15 are well docu­mented. The similarity of the asymptotical vanishing behav­ior of the bound states wave functions and the boundary conditi?ns convenient for the closed channels of scattering theory IS the starting point of the artificial channel procedure of Shapiro. 16.17 This technique, by introducing either one or two additional channels, transforms a half-collision photo­dissociation problem or a bound state problem into a scatter­ing one. The bound states energies are obtained as the real first order poles of a transition matrix element between the two artificial scattering channels coupled only via the dis­crete state manifold.

The purpose of the present paper is first to recall the coupled equations which arise in the description of a general triatomic molecule when body fixed axes are used which are more suitable to the analysis of the vibrational-rotational wave function of stable molecules than the space-fixed axis. These equations, together with bound state type boundary conditions, are then solved in the case of 160 3 using a poten­tial energy surface proposed by Barbe, Secroun, and Jouve. 18 The eigenenergies are obtained by an iterative technique based on the Fox-Goodwin propagator.19 A complete de­scription of this procedure, the reliability of which has been tested on several problems, is summarized hereafter. Results which agree well with previously published experimental as

J. Chern. Phys. 83 (4),15 August 1985 0021-9606/85/161769-09$02.10 © 1985 American Institute of Physics 1769

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1770 Atabek, Miret-Artes, and Jacon: Vibrational energy levels of ozone

well as theoretical values for the IR spectrum of ozone are presented.

II. THE GROUND STATE POTENTIAL OF OZONE

The ground state potential of ozone has been extensive­ly studied. 18.20-22 The work which is commonly referred to is that of Barbe, Secroun, and Jouve18 where a potential is de­duced from infrared absorption data as a fourth order Taylor expansion in nuclear displacements and bond angle. Actual­ly, the validity of such a representation is limited to small nuclear displacements. Murrell, Sorbie, and Varandas20 pro­pose an analytical expression which presents the advantage of reproducing the previous BSJ potential in the vicinity of the equilibrium position and has a correct asymptotic behav­ior near the dissociation limit. This potential, written as a function of D3h symmetry adapted coordinates, is made of two- and three-body interaction terms. Some of the param­eters are deduced from O2 diatomic force constants, the oth­ers being obtained by identification with the BSJ polynomial form. There are also ab initio calculations performed by Woodrow et al. 21 which are fitted by Sorbie's analytical expression. 20

Although we are not concerned in this work by a colli­sional process, we will derive our equations in term of reac­tion coordinates which are the usual ones for describing the dynamics of photodissociation. For a triatomic molecule ABC, they are denoted r for the internuclear distance BC and R for the distance between the atom A (which will sepa­rate in the dissociation process) and the center of mass of BC. 8 is the polar angle of r with respect to R. The relation between these reaction coordinates and the valence-bond co­ordinates of BSJ are obtained straightforwardly. They are

RAB = (R 2 + rr - 2yrR cos 8)1/2, (Ia)

R BC = r, (Ib)

sin 15 = R sin 8 (R 2 + rr - 2yrR cos 8 )-112, (Ic)

cos 15 = (yr - R cos 8)(R 2 + rr - 2yrR cos 8 )-1/2, (Id)

15 being the bond angle between R AB and R BC' and

mC y= (Ie)

mB +mc

The potential we are looking for only depends on the two distances rand R and on the angle 8. It can thus be derived from that of BSJ through the rigorous transforma­tions (1) and is expected to give accurate values for low lying vibrational energies. A possible solution to extrapolate this potential is to use the analytical expressions of Sorbie et al. Furthermore, to get analytical formulas, when calculating matrix elements, it is interesting to expand V(r,R,8) on a complete basis set of Legendre polynomials P k 0 (cos 8 ):

00

V(r,R,8) = L Vk (r,R) PkO(cos 8). (2) k=O

We observe that, up to 80 such polynomials are neces­sary to obtain valuable convergency. The solution we adopt deals with a least square fitting procedure with a series offive polynomials corresponding to a summation running over even values of k = 0, 2,4,6,8 (due to symmetry properties of

FIG. 1. Center of mass coordinates: (XYZ) for space-fixed axis and (xyz) for body-fixed axis. cm: center of mass ofBC, CM: center of mass of the system.

the potential surface). The domain of variation of the varia­bles, r,R,8 is restricted to be close to the equilibrium geome­try such as only the region lying within 15 000 cm - 1 above the minimum is sampled. It is to be noted that all the energy levels examined lie well below this limit. For every chosen values of rand R, an ensemble of 300 values of 8 is generated to make the fit. The maximum deviation at any point over this region is 22 cm - I. We also observe that the analytical Sorbie-Murrell potential, although accurate near the mini­mum when compared to our five polynomials fit, deviates from the BSJ values in an appreciable manner when increas­ing the nuclear separations.

III. HAMILTONIAN FOR A TRIATOMIC SYSTEM IN TERMS OF REACTION COORDINATES

One of the advantages of the coordinate system we are using is the simple expression in which results the Hamilton­ian after the separation of the center of mass motion:

H=-- -- R +--Vr + V. - fz2 [ 1 V2 1 2] 2 /-l A,BC /-lBC

(3)

In this expression, /-lA,BC = mA(mB + mc)l(mA + mB + mc) is the reduced mass for the relative motion of

A with respect to BC, while/-lBc = mBmC/(mB + mc) is the reduced mass ofBC. V~ and V; are the Laplacian operators of Rand r, respectively, and V is the Born-Oppenheimer intramolecular potential.

At that stage, two routes open which are the space-fixed theory of Arthurs and Dalgarn023 and the body-fixed theory of Curtiss and co-workers. 24 Both start out with the same coordinate system, the difference being that in the first treat­ment the coordinate axes point in the laboratory frame as for the second they are transformed to be on the molecular frame. Figure 1 represents the relative coordinates for both SF and BF systems. The SF theory leads to coupled channel equations where the complexity lies in the coupling of the angular momenta I and j of Rand r vectors, respectively, to

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Atabek, Miret-Artes, and Jacon: Vibrational energy levels of ozone 1771

produce eigenfunctions ofJ 2, Jz '/' and /2; Jbeing the quan­

tum number associated with the total rotational angular mo­mentum (J = j + I). The choice of the quantization axis Z natural for the initial state is awkward when dissociation has taken place. In the BF theory, instead of coupling spherical harmonics, one rotates the coordinates axes in such a way that the quantization axis z be coincident with R and main­tains this orientation. This rotation is obtained by choosing the Euler angles of the transformation to be (q;R,(JR,q;) with q;R and (JR the azimuthal and polar angles ofR in the labora­tory system of reference. q; is the co-called tumbling angle, i.e., rigid rotation of the triatomic system around R. More precisely it measures the angle between the two planes de­fined, respectively, by the vectors (R, Z) and (R, r). By doing so the BF y axis is left in the XY plane. It is to be noted that the same number of channels are needed in a fully converged calculation using both formalisms but the BF frame provides greater ease for the development of approximate theories25

and will be used throughout this paper. The derivation of close-coupled equations will follow that of Schatz and Kup­permann.26 We also note that this BF system is not the one defined in the usual treatments of normal modes which are based on the previously introduced valence-bond coordi­nates R AB , RBe , and 8.

According to Pack and Hirschfelder,27 the total wave function "'JM(r,R) can be expanded in terms of the elements of Wigner rotation matrix D ~n as

(4)

whereM corresponds to the projection of J on the laboratory Z axis, while fl ' is the tumbling angular momentum quan­tum number corresponding to the component of Jon Rand describes the tumbling of the triatom around R. We also note that as the component ofl along the R axis vanishes, fl ' also specifies the z component of the rotational angular momen­tumj associated to r. Finally :;Pm' is called a body-fixed wave function. Now, the only dependence of :;Pm' (r,R,(J,q;) over tp being :;Pm,(r,R,(J,tp) = ei!2'<P:;pm,(r,R,(J), this factor can re­combine with the Wigner matrix element to give28

J i!2'<p J DMn'(q;R,(JR,O)e =DMn'(tpR,(JR'tp) (5)

and Eq. (4) reads

(6)

Substitution of the expansion (6) into the Schrodinger equa­tion:

(7)

where f3 = (v,j) is the channel index denoting the intitial state of Be, after premultiplication by D ~n' and integration over the polar angles q; R , (J R , and q; yields the following set of (2/ + 1) fl-coupled equations:

(H~n -E)"'mp(r,R,(J) = L H~n' "'m'p (r,R,(J), n'~n±l

(8)

where

Hnn =-- ----+----+ J _fz2 [1 d 2 I d 2 ( I . 2 !-lA,BC dR 2 !-lBC d? !-lA,BCR 2

+ _1_)(£ + cot (J.!!..... _~) !-lBC? d(J 2 d(J sin2 (J

+ J(J + 1) ~ ;fl2] + V(r,R,(J) (9) !-lA,BC

and

HJ _ ~ 1 ll,n±l - 2 R2 [J(J+ l)-fl(fl± 1)] 12

{lA,BC

x (- fl cot (J + .!!.....). - d(J

IV. COUPLED CHANNEL EQUATIONS

(10)

We are now left with coupled equations written in terms of the three variables r,R,(J [Eq. (8)] where the different chan­nels are labeled by fl. Our goal is to eliminate two of these variables, namely (J and r, by expanding the function", m on known basis sets and obtain R-dependent coupled channel equations. As is always the case in these problems, the opti­mization of the resulting number of channels dramatically depends on the choice of the basis sets. A natural way, when dissociation is considered, is to take the Legendre polynomi­als Pjll (cos (J) which are eigenfunctions of j2 as a basis set describing the angular dependence of the diatomic internal r motion and harmonic oscillators X v (r) for the vibrational motion.6

•26

,27 But, as Shapiro and Balint-Kurti 16 have pre­viously suggested, an important improvement might be real­ized by transforming this diatomic basis set into the one which suits better with the expansion of a bound state wave functions we are looking for.

The leading idea is to construct a basis set of angular functions which diagonalizes the rotational part of the Ha­miltonian for the equilibrium configuration of the triatomic, that is for R = Re and r = re' Although channels defined in this way will be asymptotically coupled, they are more con­venient to describe a bound state for which the high ampli­tude region for the wave function lies close to the equilibrium value of R. This can be done by looking for a linear combina­tion of Legendre polynomials which diagonalizes H~n' (re ,Re,(J). The coefficients of such an expansion:

00 + J

S~w ((J) = L L 7l~w.j'n' P),ll' (cos (J) j'~ 1n'1 n'~-J

(11)

are obtained as the eigenvectors of the equation:

[H~n' (re,Ro(J) - ~w] S~w((J) = 0, (12)

where ~<" denotes the eigenvalues. The Saw's define the so-called "hindered rotor" basis29

labeled by (a, lUi which replace the initial (j, fl ) channels and are decoupled at the triatomic equilibrium configuration. Even in cases where the usefulness of the transformation j ---+ a is questionable (photodissociation being such an ex­ample where we are looking to the asymptotic behavior of a channels which remain coupled at large R, actually compro­mising the S-matrix analysis) the transformation fl ---+ lU re­sulting from a choice for a projection axis for J different from R may be considered for its own interest as diagonaliz-

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1772 Atabek, Miret-Artes, and Jacon: Vibrational energy levels of ozone

ing the Corio lis part of the Hamiltonian.1O Furthermore, these two operations may be disentangled either by perform­ing the summation over j alone or over n alone in Eq. (11). Finally, the body-fixed wave function is expanded on this basis set:

00

tPJw = I s~w(O)(U~w(r,R). (13) a=O

In a situation where Coriolis interactions are neglected (which will be the case without any approximation for calcu­lations involving J = 0) the hindered rotor basis can be ob­tained by

S~n(O) = f 'TJ~jn ~n (cos 0), j= Inl

where

p (cos 0) = 'J + v . p (cos 0) - [(2' 1)li_n),]l12 ,n 21} + n )! ,n

are the orthonormalized Legendre polynomials30 with

LT ~n ~'n sin 0 dO = 8jj,.

(14)

(15)

(16)

The 'TJ'S are orthonormalized eigenvectors solutions of equations:

f {[lfB.}'I}'+I)+ fz2 2 [J(J+l)-2n2] / = In I 2/-lA,BCR e

- E"an ]8/j + VZ(re,Re)] 'TJ~/n = 0 (17)

with 00

,,-J J ~ ~ 'TJajn 'TJaln = U aa, (18)

j= Inl

and with the following definitions:

1 Be = 2 +---,

2/-lA,Bc R e 2/-lBC "; (19)

VZ (re,Re) = LT ~'n V(re,Re,O) Pjn sin 0 dO. (20)

Weare now in a position to expand the BF wave func­tion on the orthonormalized basis set of angular functions 5 ~n with coefficients depending only over distances:

tPm(r,R,O) = IS~·n(O)(U~'n(r,R). (21) a'

These coefficients, in turn, appear as solutions of cou­pled equations obtained by substituting Eq. (21) into Eq. (7) and eliminating the angular dependance through multiplica­tion by t~n(O) and integration over O. The result is

J J " ~ fz2 (1 1 Iii 1)-J J J ( R) [h n(r,R ) - E] (Uan(r,R) = - ~ ~ -2 R 2 + ----::2 v + 'TJajn 'TJaln (Ua'n r, a' j = In I /-lA,BC /-lBC r

00 00

- I I I 7j~/n 'TJ~ln VZ(r,R )(U~'n(r,R) a' j = In I / = In I

where

h (rR)=-- ---+--J _fz2( 1 d2

1 d2

)

n , 2 /-lA,BC dR 2 /-lBC dr

fz2 + 2 [J(J + 1) - 2n2],

2/-lA.BC R (23)

;:f = [J(J + 1) - nn'] 1/2, (24)

and

;{f, = [j (j + 1) - nn'] 1/2. (25)

A final step to obtain R-coupled equations is to elimi­nate the variable r. This is done by expanding the wave func­tion on a basis of harmonic oscillator functions:

(U~n (r,R ) = I X v' (r) G ~nv' (R ). (26) v'

The X 's are not the eigenfunctions of the diatomic Ha­miltonian but solutions ofl6

(27)

where

UBC (r) = !kBdr - re)2 (28)

(22)

I and

(29)

kBC being the force constant convenient for the description of the diatomic BC when considered alone.

The already described procedure of substitution of Eq. (26) into Eq. (22), multiplication by X v (r) and integration over r leads finally to the one variable coupled channel equations defining the coefficients G (r):

[

fz2 d 2 J] J -2-- dR 2 + E - tanv - U anv.anv (R) G anv (R )

J.i.A,BC

= I I I U~nv.a'n·v,(R) G~'n'v,(R), a'-=;i:.afl'=fl±lv'::;i=v

where

The different anv channel potentials are given by

(30)

(31)

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Atabek, Miret-Artes, and Jacon: Vibrational energy levels of ozone 1773

U~I}v,aI}v(R) = 2/lA

::R 2 {[J(J + 1) - 2fl2] + j=~I} ((j + l)I1]~jI} 12}

+ I I Tj~j'I} 1]~jI} <Xv 1 VZ(r,R) 1 xV>· (32) j = II} I j' = II} I

As for the R-dependent couplings, we obtain

U~I}v,a'I} 'v' (R ) = . I fl22 ( 1 R 2 Ouv' + _1_ / X v I _~ I X v' )\;(j + 1) Tj~jI} 1]~'jI} OI}I}' }=II}I /lA,BC /lBC \ r !

It is to be noted that if one replaces the different 1]'S by Oaj' which amounts to expand the initial BF wave function on Legendre polynomials PjI} (cos 0) instead of on 5 ~I} (0 ), the coupled equations [Eqs. (30)-(33)] are identical to those of Schatz and Kuppermann,26 or Shapiro and Balint­Kurti. 16 The aflv channels we have constructed are decou­pled for R = Re [the coupling terms in Eq. (33) vanish for this equilibium distance] but remain mixed at large distance while an opposite situation is valid for the}flv channels. As long as bound states and infrared spectra constitute the pur­pose of the calculation, the aflv basis reduces considerably, as will be shown in the next section, the number of channels involved to reach convergence. In this case the vanishing asymptotic behavior of the wave functions relaxes the trou­bles which would otherwise arise by the large distance cou­pling of the channels.

The remaining calculation is that of the elements of the potential coupling matrix of Eqs. (32) and (33). This may be conveniently done by using the Legendre polynomials ex­pansion for V(r,R,O) as indicated in Eq. (2) which, when sub­stituted into Eq. (20) leads to

- I} ( 2) + 1 )112 00

Vfj(r,R) = -'f - L Vk(r,R) 2; + 1 k=O

xC Ijkj';fl Ofl)C Ijkj';OOO), (34)

where the C's are Clebsh-Gordan coefficients. 31

Only even values of k lead to nonzero terms in Eq. (34) due to the symmetry of the molecule [V(r,R,O) is symmetric about 0 = 17/2]. As a consequence of this and since30

Cfjkj'; 000) = 0 for} + k + j' = odd (35)

the potential VZ(r,R ) does not couple even rotational states to odd ones and thus reduces the number of channels by a factor of2.

Among the two families of methods used to solve the multichannel system (30), namely configuration interaction and the direct solution of close coupled equations, the first leads, through trial wave functions expressed as a superposi­tion of products of one-phonon eigenfunctions, to a vari­ational principle. The form to be given to these one-dimen­sional basis eigenfunctions is for instance suggested by the initial multichannel equations when one decouples them. Such a calculation involving harmonic oscillator basis up to 220 terms and variationally minimizing the eigenvalues of the Watson Hamiltonian written in the Eckart framework32

(33)

I has been performed by Carney et al. for the ozone mole-cule. 33 On the other hand, the body-fixed Hamiltonian writ­ten in terms of reaction coordinates and leading to the close coupled system (30) is more suitable to treat molecular half­collision processes such as predissociation or photodissocia­tion. Several methods have been given to deal with bound state type solutions ofEq. (30). We have recently derived an iterative matching procedure based on the Fox-Goodwin recursion relation. 34 This consists in propagating both outward and inward the ratio P of matrices of solutions at two adjacent points of an integration grid. The initial values of these ratios are chosen from the expected behavior of the wave function for small or large R. However, as we have also shown,35 due to some memory losts, these particular values do not limit the accuracy of the overall numerical process. The condition for the matching of the functions and their derivatives is expressed in a functional form depending on the energy:

F(E) = det 1 (lm _ 1 P~ - 213m + (lm + 1 P;" 1 = 0, (36)

where P~ and P;" represent the ratios at the matching point Rm for the outward and inward propagations and (l and 13 are the two N umerov matrices of elements 19(b) :

a .. (R) = h {o + ~ [E - U.(R )]} I} I} 12 lJ '

(37)

{ 5h 2 } (Jij(R)=h Oij-U[E-Uij(R)] , (38)

calculated at (Rm _ h)' Rm, and (Rm + h ), respectively. h is the step size. An iterative procedure based on the progressive reduction of F(E) leads to the bound states.

v. RESULTS AND DISCUSSIONS

The close coupled equations may be considerably sim­plified when the total angular momentum is zero (J = 0) as will be the case in the following. A compact matrix form is then

[~I d22 +W(R)]G(R)=O. 2/lA ,BC dR

(39)

I being the identity matrix and W(R ) a matrix which can be recast in a sum of matrices of power of R,

W(R) = AR -2 + S + CR + DR 2 + ER 3 + FR 4.(40)

The explicit form for the A, S, C, D, E, and F matrices are

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1774 Atabek, Miret-Artes, and Jacon: Vibrational energy levels of ozone

Aav,a'v' = co fz2 .. 2: -2--J(j + 1) 'TJaJ 'TJaj Dvv' ,

J = 0 /-lA,BC

(41a)

co fz2.. (111) Bav,a'v' = EDav,a'v' - .2: -2-J(j + 1) 'TJaj 'TJa'j Xv _2 Xv' - <Xv I UBdr) I Xv' )Daa, J=O /-lBc r

" ,,_'J__ 71 71 "C 2(J'2k"'0000) [VV' bvv'R + vV'R 2 d vv'R 3 VV'R4] 00 00 (2'+1)112 00

J~o/~o 2j' + 1 'fal 'faj k-::O ", a2k - 2k e C2k e - 2k e + e2k e

- EvDav,a'v' + <Xv I UBdr) I Xv)Dav,a'v" (41b)

co co (2'+1)112 00 C - ~ ~ _'J__ " C 2( °2k 0'. 000) [b vv' 2 vv' R 3d vv' R 2 vv' 3 ] av,a'v' = .£.,.£., 2'/ + 1 'TJaj' 'TJa'j £., J" , 2k - C2k e + 2k e - 4eZk R e , J=OJ =0 'J k=O

(41c)

D -" ~ _'J__ " C 2( °2k o

,. 000) [VV' 3d vv' R 6 vv' 2 co 00 (2'+1)112 00

av,a'v' = ,£" .,£" 2'/ + 1 'TJaj' 'TJa'j £., J" , C2k - 2k e + e2k R e ], (41d) J=OJ =0 'J k=O

00 00 (2j + 1 )112 00 2' ./ ' , Eav,a'v' = -.2: .,2: -2'/ 1 'TJaj' 'TJa'j 2: C (j2kJ; 000)[ d~% - 4e~% R e ],

J=OJ =0 'J + k=O (41e)

00 00 (2j +l)1I2 00 ,

Fav,a'v' = - ,2: .,2: 2'/ + 1 'TJaj' 'TJa'j 2: C2(j2kj'; OOO)e~%

J=OJ =0 'J k=O (41f)

when the matrix elements of Vk (r,R ) are developed as fourth order polynomials:

00

<Xv I V(r,R,e) I Xv,) = 2: [a~~' + b ~~'R + c~~'R 2

1=0

+ d ~{R 3 + e~{R 4] P2/(COS e). (42)

In the calculations which will be presented only the first five values of I are retained. All auxiliary matrix elements values as well as spectroscopic constants for 160 3 are collect­ed in Table I. The expansion coefficients 7J are obtained by diagonalizing the bending Hamiltonian ofEq. (17) when val­ues of j up to 50 are retained. This is necessary to reach converged results for the ten first hindered rotor basis wave functions 5 a .

Figure 2 displays some of the Uva (R ) potentials for the lowest values of channel quantum numbers v and a together with some of the observed vibrational bound states of 160 3 , It is the interaction of these one channel potentials through the R dependant nondiagonal elements of the coupling matrix W that will lead to the construction of the vibrational levels.

A remaining question is the choice of the most appro­priate (v,a) channels to fasten convergency in a given energy region. In Table II the different steps leading to such a choice are illustrated for the (0,0,0) ground vibrational level of

TABLE I. Spectroscopic constants for 160 3 taken from Ref. 18 together with matrix elements of I/r and diatomic potential Uodr) between diatom­ic basis functions,

POC = 7.9974575 amu re = 1.2717 A

k BC = 3.099 989x 10' cm- I A-2

Re = 1.658 A v

o 1 1 2 2 2

v

o o 1 o 1 2

0.62047105 - 0.420 348 40 X 10- I

0.62047478 0.30241221 X 10- 2

- 0.598 500 32 X 10- 1

0.62907893

<x,lUodr)lx,) (em-I)

0.285 889 53 X 103

0.0 0.857 668 58 X 10' 0.404 308 84 X 103

0.0 0.142944 76 Xla'

I ozone. One observes that for the case of two channels, the result is closer to the exact value (1457 cm - I) when channels (v = 0, a = 0) and (v = 0, a = 1) are retained rather than (v = 0, a = 0) and (v = 1, a = 0). An additional third chan­nel (v = 0, a = 2) does not considerably improve the result (less than 2 cm -I) but if v quantum number changes (v = 1, a = 0) the effect is - 7 cm -I. The most efficient three chan­nels are (v = 0, a = 0,1) and (v = 1, a = 0). A similar analy­sis is performed for the case of four and five channels by either adding an extra a channel for a given v or a new v channel. Convergency is obtained in this case with five chan­nels, three of them for v = ° and two for v = 1. For some other energies, nine-channel calculations are needed to reach converged results within an accuracy of 0.1 cm -I.

A great advantage of the (v, a) hindered rotor basis is that the lowest bound states ofthe channel potentials Uva (R ) are close enough to the observed levels of the triatomic to be taken as first order approximations to these levels, This, of

1,0

V=O,U=2

0.5

OL-_~ __ +-_-+~~~_~ ___ ~_~ Rcl,) 1.59 1.72

FIG. 2. U,u(R) potentials for 160 3 (J = 0) for v = 0,1 and a = 0,1,2. Also indicated are some of the experimental vibrational states.

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Atabek, Miret-Artes, and Jacon: Vibrational energy levels of ozone 1775

TABLE II. The energy of the (0,0,0) ground vibrational level of 160,(J = 0) in a multistate calculation involving the indicated (v,a) channels.

Total number of channels

2

3

4

5

Assignment

v

o o

o o o I

a

o o o

o I

o o

2 0

o 0 o I

o o o o o I

o

o o o o

o o o

o o I

o o o I

o I 2

o I o o o I 2

o I 2 3

o I 2 o o 1 o 1

o I 2 o

1489.34

1482.67

1467.84

1482.31

1481.14

1466.37

1460.95

1481.02

1466.26

1459.48

1458.79

1457. \3

course, would not be the case if (v,j) channels potential func­tions were considered. As a consequence, much more (vj) channels would be necessary to build up converged bound states. In Fig. 3, the energy positions of some one-channel levels are indicated together with the shifts which affect them as a result of nine-channel converged calculations. These shifts are expected not to be very sensitive to small changes of the positions of one-channel levels. On the other hand, the lowest of these first order levels can be considered as well separated in order to assess their efficiency in build­ing the multichannel levels. These two remarks may serve as a basis for a procedure of slightly modifying the potential [Eq. (42)] through a,b,c,d,e parameters to obtain more accu­rate results. Practically, this is done by noting the energy shift resulting from the multichannel calculation and modi­fying the parameters of the Uva (R ) potential supporting the

E(cm-1 )

3~66

3352

3226

2592

2~86

2363

1~89

v:1,Cl:1 n =0 "

V=O,Cl=1 n =1

V=O,Cl=2 n=o '

v=1.Cl=0 n-o

v=O,Cl=O n =,

v=O ,Cl=1 n =0

1-channel calculations

, , ,

, 3252

3183

2853

2561

2502

2158 - - - - --

1457

converged results ( 9 - channels)

( 110)

( 011)

( 020) -------

( 100)

(001)

(010)

(000)

experimental

FIG. 3. Calculated one-channel and nine-channel converged results for the lowest levels of 160, (J = 0) as compared with their experimental values. n = 0,1 ... labels the different vibrational levels ofa given Uva potential.

first order level fitted by a harmonic potential, in such a way that the new position of the one-channel energy corrected by the shift leads to the observed level. It is to be noted that the potential we are using results from a least square fit of BSJ potential and the differences we observed are of the order of 20 cm -1 within the limits of the integration range. This im­poses an upper bound for the adjustments affecting a,b,c,d,e parameters.

Table III displays the first vibrational levels of 160 3 on an energy range 2000 cm - 1. Weare able to reproduce with an accuracy of a few wavelengths the observed values with only nine channels. It is to be emphasized that this is due to the use of the (v,a) hindered rotor basis which diagonalizes the fixed R Hamiltonian and is very suitable for the calcula­tion of bound states. Typically less than I min computation time is required on a NAS 90-60 computer for one energy. Also are indicated comparisons with the results of the above mentioned variational method of Carney et a/.33 for which accurate converged energies are obtained. Among the var­ious couplings added to the zeroth order Watson Hamilton­ian, the Darling and Dennisson, or the Fermi resonance terms have been abundantly discussed in the literature.36 In particular, levels (0,0,2) and (2,0,0) form a pair of such Darl­ing-Dennisson resonance and their calculated splitting is of

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1776 Atabek, Miret-Artes, and Jacon: Vibrational energy levels of ozone

TABLE III. Computed vibrational energies of the infrared spectrum of 160,(J = 0) for the BSJ potential (in cm - I unit).

Assignment

V, V2 v, Experimentala FOX" VMc SMd

0 0 0 1457 1457 1457 1457 0 I 0 2158 2158 2157 2157 0 0 1 2499 2502 2504 2504

0 0 2560 2561 2564 2565 0 2 0 2848 2853 2852 0 1 3183 3183 3185 3184 1 1 0 3252 3252 3254 3253 0 0 2 3515 3522 3535 3534 0 3 0 3530 3542 3541 1 0 1 3568 3575 3588 3587 2 0 0 3658 3661 3670 3669

a Experimental data from Ref. 18. b Present work with Fox-Goodwin algorithm. CVariational method (Ref. 33). d Spectral method (Ref. 37).

the order 141 cm- I. Their evaluation does not present any

peculiarity when solving coupled equations. A reverse situa­tion is valid when looking for levels (0,0, 1) and (1,0,0) or even worse for (0,0,2) and (0,3,0) through coupled channels for­malism. These levels being separated only by 60 and 8 cm - 1,

respectively, a double cancellation of F(E) on such a narrow energy interval is only possible by a very careful choice of the initial energy for the iterative process. Otherwise only one term of these doublets would be calculated.

As a last column, the results obtained by Feit and Fleck37 and based on their so-called spectral method38 are listed. This method utilizes numerical solutions to the time­dependent Schrodinger equation through the split operator FFT technique. A subsequent Fourier analysis of the corre­lation function (t,b(R,O)It,b(R,t) reveals resonant peaks that correspond to the stationary states of the system which loca­tion leads to the eigenvalues with high accuracy.

The difference between the calculations reported in Refs. 33 and 37 and the close-coupled one, increase with the energy to reach about 12 cm - 1 for the excited levels (002), (030), (101), (200), the latest remaining closer to the experi­mental data. It is to be noticed that the overall good agree­ment which is achieved is to be understood in terms of the accuracy of the close-coupling calculation and also by taking into account its sensitiveness for excited levels to the fine adjustments of the potential surface. This may be at the ori­gin of the slight discrepancies arising between the different calculations.

As a word of conclusion, we point out the practicability of a direct coupled equations approach with appropriate boundary conditions to the determination of bound state en­ergies of a triatomic molecule. Such a method provides not only an efficient alternative to variational procedures but also by its straightforward generalization to photodissocia­tion problems leads to resonances in multichannel situa­tions. In this case, energy quantization results from different boundary conditions in the exterior region but the analogy with bound states calculations can be made closer by the use of complex rotated coordinates. Only very few such three-

dimensional exact calculations have been reported in the li­terature. 17 An application to ozone photodissociation is now in progress.

ACKNOWLEDGMENTS

We are grateful to Professor J. A. Beswick for construc­tive discussions. This work has been partly supported by a grant of computing time from the "Conseil Scientifique du Centre de Calcul Vectoriel pour la Recherche."

'M. Nicolet, Etude des Reactions Chimiques de ['Ozone dans fa Strato­sphere (Institut Royal Meteorologique de Belgique, 1978).

2J. E. Davenport, Parameters o/Ozone Photolysis as a Function o/Tempera­ture at 280-330 nm (Office of Environment and Energy, Washington, D.C., 1980).

3A. M. Bass and R. J. Paur, J. Photochem. 17,141 (1981). 4M. G. Sheppard and R. B. Walker, J. Chern. Ph,s. 78, 7191 (1983). 5J. W. Simons, R. J. Paur, H. A. Webster, and E. J. Bair, J. Chern. Phys. 59, 1203 (1973).

6For a review see, for instance, W. A. Lester, in Dynamics 0/ Molecular Collisions, edited by W. H. Miller (Plenum, New York, 1976).

7G. D. Carney and R. N. Porter, J. Chern. Phys. 65, 3547 (1976); R. I. Leroy, I. S. Carley, and J. E. Grabenstetter, Faraday Discuss. Chern. Soc. 62, 169 (1977).

MA. Lagerqvist and E. Miescher, Helv. Phys. Acta 31,221 (1958); M. Le­febvre-Brion, Can. I. Phys. 47, 541 (1969).

9A. Witkowski and W. Moffitt, I. Chern. Phys.33, 872 (1960); R. E. Merri­field, Radiat. Res. 20,154 (1963).

lOW. Moffitt and W. Thorson, Phys. Rev. 108, 1251 (1957). "I. A. Pople and H. C. Longuet-Higgins, Mol. Phys. I, 372 (1958); I. A.

Pople, ibid. 3,17 (1960). 12See for instance the special issue of Int. I. Quantum Chern. 14 (no. 6)

(1978). I3R. G. Gordon, I. Chern. Phys. 51, 14 (1969); A. M. Dunker and R. G.

Gordon, ibid. 64, 4984 (1976). 14B. R. Iohnson, I. Chern. Phys. 69, 4678 (1978). 15M. Duff', M. Rabitz, A. Askar, A. <;:akmak, and M. Ablowitz, J. Chern.

Phys.72, 1543 (1979). 16M. Shapiro and G. G. Balint-Kurti, I. Chern. Phys. 71, 1461 (1979). 17E. Segev and M. Shapiro, I. Chern. Phys. 77, 5604 (1982). '"A.Barbe, C. Secroun, and P. Iouve, I. Mol. Spectrosc. 49,171 (1974). 19(a) L. Fox, The Numerical Solution 0/ Two-Point Boundary Value Prob-

lems in Ordinary Differential Equations (Oxford University, London, 1957); (b) D. W. Norcross and M. J. Seaton, I. Phys. B 6,614 (1973).

2°K. S. Sorbie and I. N. Murrell, Mol. Phys. 29, 1387 (1975); I. N. Murrell, K. S. Sorbie, and A. I. C. Varandas, ibid. 32,1359 (1976).

21C. W. Wilson, If. and D. G. Hooper, I. Chern. Phys. 74, 595 (1981). 221. P. Hay, R. J. Pack, R. B. Walker, and E. J. Heller, J. Phys. Chern. 86,

862 (1982). 23 A. M. Arthurs and A. Dalgarno, Proc. R. Soc. London Ser. A 256, 540

(1960). 24C. F. Curtiss and T. F. Adler, I.Chem. Phys. 20, 249 (1952); C. F. Curtiss,

ibid. 21, 2045 (1953); G. Gioumoussis and C. F. Curtiss, I. Math. Phys. 2, 96 (1961); c. F. Curtiss, I. Chern. Phys. 49,1952 (1968); L. W. Hunter and C. F. Curtiss, ibid. 58, 3884 (1973), and references therein.

25R. T Pack, I. Chern. Phys. 60, 633 (1974). 26G. C. Schatz and A. Kuppermann, I. Chern. Phys. 65, 4642 (1976). 27R. T Pack and I. O. Hirschfe1der, I. Chern. Phys. 49, 4009 (1968); 52, 521

(1970). 28 A. R. Edmond, Angular Momentum in Quantum Mechanics (Princeton

University, Princeton, 1960). 291. F. Kidd, G. G. Balint-Kurti, and M. Shapiro, Faraday Discuss. Chern.

Soc. 71, 287 (1981). 'OM. Abramowitz and I. A. Stegun, Handbook 0/ Mathematical Functions

(Dover, New York, 1965). 31M. E. Rose, Elementary Theory 0/ Angular Momentum (Wiley, New

York, 1957). 321. K. G. Watson, Mol. Phys. 15,479 (1968). "G. D. Carney and R. N. Porter, 27th Symposium on Molecular Structure

and Spectroscopy. The Ohio State University, Colombus, Ohio (1972), pa-

J. Chem. Phys., Vol. 83, No.4, 15 August 1985 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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Atabek, Miret-Artes, and Jacon: Vibrational energy levels of ozone 1777

per Y5; G. D. Carney, Ph.D. thesis, University of Arkansas, 1973; Disser­tation Abstracts International B 34, 1956 (1973). G. D. Carney and C. W. Kern, in Proceedings of the Ninth Quantum Chemistry Symposium (Wiley, New York, 1975); G. D. Carney, L. A. Curtiss, andS. R. Langhoff,J. Mol. Spectrosc. 61, 371 (1976).

340. Atabekand R. Lefebvre, Phys. Rev. A22, 1817 (1980); Chern. Phys. 52, 199 (1980).

350. Atabek, R. Lefebvre, and M. Jacon, J. Phys. B 15, 2689 (1982); O. Ata-bek and R. Lefebvre, II Nuovo Cirnento B 76, 176 (1983).

36B. T. Darling and D. M. Dennison, Phys. Rev. 57, 128 (1940). 37M. D. Feit and J. A. Fleck, Jr., J. Chern. Phys. 78, 301 (1983). 38M. D. Feit and J. A. Fleck, Jr. Appl. Opt. 19.2240 (1980); J. Opt. Soc. Am.

17. 1361 (1981).

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