- Email: [email protected]
Localized natural orbitals of unstable molecules: ozone
Volume
50. nurnbcr
LOCALIZED
2
CHkMICAL
NATURAL
ORBITALS
PHYSICS
OF UNSTABLE
Kizashi YAMAGUCHI, Koji OHTA and Takayuki Dqartmcnr Toyorlaka. Kcccived
of Chonirt.y. Faculty of Enguxwing Osaka 560, Japan 15 June
1 Scptcmbcr
LITTCRS
MOLECULES:
1977
OZONE
FUENO
ScrcncL: Osaka Universify,
1977
Use ot locdlved natural orbltdls (LNOX) IS proposed for the elucidation of chcmicnl vaIcncc of unstable ground-stdtc rnolccuks. TIIC LNO’\ of~ronc as an evarnplc hnvc been derived from itsgcncralircd Ilartrcc-Fock (Glfr) solutions It is shown that the LNO de\aption\ of ozone are much in Imc with the results which Goddard cd al. have obtained using the gcncralircxl valence bond (GVB) rncthod.
1. Introduction In a previous paper [I] the clcctronic structure of the trimcthylcne diradical was investigated on the basis of natural orbitals (NO’s) derived from the generali7cd Hartree-Fock (GIIF) so!utions. It was found that the molecular orbital concept can be extended successfully even to such a strongly correlative rnolccule by the aid of the !imited NO configuration interaction (CI) procedure. Since the natural orbit& belong to irreducible representations of a point group, they are also well suited for discussions of the symmetry properties of the electronic states of unstable molecules. Howcvcr, the delocalized natural orbitals (DNO’s) arc not particularly useful for the elucidation of chemical valence problems. The theory of localized natural orbitals (LNO’s) [21, on the other hand, appears to be of much value to discussions of the nature or the inner-shell, lone-pair and bond orbitals in molecules 13-51. The purpose of this work is to assess thesignificance of the LNO’s derived from the CHF solutrons for unstable molecules. The electronic structure ofozone ~111be dealt with as an example.
2. Localized natural molecular
orbit&
The restricted ILrtrec-Fock (RilP) solutions ot unstable molecules are triplet-unstable [7 ] and spin266
flipping unstable [8] in the singlet and nonsinglet states, respectively. They arc reorganized into more stable unrestricted Hartree-Fock (DODS) wavefunctions, which arc given by
where xi and vi are the paired corresponding orbit& [9,10] and ho, denotes an unpaired natural orbital. The former orbit& are related to the natural orbitals hi and pi of the DODS solutions by x, = (cos q/2)$
+ (sin wi/2)cc,.
vi = (cos w,/2)h,
- (sin w,/2)i.+,
(2) the RIIF solutions being given by setting wI = 0 (i = 1, . . .,12). The first-order spinless densrty matrix is now expressed as
+,I$
Xo,(r)Aoj(r) =
‘3 k=l
“k uk
tr) uk (d,
(3)
where N+ = (1 + cos Wi), TIC
form ofp(‘)(r)
rzy = (1 - cos Wi).
is invariant to the non-orthogonal
(4)
Volume 50, number
transformation
2
CliEXIICAL
PIIYSICS
which yields the localized natural orbit&
ui [2]:
1 Scptelllller
LIT-l LRS
projection is necessary function is +;;;I
(5)
=5& 1LL1 . . - ulvl+
’ u01L’02u03 “‘“O~ The density matrix can then be rcwrittcn
[l I] . The projected
1977
HF wave-
1VI+ 1 ---X,,%
ccco 1’ j=,
I I
as
An essentral feature of the above transformation is that the form of p’*)(r) is preserved as a sum of square terms a¶one. The transformation matrix T = (T&) is determincd, for example, by the energy-Iocalizcd criterion [2,3]. When ‘I$$ involves only tbc doubly occupied orbitals, i.e., nk = 2 for k = I,. _., tz and irk = 0 for k=2n+l , . _., 2rz + p, the transfonnatron is equivalent to the Edmiston-Ruedenberg localization procedure f3]. Since the RHF wavcfunctions of nonsinglet species also involve n doubly occupied orbitals, localuation of the closed-shell part gives n Hartrce-like NO’s which can be regarded as tz singlet pair bonds, However, the number of the Ilartree-like natural orbitals is no longer equal to that of the singlet pair bonds if G$ involves split orbital pairs (wi # 0). In such cases, the transformation cq. (5) cannot be directly applied to discussions of chemical valency of unstable molecules. As shown previously [ 1,8,10] , there are only a few split orbitals in the ground HF solutions of unstable molecules. The corresponding orbitals Xi and TQof a split pair are inbercntly more or less localized to reduce the strong intrashcll Coulomb repulsion. Thus it seems reasonable to leave the split pairs out of the transformation procedure. Such a localization procedure will maximize the sum of the intrapair Coulomb repulsions of tight pairs, preserving the number of the pair bonds. ‘Ihc unpaired natural orbitals {hoi) can also be transformed into the Hartrce-like unpaired natural orbitals IUui}. The HF wavefunction rewritten in conformity to the above localization principle is ,I,@-) = lu u --.ffIuIXl+1 e 1+1 -- -X,?JQ,~()2 iIF r 1
- - -uQJ,
where dIis the antisymmetrizcr and 0, ary the orthogonal spin functions. The spin coupling coefficients Ci are determined without reoptimiring the Gl-lF orbit& [l l] . This reduces the number of the spin coupling parameters since the tight pairs are singlet-coupled. As has been shown previously [lo] , singlet diradical species involve only one split pair which may be referred to as the reaction orbitals (RO’s). Therefore, their projected HF wavefunctions can easily be obtained by unng the singlet-coupled spin function only (Cl = 1 whi\eCi=O fori+ 1) [12- IS]: $$[
=92’u, 24@ . . . q&3,
(9
whcrc the split pair is given by
The spatial overlap TRo and the normalizing are given, respectively, by TRO =,! kR09R0
(17 = cos OR0
factor N (11)
and N= [2(1 + 7$)]
-“I.
(12)
The strong orthogonality between singlet pairs is preserved because of the orthononnal condition of natural orbrtals j12,15]. Expression (10) is similar to that used for the IlcrtlerLondon geminal [12]. [lowever, the spatial functions need no longer be atomic orbitals. In fact Xlio and r+10 are a little delocalized because of the mclusion of ionic structures in the simple Vl3 Cl terminology. In this SCrlsC the corrcspondmg reaction orbrtals sl~oulcl bc simrI:n to the generalized valence bond &VU) orbitals [ 151 in the strong correlation region (T,, < 1).
(7) which involves I tight pairs, (n - I) split pairs and p unpaired orbitals. Since the split pairs do not possess the
pure spin symmetry concerning the S2 operator, spin
3. Ozone The electronic
structure
of ozone has already received 267
volume
so,
nuIrlber
CIII-MIC’AL PHYSICS
2
LI:TTI:RS
1 September
1977
considerable attention [ 16 - 201. Therefore, it seems to be one of the most appropriate molecules that deserves investigation by the present localization! theory. IIerc, WC will examine the ring-opened ozone [18] as an example (fig. 1). The corrcspondmg orbit& were dctermined by the method described previously [lo]. Only the highest occupied n-orbital is split in the ground singlet (IA,) state. The remaining eight delocalized orbitals are transformed into five non-bonding (lone pair) orbit&, two o-bonding orbitals and one delocalized norbital. This last n-orbital remains intact after the localization procedure. The projected HF solution of Ozone can thus bc rcprcsented as
(13) x G&0 + G&c& where the subscripts b and n denote the bonding and nonbonding orbit&, respectively. In what follows, characteristics of ali these orbitals ~111be illustrated by use of the contour maps. There are three natural rr-orbitah, i.e., X:(S), hrr(A) and ,u”(Sj in the cases of three-center (isosceles) systems such as ozone. The lowest r-r-orbital, hi(S), is symmetric and delocalized over the whole molecule. fn the case of ozone, h;(S) is nothmg but the rrb. Its contour map is as depicted in fig. 2. Fig. 2 clearly shows the strong nbonding property of the Xt_ The anti-sjrmmetrlc NO, A”(A), is the nonbonding orbIta which has a node line on the central atom 0,. Howcvcr, the energy gap between this NO and the antibonding NO, pZ(S), is considerably small in the case of ozone. This leads to the instability of the RHF solution [2 11, which should be reorganized into the more stable DODS solution through the mixing between h”(A) and r_l”(S). The resulting corresponding orbitals x:I, and r&, no longer belong to the irreducible representations of the C,, group, thus being of the broken-symmetry.
1
I-
LI’ig. 1. Geometry 268
_----old coordmdtc
1 axes of ozone.
I’ig. 2. Contour nldp of the .syIllmetriC IT-NO.AZ(S) [Q] _ The orbital IS dcplcted on the zx-plane BS lllustratcd m fig. 1. The mark x dcnotcs the posltwn of an 0 atom. The contour incrcmcnt is 0.1 au (throughout this paper). Yet, the singlet geminal constructed of xx, and vyIO retains the correct A, spatial symmetry, as can be recognized from eq. (IO). Fig. 3 shows tile contour maps of the singlet-paired but spatially split corresponding orbitals xyio and vi0 of ozone. These orbitals arc considerably localized on either of the left (Q) and right (r) 0 atoms, in accord with the diradical property of ozone. However, the upspin electron which is localized primarily on the 0, atom is also populated to some extent on the 0, and 0, atoms (and vice versa). The leads to a relatively large overlap (5-W = 0.25) between the corresponding pair orbitals. The diradical character of ozone should accordingly be only moderate (52%) [lo]. The u-bonding orbitals of ozone arc the 2po-2~0 overlap type which is localized in the O-O bond regions, as is illustrated in fig. 4A. The 2pa-nonbonding orbit& are localized essentially around either 0, and 0,, as is shown in fig. 4B. However, the latter orbitais are slightly delocalized on 0, because of the direct (through-space) interactions. On the other hand, the node line indicates the repulsive interaction between the terminal atoms 0, and 0,. The remaining terminal lone pairs are both the 2stype which is localized on Or and 0, (fig. 5B). The lone pair on 0, has r~ snail sp-hydrid character with the greater lobe projected cutsidc tllc 0,-0,-O, triangle (fig. 5A). This must be an origin of the strong atomic
CEIEMICAL I’llYSICS L1.T-I‘I-KS
Volume 50. number 2
0,
0,
or
x
1 ig. 3. Contour maps of the split corresponding YY-
--
0, n-n~olccular orbhIs,
1
y
I
_-
-_I_-1
x
Y
r--
1 Scptcmbcr
9 x:,0
0,
1977
x
and qyiO of 08c)nc.
_--IN
_* *4Ea f?l \ L__ --
L
I*&. 4. Contour maps of the 2pa-bondmg arbital [2pab] (A) and of one of the two Zpo-nonbonding (loncpau) orbit& [2po,] (B) ot ozone. force [22,23] which is responsibIe for the 0,-0,-O, bending. Since the localization has been performed so as to retain the strong orthogonality of the gemenal-like projected HF wnvefunction, it seems interesting to compare the present localized MO’s with the GVB orbit&
---.
F:lg. 5. Contour maps of the Zs-nonbondmg (lone-patr) orbit& W’,, 2s,,] of the central (A) and ryJ]t (B) 0 awn\
reported by Goddard et al. [ lG,17]. The split rr-orbitals (fig. 3) are very slrnilar to those of the GVB tbcory (fig. 8 in ref. [ 181). The lower n-orbital and the five nonbonding orbitals (figs. 45,5A and 5B) are also quite similar. As for the u-bonding orbit&, however, the GVI solutions are slightly split in contrast to our HF solution
Volume
50. number
2
CHEMICAL
PIIYSICS
(fig. 4A). Ycrhaps our HF solution of ozone does not involve the dynamical correlation effect to such an extent as the GVB solution does.
LL-I-1-ERS
1 Septcmbcr
processes which involve the dissociation bonds.
1977
of covalent
References 4. Discussion and concluding
remarks
The present results suggest that there are basically two different types of pairs. One is a tight pair, whi!e the other is a pair with near dcgencracy. The 2s lone pair (2sn)’
and the lower 7r-orhit:d
(A;)’
of osone,
in
which two electrons are tightly packed into one orbital, belong to the former type, whcrcas the highest occupied n-orbltal of ozone, where electrons arc not tightly packed together, falls under tile latter class. The distmction betwcen tight and loose pairs is closely related with the orbital correction made by taking account of the correlation effects [24]. The correction is small for the RHF orbital which is responsible for the tight pair; the correlation effect remains dymrnical in nature in this case [24]. On the other hand, the orbital correction is large in the near-degeneracy case, whele the correlation is nondynamical [24] _ 1x1 fact, the RHF orbitals ale rcorganized into DODS orbit&, i.e., split pairs. The extended SCF orbitals involve the orbital corrections pecially,
clue to the clynarnical correlation effects. Esthe n-orbit& in the CVB method ale more
or less localized (i.e., diradicaloid) even in the case of stable molecules such as ethylene and acetylene [ 15 J _ Thus the GVB orbitals should differ considerably from the GHF orbitals in the domain of weak correlation. In the strong-correlation molecules such as ozone, however, both types of orbitals are mutually similar, as has been stated above. The localization procedure presented here provides lone pairs, electron-pair bonds and split (Le.. broken) bonds
of unstable
molecules.
It is useful
enough
for
dlscussions of chemical valency of unstable molecules in the framework of the molecular orbitlll (MO) concept. It is hoped that the LNO descriptions ~111be promising for the MO thcorctlcal tracing of chemical
270
[ I] K. YarnaguchI. Pbys. Letters
K. Obta, S. Yabuzillta
and T. r’ucno, Chem
49 (1977) 555.
[2] P.R. Ccrtam and J 0. Hirschfcldcr,
Chcm. Pbyr. Letters 2 (1968) 274. [ 3 J C. Edm1ston and K. liuedcnbcrg. Rev. Mod. Pbys 35 (1963) 457. [4] D. Peters. J.Cbem. Phys. 39 (1963) 2003. [S] V. h¶agnxco and A Periso. J. Chem. Phyc. 47 (1967) 971. (61 II. Wcmstcin, R. PJIIIICL and bl. Cohen, Advan. At. Mol. Phys. 7 (1971) 97. (71 K. Yamagurhi, ‘I’. I ucnu and II. fukutome. Chem. Pbys. Lcttcrs 22 (1973)461. [S] K. Yamtiguchi and 1’. l-ueno, Cbern. Pbys. Lctfers 38 (1976) 47,52. [9] AT Amos nnd G.G. IIall. Proc. Roy. Sot. A263 (1961) 483. [ 1OJ K. Yamnguch1,Chem. Pbys. Letters 33 (1975) 330. [Ii J K. Ynm;rgucb1. Y. Yoshiokn and T. Fueno. Cbem. l’by\. Letters 46 (1977) 360. [ 121 A C. Hurley, J-L. Lennxd-Jonc\ .md J.A. Poplc, Proc. Ray. SW. A220 (1953) 446. [ 131 \V. Kut/elnig+ J. Chcm. Ptiys 40 (1964) 3640. 1141 I) hf. S1lvcr, t.L. hlchler and K. Ruedenberg, J. Chum. l’hy\. 52 (1970) 1174. [IS] W-J. Ilunt, P J. Ilay .md W-A. Goddard III, J. Cbcm. Pbys. 57 (1972) 738. [ 161 W.K. Wadt and W.A. Goddard III, J. Am. Chem. Sot. 96 (1974) 1689. [ 171 D. Grimbert and A. Dcvdquct, Mol. Whys. 27 (1974) 831. [18] W A. Goddard 111,J.H. Dunnmg. W.J. Hlrnt and P.J Ilay. AccountsChcn1. Res. 6 (1973) 368. [ 191 R.P. hiessmer and D.R. Salahub, J. Chcm. Phys. 65 (1976) 779. [20] h1.J.S. Dewar, S. OlivelIe and II S. R7cpa, Chcm. Phy\. Lettcrc47 (1977) 80 (211 R. Yamquchi and T. I’ueno, Cbem. Phys. Letters 23 (l973)471. [22] K.T.W. Bader, I. Keavcny and P.L..Cade, J. Cilem. Phyr. 47 (1967) 3381. [23i H. Nnkatwji, J. Am.Cbcm. Sot. 95 (1973) 345. [24] 0. Sinanoglu, Advan. C’hcm. Phys. 14 (1968) 237.
50. nurnbcr
LOCALIZED
2
CHkMICAL
NATURAL
ORBITALS
PHYSICS
OF UNSTABLE
Kizashi YAMAGUCHI, Koji OHTA and Takayuki Dqartmcnr Toyorlaka. Kcccived
of Chonirt.y. Faculty of Enguxwing Osaka 560, Japan 15 June
1 Scptcmbcr
LITTCRS
MOLECULES:
1977
OZONE
FUENO
ScrcncL: Osaka Universify,
1977
Use ot locdlved natural orbltdls (LNOX) IS proposed for the elucidation of chcmicnl vaIcncc of unstable ground-stdtc rnolccuks. TIIC LNO’\ of~ronc as an evarnplc hnvc been derived from itsgcncralircd Ilartrcc-Fock (Glfr) solutions It is shown that the LNO de\aption\ of ozone are much in Imc with the results which Goddard cd al. have obtained using the gcncralircxl valence bond (GVB) rncthod.
1. Introduction In a previous paper [I] the clcctronic structure of the trimcthylcne diradical was investigated on the basis of natural orbitals (NO’s) derived from the generali7cd Hartree-Fock (GIIF) so!utions. It was found that the molecular orbital concept can be extended successfully even to such a strongly correlative rnolccule by the aid of the !imited NO configuration interaction (CI) procedure. Since the natural orbit& belong to irreducible representations of a point group, they are also well suited for discussions of the symmetry properties of the electronic states of unstable molecules. Howcvcr, the delocalized natural orbitals (DNO’s) arc not particularly useful for the elucidation of chemical valence problems. The theory of localized natural orbitals (LNO’s) [21, on the other hand, appears to be of much value to discussions of the nature or the inner-shell, lone-pair and bond orbitals in molecules 13-51. The purpose of this work is to assess thesignificance of the LNO’s derived from the CHF solutrons for unstable molecules. The electronic structure ofozone ~111be dealt with as an example.
2. Localized natural molecular
orbit&
The restricted ILrtrec-Fock (RilP) solutions ot unstable molecules are triplet-unstable [7 ] and spin266
flipping unstable [8] in the singlet and nonsinglet states, respectively. They arc reorganized into more stable unrestricted Hartree-Fock (DODS) wavefunctions, which arc given by
where xi and vi are the paired corresponding orbit& [9,10] and ho, denotes an unpaired natural orbital. The former orbit& are related to the natural orbitals hi and pi of the DODS solutions by x, = (cos q/2)$
+ (sin wi/2)cc,.
vi = (cos w,/2)h,
- (sin w,/2)i.+,
(2) the RIIF solutions being given by setting wI = 0 (i = 1, . . .,12). The first-order spinless densrty matrix is now expressed as
+,I$
Xo,(r)Aoj(r) =
‘3 k=l
“k uk
tr) uk (d,
(3)
where N+ = (1 + cos Wi), TIC
form ofp(‘)(r)
rzy = (1 - cos Wi).
is invariant to the non-orthogonal
(4)
Volume 50, number
transformation
2
CliEXIICAL
PIIYSICS
which yields the localized natural orbit&
ui [2]:
1 Scptelllller
LIT-l LRS
projection is necessary function is +;;;I
(5)
=5& 1LL1 . . - ulvl+
’ u01L’02u03 “‘“O~ The density matrix can then be rcwrittcn
[l I] . The projected
1977
HF wave-
1VI+ 1 ---X,,%
ccco 1’ j=,
I I
as
An essentral feature of the above transformation is that the form of p’*)(r) is preserved as a sum of square terms a¶one. The transformation matrix T = (T&) is determincd, for example, by the energy-Iocalizcd criterion [2,3]. When ‘I$$ involves only tbc doubly occupied orbitals, i.e., nk = 2 for k = I,. _., tz and irk = 0 for k=2n+l , . _., 2rz + p, the transfonnatron is equivalent to the Edmiston-Ruedenberg localization procedure f3]. Since the RHF wavcfunctions of nonsinglet species also involve n doubly occupied orbitals, localuation of the closed-shell part gives n Hartrce-like NO’s which can be regarded as tz singlet pair bonds, However, the number of the Ilartree-like natural orbitals is no longer equal to that of the singlet pair bonds if G$ involves split orbital pairs (wi # 0). In such cases, the transformation cq. (5) cannot be directly applied to discussions of chemical valency of unstable molecules. As shown previously [ 1,8,10] , there are only a few split orbitals in the ground HF solutions of unstable molecules. The corresponding orbitals Xi and TQof a split pair are inbercntly more or less localized to reduce the strong intrashcll Coulomb repulsion. Thus it seems reasonable to leave the split pairs out of the transformation procedure. Such a localization procedure will maximize the sum of the intrapair Coulomb repulsions of tight pairs, preserving the number of the pair bonds. ‘Ihc unpaired natural orbitals {hoi) can also be transformed into the Hartrce-like unpaired natural orbitals IUui}. The HF wavefunction rewritten in conformity to the above localization principle is ,I,@-) = lu u --.ffIuIXl+1 e 1+1 -- -X,?JQ,~()2 iIF r 1
- - -uQJ,
where dIis the antisymmetrizcr and 0, ary the orthogonal spin functions. The spin coupling coefficients Ci are determined without reoptimiring the Gl-lF orbit& [l l] . This reduces the number of the spin coupling parameters since the tight pairs are singlet-coupled. As has been shown previously [lo] , singlet diradical species involve only one split pair which may be referred to as the reaction orbitals (RO’s). Therefore, their projected HF wavefunctions can easily be obtained by unng the singlet-coupled spin function only (Cl = 1 whi\eCi=O fori+ 1) [12- IS]: $$[
=92’u, 24@ . . . q&3,
(9
whcrc the split pair is given by
The spatial overlap TRo and the normalizing are given, respectively, by TRO =,! kR09R0
(17 = cos OR0
factor N (11)
and N= [2(1 + 7$)]
-“I.
(12)
The strong orthogonality between singlet pairs is preserved because of the orthononnal condition of natural orbrtals j12,15]. Expression (10) is similar to that used for the IlcrtlerLondon geminal [12]. [lowever, the spatial functions need no longer be atomic orbitals. In fact Xlio and r+10 are a little delocalized because of the mclusion of ionic structures in the simple Vl3 Cl terminology. In this SCrlsC the corrcspondmg reaction orbrtals sl~oulcl bc simrI:n to the generalized valence bond &VU) orbitals [ 151 in the strong correlation region (T,, < 1).
(7) which involves I tight pairs, (n - I) split pairs and p unpaired orbitals. Since the split pairs do not possess the
pure spin symmetry concerning the S2 operator, spin
3. Ozone The electronic
structure
of ozone has already received 267
volume
so,
nuIrlber
CIII-MIC’AL PHYSICS
2
LI:TTI:RS
1 September
1977
considerable attention [ 16 - 201. Therefore, it seems to be one of the most appropriate molecules that deserves investigation by the present localization! theory. IIerc, WC will examine the ring-opened ozone [18] as an example (fig. 1). The corrcspondmg orbit& were dctermined by the method described previously [lo]. Only the highest occupied n-orbital is split in the ground singlet (IA,) state. The remaining eight delocalized orbitals are transformed into five non-bonding (lone pair) orbit&, two o-bonding orbitals and one delocalized norbital. This last n-orbital remains intact after the localization procedure. The projected HF solution of Ozone can thus bc rcprcsented as
(13) x G&0 + G&c& where the subscripts b and n denote the bonding and nonbonding orbit&, respectively. In what follows, characteristics of ali these orbitals ~111be illustrated by use of the contour maps. There are three natural rr-orbitah, i.e., X:(S), hrr(A) and ,u”(Sj in the cases of three-center (isosceles) systems such as ozone. The lowest r-r-orbital, hi(S), is symmetric and delocalized over the whole molecule. fn the case of ozone, h;(S) is nothmg but the rrb. Its contour map is as depicted in fig. 2. Fig. 2 clearly shows the strong nbonding property of the Xt_ The anti-sjrmmetrlc NO, A”(A), is the nonbonding orbIta which has a node line on the central atom 0,. Howcvcr, the energy gap between this NO and the antibonding NO, pZ(S), is considerably small in the case of ozone. This leads to the instability of the RHF solution [2 11, which should be reorganized into the more stable DODS solution through the mixing between h”(A) and r_l”(S). The resulting corresponding orbitals x:I, and r&, no longer belong to the irreducible representations of the C,, group, thus being of the broken-symmetry.
1
I-
LI’ig. 1. Geometry 268
_----old coordmdtc
1 axes of ozone.
I’ig. 2. Contour nldp of the .syIllmetriC IT-NO.AZ(S) [Q] _ The orbital IS dcplcted on the zx-plane BS lllustratcd m fig. 1. The mark x dcnotcs the posltwn of an 0 atom. The contour incrcmcnt is 0.1 au (throughout this paper). Yet, the singlet geminal constructed of xx, and vyIO retains the correct A, spatial symmetry, as can be recognized from eq. (IO). Fig. 3 shows tile contour maps of the singlet-paired but spatially split corresponding orbitals xyio and vi0 of ozone. These orbitals arc considerably localized on either of the left (Q) and right (r) 0 atoms, in accord with the diradical property of ozone. However, the upspin electron which is localized primarily on the 0, atom is also populated to some extent on the 0, and 0, atoms (and vice versa). The leads to a relatively large overlap (5-W = 0.25) between the corresponding pair orbitals. The diradical character of ozone should accordingly be only moderate (52%) [lo]. The u-bonding orbitals of ozone arc the 2po-2~0 overlap type which is localized in the O-O bond regions, as is illustrated in fig. 4A. The 2pa-nonbonding orbit& are localized essentially around either 0, and 0,, as is shown in fig. 4B. However, the latter orbitais are slightly delocalized on 0, because of the direct (through-space) interactions. On the other hand, the node line indicates the repulsive interaction between the terminal atoms 0, and 0,. The remaining terminal lone pairs are both the 2stype which is localized on Or and 0, (fig. 5B). The lone pair on 0, has r~ snail sp-hydrid character with the greater lobe projected cutsidc tllc 0,-0,-O, triangle (fig. 5A). This must be an origin of the strong atomic
CEIEMICAL I’llYSICS L1.T-I‘I-KS
Volume 50. number 2
0,
0,
or
x
1 ig. 3. Contour maps of the split corresponding YY-
--
0, n-n~olccular orbhIs,
1
y
I
_-
-_I_-1
x
Y
r--
1 Scptcmbcr
9 x:,0
0,
1977
x
and qyiO of 08c)nc.
_--IN
_* *4Ea f?l \ L__ --
L
I*&. 4. Contour maps of the 2pa-bondmg arbital [2pab] (A) and of one of the two Zpo-nonbonding (loncpau) orbit& [2po,] (B) ot ozone. force [22,23] which is responsibIe for the 0,-0,-O, bending. Since the localization has been performed so as to retain the strong orthogonality of the gemenal-like projected HF wnvefunction, it seems interesting to compare the present localized MO’s with the GVB orbit&
---.
F:lg. 5. Contour maps of the Zs-nonbondmg (lone-patr) orbit& W’,, 2s,,] of the central (A) and ryJ]t (B) 0 awn\
reported by Goddard et al. [ lG,17]. The split rr-orbitals (fig. 3) are very slrnilar to those of the GVB tbcory (fig. 8 in ref. [ 181). The lower n-orbital and the five nonbonding orbitals (figs. 45,5A and 5B) are also quite similar. As for the u-bonding orbit&, however, the GVI solutions are slightly split in contrast to our HF solution
Volume
50. number
2
CHEMICAL
PIIYSICS
(fig. 4A). Ycrhaps our HF solution of ozone does not involve the dynamical correlation effect to such an extent as the GVB solution does.
LL-I-1-ERS
1 Septcmbcr
processes which involve the dissociation bonds.
1977
of covalent
References 4. Discussion and concluding
remarks
The present results suggest that there are basically two different types of pairs. One is a tight pair, whi!e the other is a pair with near dcgencracy. The 2s lone pair (2sn)’
and the lower 7r-orhit:d
(A;)’
of osone,
in
which two electrons are tightly packed into one orbital, belong to the former type, whcrcas the highest occupied n-orbltal of ozone, where electrons arc not tightly packed together, falls under tile latter class. The distmction betwcen tight and loose pairs is closely related with the orbital correction made by taking account of the correlation effects [24]. The correction is small for the RHF orbital which is responsible for the tight pair; the correlation effect remains dymrnical in nature in this case [24]. On the other hand, the orbital correction is large in the near-degeneracy case, whele the correlation is nondynamical [24] _ 1x1 fact, the RHF orbitals ale rcorganized into DODS orbit&, i.e., split pairs. The extended SCF orbitals involve the orbital corrections pecially,
clue to the clynarnical correlation effects. Esthe n-orbit& in the CVB method ale more
or less localized (i.e., diradicaloid) even in the case of stable molecules such as ethylene and acetylene [ 15 J _ Thus the GVB orbitals should differ considerably from the GHF orbitals in the domain of weak correlation. In the strong-correlation molecules such as ozone, however, both types of orbitals are mutually similar, as has been stated above. The localization procedure presented here provides lone pairs, electron-pair bonds and split (Le.. broken) bonds
of unstable
molecules.
It is useful
enough
for
dlscussions of chemical valency of unstable molecules in the framework of the molecular orbitlll (MO) concept. It is hoped that the LNO descriptions ~111be promising for the MO thcorctlcal tracing of chemical
270
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