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Prediction of a metastable D3h form of tetra oxygen
Volume 157, number 5
PREDICTION I. R0EGGEN
19May 1989
CHEMICAL PHYSICS LETTERS
OF A METASTABLE and E. WISLQFF
D3,, FORM OF TETRA OXYGEN
NILSSEN
Institute of Mathematical and Physical Sciences. Universityof Trams@,P.O. Box 953, N-9001 Trams@,Norway Received 3 June 1986; in final form 8 March 19S9
Ab initio calculations based on an extended geminal model have shown that a branched form of O,, i.e. OOXcharacterized by a central atom and three equivalent ligand atoms, can exist in a metastable D3,,form. The equilibrium bond length of this form is 1.330 A. The energy of 04(Dj,) is 9.81 kJ/mol higher than the energy of isolated ozone and oxygen in the ‘D state. There is an energy barrier (measured from O,( D,,) ) of 127.28 kJ/mol separating tetra oxygen fromozone and oxygen (ID).
1. Introduction Tetra oxygen is known experimentally as a dimer [ 1,2 1. The structure of the molecule is of the van der Waals type with very low stability ( z 1.0 kJ/mol ) and large 02-0, equilibrium distance ( z 4 A). More recently the theoretical work of Adamantides and coworkers [ 31 points to the existence of a metastable covalent molecule 0, completely different from the van der Waals structure, (O,),, detected experimentally. At its equilibrium geometry the molecule is quasisquare, slightly twisted out of plane, corresponding to the symmetry group DZd. To our knowledge no other stable or metastable form of tetra oxygen has been reported. In this work we are concerned with the possible existence of a branched 04, i.e. 003, characterized by a central atom and three equivalent ligand atoms. The symmetry group of the molecule is in this case D3,,. Our interest in this form of the molecule can be traced back to the following very simple idea. The isoelectronic ions SO:-, CO:- and NO, are known to be stable. Since 003 is just the continuation of this sequence, it might therefore exist in a metastable form.
2. Method of study The method which we adopt in this study is a slight modification of a recently proposed approximation 0 009-2614/89/$ (North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
scheme denoted the EXRHF2 model [ 41. The simplification introduced by using a restricted HartreeFock wavefunction as the rootfunction in the model rather than a proper antisymmetric product of strongly orthogonal geminals (APSG) function, can be justified by the fact that there is no splitting of electron pairs when O4 dissociates into ozone and atomic oxygen ( ‘D ). The main features of the model are as follows: For a 2N-electron closed-shell system the ansatz for the electronic wavefunction is expressed by the equation @EXRHF
= gjR”F
+fM
wJl&&Nl{~
+fM K= I P-2.21&
WI{@
;;7&41
w&WQjc21]
Q~W)
K
=QRHF+
5
ylK+
K=l
f
YK,.
(1)
K
The basic approximation in this model, i.e. @ RHF,is a restricted Hartree-Fock wavefunction: ,$InHF=+jQ~l~W’l
jj,
/j;Hi”‘,
(2)
where /i~HF(rl~,,r2u~)=V1K(r,)~K(rZ)eO((T1,4). The localized orbitals {@} are determined B.V.
(3) by min409
Volume 157, number 5
CHEMICAL
imizing the Coulomb repulsion between the associated electron pairs. The interpretation of the terms in eq. (2) is very simple. The RHF function plus the single-pair correction term YK describe the correlated motion of the electron pair K when the rest of the system is described by an RHF approximation. Similarly, YtK,Ll=@HF+
YKt YL t YKL
(4)
describes the correlated motion of the electron pairs K and L when the residual system is represented by an RHF function. As for the wavefunction YEXRHFwe adopt intermediate normalization, i.e. (@RHF] !JJEXRHF)= 1 .
@RHF
-(
=ERHF+
I
H~EXRHF
f c,+ R= I
2
cKL,
RHFlHVK)
RHFl
(7)
H!PKL) .
410
, Kc{l,2,
A&=&,
ie{l, 2, 3).
(11)
$nAI, A& AIS .
(12)
In previous work [ 51, we introduced the concept of fragment analysis of molecular electronic energies. By considering the system as composed of fragments, the total electronic energy, i.e. the total energy in the absence of nuclear motion, can be written
(8)
Tn eqs. (7), (8), H is the electronic Hamiltonian in question. The double-pair correction terms {tKL} are determined by full CI calculations using a restricted virtual orbital space. The procedure is identical to the one defined for the EXRHF2 model [ 41, but we neglect the terms t@ and S& in the formulas defining eKL.The reason for this modification of the model is that the second-order perturbation type corrections c& and S& might not be sufficiently reliable for the problem in question. Test calculations on water suggest that this model yields a correlation energy which is in error by roughly 2-3% compared with the full CI result, The localization of the RHF geminals is in this work described by charge centroids and charge ellipsoids. The charge centroids are given by the familiar expressions rK=(ffIrpK)
where xf’ is the rth component of the charge centroid rK defined in eq. (9). The diagonalization of the variance matrix yields the charge ellipsoid. The eigenvalues {a,, a2, as} of the matrix (MrS) correspond to the squares of the half-axes of the ellipsoid. The standard deviations in three orthogonal directions are therefore given by
V=
and GL=<@
(10)
(6)
where t,=(@
f-,SE{13253) 3
(or variance matrix) are
The quantities {A/,} can then be used as a measure of the extension of the geminal one-electron density. Furthermore, we may also use the volume of the ek lipsoid as a single number for the extension of the geminal one-electron density:
>
K
The second-order moments defined by the relations
1989
(5)
The energy is then given by EEXRHF_
19May
PHYSICS LETTERS
.... N}.
(9)
(13) where A4 is the number of fragments. The intra- and inter-fragment energies can be further partitioned
151.
3. Computational details The orbital basis is a set of (9s,5p, Id) GTOs contracted to [ 6s,4p, I d 1. The uncontracted s- and p-type functions are taken from the work of van Duijneveldt [ 61. The expansion coefficients of the atomic 1s and 2p orbitals are used as contraction coefficients. The exponents of the polarization functions are set equal to 1.0. Calculations on Nz, C,H, and C,H, using the APSG model demonstrated that the most appropriate description of multiple bonds are characterized by a (3-7~structure rather than equivalent bent bonds
Volume 157, number 5
CHEMICAL PHYSICS LETTERS
[ 71, If we replace the RHF function with an APSG function, we would therefore anticipate a bonding picture characterized by o-x-type double bonds in ozone and tetra oxygen. A proper description should therefore imply a partitioning into cr- and n-type geminals. Such a partitioning was indeed obtained by using energy localization of the RHF orbitals. In the calculation of the single-pair correction terms {Q}, we used 42 natural orbitals for ozone and tetra oxygen. For isolated oxygen we naturally had to restrict the number of orbitals to the maximum ,number available, i.e. 20. The adopted procedure for the calculation of the double-pair correction terms {tKL} requires that we pick out the rrzKand m, natural or_bitals with the highest occupation coefficients in the improved description of electron pairs K and L, respectively [ 41. In this work we selected for the double-pair correction terms mK equal to 10, 14 and 16 for core geminals, o-type valence geminals and n-type geminals, respectively. Our largest full CI expansion for the four-electron problem in question was therefore constructed from 32 spatial orbitals. It comprised 2460 16 Slater determinants, Furthermore, for a given geometry of tetra oxygen there are 120 full CI four-electron problems to be solved. However, due to symmetry, the number of computed double-pair correction terms is considerably smaller.
4. Results and discussion For isolated ozone we use the experimental geometry [ 81 corresponding to a bond length of 2.38 au and a bond angle of 117”. Due to the highly correlated wavefunction adopted in this work, a complete geometry optimization is expected to yield only a minor energy change compared with the calculated value for the experimental geometry. The energy localizations of the orbitals gives as a result a bonding picture for ozone characterized by two o-x-type double bonds. The volumes of the corresponding charge ellipsoids are respectively 1.1504 and 4.4307 au for the o- and n-type geminals (table 1). Oxygen in the ‘D state is characterized by three equivalent valence electron pairs with charge centroids in a plane through the nucleus. The volumes of the associated charge ellipsoids are 1.3995 au. The total energies of isolated ozone and ‘D oxygen are in table 2.
19 May 1989
The optimized bond length for the D3,, form of tetra oxygen is obtained by a parabolic fit based on calculations for the bond lengths 2.47,2.52 and 2.57 au. The bond length corresponding to the energy minimum is equal to 2.5 14 au. The energy localization of the RHF geminals yields a bonding picture characterized by three equivalent o-x-type double bonds. In fig. 1 we display charge centroids and the intersection of the charge ellipsoids of the geminals with the molecular plane. The volumes of the charge ellipsoids are 1.3498 and 4.6763 au, respectively, for the o- and a-type bond pair geminals. The total energy of the Dji, form of tetra oxygen for the optimized geometry is 0.003739 au higher than the sum of energies of isolated ozone and oxygen in the ‘D state. However, this result does not exclude the existence of an energy barrier separating the Dlh form of tetra oxygen from isolated ozone and oxygen in the ‘D state. To determine a possible transition state the following procedure was adopted. First, a transition state of 0, is assumed to have a planar geometry. Second, the change in the bond angle, ~020103, from 117o in isolated ozone to 120’ in tetra oxygen, is assumed to have only a minor effect on the energy changes during a reaction between ozone and atomic oxygen, The validity of this assumption is checked by calculating the energy change for isolated ozone when the bond angle was increased from 117” to 120”. The energy change was only 0.000020 au. For three fixed 01-04 distances, i.e. {3.014 au, 3.5 14 au, 4.014 au}, the bond length in the ozone-like fragment was optimized. By a parabolic tit we then obtained the following geometry for the transition state: R(Ol-04)
~3.476 au,
R(Ol-02)=R(Ol-03)=2.436 ~020103=
au,
120” (fixed) .
The interpolated value of the energy is - 300.042622 au to be compared with the calculated value of - 300.042664 au for the nuclear configuration of the transition state. We therefore conclude that the calculated energy for the transition state is very close to the value that can be obtained by a complete geometry optimization. The assumption of no splitting of electron pairs during a reaction of O4 to 03+0( ‘D) is supported 411
CHEMICAL PHYSICS LETTERS
Volume 157, number 5
19 May 1989
Table 1 Volumes of selected geminal charge ellipsoids of 0, for three different geometries a.bl
03+O(‘D) transition State d,
O.dh,) e,
‘)
a(Ol-02)
n(Ol-02)
o(Ol-04)
1.1504
4.4307
1.1755 f’
I .2758 1.3498
4.6014 4.6763
1.3498
1.472I
a(Ol-04)
2.1531 4.6763
a’ Volumes in atomic units ‘) 0 1 is the central nucleus, 02 and 03 are the ligand nuclei of the ozone fragment. c, Isolated ozone and oxygen. Experimental geometry is adopted for ozone [ 81. d, Thegeometryofthe transitionstate: R(Ol-02)=R(Ol-03)=2.436au,R(Ol-04)=3.476 for numbering of atoms. ‘) Bond length R= 2.52 au. ‘) Transforms to a u-type bond pair in O,,.
Lone pair 1 (02)
Lone pair 2 (02)
Lone pair (04)
1.3961
1.3513
1.3995
1.4028 1.3964
1.3998 1.3964
1.2721 1.3964
au, L020103=
120’. See
figs.1and 2
Table 2 Total energy and energy partitioning of O4 for selected geometries ‘)
03tO(‘D)
U)
transition state d, O,(D,,) ‘I
0.0 (-300.094903) 0.052237 0.003761
0.0 (-225.138387) 0.172399 1.638716
0.0 (-74.956516) 0.140559 1.736001
-0.260720 -3.370956
a) Energies in atomic units. b, Isolated ozone and oxygen. Experimental geometry is adopted for ozone [ 81. c) The values of the energy components for this geometry are used as zero levels. Calculated values in parentheses. d, Geometry as defined in footnoted to table 1. e, This partitioning is for a bond length of Rz2.52 au. The total energy is 0.000022 au higher than the value corresponding to the optimized bond length of R = 2.5 14 au.
Fig. 1,Intersection between the xy plane (molecular plane) and the charge ellipsoids of O4 in Dlh symmetry. The bond length is 2.52 au. The charge centroids of the o- and n-type geminals are marked with the symbols x and q, respectively.
412
Fig.2. Intersection between the xy plane (molecular plane) and the charge ellipsoids of Oq in the transition state. See footnote d, table 1,for the specification of the geometry of the transition state. The charge centroids of the o- and x-type gemiaals are marked with the symbols x and El, respectively.
Volume 157, number 5
CHEMICAL PHYSICS LETTERS
by these calculations. A possible bond-breaking process for the four-electron system associated with the 0 104 double bond would imply large values for the corresponding double-pair correction term tKL,since eKLis determined by the lowest eigenvalue of the full CI matrix in question. In fact, the largest magnitude of cKLis obtained for the smallest value of the 0 104 distance, If we consider a reaction starting from ozone and oxygen in the ‘D state, the charge centroids of the geminals, table 3, give the following picture of the reaction. At the transition state the oxygen atom has made itself “ready” for accepting the lone pair of the central oxygen of the ozone molecule, as a dative otype bond. One of the three equivalent valence geminals of the oxygen atom is changed into a pure pr type geminal with respect to the molecular plane (zaxis orthogonal to the molecular plane). This particular geminal is transformed into a x-type bond pair geminal during the reaction. The two other valence geminals of the oxygen fragment have charge centroid vectors in the molecular plane and pointing almost exactly in opposite directions (fig. 2). In the ozone fragment the lone pair of the central oxygen atom is shifted towards the approaching oxygen atom. As a result of this shift, there is more space available for the o-type bond pairs in the vicinity of the central nucleus and they are shifted towards this nucleus. The approaching x-type electron pair of the oxygen fragment reduces the available space for the ozone Relectrons in the vicinity of the central oxygen nucleus and as a result the x-electron pairs of the ozone fragment are shifted away from the central nucleus. As the system passes through the transition state towards 04(Djh), the same trend in the shifts of the
19 May 1989
charge centroids is observed, but the shifts are more pronounced. The internal rearrangement of the nuclei and the electrons of the fragments during the reaction leads to higher intrafragment energies. This effect is displayed in table 2. We notice from table 2 that in the transition state, the distortion energies, i.e. the energy changes due to the presence of the other system, and the interaction energy between the fragments, are all of the same magnitude as the net energy increase. On the other hand, when the system passes the saddle point and transforms into the Djt, form of tetra oxygen, huge changes are taking place with respect to the energy components. For the equilibrium structure, we find large distortion energies which are almost compensated by a corresponding large interaction energy. The magnitudes of the energy components are in this case practically three orders of magnitude larger than the net energy change. 5. Conclusion This work indicates that tetra oxygen can exist in a metastable DShform. The energy of tetra oxygen is 9.8 1 kJ/mol higher than the energy of isolated ozone and oxygen in the ‘D state. There is an energy barrier (measured from 04(D3,,) ) of 127.28 kJ/mol separating tetra oxygen from ozone and oxygen ( ‘D). Due to this barrier, a reaction between ozone and oxygen ( ‘D) yielding 0, (Ds,,) as reaction product, can only take place in exceptional circumstances. However, if 04(Dsh) is formed, then this form should be kinetically stable.
Table 3 Distances between selected charge centroids of 0, and the central oxygen nucleus for three different geometries aSb)
Ox+O(‘D) =) transition state d, O4Uh)
‘)
R(a(Ol-02))
R(n(Ol-02))
R(o(Ol-04))
1.0810 1.0741 1.0055
1.5286 1.5473 1.8295
0.6208 ‘) 0.7224 1.0055
‘) Distances in atomic units. ‘) 01 is the central nucleus, 02 and 03 are the Iigand nuclei of the ozone fragment. c) Isolated ozone and oxygen. Experimental geometry is adopted for ozone [ 81. d, Geometry as defined in footnote d, table 1. ‘1 Bond length R~2.52 au. f, Transforms to a o-type bond pair in O+
413
Volume 157, number 5
CHEMICAL PHYSICS LETTERS
Acknowledgement
The authors are grateful to Professor Asbjom Hordvik for suggesting the problem. The calculations have been performed on the CRAY X-MP28 supercomputer, SINTEF, Trondheim, Norway. References [ 1 ] S.J. Arnold, E.A. Ogryzlo and H. Witzke, J. Chem. Phys. 40 (1964) 1769.
414
19 May 1989
[2 ] D.L. Huestis, G. Black, S.A. Edelstein and R.L. Blunt, J. Chem. Phys. 60 (1974) 4471. [ 3 ] V. Adamantides, N. Neisius and G. Verhaegen, Chem. Phys. 48 (1980) 215. [4] I. Reeggen, J. Chem. Phys. 89 (1988) 441. [ 5] 1.Rseggen and E. Wisleff-Nilssen, J. Chem. Phys. 86 ( 1986) 2869. [6] F.B. van Duijneveldt, IBM Research Report RJ 945 (1971). [ 7 ] I. Reeggen, unpublished calculations. [8] T. Tanaka and Y. Monno, J. Mol. Spectry. 33 (1970) 538.
PREDICTION I. R0EGGEN
19May 1989
CHEMICAL PHYSICS LETTERS
OF A METASTABLE and E. WISLQFF
D3,, FORM OF TETRA OXYGEN
NILSSEN
Institute of Mathematical and Physical Sciences. Universityof Trams@,P.O. Box 953, N-9001 Trams@,Norway Received 3 June 1986; in final form 8 March 19S9
Ab initio calculations based on an extended geminal model have shown that a branched form of O,, i.e. OOXcharacterized by a central atom and three equivalent ligand atoms, can exist in a metastable D3,,form. The equilibrium bond length of this form is 1.330 A. The energy of 04(Dj,) is 9.81 kJ/mol higher than the energy of isolated ozone and oxygen in the ‘D state. There is an energy barrier (measured from O,( D,,) ) of 127.28 kJ/mol separating tetra oxygen fromozone and oxygen (ID).
1. Introduction Tetra oxygen is known experimentally as a dimer [ 1,2 1. The structure of the molecule is of the van der Waals type with very low stability ( z 1.0 kJ/mol ) and large 02-0, equilibrium distance ( z 4 A). More recently the theoretical work of Adamantides and coworkers [ 31 points to the existence of a metastable covalent molecule 0, completely different from the van der Waals structure, (O,),, detected experimentally. At its equilibrium geometry the molecule is quasisquare, slightly twisted out of plane, corresponding to the symmetry group DZd. To our knowledge no other stable or metastable form of tetra oxygen has been reported. In this work we are concerned with the possible existence of a branched 04, i.e. 003, characterized by a central atom and three equivalent ligand atoms. The symmetry group of the molecule is in this case D3,,. Our interest in this form of the molecule can be traced back to the following very simple idea. The isoelectronic ions SO:-, CO:- and NO, are known to be stable. Since 003 is just the continuation of this sequence, it might therefore exist in a metastable form.
2. Method of study The method which we adopt in this study is a slight modification of a recently proposed approximation 0 009-2614/89/$ (North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
scheme denoted the EXRHF2 model [ 41. The simplification introduced by using a restricted HartreeFock wavefunction as the rootfunction in the model rather than a proper antisymmetric product of strongly orthogonal geminals (APSG) function, can be justified by the fact that there is no splitting of electron pairs when O4 dissociates into ozone and atomic oxygen ( ‘D ). The main features of the model are as follows: For a 2N-electron closed-shell system the ansatz for the electronic wavefunction is expressed by the equation @EXRHF
= gjR”F
+fM
wJl&&Nl{~
+fM K= I P-2.21&
WI{@
;;7&41
w&WQjc21]
Q~W)
K
=QRHF+
5
ylK+
K=l
f
YK,.
(1)
K
The basic approximation in this model, i.e. @ RHF,is a restricted Hartree-Fock wavefunction: ,$InHF=+jQ~l~W’l
jj,
/j;Hi”‘,
(2)
where /i~HF(rl~,,r2u~)=V1K(r,)~K(rZ)eO((T1,4). The localized orbitals {@} are determined B.V.
(3) by min409
Volume 157, number 5
CHEMICAL
imizing the Coulomb repulsion between the associated electron pairs. The interpretation of the terms in eq. (2) is very simple. The RHF function plus the single-pair correction term YK describe the correlated motion of the electron pair K when the rest of the system is described by an RHF approximation. Similarly, YtK,Ll=@HF+
YKt YL t YKL
(4)
describes the correlated motion of the electron pairs K and L when the residual system is represented by an RHF function. As for the wavefunction YEXRHFwe adopt intermediate normalization, i.e. (@RHF] !JJEXRHF)= 1 .
@RHF
-(
=ERHF+
I
H~EXRHF
f c,+ R= I
2
cKL,
RHFlHVK)
RHFl
(7)
H!PKL) .
410
, Kc{l,2,
A&=&,
ie{l, 2, 3).
(11)
$nAI, A& AIS .
(12)
In previous work [ 51, we introduced the concept of fragment analysis of molecular electronic energies. By considering the system as composed of fragments, the total electronic energy, i.e. the total energy in the absence of nuclear motion, can be written
(8)
Tn eqs. (7), (8), H is the electronic Hamiltonian in question. The double-pair correction terms {tKL} are determined by full CI calculations using a restricted virtual orbital space. The procedure is identical to the one defined for the EXRHF2 model [ 41, but we neglect the terms t@ and S& in the formulas defining eKL.The reason for this modification of the model is that the second-order perturbation type corrections c& and S& might not be sufficiently reliable for the problem in question. Test calculations on water suggest that this model yields a correlation energy which is in error by roughly 2-3% compared with the full CI result, The localization of the RHF geminals is in this work described by charge centroids and charge ellipsoids. The charge centroids are given by the familiar expressions rK=(ffIrpK)
where xf’ is the rth component of the charge centroid rK defined in eq. (9). The diagonalization of the variance matrix yields the charge ellipsoid. The eigenvalues {a,, a2, as} of the matrix (MrS) correspond to the squares of the half-axes of the ellipsoid. The standard deviations in three orthogonal directions are therefore given by
V=
and GL=<@
(10)
(6)
where t,=(@
f-,SE{13253) 3
(or variance matrix) are
The quantities {A/,} can then be used as a measure of the extension of the geminal one-electron density. Furthermore, we may also use the volume of the ek lipsoid as a single number for the extension of the geminal one-electron density:
>
K
The second-order moments defined by the relations
1989
(5)
The energy is then given by EEXRHF_
19May
PHYSICS LETTERS
.... N}.
(9)
(13) where A4 is the number of fragments. The intra- and inter-fragment energies can be further partitioned
151.
3. Computational details The orbital basis is a set of (9s,5p, Id) GTOs contracted to [ 6s,4p, I d 1. The uncontracted s- and p-type functions are taken from the work of van Duijneveldt [ 61. The expansion coefficients of the atomic 1s and 2p orbitals are used as contraction coefficients. The exponents of the polarization functions are set equal to 1.0. Calculations on Nz, C,H, and C,H, using the APSG model demonstrated that the most appropriate description of multiple bonds are characterized by a (3-7~structure rather than equivalent bent bonds
Volume 157, number 5
CHEMICAL PHYSICS LETTERS
[ 71, If we replace the RHF function with an APSG function, we would therefore anticipate a bonding picture characterized by o-x-type double bonds in ozone and tetra oxygen. A proper description should therefore imply a partitioning into cr- and n-type geminals. Such a partitioning was indeed obtained by using energy localization of the RHF orbitals. In the calculation of the single-pair correction terms {Q}, we used 42 natural orbitals for ozone and tetra oxygen. For isolated oxygen we naturally had to restrict the number of orbitals to the maximum ,number available, i.e. 20. The adopted procedure for the calculation of the double-pair correction terms {tKL} requires that we pick out the rrzKand m, natural or_bitals with the highest occupation coefficients in the improved description of electron pairs K and L, respectively [ 41. In this work we selected for the double-pair correction terms mK equal to 10, 14 and 16 for core geminals, o-type valence geminals and n-type geminals, respectively. Our largest full CI expansion for the four-electron problem in question was therefore constructed from 32 spatial orbitals. It comprised 2460 16 Slater determinants, Furthermore, for a given geometry of tetra oxygen there are 120 full CI four-electron problems to be solved. However, due to symmetry, the number of computed double-pair correction terms is considerably smaller.
4. Results and discussion For isolated ozone we use the experimental geometry [ 81 corresponding to a bond length of 2.38 au and a bond angle of 117”. Due to the highly correlated wavefunction adopted in this work, a complete geometry optimization is expected to yield only a minor energy change compared with the calculated value for the experimental geometry. The energy localizations of the orbitals gives as a result a bonding picture for ozone characterized by two o-x-type double bonds. The volumes of the corresponding charge ellipsoids are respectively 1.1504 and 4.4307 au for the o- and n-type geminals (table 1). Oxygen in the ‘D state is characterized by three equivalent valence electron pairs with charge centroids in a plane through the nucleus. The volumes of the associated charge ellipsoids are 1.3995 au. The total energies of isolated ozone and ‘D oxygen are in table 2.
19 May 1989
The optimized bond length for the D3,, form of tetra oxygen is obtained by a parabolic fit based on calculations for the bond lengths 2.47,2.52 and 2.57 au. The bond length corresponding to the energy minimum is equal to 2.5 14 au. The energy localization of the RHF geminals yields a bonding picture characterized by three equivalent o-x-type double bonds. In fig. 1 we display charge centroids and the intersection of the charge ellipsoids of the geminals with the molecular plane. The volumes of the charge ellipsoids are 1.3498 and 4.6763 au, respectively, for the o- and a-type bond pair geminals. The total energy of the Dji, form of tetra oxygen for the optimized geometry is 0.003739 au higher than the sum of energies of isolated ozone and oxygen in the ‘D state. However, this result does not exclude the existence of an energy barrier separating the Dlh form of tetra oxygen from isolated ozone and oxygen in the ‘D state. To determine a possible transition state the following procedure was adopted. First, a transition state of 0, is assumed to have a planar geometry. Second, the change in the bond angle, ~020103, from 117o in isolated ozone to 120’ in tetra oxygen, is assumed to have only a minor effect on the energy changes during a reaction between ozone and atomic oxygen, The validity of this assumption is checked by calculating the energy change for isolated ozone when the bond angle was increased from 117” to 120”. The energy change was only 0.000020 au. For three fixed 01-04 distances, i.e. {3.014 au, 3.5 14 au, 4.014 au}, the bond length in the ozone-like fragment was optimized. By a parabolic tit we then obtained the following geometry for the transition state: R(Ol-04)
~3.476 au,
R(Ol-02)=R(Ol-03)=2.436 ~020103=
au,
120” (fixed) .
The interpolated value of the energy is - 300.042622 au to be compared with the calculated value of - 300.042664 au for the nuclear configuration of the transition state. We therefore conclude that the calculated energy for the transition state is very close to the value that can be obtained by a complete geometry optimization. The assumption of no splitting of electron pairs during a reaction of O4 to 03+0( ‘D) is supported 411
CHEMICAL PHYSICS LETTERS
Volume 157, number 5
19 May 1989
Table 1 Volumes of selected geminal charge ellipsoids of 0, for three different geometries a.bl
03+O(‘D) transition State d,
O.dh,) e,
‘)
a(Ol-02)
n(Ol-02)
o(Ol-04)
1.1504
4.4307
1.1755 f’
I .2758 1.3498
4.6014 4.6763
1.3498
1.472I
a(Ol-04)
2.1531 4.6763
a’ Volumes in atomic units ‘) 0 1 is the central nucleus, 02 and 03 are the ligand nuclei of the ozone fragment. c, Isolated ozone and oxygen. Experimental geometry is adopted for ozone [ 81. d, Thegeometryofthe transitionstate: R(Ol-02)=R(Ol-03)=2.436au,R(Ol-04)=3.476 for numbering of atoms. ‘) Bond length R= 2.52 au. ‘) Transforms to a u-type bond pair in O,,.
Lone pair 1 (02)
Lone pair 2 (02)
Lone pair (04)
1.3961
1.3513
1.3995
1.4028 1.3964
1.3998 1.3964
1.2721 1.3964
au, L020103=
120’. See
figs.1and 2
Table 2 Total energy and energy partitioning of O4 for selected geometries ‘)
03tO(‘D)
U)
transition state d, O,(D,,) ‘I
0.0 (-300.094903) 0.052237 0.003761
0.0 (-225.138387) 0.172399 1.638716
0.0 (-74.956516) 0.140559 1.736001
-0.260720 -3.370956
a) Energies in atomic units. b, Isolated ozone and oxygen. Experimental geometry is adopted for ozone [ 81. c) The values of the energy components for this geometry are used as zero levels. Calculated values in parentheses. d, Geometry as defined in footnoted to table 1. e, This partitioning is for a bond length of Rz2.52 au. The total energy is 0.000022 au higher than the value corresponding to the optimized bond length of R = 2.5 14 au.
Fig. 1,Intersection between the xy plane (molecular plane) and the charge ellipsoids of O4 in Dlh symmetry. The bond length is 2.52 au. The charge centroids of the o- and n-type geminals are marked with the symbols x and q, respectively.
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Fig.2. Intersection between the xy plane (molecular plane) and the charge ellipsoids of Oq in the transition state. See footnote d, table 1,for the specification of the geometry of the transition state. The charge centroids of the o- and x-type gemiaals are marked with the symbols x and El, respectively.
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by these calculations. A possible bond-breaking process for the four-electron system associated with the 0 104 double bond would imply large values for the corresponding double-pair correction term tKL,since eKLis determined by the lowest eigenvalue of the full CI matrix in question. In fact, the largest magnitude of cKLis obtained for the smallest value of the 0 104 distance, If we consider a reaction starting from ozone and oxygen in the ‘D state, the charge centroids of the geminals, table 3, give the following picture of the reaction. At the transition state the oxygen atom has made itself “ready” for accepting the lone pair of the central oxygen of the ozone molecule, as a dative otype bond. One of the three equivalent valence geminals of the oxygen atom is changed into a pure pr type geminal with respect to the molecular plane (zaxis orthogonal to the molecular plane). This particular geminal is transformed into a x-type bond pair geminal during the reaction. The two other valence geminals of the oxygen fragment have charge centroid vectors in the molecular plane and pointing almost exactly in opposite directions (fig. 2). In the ozone fragment the lone pair of the central oxygen atom is shifted towards the approaching oxygen atom. As a result of this shift, there is more space available for the o-type bond pairs in the vicinity of the central nucleus and they are shifted towards this nucleus. The approaching x-type electron pair of the oxygen fragment reduces the available space for the ozone Relectrons in the vicinity of the central oxygen nucleus and as a result the x-electron pairs of the ozone fragment are shifted away from the central nucleus. As the system passes through the transition state towards 04(Djh), the same trend in the shifts of the
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charge centroids is observed, but the shifts are more pronounced. The internal rearrangement of the nuclei and the electrons of the fragments during the reaction leads to higher intrafragment energies. This effect is displayed in table 2. We notice from table 2 that in the transition state, the distortion energies, i.e. the energy changes due to the presence of the other system, and the interaction energy between the fragments, are all of the same magnitude as the net energy increase. On the other hand, when the system passes the saddle point and transforms into the Djt, form of tetra oxygen, huge changes are taking place with respect to the energy components. For the equilibrium structure, we find large distortion energies which are almost compensated by a corresponding large interaction energy. The magnitudes of the energy components are in this case practically three orders of magnitude larger than the net energy change. 5. Conclusion This work indicates that tetra oxygen can exist in a metastable DShform. The energy of tetra oxygen is 9.8 1 kJ/mol higher than the energy of isolated ozone and oxygen in the ‘D state. There is an energy barrier (measured from 04(D3,,) ) of 127.28 kJ/mol separating tetra oxygen from ozone and oxygen ( ‘D). Due to this barrier, a reaction between ozone and oxygen ( ‘D) yielding 0, (Ds,,) as reaction product, can only take place in exceptional circumstances. However, if 04(Dsh) is formed, then this form should be kinetically stable.
Table 3 Distances between selected charge centroids of 0, and the central oxygen nucleus for three different geometries aSb)
Ox+O(‘D) =) transition state d, O4Uh)
‘)
R(a(Ol-02))
R(n(Ol-02))
R(o(Ol-04))
1.0810 1.0741 1.0055
1.5286 1.5473 1.8295
0.6208 ‘) 0.7224 1.0055
‘) Distances in atomic units. ‘) 01 is the central nucleus, 02 and 03 are the Iigand nuclei of the ozone fragment. c) Isolated ozone and oxygen. Experimental geometry is adopted for ozone [ 81. d, Geometry as defined in footnote d, table 1. ‘1 Bond length R~2.52 au. f, Transforms to a o-type bond pair in O+
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Acknowledgement
The authors are grateful to Professor Asbjom Hordvik for suggesting the problem. The calculations have been performed on the CRAY X-MP28 supercomputer, SINTEF, Trondheim, Norway. References [ 1 ] S.J. Arnold, E.A. Ogryzlo and H. Witzke, J. Chem. Phys. 40 (1964) 1769.
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[2 ] D.L. Huestis, G. Black, S.A. Edelstein and R.L. Blunt, J. Chem. Phys. 60 (1974) 4471. [ 3 ] V. Adamantides, N. Neisius and G. Verhaegen, Chem. Phys. 48 (1980) 215. [4] I. Reeggen, J. Chem. Phys. 89 (1988) 441. [ 5] 1.Rseggen and E. Wisleff-Nilssen, J. Chem. Phys. 86 ( 1986) 2869. [6] F.B. van Duijneveldt, IBM Research Report RJ 945 (1971). [ 7 ] I. Reeggen, unpublished calculations. [8] T. Tanaka and Y. Monno, J. Mol. Spectry. 33 (1970) 538.