Indian Journal of Chemistry Vol. 4 I A, October 2002, pp. 1995-2007

A localized molecular orbital study of the structure and bonding of ozone

Dubl C Ghosh* & Arindam Chakraborty

Department of Chemi stry ,U ni vers ity of Kalyani , Kalyani 74 1 235, India

E-mail: dulal @k lyuinv.erneLin

Received 27 March 2002; revised i5 Jilly 2002

The loca lized molecular o rbital theory and energy partiti on ing formali sm have been invoked to study the structure and bonding in ozone molecule. The range of in vestigati o n covers a large number o f conformations generated theore ti cally over

the w ide range of apical angles between angular (C21.) to linear (0 =11 ) shapes. The results demonstrate that. s imilar to the bonding in diborane, the re are two equi valent three-cente red bonds in ozone embracing three oxygen atoms in all the conformations. The possibility of a triangu lar struc ture of ozone is ruled out because the computed locali zed mo lecular orbitals demonstrate that there is no bonding between the termin al oxygen atoms and the decomposed e nergy component s show that the inte raction be tween terminal oxygen atoms is strong ly repul s ive. Charge density di stribution is asy mmetric in the homonuclear molecule and its dipole moment is an al gebraic sum of bond moment and lone pair mo ment and lone pair on the apical oxygen ato m contributes significantl y to the dipo le mo ment. It is demonstrated that any atlempt of calculating the apical angle from the ex perimentall y determined dipole moment would be e rroneous and mi slead ing. Variation of dipole moment as a function of angular to linear reorgani za tion of mo lecular shape is corre lated in terms of computed quantum mec hanical hybridi zat ion of the lone pair on the apical oxygen a tom. The barrier to in vers ion of ozone th rough the linear

(0=1, ) transiti on state originates from a subtle inte rplay of one- and two- cente r energy components over the ent ire skeleton of the molecule.

o o

o o o o

Ozone, 0 3. has been the subject of numerous ex peri mental '-

' 7 and theoretical 18-26 investi gations over a long period of time. The knowledge of the structure and bonding in the molecule is absolutely necessary to correlate and understand the absorption of solar radiation by the atmospheric ozone layer acting as a shield to protect the I ife processes on earth from the harmful radiation generated from various cosmological processes. The equilibrium geometry of ozone has been a puzzling problem for quite a long time and has attracted the attention of both experimental and theoretical chemists. The molecule is dipo lar in nature and has a number of unusual bands in the photoelectron spectra l

-4. The experimental determination of the dipole moment and correlation of the shape in terms of dipole moment, the spectroscopic investigations into the shape of the molecule and al so an analysis of chemical bonding in ozone have been an enticing problem of chemistry and is an active field of research since 1933. Sophisticated ab initio calculations viz. MBPT and CC22

.2J

, multi-reference CC24, CASSCF and CASSCF second order perturbation theory (CASPT2)2s.26 in to the problem of structure and bonding in ozone are also reported. One group of workers proposed an angular (C2v) structure while other group proposed a planar triangular (D.l/J structure shown in Fig. 1.

(a) Angular form (C11·).

(b) Triangul ar fo rzm ( D 'h)

Fig. I-Angular (a) and triangu lar (b) shapes o f the ozone mol ecul e.

Ozone is a triatomic molecule hence can form a linear, an angular or a plane triangular geometry. A plane triangular structure of the molecule, however, can be formed with a large amount of Bayer stra in in the structure. The angular (C21.) versus plane triangular (D3,,) structure of ozone has been a long controversy . The ground state of the ozone molecule, in particular, its equilibrium geometry and the vibrational frequencies have been carefully investigated by both experimentai s. '7 and theoretical lS

-21 methods . More

involved and sophisticated quantum chemical calculations for both ground and excited states of the molecule are reported22

-26

. The long and controversial history of the elucidation of the equilibrium shape of ozone molecule by a large number of experimental and theoretical effortsS

-26 is found to propose the

1996 INDIAN J CHEM, SEC A, OCTOBER 2002

-II III IV

Fig.2-The resonance in ozone

< 000 bond angle ranging from 34° to 140° and the 0 - 0 bond length between that of double and single bond.

The molecule falls into the general category of AB2 systems, the geometry of which was discussed by means of empirical MO schemes of Mulliken l2 and Walsh27. Peyerimhoff28 put forward a Hartree-Fock justification of the empirical rules of Walsh27 and Mulliken l2. However, extensive theoretical calculations have settled the issue of angular vs. linear structure of ozone. The equilibrium structure of the molecule is the open C21' angular one and a ring structure with D 3h symmetry lies above the C2Io

'1' b . 19 29 4) Th . . equI I num structure' - ". e quantItatIve evaluation of this ring minimum above the open minimum is still a subject of debate and as many as 13 different values ranging between 12 to 38.9 kcallmole are reported. A number of reports on protonation of ozone are also available43

-47 (Fig. 2). Shand and Spurr l3, in order to correlate the high

explosive nature and chemical reactivity of ozone, suggested a resonance between structures I-IV, of which structures I and II are important.

Trambarulo et al. 17 seem to have supported the angular structure on the basis of microwave study and suggested a bonding scheme of the molecule following the method of valence bond resonance of Pauling. However, Gould and Linnett48 later on discounted the resonance in ozone proposed by Shand and Spurr l3 and proposed a two center three-electron bond in the electronic structure of the molecule.

In the valence shell of oxygen atom there are six electrons in four orbitals, i.e. two orbitals are available for a normal covalent bonding. Thus, if a triangular ring is formed, equilateral or isosceles, each O-atom can form two covalent bonds but there will be large Baeyer strain in the structure so formed. But if the open angular structure is assumed to be the shape of the molecule, it is difficult to explain the bonding scheme of the molecule in terms of normal covalent binding. The formation of two single covalent bonds by the two terminal atoms with the central atom can be the only binding scheme in terms of normal

covalent bonding and this leaves two terminal atoms as radicals making the system too unstable and such a type of bonding can hardly develop any dipole moment in the molecule. So, in order to explain the structure, stability, and dipole moment of the molecule, an appropriate scheme of bonding in ozone is required. One may cite the classic example of the problem of bonding in diborane molecule. The chemists were compelled to broaden their views of chemical bonding for electron deficient molecules and suggested a scheme of localized three-centered bonding49

. Lipscombso, however, demonstrated that the localized molecular orbitals generated from the ab initio self-consistent field (SCF) canonical molecular orbitals, CMO's, quantitatively reproduce the structure of diborane just envisaged in the qualitative theory of n-centered bonding and established that the localized molecular orbital, LMO approach is a quite suitable quantum mechanical paradigm to study the electronic structure of electron deficient compounds. Since ozone is an electron deficient molecule like diborane, the quantum mechanical locali zed molecular orbital method will be the appropriate theoretical paradigm to deal with the electronic structure and bonding in ozone.

The localized lIIoleclIlar orbital (LMO) method of study of bonding ill molecules

The Hartree-Fock-Roothaan'ssl method generate~ canonical or spectroscopic molecular orbitals (CMO's or SMO'S)52 where the concept of lone pair and bond pair vanishes. However, the freedom of the unitary transformation in Hartree-Fock space has been conveniently exploited to generate orbitals, which are localized in the various regions of the molecule and the familiar concept of the lone pair and bond pair is quantum mechanically restored51.54

. The LMO's 54 generated through the method of Sinanogl u not only

restores the conceptual electronic structure of chemistry with lone-pair and bond-pair but also maintains the n-bond and cr-bond separation. While Pauling's hybridization is conceived for some fixed geometry of known point groups only, the quantum

GHOSH el al.: MOLECULAR ORBITAL STUDY OF STRUCTURE & BONDING OF OZONE 1997

mechanical hybridization can be computed for all and f . 50.55.56.57 W 56 h any molecular con ormatlons . eave

recentl y shown that, during the physical process of the evolution of geometry of molecules from equilibrium confo rmation towards non-equilibrium one, the bond energy and hybridization vary according to a suggestion of Coulson58. The theoretical es timation of bond energy is feasible by the energy partiti oning analys is of Kollmar and Fischer5

,! .

It is already mentioned that ozone possesses dipole moment. Trambarulo et al. 17 pointed out that in the bonding mechanism of ozone one lone pair must be left on the apex oxygen atom and thi s lone pair will contribute to the dipole moment of the molecule. Coulson58 pointed out that molecules with angul ar shape with lone pair at the apex atom must have dipole moment contributed by the lone pair on the apex atom. Thus the dipole moment of ozone is expected to have two contributions - one from bond moment and other from the lone pair moment of the central atom. Ghosh56.57 et al. have recently establi shed that the evolution of dipole moment with the evolution of molecular geometry of the molecules having lone pair in the central atom can be calculated and correlated conveniently in terms of the localized molecular orbitals, LMO's. It is also demonstrateds6. 57 that if the variation of dipole moment of a molecule containing lone pairs on the central atom is studied through LMO' s as a function of geometry evolution, the contributing components of the dipole moment and their relative magnitudes become self evident.

In order to explore whether the dipole moment of the molecule has contributing components we have studi ed the bonding in ozone at its equilibrium geometry. A number of conformat ions are generated by varying the < 000 angle below and above the equilibrium shape such that the molecul e evolves ultimately to the linear (D=,,) form starting from very close angular (C2l.) form. The lone pair moment develops from the asymmetry of charge di stribution in the lone pair arising out of the mi xing of sand p orbitals and the dipole moment of lone pair in a pure s or p orbital vanishes identically to zeroS8.60

. The extent of mi xing of sand p orbitals and the symmetry or asy mmetry aspect of charge distribution in lone pairs is transparent and straightforward in the hybridizati on of the lone pair. In terms of Pauling's hybridization, the angular shape is Sp2 and the linear form should be sp. Thus, in the linear form the lone pair on the apical atom is in a pure p-orbital. In the linear form, the dipole moment is expected to be zero because, both

the bond moment and the lone pair moment should be zero by symmetry. Now, if the canonical molecul ar orbitals, CMO's for energy minimum structure is crenerated and such CMO's are locali zed, the b

electronic structure in terms of lone pair and bond pair will be generated in a quantum mechani cal way. The question whether the molecule is triangular or electron deficient angul ar may be straightfo rward in the LMO's. It may be further pointed out that the energy partitioning analysisS9 provides with the scope of calculating the energy of interaction fo r bonded or non-bonded atom pairs including the one-center term. The energy of interaction between the two terminal atoms of ozone can once again decide the issue of the open angular versus triangul ar structure of the molecule.

Procedure of Calculation Whereas Sinanoglu's54 method of locali zation starts

with CMO's generated through the all -valence electron method of Pople and co-workers61 and the attempt of energy partitioning has been successful in the approximate SCF formali sm of Pople and co­workers61 , we have invoked the CNDO/2 formali sm in the present investi gation. Standard parameters61 are used. The overlap and electron repulsion integrals are computed through the explicit analytical formulae laid down by Roothaan62

. Initi ally the molecul e is assumed to be plane angular in shape. The equilibrium geometry is first determined by optimizing the 0-0 bond length and the < 000 angle. Then a sufficient number of conformations are generated below and above the equil ibrium geo metry at regular interval s of < 000 angles between 110° to linear form (1 80°). The molecular di storti on is initiated by altering the < 000 angle and the bond length of the generated confo rmation is optimized at each stage. The wave fun ction, energy, charge di stribution and dipole moment are calculated at each of the conformer at optimized bond length. The energy of each conformer is then partitioned into one­and two-center components through the explicit formulae laid down by Kollmar and Fischers'!

(Eqs 1-3). The detail s of the algorithm are di scussed else whereS6.57.59. The generated CMO's are locali zed through the procedure developed by Sinanoglu5

'+ . A bonding analysi s of all the generated conformations of the molecule is attempted in terms of the LMO's.

The total CNDO energy of a system can be written as sum of one-center and two-center terms as follows:

. . . (l )

1998 INDIAN J CHEM, SEC A, OCTOBER 2002

-5522 r----------------------------_,

t -55.23

-55.24

::i -55.2S ~ I:: -55.35

6ri -55.27 ..... (l)

5 -55.28

fl -55.21 .9 (l) -55.3

..c: t-< -55.31

1l-e rurrters irdicae apcal m-gles in ~ If{)

-55.32 L... __________________________ -'

The angles of reorgan ization in degrees --.

Fig. 3-Plot of total energy as a function of angles of reorgan ization of ozone

where EA are monatomic terms and EAB are diatomic terms. The monatomic terms EA and the diatomic terms EAS can be further broken down into physically meaningful components as follows:

... (2)

where, EA U , E/ and EA K are total monatomic orbital energy, electron-electron repulsion energy and non­classical exchange energy respectively.

... (3)

where EA/ is the contribution of the resonance integrals to the energy of A-B bond and is the principal feature of covalent bond, EA/ signifies the total potential attraction of all electrons of A in the field of the nucleus of B plus those of B in the field of the nucleus of A, EABl estimates the total electron­electron repulsion energy between two centres A and B, whi le EA/ stands for nuclear repulsion and EAB K

defi nes the total exchange energy arising out of quantum mechanical exchange effect between electrons of A and B and is an important quantity in covalent bonding.

Results and Discussion The results are shown in Tables and/or diagrams

fo r a clear visualization of computed data. The data is plotted as a function of < 000 angles, the reaction coordinates. The potential energy diagram of the molecule is obtained by plotting the total energy as a function of < 000 angles of reorganization in Fig. 3.

The report of the experimental determination of the apical angles has a wide range between 34° to ] 40° but the accurate determination decides the value -116°45". Figure 3 shows that the equilibrium value of the apex angle is around 120°. Thus we see that the value of the apical angle of the present calculation is close to that of accurate experimental determination. The optimized 0-0 bond lengths and the computed reorganization energy, the difference of energies of the equilibrium form and reorganized form , are plotted as functions of < 000 angles of all the conformations in Figs. 4 and 5 respectively. From Fig. 4 it is distinct that as the conformation of the molecule evolves with the variation of < 000 angle, the 0-0 bond length undergoes a steady shortening and the rate of shortening is a bit sharp below the equilibrium shape but very slow above the equilibrium form. Figure 5 demonstrates that the energetics of the physical process of reorganization of the molecular structure of ozone is symmetrical below and above the equilibrium form up to a certain extent of angular deformation and the process becomes more and more energetically difficult above and over 130° of < 000 angle. Thus the report of experimental determination of very high value of < 000 angle12 is apparently fortuitous. We may venture to point out one possible source of error in computing the < 000 angle on the basis of experimental d ipole moment. Table 1 shows that although the molecule is homonuclear, the charge distribution in the molecule is asymmetric and the gross atomic charges on the apical and the terminal atoms are different at all conformations between angular to linear shapes and

GHOSH et al. : MOLECULAR ORBITAL STUDY OF STRUCTURE & BONDING OF OZONE 1999

i

110 11 2 114 116 11 8 120 125 130 140 150 160 180

Reaction Coordinates in degrees, Q ----l.~

Fig. 4- The optimized 0-0 bond length as a function of angular to linear geometry reorganization of ozone

i 001

---:- 0.06 ::I «i ~ 0.05 ;>,

2.'l '" G:i 0.04

'" .g 0.03 co; N '2 ~ 0.02 .... o '" ~ 0.01

110 11 2 114 116 118 120 125 130 140 160 180

Reaction Coordinates in degrees, Q 150

• Fig. 5-Plot of the angular to linear reorganization energy of ozone as a function of reaction coordinates

hence the molecule is dipolar at all conformations. The dipole moment of ozone decreases continuously and sharply with the angle of reorganization from 1100 to 1800 and becomes zero at the linear form .

Table I demonstrates that as the structure evolves from < 000 angles of 1100 to linear form, th~ charge density is depleted from the apical atom and piled up more and more on the terminal atoms and up to the < 000 angles of 1180 and there after the charge distribution pattern is reversed and more and more charge is piled up on the apical atom and depleted from the terminal atoms and the process is continued till the linear form is reached. It may also be noted that the process of the charge reorganization below the equilibrium shape occurs at much slower rate than

that above the equilibrium shape of the molecule. In order to make a comparative study of the variation of charge density distribution vis-a-vis the variat ion of dipole moment, the atomic charges and the dipole moment are plotted in Fig. 6. The profiles of the charge density and the dipole moment (Fig. 6) ex hi bit that the net atomic charge differences on O-atoms fluctuates below and above the equilibrium shape but the dipole moment decreases steadi ly and monotonically with the physical process of evolution of shape of the molecule. Thus, since during the evolution of molecular structure from angular to linear form, the length of 0 - 0 bond decreases very slowly and the charge density profile turns near the equilibrium shape, the computed nature of variation

2000 INDIAN] CHEM, SEC A, OCTOBER 2002

1.8 80 The numbers indicate the apical ang les in degrees

i 1.6 110 180 t 70

1.4 Percentage of s-character o f

110 112 lone pair of apical O-atom 60.;, 0

c: ..... " 0 1.2 0

1..) .~

.~ 116

50 g,o 0-

'" c: c: -5 " " "0 U ~ c:

~.o E 110 112 11 4 116

/120 40 & ~ a 9 " .--;:; 608 -5 K E "0 " 8

;!- Chilrge density on terminal O-atom c: c: '" OJ '" 30t'l:! ..2 c:

5§O6 ,., " .~ -5

g S c: .....

" 0

E 20~ ~ " 0.4

Charge densit y on apical Q-atom eo u (3 ~ '" '" ~ 0- ..<:: '" is IOU

or:;

0.2 u

130 114 116 120 125

118

110 112 114 116 118 120 125 130 140 150 160 180

Reaction Coordinates in degrees,Q

Fig . 6- Plot of dipo le moment, the charge densities on apical and termina l 0 atoms and the percentage of s-character of the lone pair on the apical O-atom as a function angular to linear geometry reorganization of ozone

Tab le I--The ch <l rge densities on the apical and terminal oxygen atoms and the d ipole moment as a function o f angles of

reorganization of ozone.

LOOO Dipole Charge den sity Charge angles moment on (apical density on

in degrees (D) O-atom) (terminal O-atom)

110 1.5645 5.6097 6.195 1

112 1.4948 5 .6085 6.1957

114 1.4280 5.6064 6. 1968

116 1.3554 5.6057 6. 197 1

118 1.2819 5.6053 6.1973

120 1.2072 5.6063 6.1968

125 1.01 83 5.6075 6.1962

130 0 .8319 5.6092 6.1954

140 0.4804 5.6186 6.1907

150 0.1854 5.6335 6.1832

160 0.0107 5.6487 6.1756

180 0.0000 5.6688 6.1656

of the dipole moment of the molecule cannot be 3ccounted for in terms of bond moment only . In all probability, the dipole moment of ozone molecule must have two components-the bond moment and the lone pair moment. We have, therefore, theoretically ;lj ,>sected the dipole moment of ozone in to two contributing components -the bond moment and the lone pair moment and the < 000 angles are calculated first in terms of bond moments and then in te rms of total moment assumed to be the only bond mo ment. The co mputed data are shown in Table 2 \vhich distinctly demonstrate that < 000 angle for

equilibrium geometry calculated from bond moment

only is 119.94° whereas that calculated in terms of the total moment assumed to be the bond moment is 113.60°. The columns (iv) and (v) of Table 2 reveal that values of < 000 angles are widely divergent and hence any calcu lation of the equilibrium < 000 angles from the experimental dipole moment. assumed to be the bond moment only, must be wrong. It is not possible to ascertain experimentally whether the bond moment and/or the lone pair moment create the dipole moment. Thus it is apparent that the earlier calculations of equilibrium geometry of ozone in terms of experimentally determined dipole moment was erroneous because of wrong assumption regarding the origin of dipole moment of ozone.

Analysis of bindin.g We have generated as many as twelve LMO's fo r

the generated conformations of the molecule from < 000 angles 110° to 180°. The LMO's for the equilibrium geometry are shown in Table 3. From a close analysis of the generated LMO's we have found that there are two equivalent three-center bonds embracing the three O-atoms in all the generated conformations of the molecule and there is no whisper of bonding between the terminal oxygen atoms in any of the conformations of the molecule at all. A critical examination of the LMO's has revealed the following general pattern of bonding in all the conformations of ozone: i) There are two equivalent three-center bonds embracing the three O-atoms; ii) the apical O-atom is

GHOSH et al.: MOLECULAR ORBITAL STUDY OF STRUCTURE & BONDING OF OZONE 2001

Table 2-The di ssected components of the total moment and computation of < 000 angles in terms of bond moment and the total moment as a function of angles of reorganization of ozone.

Reorganization ang les in degrees

Bond Moment Lone pair moment < 000 in degrees < 000 in degrees (from total moment) (D) (D) (from bond moment)

110

112

114

\16

118

120

125

130

140

ISO

160

180

Bond angle Sigma bond

1100 /\ 120· A 130· "

140· ~

15~ ~

180. ~O-..o

1.2588

1.2299

1.2035

1.1718

1.1392

1.1033

1.0150

0.9233

0.7285

0.5302

0.3408

0.0000

Lone pair

0 if \)

6~~" .. 0

0.3057

0.2649

0.2245

0.1836

0.1427

0.1039

0.0033

-0.0914

-0.2481

-0.3448

-0.3514

0.0000

Three-centered bond

~

~ ~ ~ ~

0.....-----0 ___... .. ~ .. 9~

:8-8-8: C§ =0

Fig. 7-Conlour plot of bonding in ozone in dominant conformations

bonded to the terminal O-atoms by two a-bonds and iii ) there is one lone pair on the apical oxygen and two-lone pairs on each of the terminal oxygen atom. But in the linear form, each oxygen atom has two lone pairs- one in a hybrid and the other in a p-orbital. Thus the bonding pattern in ozone mimics the bonding in diborane5o as regards the three-center localized bond. We have drawn a contour plot of the bonding pattern in some representative conformations of ozone in Fig. 7.

109.95 88.98

111.95 94.31

113.96 99.43

115.96 104.34

117.95 109.11

119.94 113.60

124.95 124.75

129.97 135.21

139.98 153.92

149.98 169.61

159.98 179.37

180 180

We have calculated the quantum mechanical hybridizations of the lone pairs and the bond pairs of the oxygen atoms of all conformations and are shown in Table 4. From Table 3 it is evident that the quantum mechanical hybridization of the orbitals of the apical O-atom to form the a-bonds with terminal O-atoms in the equilibrium geometry is Sp 3 14 while the terminal O-atoms offer spo type of hybrids to form the same a-bond. The lone pair of the apical O-atom is Sp09 whereas the two lone pairs on the terminal 0-atoms are Sp0 23 and sp2 1.9 type respectively. Thus the lone pair in the apical atom is almost 1: I mixture of s and p-orbitals whereas the lone pairs on the terminal atom are predominantly .Hype and p-type. The subtle variation in the bonding pattern as a function of the evolution of geometry is evident from Table 4 which shows that as the molecular geometry evolves below and above the equilibrium shape, the hybridization in the bond pairs in the apical atom changes significantly but that in the terminal atoms remain virtually constant. The hybridization of the bond pair of the apical O-atom has the range Sp376 (the percentage of s-character is -21) at apical angle 110° and through equilibrium it passes to Sp0 77 (the percentage of s ­

character 56.6.) at the (D=,,) form. The bond pair of the terminal O-atoms on the other hand starts with Sp 6 04 (the percentage of s-character ;~ i 4.21) at the beginning and S/06 (the percentage of s-character 12.41) at the linear form. The lone pair of the apical O-atom is Sp075 (the percentage of s-character 57.1 ) at the 110° and pure p-orbi tal at the 180° of the < 000 angles . The lone pairs on the terminal O-atoms undergo a drastic and anomalous change in the pattern

2002 INDIAN J CHEM, SEC A, OCTOBER 2002

Table 3-The LMO's of 0 ) at < 000 = 1200.

LMO's 1.p.10(" a(OI-O.l) I.p. 20 (3) l.p.IO(2) n-centered a(01-02) I.p. 20 (1) l.p.IOI I) n-centered AO's bond bond O~,

I -0.0259 0.3863 0.0000 - 0.0000 0.0000 0.3863 0.0259 -0.7223 0.0000

O lpx I -0.0382 -0.5527 -0.0000 0.0000 0.0000 0.5527 -0.0382 -0.0000 0.0000

0 21 ).1' I 0.0000 0.0000 0.0000 0.0000 0.5303 0.0000 0.0000 0.0000 0.5303

O~I': I 0.0270 - 0.4031 0.0001 -0.0000 0.0000 - 0.4034 - 0.0270 -0.6916 0.0000

O2.\"'2 0.0228 -0.0009 - 0.0000 0.9019 0.0000 0.2337 -0.2083 -0.0000 0.0000

0 ]/1./- - 0.0493 0.0067 0.0000 0.2202 0.0000 -0.5064 -0.5283 -0.0000 0.0000

O lP.""2 0.0000 0.0000 0.0000 0.0000 0.8307 0.0000 0.0000 0.0000 -0.1693

0 21'/ 0.0261 0.0017 0.0000 -0.3714 0.0000 0.2673 - 0.8191 -0.0000 0.0000

O~/ 0.2082 0.2337 0.9020 -0.0000 0.0000 -0.0008 -0.0227 0.0001 0.0000

0 11''' .1 -0.5283 0.5064 -0.2202 0.0000 0.0000 - 0.0068 -0.0493 0.0000 0.0000

O ']I'Y :1 0.0000 0.0000 0.0000 0.0000 -0,1693 0.0000 0.0000 0.0000 0.8307

0 21" :1 0.8192 0.2673 -0.3713 -0.0000 0.0000 0.0016 -0.0261 -0.0000 0.0000

Table 4-The 0 - 0 bond length, the quantum mechanical hybridization of the lone pair and bond pair at the apical O-atom and the bond pair of terminal O-atom of ozone and the percentage of s-character of the hybrids as a function of angles of reorganization.

< 000 in 0-0 bond Hybridization % of s- Hybridization % of s- Hybridization % of s-degrees length of the lone pair of character of the of the bond pair of character of the of the bond pair of character of

(A) apical oxygen lone pair apical oxygen bond pair terminal oxygen the bond pair

110 1.171 SpO.75 57.110

112 1.170 .lpO.78 56.211

114 1.169 Sp08 1 55.249

116 1.168 SpO.84 54.259

118 1.167 SpO.88 53.220

120 1.167 SpO.91 52.165

125 1.166 sp 1.01 49.383

130 1.164 s p 1.1 7 46.168

140 1.163 Spl.60 38.454

150 1.164 S/48 28.703

160 1.163 S/·91 16.918

180 1.163 py 0.0000

of their hybridization with the physical process of the geometry evolution of ozone between 110° to 180° of < 000 angles. The variation of the pattern of hybridization and the percentage of s-character of the bond pairs on the apical and terminal O-atoms as a fun ction of the evolution of molecular shape are self evident from Fig. 8 which shows that the percentage of s-character of the hybrids forming the 0-0 bond remains virtually constant over a wide range of < 000 angles between 110° to 125° and thereafter the percentage of s-character of the hybrid on apicai oxygen increases and that on terminal oxygen decreases without any turn. Thus, the non-equivalence or asymmetric nature of the oxygen atoms of the molecule with regard to charge density distribution is also reflected in the pattern of hybridization of the

Sp3.76 20.986 sp6.04 14.206 Sp 3.63 21.575 Sp 6.0 1 14.259 Sp3.50 22.217 Sp 6.00 14.275 s/38 22.831 sp

5.99 14.296 S/ .16 23.491 .1'/ .98 14.316 .1'/14 24.172 s p

6.00 14.279 Spl .85 25.947 Sp6.0:1 14.23 1 Sp257 27.964 s p 6.06 14.164 Sp205 32.787 Sp 6.2:1 13.824 Spl.57 38.880 s p 6.52 13.305

sp 1.17 46.147 sp6.77 12.877 0.77 56.625 7.06 12.405 sp s p

lone pairs and bond pairs of the atoms. Figure 7 and Table 4 shows that each oxygen atom has one lone pair in a pure p-orbital. Thus, the electronic structure of the linear form is as par prediction in terms of Pauling's scheme of hybridization .

Correlatioll of dipole moment variatioll with geometry evolution

Figure 6 shows that the dipole moment of ozone is higher below and lower above the equilibrium api cal angle. Thus, the charge density di stribution in ozone is more asymmetric in conformation with apical angles below the equilibrium value but the asymmetry is destroyed with movement of the molecule from equilibrium shape towards the linear transition state for inversion. From Table 2 it is evident that the

GHOSH et al.: MOLECULAR ORBITAL STUDY OF STRUCTURE & BONDING OF OZONE 2003

60 -1.36

i 50

The numbers indicate the apical angles in degrees 180

i -1.38

~ ·iO 0-

-1.4 "'" 40 % of S-character of bond ::J

c: «i .2 pair on apical O-atom .s '- >. 0 e." ~ 30 -1.42 " c: u " e

II 118 u '" 112 114 .§ ..<:: 110 u % of S-characler of bond .9 ~ 20 114 116 118

-1.44 " 0 110 112 120 125 :a " -.; "" § " C 112

" 10 114 116 -1.46 " 1:: 118 120 125 ..<::

" F-

e..

- 1.48

110 112 114 116 118 120 125 130 140 150 160 180

The angular reorganization in degrees

Fig. 8-Plot of the percentage of s-characters of bond pairs On apical and te rminal O-atoms and the two-centre bonded energy as a funct ion of the angular to linear geometry reorganization of ozone

variation of the bond moment part is not as sharp as the lone pair moment with the geometry evolution. The lone pair moment arises from the asymmetry of charge distribution by mixing sand p_orbitals56-58.6o and the higher the mixing the higher the dipole moment. The dipole moment of a lone pair housed in a pure p or pure s orbital vanishes identically to zero58.60

. Table 4 shows that as the molecule evolves from below the equilibrium shape the percentage of s­character of the lone pair hybrid on the apical oxygen decreases and identically vanishes at the (D=,,) form. Table 2 demonstrates that the dipole moment is also a monotone decreasi ng function of the angles of reorganization from angular to linear form and vanishes at the linear (D=,,) form . The resultant of the bond moments at the linear (D=,,) form is zero. Figure 7 demonstrates that the lone pair on the apical o res ides in a pure p-orbital. Hence the vanishing dipole moment of ozone at the (D=,, ) form is due to placing the lone pair electron on the apical oxygen atom in pure p orbital and the mutual cancellation of equal and opposite bond moments. Since, it is also evident from Fig.6 that dipole moment and the percentage of s-character of the lone pair on the apical oxygen decrease hand in hand and the bond moment variation with the geometry evolution is not sharp, the sharp variation of the dipole moment with the evolution of molecular shape is principally due to the sharp destruction of the asymmetry of charge distribution in the lone pair of the apical oxygen atom by eliminating the contribution of s-orbital to the lone

pair. This study of variation of dipole moment of ozone as a function ·of variation of shape of the molecule proves unequivocally that the dipole moment of ozone is made of two components - the bond moment and the lone pair moment.

Energy partitioning study of the open angular liS.

triangular structure The total energy of each of the generated

conformation is partitioned into one- and two- center components according to (Eqs 1-3). The total energies of the one-center components on the apical and terminal oxygen atoms are plotted in Figs 9 and 10 respectively. The total two center non-bonded energy is plotted in Fig. 11. The two center bonded energy and its physical components are shown in Table s. The two center bonded interaction is plotted along with some other quantities in Fig. 8. The two center bonded component is the measure of the bond energi9 and Coulson58 pointed out that the variation of the energy of a bond is linearly related to the percentage of s-character of the hybrids forming the bond.

A close examination of the nature of the curves in Figs 8-11 and Table 5 reveals the following general observations. The two-centered bonded component is most stable at the equilibrium shape but it increases below and above the equilibrium geometry monotonically. Thus, although the 0-0 bond undergoes a steady but slow shortening in length , the energy of the bond does not decrease with the

2004 INDIAN J CHEM, SEC A, OCTOBER 2002

E 8 co 6 -; u '0.. co "­o

-17.094

-17.0906

110 112 114 116 11 8 120 125 130 140 150 160 180 Reaction Coordinates in degrees,Q

Fig. 9-Plot of the total monatomic energy of apical O-atom of ozone as a fun ction of angles of reorgani za ti on

-17.685

-17.6763

-17.6697

-17.6635

· 17.6593 -17.6588 -17.6588 -17.6583 -17.6581 .1 7.6584 - 17.6586 -17.6594

110 112 114 116 118 120 t25 130 140 150 160 180

Reaction Coordinates in degrees,Q ~

Fig. l O-Plot of total monatom ic energy of terminal O-atom of ozone as a function of angles of reorgan izat ion

shortening of length. The energy of 0-0 bond and the percentage of s-character of the hybrids on the terminal and apical oxygen atoms are plotted as a function of < 000 angles in Fig. 8. It is obvious from Fig. 8 that 0-0 bond strength increases from ! 10° up to the equilibrium shape and thereafter it begins to decrease. But the percentage of s-character of the hybrids on the apical and terminal O-atoms forming thi s bond exhibit anomalous variation . The percentage of s-character of the apical oxygen increases but that on the terminal oxygen decreases as

a function of reaction coordinates. Thus the nature of variation of hybrid on the terminal oxygen is in accordance with Coulson's suggestion while that on the apical oxygen shows the reverse trend .

A rationale of the above nature of variation of the two center energy term is straightforward from an analysis of the decomposed components of the two center bonded energy from Table 5. It is di sti nct from Table 5 that, with the gradual shortening of bond length, the EA/ and EA/ terms increase monotonically with the evolution of the geometry

GHOSH et al.: MOLECULAR ORBITAL STUDY OF STRUCTURE & BONDING OF OZONE 2005

0.008

0.006 ::i cd >. 0.004 01) .... 0) c:: 0)

0.002 "0 0)

"0 c:: 0 0 .D <: 0 c::

-0.002 .... ~ c:: 0) u -0.004 6 ~

E--0.006 150

180

160

-0.008

The angles of reorganization in degrees

Fig. II-Plot of total two-centre non-bonded energy of ozone as a function of angles of reorganization

from a close angular (C2,,) position to the linear (Doo/,) form; but the EA/' EA/ terms decrease monotonically while and EA/ term decreases initially but takes a turn at 1400 and slowly increases thereafter. But the resultant effect of these five contributing factors, the 0 - 0 bond energy decreases steadily with evolution of geometry from the close angular shape up to 1200 of the < 000 angle and thereafter the bond energy increases steadily. Thus, the subtle interplay of the diverse nature of the variation of the components of the bond energy is responsible for the anomalous nature of the profile of the energy of 0-0 bond with the evolution of the conformation from a close angular position to the linear form. From Fig. II it is evident that the two-centered non-bonded interaction is repulsive at all angles between 1100 to 125 0 and there after above 1250

, a loose attraction develops between the two terminal O-atoms. A close examination of the variation of energy components of the two center non-bonded interaction with geometry evolution shows that a sharp decrease in electron­electron and nuclear-nuclear repulsion with increasing distance of separation between the terminal oxygen atoms make the sum of the components negative. The pattern variation of one-center terms is quite interesting (Figs 9 and 10). The one-center energy term on apical oxygen (Fig. 9) increases from the apical angle from 1100 to 1400 and thereafter it takes a turn to decrease. The one-centre terms on the terminal oxygen, on the other hand (Fig. 10) increases from 1100 to 1180 and thereafter it takes a turn to decrease monotonical I y.

Thus having established that the angular structure (C2,,) is the equilibrium shape of ozone, a simple continuation of normal mode of vibrat ion can con vert the angular structure to linear (D oo/, ) form , the transition state for inversion. After attaining the transition state the molecule can undergo inversion as well as retention of configuration. The height of the inversion barrier is 0.0744 a.u . or 195 .2 1 kJ/mole. Although such inversion processes takes place through quantum mechanical tunneling effect, still the quest for the origin and development of barrier for such physical processes is interesting and enti cing problem of theoretical chemistry. In a quest of elucidating the origin of barrier of umbrella inversion of molecules, Ghosh et al. 56.57 have found that the barrier does not lie at a particular center or atom in the molecule rather the barrier develops from the entire skeleton of the molecule. In the present case we see that as the molecular shape evolves from the equilibrium geometry, the physical process of geometry reorganization is retarded by the two-center bonded interaction and accelerated by the two center non-bonded interactions. The one-center terms on the apical 0 atom initially retard the process and the retardation continues from 1100 to 1400 of < 000 angle and thereafter, it accelerates the process. The one-center energy term on the terminal O-atoms accelerates the physical process of attaining the transition state monotonically. Thus, it transpires from the complex and anomalous nature of variation of one and two center energy terms as function of (C21·) to (Doo/,) geometry reorganization of ozone that the origin

2006 INDIAN J CHEM, SEC A, OCTOBER 2002

Table 5- The computation and partitioning of the two-center bonded energy terms into its physical components (a.u .) as a function of < 000 angle.

< 000 EA/ EAB N EAB V EAB K EAB R EAB (degrees)

110 15.2947 16.2689 -31.1717 -0.3154 -1.5257 -1.4492

112 15.3045 16.2833 -31.1938 -0.3158 - 1.5303 - 1.4521 114 15.3115 16.2959 -31.2116 -0.3161 - 1.5341 -1.4544 116 15.3222 16.3105 -31.2347 -0.3164 -1.5380 - 1.4564 118 15.3327 16.3244 -3 1.2568 -0.3166 -1.5414 -1.4577 120 15.3338 16.3239 -3 1.2573 -{).3165 -1.5416 -1.4577 125 15.3468 16.3380 -31.2818 -0.3163 -1.5443 -1.4576 130 15.3719 16.3660

-31.3300 -0.3162 -1 .5482 -1.4565 140 15.3973 16.3801

-31.3655 -0.3141 - 1.5439 -1.4461 150 15.4082 16.3660

-31.3622 -0.3103 -1.5311 -1.4294 160 15.4421 16.380 1

180 15.4719 16.3801 -3 1.4054 - 0.3063 -1.5238 - 1.4 133

-31.4321 -0.3004 -1.5132 -1.3937

of barrier of inversion of ozone should develop not from a particular region rather barrier develops and originates from the entire skeleton of the molecule.

Conclusion We have reported a study of the elucidation of

structure, bonding and the origin of dipole moment of ozone. The quest into the puzzling problem of angular vs. triangular structure of ozone has been made by invoking suitable theoretical apparatus like energy partitioning and quantum chemical localized molecular orbital methods. The range of study covers a good number of conformations between close angular shapes to linear transition state for the physical process of inversion. The results seem to conclude, once again, that the equilibrium structure of the ozone molecule is not a triangle rather it has a planar angular shape. The computed values of charge density distribution shows that although ozone is a homonuclear triatomic molecule, the atoms are non­equivalent and structure is dipolar in nature. It is also demonstrated that the dipole moment of the molecule has two components-lone pair moment on the apical atom and the bond moment. The lone pair on the apical oxygen contributes significantly to the dipole moment of the molecule. It is also demonstrated that the calculation of the apical angle on the basis of experimentally determined dipole moment was erroneous because of the fact that the dipole moment of the molecule is not due to the bond moment only and it is not possible to identify and dissect experimental dipole moment into its contributing components. Since the shape of the molecule is not a

plane triangle rather an angular one, its electronic structure is of electron deficient nature from the viewpoint of chemical bonding. The quantum chemical localized molecular orbital calculation demonstrates that there are two equivalent three­center bonds embracing all the three oxygen atoms in ozone. The bonding pattern of ozone mimics the pattern of bonding in diborane. The computed quantum mechanical hybridization, in terms of the generated LMO's, of the bond pairs and the lone pairs on the apical and terminal oxygen atoms for all the conformations demonstrate that the hybridization in the bond pairs and the lone pairs in terminal and apical O-atoms are inherently different. The sharp decrease of dipole moment with the evolution of structure from angular to linear shape is nicely correlated in terms of the variation of quantum mechanical hybridization and estimation of the destruction of asymmetry in the charge distribution in the lone pair on the apical atom of ozone. The physical process of geometry reorgani zation converting the angular shape (C2,.) to linear form (Doo/J is studied by energy partitioning method and it is concluded that the inversion barrier of ozone originates not from a particular region of the molecu le rather it originates from the entire molecular constitution by a subtle interplay of one and two center energy components. It is demonstrated that there is no possibility of formation of bond between the terminal oxygen atoms because the interacti on between the terminal oxygen atoms is either repul sive at closer < 000 angles or very loosely attractive at larger < 000 angles. Thus, there is no whisper of

GHOSH et al.: MOLECULAR ORBITAL STUDY OF STRUCTURE & BONDING OF OZONE 2007

bond formation between the terminal atoms. Hence, the possibility of any triangular structure is ruled out.

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