J. Phys. Chem. 1993,97, 40234030
4023
Strain Energies in Cyclic O n , n = 3-8 Ming Zhao and Benjamin M. Cimarc' Department of Chemistry and Biochemistry, University of South Carolina, Columbia, South Carolina 29208 Received: August 18, 1992
We report the results of geometry-optimized a b initio SCF M O calculations of total energies and structures of 0, rings, n = 2-8, and H 2 0 , chains, m = 0-5, at RHF and MP2 levels using the 6-31G** basis set. From the total energies we calculate 0, strain energies on the basis of energy changes for several model reactions that convert 0, rings into H 2 0 , chains. We introduce a generalization of the homodesmotic reaction that we call the s-homodesmotic reaction, 0, nH20s+I nH20s+2, where s = -1 is an isogyric reaction, s = 0 is isodesmic, s = 1 is homodesmotic, s = 2 is hyperhomodesmotic, and so on. Larger values of s permit the conversion of 0, rings into longer H20s+2 chains which should contain conformational effects not present in short chains with small s. For large s, calculated strain energies approach a constant value and results based on RHF energies approach those from MP2 energies. Unlike the cycloalkanes for which strain energies decline quickly with increasing ring size, we find the strain energy of O4 to be larger than that of O3for all models tested. Beyond 04, strain energies of larger rings decline through the series as expected. The large strain energy of 0 4 is probably due to large lone pair-lone pair interactions which are not present in the cycloalkanes. We estimate heats of formation for 0, rings and H 2 0 , chains. For m 1 3 total energies of H20m+l chains appear to be adequately approximated by a constant plus the total energy of H20,.
+
Introduction Strain is an important concept in structural organicchemistry.Is2 Deviations of bond angles at carbons from their preferred valence angles imply a strained structure with increased molecular energy. Strain energy can be defined as the difference between energies for a process that releases strain as determined by experiment and as obtained from a model that does not involve strain. The strain energy of cyclopropane, C3H6, has been determined to be 27.5 kcal/mol, representing an energy increase or destabilization compared to an acyclic structure because the 60' angles at carbons in the ring are severely displaced from the value of 109.5' expected for sp) hybridized carbon. As additional methylene groups are added to the ring, strain energies decrease through the series of cycloalkanes to 26.5 kcal/mol for cyclobutane, to 6.2 kcal/mol for cyclopentane, and to 0 kcal/mol for cyclohexane as larger rings allow angles at carbons to relax toward the preferred value. Beyond n = 6, the strain energies of cycloalkanesincrease. These increases have been attributed to transannular repulsions among hydrogen substituents.l.2 A much larger decrease ih strain energy has been observed for the analogous cyclosilanes, from 41 kcal/ mol for Si3-cyclopropaneto 23 kcal/mol for Si4-cy~lobutane.~ In this paper we report strain energies of oxygen rings 0,, n = 2-8. As with cyclic alkanes and silanes, strain energies of 0, rings decline for larger n, with the exception of n = 4; we find that the strain energy of O4is larger than that of 03.Strain energies of 0, rings may not be determined by distortions from preferred bond angles alone. Although cyclic 0, and C,H2, are isoelectronic, there are significant differences between the two ring systems. The unshared electron pairs on oxygens that replace the hydrogen substituents of the carbons should produce quite different interactions within the ring. Although 0, rings are unknown, sulfur rings containing five or more atoms have been prepared. The hypothetical 0, series provides a simple model for comparisons of the concept of ring strain between valence isoelectronic series CnH2n and S,. No cyclic 0, molecules have ever been reported. Electronic structure considerations suggest that a cyclic form of 0,could exist not many kilocalories per mole above the normal bent or open ground state of ozone.4-14 The best ab initio calculations give estimates that cyclic O3is bound and 28-29 kcal/mol higher in energy than the bent form. Although this energy gap is a 0022-3654/93/2091-4023%04.00/0
-
direct comparisonof energiesof cyclic and acyclic forms, it would be inappropriate to call it the strain energy of O3because cyclic and open forms have different 00 bond types. Calculated bond lengths in cyclic O3are comparable to the 0-0 single-bond distance in HzOz, while 00 bonds in bent O3have considerable double-bond character. The energy gap between cyclic and open forms of ozone is slightly larger than the 26.1 kcal/mol required to dissociate ground-state 0 3 into 02 + 0.14Calculations by Schaefer and co-workers support the notion that O4and O6may be bound molecules but high in energy above 202 and 302, respe~tively.l~-~~ As Schaefer points out, the source of this extra energy for 02, rings has less to do with ring strain than it does with differences in bond energies between 00 single and double bonds. The 00 double bond, at 118 kcal/mol, is over 3 times stronger than the 00 single bond at 35 kcal/mol. From bond additivity considerations alone, cyclization of n02 replaces n double bonds by 2n single bonds for an energy increase of n( 1 18 - (2 X 35)) kcal/mol. For O4 and 06 these increases would be 96 and 144 kcal/mol, respectively. Ring strain considerations would further increase ring energies. In contrast to oxygen,sulfur is known to form rings, at least for n 1 5 . This is because the S-S double bond (100 kcal/mol) is weaker than two S-S single bonds (64 kcal/mol each). From bond energy considerations alone, formation of S2, rings from nS2 should be exothermic by n( 100- (2 X 64)) kcal/mol or -56 and -84 kcal/mol, respectively, for S4 and Sg. These values would be raised if S, ring strain energies were included. We proposethat strain energies of 0, rings are related to energy changes AJ% for reactions such as those in isogyric: isodesmic: homodesmotic:
0, + H 2
-
HO,H (acyclic)
0, + n H 2 0
-
0, + n H 2 0 2
nHOOH nHOOOH
(1) (2)
(3)
The most obvious solution to the relief of ring strain in 0, is the hydrogenation reaction, eq 1. This is an example of an isogyric reaction in which the number of electron pairs is the same on either side of the equation.lsa According to eq 1 one 0-0 bond breaks to open the ring, the H-H bond breaks, and the two 0 1993 American Chemical Society
4024
Zhao and Gimarc
The Journal of Physical Chemistry, Vol. 97,No. 16, 1993
hydrogens add to the ends of the 0, chain. The bond additivity model, which includes nothing of the concept of strain, gives an energy change for eq 1 as MI’= D(O-0)+ D(H-H) - 2D(0H). Choosing values of standard bond energies ( 0 4 3 5 kcal/ mol; H-H, 104 kcal/mol; 0-H, 111 kcal/mol) yields MI’= -83 kcal/mol, independent of ring size n. Since we expect ring strain to be relieved in eq 1, we anticipate that MIas determined from experiment would be even more exothermic than -83 kcal/mol. Therefore, any difference between AEI and MI’should be a measure of the strain energy of 0,. The quantity MI’depends on our choice of standard bond energies and as a consequence so does our estimated strain energy. But we can expect that the error introduced by any reasonable choice of standard bond energies should be at most only a few kilocalories per mole and the same for all n so it should not affect strain energy trends. Furthermore, there are less arbitrary ways to evaluate MI’,to be described in a later section. Equation 2 is an example of an isodesmicreaction.18bNumbers of bonds of different types are conserved from reactants to products. Since numbers of 0-0and 0-H bonds are the same on either side of the equation, the bond additivity model gives Mi = 0. But the valence environment around each atom is not preserved. For example, each oxygen atom in 0, is bonded to two other oxygens, but in the HOOH product no oxygen has such an environment. Each oxygen in HOOH is bonded to a hydrogen and to another oxygen, but no oxygen in the reactants has this arrangement. Still, the exact cancellation of bond energy terms makes eq 2 independent of bond energy choices, and the experimentally determined energy change MZ should be a direct measure of strain energy. Equation 3 is an example of a homodesmoticreaction in which not only are numbers of bonds of various types conserved but the valence environment around each atom is preserved as well.19 For example, each oxygen atom in the 0, reactant ends up between two oxygens in the acyclic product HOOOH, and each oxygen in the reactant HOOH becomes one of the end oxygens in the HOOOH product chain. From the bond additivity model, Mi = 0 and the experimental M3 gives the strain energy directly. As just described, we anticipate eqs 1-3 to be exothermic in the directionwritten, or AE, < 0. We wish to definestrain energy as a positive quantity. Therefore, we relate strain energy with -M, for eqs 2 and 3 and -(83 M I )for eq 1. Although energy changes for eqs 2 and 3 give strain energies without requiring correction terms, they may do so at a price. Each reaction takes a ring and breaks it up into a set of short chains. Equation 1 perhaps more accurately reflects the original concept of ring strain because it converts cyclic 0, into a single acyclic HO,H product. It has not been established that conformational effects that operate in the general HO,H chain are additive accumulations from those in shorter HOOH and HOOOH chains. We can alleviate this difficulty by modifying the homodesmotic reactions to give longer product chains. For example, in their calculation of the strain energy of 0 6 , Blahous and Schaefer‘?introduced a product with a chain of four oxygens, HOOOOH. Equation 4 describes such a process for the n-membered cycle:
+
0, + nH203
nH204
(4) Following Hess and Schaad,20 Blahous and Schaefer called something like eq 4 a hyperhomodesmoticreaction. We wish to generalize the concept by proposing eq 5 , which we call the s-homodesmotic reaction: -+
s-homodesmotic:
-
0, + nH20s+l
nH,O,+,
(5)
The indices are chosen such that for s = 0, eq 5 reduces to the isodesmic reaction; for s = 1, it is homodesmotic; s = 2,
hyperhomodesmotic; and so on. The energy change for eq 5 is
m,)
4 = n[~(H,O,,,) - E(H2O5,I)I In later sections we show that the difference M ( s + l ) = E(H20s+2) - E(H20,+1) approaches a constant value for s 1 1 3. Equation 5 shows the relationship between isodesmic, homodesmotic, and s-homodesmotic reactions. Consider two reactions of successive homodesmoticity s and s + 1. The difference between these two reactions is
+
2nH20s+2
-
The energy change for eq 6 is U mfj
+ nH20~+3
nH20s+l
(6)
S :
= n([E(H20,+3) - E(H20,+2)1 - [E(H20s+2) E(H20S+l)l1 = n[hE(s+2) - AE(s+l)]
If for long chains (large s + 3) the energy differences between successive chains approach a constant, then successive difference pairs cancel and M 6 = 0. For s = -1, eq 5 reduces to
0, + nH,
-
nH,O
(7) in which the 0, ring is converted into a set of n water molecules. The water-formingreaction of eq 7 is isogyric but different from the isogyric hydrogenation reaction eq 1. From the bond additivity model the energy change for eq 7 is just n times that for eq 1 or M7’ = n u l ‘ or A E 7 ’ = -83n kcal/mol. The @trainenergy of 0,is then the difference between experimental M 7 and M{. Since 0, rings have never been prepared, no experimentaldata are available for determining strain energies from any of the equations proposed above. In this paper we depend on total energies from ab initio calculationsfor On,H20,, and Hz to give energy changes that measure strain energies. calculations When we started thinking about strain energies in 0, rings, we suspected that enough results from ab initio calculationswere already in the literature to allow us to make some quick estimates of 0, strain energies. Indeed, Hz and H20 are extremely important molecules in chemical valence theory,21H202is the simplest molecule that can undergo internal r o t a t i ~ n , HOOOH ~~-~~ and HOOOOH have been proposed as intermediates in reactions involving HO, radical^,^^,^"^^ and several studies have been published concerning the possibility of stable 03,04, and Os rings.”’ The wide availability and usage of ab initio program packages that include standard basis sets suggested that we might be able to obtain all of the numbers we needed from previously published work. In fact, calculations have been done for all of the molecules that appear in eqs 1-3, with the exception of 05. But because of nonmatching choices of basis sets, incomplete optimizations of molecular geometries, or occasional variations in standard orbital exponents, no complete set of results at the same level of approximationwas availablefor all of the molecules of interest here. We have carried out ab initio SCF MO calculations for 0, rings and H20, chains using the GAUSSIAN 82 and GAUSSIAN 90 program packagesz9JO and performed on the IBM 3084 and RS6000 computers at the Computer Services Division of the University of South Carolina. We report results at two levels of theory. First, we present geometry-optimized RHF calculations employing the 6-31G* basis set for On,n = 2-8, and with the 6-31G** basis set for H20,, m = 0-5. These basis sets contain split-valence shell atomic orbitals with d-type polarization functions on oxygens and p-type polarization functions on hydrogens.21 Allen and co-workers have recently reported calculations of strain energies for three- and four-membered cycloalkanes and silanes using the 6-31G* basis set and ho-
Strain Energies in Cyclic 0,
The Journal of Physical Chemistry, Vol. 97, No. 16, I993 4025
TABLE I: Total Energies (hartrees) and Structural Parameters of 0, and H20,Obtained with 6-31G5 and 6-31G5* Basis Sets and Geometry Optimized at Both RHF and MP2 Levels molecule
RHF
MP2
-149.61791 1.1677
-149.95432 1.2460
-224.26144 1.2043 119.01
-224.87675 1.2996 116.32
-224.24479 1.3728 60.
-224.82052 1.4771 60.
-298.98461 1.3989 88.23 19.94
-299.73834 1.4985 86.31 28.48
-373.78983 1.3598 1.3770 1.4100 98.62 101.02 103.70 32.59 0.0 -53.38
O7(C$,chair)
-373.78980 1.3985 1.3637 1.3593 102.70 99.35 104.00 -45.05 -56.26 17.10
-374.73325 1.5028 1.4179 1.4491 102.09 92.79 101.52 -49.38 -62.40 19.09
-448.56858 1.3680 104.52 70.45
-449.69991 1.4466 103.10 72.96
-523.32895 1.3586 1.3699 1.3380 1.4252 105.72 107.46 111.56 107.40 -100.87 -77.63 0.0 -75.23 -523.3 1069 1.3513 1.4046 1.3222 1.4435 107.58 109.49 109.25 110.20 -36.93 80.61 0.0 -67.07 -523.3107 1 1.3746 1.3346 1.4321 1.3172 109.85 108.62 109.44 106.99 -76.47 -18.70 81.39 53.40
4026
Zhao and Gimarc
The Journal of Physical Chemistry, Vol. 97, No. 16, 1993
TABLE I (Continued) molecule
modesmotic reactions related to eq 3.3 Their results are in good agreement with experimental strain energies. Second, we have included some of the effects of electron correlation by doing geometry-optimized calculationsthat include second-order MdlerPlesset (MP2) perturbation corrections utilizing the 6-3 lG* and 6-31G** wavefunctions. TheRHF/6-31G** energiesarealittle higher than those from calculations employing Dunning's 9s,5p double-zeta plus polarization function basis set (DZP). The MP2/ 6-31G** energies are a little lower than those from calculations that use DZP plus configurationinteraction with single and double excitations (CISD). In carrying out calculations on 0, rings and H20, chains we optimized geometries within the symmetry constraints indicated in Table I. Except as noted below for the 0 5 and 0 7 rings, we did not calculatevibrational frequenciesto establish whether these structures are indeed minima on their respective energy surfaces. The structures of H20, chains, m L 2, are helical chains of C2 symmetry with 00 bonds that appear to be normal to 0-0single bonds. Our structures for cyclic 0 3 and 0 4 are very similar to thoseobtained by others. For 0 6 we chose toperform calculations only for the D 3 d (chair) structure, which Blahous and Schaefer found to be lower in energy than either the boat or the twist
RHF -523.32921 1.3772 1.3550 1.3647 1.3593 110.01 104.70 107.56 110.97 -100.65 86.82 -69.01 43.04
MP2 -524.65262 1.4469 1.4051 1.4721 1.4040 109.89 102.65 106.79 109.50 -101.46 88.97 -7 1.47 44.50
-598.10661 1.3595 108.21 98.51
-599.61674 1.4339 107.68 99.14
-1.13 133 0.7329
-1.15766 0.7339
-76.02362 0.9431 105.97
-76.22245 0.9608 103.87
-150.77696 1.3953 0.9456 102.34 115.27
-15 1 . 1 5709 1.4664 0.9679 98.65 120.26
-225.54584 1.3726 0.9485 107.38 103.51 80.70
-226.1 1275 1.4408 0.9715 106.22 100.23 78.63
-300.31050 1.3588 1.3703 0.9487 107.99 103.51 81.57 78.53
-301.065 16 1.4342 1.4353 0.9719 107.08 100.45 79.33 76.27
-375.07504 0.9487 1.3700 1.3586 107.60 108.10 103.54 79.90 84.61
-376.01795 0.9720 1.4344 1.4334 106.71 107.12 100.55 77.65 82.36
8 we considered only the D 4 d (crown) conformations. For 0 conformation, which is the known structure of the isoelectronic SSring. For O5we attempted calculations for both envelope (C,) and twist (C2) conformations. At the RHF level these two structures have virtually identical energies. However,vibrational frequency calculations gave all real frequencies for C, and one imaginary frequency for C2. Although we obtained geometry optimization for the twist (C2) structure from our MP2 calculations, geometry optimization of the envelope (C,)conformation failed to converge at the MP2 level, despite choices of different 2-matrices and different optimization procedures. The attempt seemed to be leading to the dissociation of 05 to 202 + 0. For the 0 7 ring we examined two C, structures (chair and boat) and two conformations of C2 symmetry. At the RHF level, chair and boat structures each gave a singleimaginaryvibrationalfrequency, indicating that these structures correspond to transition states on the energy surface. At the MP2 level, we could not achieve geometry optimization for either chair or boat, these structures appearing to dissociate into 0 3 and 202 fragments. Both C2 conformers are real minima at the RHF level. We were successful in obtaining MP2 convergence of geometry optimization for one
Strain Energies in Cyclic 0,
The Journal of Physical Chemistry, Vol. 97, No. 16, 1993 4027
TABLE Ik Strain Energies of Cyclic 0, (kilocalories per mole), RHF/631G* Results isog ric hydl-0- isogyric water genation forming, s = -1 -65.6 13.1 23.5 39.1 13.6 4.6 7.1 18.7 18.7 7.1 -1.0
-61 12 22 35 6 -5
isodesmic homodesmotic s=l s=2 s=3
's=O
-69.8 -50.3 -55.6 -55.7 -0.9 28.4 20.5 20.2 9.6 38.8 30.9 30.6 18.0 57.0 46.5 46.2 -14.5 34.2 21.0 20.6 -30.5 28.0 12.2 11.7 -34.9 33.4 14.8 14.3 -27.4 40.9 22.3 21.8 -27.4 40.9 22.3 21.8 -34.9 33.4 14.8 14.3 -50.1 27.9 6.7 6.1
-6 L
-2 4 -17
TABLE III: Strain Energies of Cyclic 0, (kilocalories per mole), MP2/631G* Results isog hy&oric
0,)
isogyricwater
isodesmic homodesmotic s =0
aenation f0rming.s = - 1 -54.7 -33.9 1.4 23.1 -3.3 -12.0 -11.9 -19.0
s= 1
-53.4 -27.0 4 5 . 7 -6.1 -10.4 29.2 0.1 52.9 -37.7 28.3 -57.8 21.4 -69.1 23.2 -87.6 17.9
-56 -50 -14 -5 -44 -66 -7 8 -98
s=2 s=3 -31.1 -30.6 -12.3 -11.5 23.0 23.8 44.7 45.7 18.1 19.3 9.1 10.6 8.9 10.6 1.6 3.5
-
TABLE 1% Total Energies (hartrees) of Reactants and Products and Energy Changes A& (kilocalories per mole) for Equation 8, HZ+ H2O2 2Hz0, at Various Levels of Approximation total energies level of theory
H2
H202
H20
u s
RHF MP2 MP3 MP4 (SDQ) exptl AHfo (0 K)," kcal/mol
-1.13133 -1.15766 -1.16316 -1.16457 0
-150.77696 -151.15709 -1 51.15842 -151.16456 -31.03
-76.02362 -76.22245 -76.22610 -76.22848 -57.10
-87.19 -81.67 -81.96 -80.21 -83.17
30
I
20
c
:
5.
10
8
a
Y
b
o
W
-10
5
.-c
:
*
-20
-30
- 0
-50
s irogyric hydrogenotion
I-1
0
irogyric irodormk wotor
-
1 I
2 homodormotic
3
forming
Figure 1. 0,ringstrain energiesat the RHF/6-31G** levelascalculated from various ring strain relieving processes: hydrogenation, eq 1, and s-homodesmotic reactions, eq 5, for s 1 -1.
MP2 level or higher. In correcting AI?,to give strain energies according to eq 1, we used the average bond energy value of MI' = -83 kcal/mol.
Discussion
Reference 3 1 .
of the Cz conformers, the one with the lowest energy of all 0 7 structures at the RHF level. It has long been argued that reactions such as eqs 1-7 allow much of the error in electron correlation to cancel as energy differences are taken between products and reactants. Isogyric reactions, by conserving numbers of electron pairs, should allow the largest correlation errors to cancel. The situation should be even better as bond types and then valence environment are conserved in isodesmic and homodesmoticreactions and beyond. Table I contains total energies and optimized structural parameters of molecules involved in eqs 1-7. Table I1 presents strain energies (kilocalories per mole) calculated according to eqs 1-5 and 7 using RHF/6-31G** results from Table I. Table I11 gives comparable strain energies obtained from the MP2/ 6-31G** results. According to the bond additivity model, the energy change for eq (8) is U S ' = D(O-0) D(H-H) - 20(0-H), exactly the
+
H,
+ H,O,
-
2H,O
(8)
same as MI'for eq 1. Both theoretical and empirical values for A,??*can be used to estimate or approximate the bond additivity difference MI'.Table IV contains geometry-optimizedab initio total energies of reactants and products of eq 8 and corresponding energy changes h E 8 calculated at RHF, MP2, MP3, and MP4 levels with 6-3 1G**wave functions. Standard heats of formation for H2O(g) and H2O2(g)3Igive a thermochemical value of MS = -83.17 kcal/mol, almost identical to the average bond energy values MI' = A&' = -83 kcal/mol. The ab initio SCF MO results for A E s are within 1 or 2 kcal/mol of these values at the
Consider the strain energies in Tables I1 and 111. Ignore for a moment the wide variation of strain energies according to the reaction on which they are based and the fact that some values even turn out to be negative. Instead, look at strain energy trends. The most significant observation from Tables I1 and I11 is that the strain energy of cyclic 0 4 is greater than that of cyclic 03, a feature that is maintained for all processes and at both RHF and MP2 levels of theory. This peak in strain energy at n = 4, depicted in Figures 1 and 2, is absent for cycloalkanes and cyclosilanes. We find a similar n = 4 peak in calculated strain energies of S, rings.32 Strain energies of 0 7 are higher than those of O6at the RHF level, but the two rings have essentially equal strain energies at the MP2 level. O8has the lowest strain energy of all 0, rings up to n = 8. For the cycloalkanes (CH& the minimum strain energy occurs at n = 6. Strain energies in the 0, and (CHz),, series are compared in Figure 3. In all of the results, Tables I1 and 111, the MP2 correlation corrected strain energies are smaller than those obtained at the RHF level. This suggests that electron correlation selectively stabilizes rings relative to chains. Indeed, correlation corrections are larger for 0, rings than for the corresponding HzO, chains which have an additional electron pair. There are other trends worthy of note in Figures 1 and 2. For larger values of s (of s-homodesmoticity,eq 5 ) , the strain energy of each 0, approaches a constant value. Furthermore, for larger s, the RHF strain energy for each 0, approachesthe value obtained from MP2 calculations. But this is not true for reactions designated by s = 0 and -1. Therefore, there must be different correlation corrections for reactants and products of reactions specified by s = 0 and -1. Surely, correlation errors cannot cancel
'
Zhao and Gimarc
4028 The Journal of Physical Chemistry, Vol. 97, No. 16, I993 40
so
40 40
20
t
c
I
e
o
2aE 30
3
P f
Y
Y
A
5
-?O
0
Y
Y
f -40
e 10
H
0
L
5
d 0
-60
10 -00
-100
I
1
r
a
-1
0
irogyrlc irogyric isodormlc hydrogen- water ation forming
1 1 I homodosmotic
-
1
3
0
Rlng
SI;.
n
Fipre2. 0, ring strain energies a t the MP2/6-3 1G** level as calculated from the hydrogenation reaction, eq 1, and s-homodesmotic reactions, eq 5 , for s 2 -1.
Figure 3. Strain energies (kilocalories per mole) for 0, (MP2, s = 3) and (CHz), (from ref 2). In contrast to the cycloalkanes, strain energies of 0, rings show a sharp maximum at n = 4 and reach their lowest value at n = 8.
exactly for s = -1 because reactants and products involve different bond types or for s = 0 because of different valence environments around some atoms of reactants and products. Even for the homodesmoticreaction (s = 1 ),traditionally considered a situation for which correlation errors can usually be neglected, we still need to account for differences in conformations or next-toneighbor bond interactions. These effects can be used to explain the large difference between strain energies at RHF and MP2 levels for the O3ring. Compared to s = 1-, 2-, and 3-homodesmotic strain energies, the s = -1 and s = 0 values are very low, often negative. The s = -1 and 0 results derive from reactions that yield products HzO and HzOz, respectively. It is not surprising that these molecules are poor representatives of the longer chains that should serve as acyclic reference structures. Using a DZP basis set for RHF calculations, Blahous and Schaefer obtained 0 6 strain energies of 26.8 and 14.5 kcal/mol, respectively, from homodesmotic and hyperhomodesmotic reaction~.'~ Their numbers are in reasonable agreement with our comparable results of 28.0 and 12.2 kcal/mol for s = 1 and 2, respectively, in Table 11. The microwave spectrum of O3reveals it to be a bent, symmetric molecule with a bond angle of 117.8O at the central oxygen and 00 bond distances of 1.272 This rather short distance indicates partial double-bond character (compare with 1.2075 A for O=O) as expected from elementary valence theory. The suggests calculated distances in cyclic O3 (and other cyclic 0,) normal 0-0single bonds similar to those observed in acyclic HOOH (1.475 A)34and calculated for HOOOH. Therefore, the conversion of 0, ring to H20, chain does not involve a change of bond type. Our results in Table I show that the cyclic form of 0 3 is 10.45 kcal/mol higher than bent O3at the RHF level and 35.28 kcal/mol higher at the MP2 level. The latter result is too large by 6-7 kcal/mol, but it is the larger gap between cyclic and bent forms of O3that drives down bent O3strain energy
TABLE V Energy Differences AE(m) = E(H20acl)
-
E(H20,) (in hartrees) for Adding Another Oxygen to the H?O....chain m
RHF/6-31Gt*
MP2/6-31G8*
0 1 2 3 4
-74.89229 -14.15334 -14.16888 -14.16466 -14.16454
-15.06419 -74.93464 -14.95566 -14.95241 -14.95219
and in fact gives it a small negative value. The increased gap is entirely a result of the fact that the correlation correction for bent O3is greater than that for cyclic 03.Goddard and coworkers5 as well as others35have shown that bent O3has a sizable diradical character; hence, its electronic structure is not well represented by a single configuration. This is apparently the origin of the larger correlation correction for bent 03. Tables 11 and I11 also include values of -hE for eqs 1-5 for 02 and bent 03. Such results would be strain energies for cyclic On, but for acyclic 02 and Os, these values, uniformly negative, indicate that 02and bent 03, with full or partial 00 double-bond character, are more stable than the corresponding HzO,chains with which they are compared. Table I contains calculated total energies of several H20, molecules, m = 0-5. Energy differences, hE(m) = E(H20,+1) - E(H20,), show the effect of adding another oxygen to the H20, chain. Table V lists these differences. For m = 0, AE(0) represents the energy change that results when an oxygen atom is inserted between the two hydrogens of H2, eliminating the H-H bond and forming two 0-H bonds. For m = 1 and further changes in m, M ( m ) is the energy of adding another oxygen atom and another 0-0 bond. Values of hE(m) approach a constant for larger m. The differencebetween hE(3) and AE(4) is less than 0.3 kcal/mol at the MP2 level and even smaller at
Strain Energies in Cyclic 0,
The Journal of Physical Chemistry, Vol. 97,No. 16, 1993 4029 zoo
TABLE M: Heats of Formation (in kalocalories per mole) for H20, Chains direct calculation, eq 9
H20,
RHF
MP2
H20 H202
-52.29 -17.39 7.75 35.55 63.42
-54.99 -28.31 -14.82 0.72 16.01
H203 H204 H2O5
isodesmic reaction, eq 11 RHF
-14.72 4.25 23.29
MP2
expt
-18.16 -3.24 11.44
-57.10" -31.03" -15.7b l.lb 17.9b
t
-., >
Reference 31. Reference 34.
TABLE VII: Heats of Formation (in kilocalories per mole) for 0. Rings
a
100
s-homodesmotic reaction, ea 5 s=2 s=3 s=l
direct calculation, eq 10
s=o
114.3 69.6 157.6 106.7 159.6 95.7 178.9 102.3 209.3 117.7 229.1 125.8
87.8 67.8 122.3 104.4 115.9 92.7 125.9 98.6 147.6 113.4 158.5 121.0
RHF MP2 RHF MP2 RHF MP2 RHF MP2 RHF MP2 RHF MP2
IS0
-
77.3 67.7 108.4 104.3 98.4 92.5 105.0 98.4 123.5 113.4 130.9 121.0
75.7 67.8 106.2 104.4 95.6 92.7 101.7 98.6 119.2 113.4 126.1 121.0
74.6 67.8 104.9 104.4 94.0 92.7 99.8 98.6 117.1 113.4 123.5 121.0
the RHF level. Presumably, changes in A,??(m)are negligible form = 5 and larger, permitting an accurate estimate of energies of longer HzO, chains by adding M ( 4 ) for each new oxygen.
Heats of Formation The straightforward or direct calculation of heats of formation of 0, rings and H20, chains is based on energy changes for
p2+ H, - H,O,
(9)
Since the 02 ground state is a triplet with two unpaired electrons, while H2, On,and H20, have all electrons paired, eqs 9 and 10 are not even isogyric and we would expect poor results because electron correlation errors are unlikely to cancel. But the total energies of all reactants and products are available in Table I, so appropriate energy differences are easily taken. Differences between zero-pointvibrational energiesof products and reactants should also be included in these heat of formation estimates, as they should in our strain energy calculations as well. We did not calculate vibrational frequencies for most 0, rings, and previous experience shows that zero-point and thermal corrections are generally small compared to other inherent errors in our results such as basis set and electron correlation errors. Therefore, we have neglected zero-point and thermal corrections in these and later estimates of heats of formation. The first two columns of Table VI contain heats of formation for H20, calculated directly from eq 9 using RHF and MP2 total energies, respectively. Experimental heats of formation of H20and H202 are well established,jI and values for H2O3, H204,and H205 have been estimated empirically.36 Compared to these experimental or empirical values, the directly calculated, eq 9, MP2 results are somewhat higher (less negative); the RHF values are higher still. The agreement between directly calculated MP2 results and the experimentalquantitiesis surprisinglygood. The left-most column in Table VI1 contains heats of formation for 0, calculated from eq 10. No experimental data are available for comparison. Equation 11 provides another option for calculating heats of formation of H20,, n = 3-5, starting from the well established
0
3
4
I Ring
6
Sir.
7
a
n
Figure 4. Calculated heats of formation, AEr (MP2, s = 3, Table VII) for 0, rings compared to the estimate of bond additivity differences plus strain energy, 2411 + SE.
experimental heats of formation of H20 and H202 and A,??, the
H,O,
+ (n - 2)HZO
-
(n - 1)HZOz
(11)
energy change calculated from ab initio total energies of reactants and products. Equation 11 is isodesmic, and the resulting isodesmic heats of formation of HzO,,, n = 3-5, appear in Table VI. As expected, both RHF and MP2 values from the isodesmic process are closer to the empirical estimates than are the RHF values from direct calculation based on eq 9. RHF and MP2 values from isodesmic eq 11 bracket the empirical estimates. Table VI1 also includes heats of formation of 0, rings based on the s-homodesmotic reactions, eq 5. If we use heats of formation of H20, from the isodesmic reaction, eq 11, Table IV, in the s-homodesmoticreaction, eq 5, we can show that the results are s-independent and are in fact just those calculated from the isodesmic reaction, eq 2, or eq 5, s = 0. This accounts for the identical listings of ring strain energies at the MP2 level for s = 0-3. If we use MP2 heats of formation of H20, from Table VI, eq 11, with RHF values of energies for 0, rings in eq 5, we can force a dependence of strain energy on s, and in Table VI1 these mixed RHF/MP2 results for s 1 1 approach the s-independent MP2 values. Blahous and Schaefer reported heats of formation for O6 of 128.8 and 74.5 kcal/mol based on homodesmotic and hyperhomodesmotic reactions, respectively, using DZP wave functions at the RHF level." These should be compared with o u r s = 1 and 2 RHF results of 105.3 and 101.7 kcal/mol. Finally, it is interesting to see how well bond energy terms plus strain energy corrections approximate the calculated heats of formation of 0, rings in Table VII. Following eq 10 and using the standard bond energies mentioned earlier, the bond energy change AElo' = 118n/2 - 35n = 24n kcal/mol. But the heat of formation of an 0, ring should be hElo'(n) plus the appropriate strain energy. Using s = 3, MP2 strain energies from Table I1 plus 24n, we get values plotted in Figure 4 compared with s = 3, MP2 heats of formation from Table VII. Quantitative agreement is poor, and the gap between the two curves widens for larger rings but the two curves follow a similar trend.
Summary We have presented the results of ab initio SCF MO calculations for a series of 0, rings and H20, chains. The calculations were
4030 The Journal of Physical Chemistry, Vol. 97, No. 16, 1993
done with a basis set composed of split valence shell orbitals plus polarization functions on both oxygen and hydrogen. The calculations were carried out at the RHF level and at the MP2 level, which includes some of the effects of electron correlation. Geometry optimizations were executed at both RHF and MP2 levels under symmetry constraints but, except in the cases of the OS and 0 7 rings, no vibrational frequency calculations were performed. The rings turned out to be nonplanar, except of course for 03.The HzO, structures were helical chains for m 2 2. Calculated 0-0bond distances indicated normal single bonds in both rings and chains. We have used the total energies to obtain 0, strain energies as energy changes calculated for several different model reactions that convert 0, rings into HzO, chains. Ideally, one would prefer to compare energies of n-membered rings with those of chains containing the same number of maingroup atoms. In practice this is rarely done, the chains usually being rather short. The comparison of large rings with the short chains may neglect conformational effects or introduce inappropriate ones. Therefore, we have proposed a generalization of the homodesmotic reaction which we have called the s-homodesmotic reaction, eq 5 , such that for s = -1, the reaction is isogyric, for s = 0, it is isodesmic, for s = +1, it is homodesmotic, for s = 2, it is hyperhomodesmotic, and so on. This flexible extension of homodesmoticity allows, at larger values of s, the conversion of 0, rings into longer H20,+2 chains. We find that for large s calculated strain energies approach a constant value and results based on RHF energies approach those from MP2 energies, indicating that the anticipated cancellation of correlation errors has been achieved. Unlike the cycloalkanes for which strain energies decline continuously with increasing ring size, we find the strain energy of 0 4 to be larger than that of O3for all model reactions tested. Strain energies of OSand 0 6 are smaller than that of 0 3 as expected. The large strain energy of 0 4 is probably due to lone pair-lone pair interactions which are not present in cycloalkanes. Strain energies of O6and O7are comparable and 08 has the lowest strain energy among the rings n = 3-8. Strain energies at 0, are compared with those of cycloalkanes (CHz), in Figure 3. We estimated heats of formation for 0, rings and HzO, chains. For m 1 3, total energies of H20m+lchains appear to be adequately approximated by a constant plus the total energy of H20,. This result allows the practical use of s-homodesmotic reactions to calculate the strain energy of 0, for s such that s + 2 < n. Acknowledgment. We are grateful to the National Science Foundation for partial support of this research through Grant CHE-9012216 to the University of South Carolina. References and Notes (1) Cox, J. D.; Pilcher, G. Thermochemistry of Organic and Organometallic Compounds; Academic Press: New York, 1970; p 571.
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