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J . Phys. Chem. 1988, 92, 4306-4313
functions, the general shape of these quantities (leading to the observed v-dependence of the trans barrier and equilibrium torsional angle) is adequately represented at much lower basis set levels. Thus, overtone enhancement of the torsional barrier is indicated to be a fairly low-level qualitative feature of the H202 potential energy surface.
In the following paper, we employ the natural bond orbital (NBO) procedure to analyze the major electronic interaction that leads to the trans barrier and how that interaction is strengthened by stretching one of the OH bonds. Registry No. HzOZ, 7722-84- 1.
Torsion-Vibration Interactions in Hydrogen Peroxide. 2. Natural Bond Orbital Analysis J. E. Carpenter and F. Weinhold* Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin-Madison, Wisconsin 53706 (Received: November 2, 1987)
Madison,
In the preceding paper, we showed how the observed increase of the trans barrier in hydrogen peroxide upon OH overtone excitation can be calculated on the basis of potential energy coupling terms (of electronic origin) between the torsional and local OH stretching modes. In this paper, we investigate the nature of the potential coupling terms via the natural bond orbital procedure. The trans barrier and its vibrational enhancement are found to be primarily associated with charge-transfer interactions (hyperconjugation) between oxygen lone pairs and vicinal OH antibonds, as previous work on other ethane-type barriers would suggest. However, we also find an unusual contribution to the torsional potential (particularly, the cis barrier) which can be associated with electronic strain or “bond bending” in the axial 0-0 bond, and whose description is very sensitive to the inclusion of d functions. The d-orbital contribution to the 0-0 bond “kink” in turn has an indirect effect on the trans barrier. This appears to resolve the puzzling dependence of the peroxide barrier on d functions, which are necessary for the appearance of a trans barrier in the ground vibrational state but contribute negligibly to its vibrational enhancement.
I. Introduction Recent experiments have shown that the trans barrier in hydrogen peroxide increases upon excitation of the OH overtone vibration.’ In the preceding paper2 (referred to as paper 1 here), we showed that this surprising observation can be numerically reproduced on the basis of potential energy coupling alone within a Born-Oppenheimer separation framework, thus providing support for the general separability of the torsion from the other higher frequency vibrations as well as for a local mode model of the OH overtone states. In this paper, we discuss the electronic nature of the potential energy coupling via the natural bond orbital (NBO) p r o c e d ~ r e . ~This procedure provides an efficient method for finding a compact set of bonds and lone pairs for a molecule from the density obtained by ab initio calculations. From our previous work and the work of others: it is known that the shape of calculated torsional potential (and hence the trans barrier) is strongly dependent upon the composition of the basis set used, particularly the number of polarization functions. Figure 1 shows the torsional potential P ( x ) computed with the R H F method at three basis set levels: 6-31 lG, 6-31 1G(d), and 6-31 l G ( 2 d , ~ ) .Table ~ I gives the optimized geometries and total energies found by the calculations used to generate the curves. The strong dependence of the torsional potential on the number of d functions is clearly evident in the figure. It is a striking feature of H202that the extreme sensitivity of the torsional potential to the basis set is reproduced at all levels of theory-both correlated and uncorrelated. This suggests that polarization functions enter (1) Dtibal, H.-R.; Crim, F. F. J. Chem. Phys. 1985,83, 3863-3872. (2) Carpenter, J.; Weinhold, F. J . Phys. Chem., preceding paper in this issue. (3) Foster, J. P.; Weinhold, F. J. Am. Chem. SOC.1980,102, 7211-7218. Reed, A. E.; Weinhold, F. J. Chem. Phys. 1983, 78, 4066-4073. Reed, A. E.; Weinstock, R. B.;Weinhold, F. J . Chem. Phys. 1985, 83, 735-746. (4) Dunning, T. H.; Winter, N. W. J . Chem. Phys. 1975,63, 1847-1855. See also Cremer, D. J . Chem. Phys. 1978, 69, 4440-4455. ( 5 ) All calculations were carried out with the VAX (revision H) version of GAUSSIAN 82. This programs was developed by J. S. Binkley, M. J. Frisch, D. J. Defrees, K.Ravhavachri, R.A. Whitaide, H. B.Schlegal, E. M. Fluder, and J. A. Pople (Carnegie-Mellon University, Pittsburgh, PA). 6-31 1G: Krishnan, R.; Binkley, J. S.; Seeger, R.;Pople, J. A. J. Chem. Phys. 1980, 72, 650-654. (df,p) supplementary functions: Frisch, M. J.; Pople, J. A.; Binkley, J. S. J . Chem. Phys. 1984, 80, 3265-3269.
TABLE I: RHF Optimized Geometrical Parameters and Total and Relative Energies [Giving the Torsional Potentid V*l(x)]of H2O2as a Function of the Torsional Annle .xa I
x , deg
A ROH,A
W X h
0.0 30.0 60.0 90.0 20.0 50.0 56.0 80.0
a m H ,deg RHF/6-3 11G Optimized 1.4396 0.9531 lb8.99 1.4365 0.9531 108.22 1.4310 0.9528 106.52 1.4297 0.9516 104.81 1.4345 0.9503 103.33 1.4394 0.9494 102.32 1.4401 0.9493 102.19 1.4414 0.9492 101.96
0.0 30.0 60.0 90.0 17.3 20.0 50.0 80.0
RHF/6-3 11G(d) Optiknized Parameters 1.3910 0.9416 107.85 -150.7908500 1.3876 0.9419 107.15 -150.793 8958 1.3818 0.9422 105.64 -150.800377 3 1.3806 0.9416 104.17 -150.805 2972 1.3845 0.9407 103.02 -150.806 567 0 1.3851 0.9406 102.91 -150.806558 5 1.391 1 0.9400 101.92 -150.805 792 5 1.3936 0.9399 101.53 -150.8052600
0.0 30.0 60.0 90.0 112.1 120.0 150.0 180.0
RHF/6-311G(2d,p) Optimized Parameters 1.3965 0.9429 106.76 -150.811 6800 1.3935 0.9434 106.14 -150.814265 3 1.3881 0.9441 104.79 -150.8197384 1.3870 0.9438 103.43 -150.823 8250 1.3894 0.9432 102.53 -150.824671 4 1.3904 0.9430 102.27 -150.824587 3 1.3964 0.9425 101.39 -150.823 4468 1.3985 0.9424 101.01 -150.8227447
R-,
total energy, au kcal/mol Parameters -150.7420150 -150.744818 3 -150.751 059 1 -150.756490 1 -150.7590184 -150.759 534 5 -150.759541 1 -150.7595246
10.99 9.24 5.32 1.91 0.32 0.00
O.O* 0.01 9.86 7.95 3.88 0.80
O.Ob 0.01 0.49 0.82 8.17 6.53 3.10 0.53
O.Ob 0.05 0.77 1.21
Values are given for three basis set levels. Fully optimized structure.
the electronic structure of the molecule in a very basic way. One of the questions we address in this work is exactly what role these functions play in determining the torsional potential. Figure 2 shows the RHF/6-3 11G(2d,p) torsional potential as compared with the experimental torsional potential found by Hunt and co-workers.6 It is clear that the calculated potential is in
0022-365418812092-4306$01.50/0 0 1988 American Chemical Society
Torsion-Vibration Interactions in Hydrogen Peroxide
The Journal of Physical Chemistry, Vol. 92, No. 15, I988
4307
set of atomic orbitals (NAOs), which are then unitarily 7normal transformed to orthornormal hybrid orbitals (NHO’s) and bond l4 12 10
t
o 6-311G A
6-311G(d)
-
8-
64 -
2 0 -
0
45
90
135 180 225 270 315 360
X (degrees)
Figure 1. Torsional potential, F(x),in H202computed with the RHF
method at three basis set levels.
0
6-311G(Zd.p)
+ experiment
0
45
90 135 180 225 270 315 360 X (degrees)
Figure 2. Comparison of the RHF/6-3 1 lG(Zd,p) torsional potential W ( x )in H202with the experimental curve (taken from ref 6 ) .
fairly good agreement with the experimental curve, but as was made clear in paper 1, a much higher level of theory is needed to gain the near-quantitative accuracy necessary to reproduce the detailed level of potential information found in high-resolution spectroscopic studies. The emphasis in this paper is on analyzing the basic electronic interactions which determine the form of the potential. The quality of the RHF calculations is sufficient for that purpose. Indeed, we will see that the interaction responsible for the trans barrier is present in the very lowest level calculations and that the effect of improving the computational method is simply to modulate the way that interaction comes into the full potential. The outline of the paper is as follows: section I1 describes the method of natural bond orbital (NBO) analysis and presents an application of this analysis to the orbital interactions which determine the torsional potential. Section I11 shows how OH overtone excitation enhances the charge-transfer interaction between the oxygen lone pairs with the vicinal OH antibond, thus increasing the trans barrier height. Section IV presents a discussion of how polarization functions come in to modify the value of the trans barrier without changing the basic interaction which gives rise to it. Section V presents a final summary of the work. 11. Natural Bond Orbital Analysis of the Stereoelectronic Effects Governing the Form of the Torsional Potential A . Summary of the Natural Bond Orbital Method. The natural bond orbital (NBO) procedure for analyzing the electronic structure of molecules has been described in detail e l ~ e w h e r e . ~ Briefly, the procedure consists of determining an optimal ortho( 6 ) Hunt, R. H.; Leacock, R. A.; Peters, C.W.; Hecht, K.T.J . Chem. P h p . 1965,42, 1931-1946.
orbitals (NBOs) in an optimal way. “Optimal” refers to a criterion of maximum occupancy, as determined by the 1-electron density matrix. These orbitals therefore have “natural” character analogous to that of Lowdin’s natural orbitals (NO’S).’ Unlike NOS, which are eigenvectors of the full 1-electron density matrix, N B O s correspond to localized eigenvectors of 1- and 2-center blocks of the density matrix which describe 1-center core and nonbonding (lone pair) orbitals and 2-center bonds and antibonds. The minimal set of valence NBO’s (usually bond, core, and lone pair orbitals) contain almost all of the electron density, thus forming an accurate representation of the density in terms of a natural Lewis structure. The remaining NBOs, 1-center Rydberg orbitals and 2-center antibonds (non-Lewis orbitals), usually contain only a small amount of electron density, but often play a key role in determining the features of the potential energy surface. The full set of NBOs, both the Lewis structure orbitals and the non-Lewis orbitals, are needed to completely span the space of the occupied molecular orbitals. Charge-transfer interactions (“delocalization”) between the Lewis structure and the non-Lewis NBO’s can therefore directly lower the energies of the occupied molecular orbitals. The total energy of the molecule, E , can be decomposed perturbatively into terms arising from the Lewis structure E,, and from charge-transfer E,,. E = E,, + E,,, (1) The “bonds only” or Lewis term E,, contains almost all of the total energy and represents all steric, electrostatic, and other interactions between the filled Lewis structure orbitals alone, while the “hyperconjugative” or non-Lewis term E,,, is orders of magnitude less. The actual magnitude of E,, can be measured, in principle, by removing the acceptor orbitals from the basis set and then recalculating the total energy. The magnitude of E,,, is obtained by difference. Such a procedure is independent of the a b initio method used. Because of the small magnitude of the E,,. term, this procedure can be carried out approximately for SCF wave functions by transforming the converged Fock matrix from the original A 0 basis to the NBO basis and then deleting the off-diagonal elements coupling an acceptor orbital to a donor orbital. The new density matrix (with the hyperconjugative interaction removed) is obtained by diagonalizing the deleted Fock matrix and the total energy of that density is evaluated. The change in energy gives the strength of the deleted interaction. Decomposing the Lewis term E,, into its component interactions is intrinsically more difficult due to the large energies involved, preventing any simple perturbative decomposition. Thus, we do not attempt in this work to determine the relative roles of electrostatic, steric, or other interactions involved in E,,. In the following discussion, we refer to the interactions contributing to E,, as Lewis interactions and the interactions contributing to the charge-transfer term E,,. as hyperconjugative interactions. B. Natural Bond Orbitals of Hydrogen Peroxide. The NBO analysis was applied to the R H F optimized structures of H 2 0 2 obtained at the 6-31 lG, 6-311G(d), and 6-311G(2d,p) basis set levels. Table I1 gives the occupancies, hybridization and polarization parameters of the valence NBO’s, and the nonnegligible (occupancy > 0.001e) Rydberg NBO’s for each of three basis sets. The occupancies of the Lewis structure NBOs indicate that H202 is well described by a single Lewis structure, although a small amount of density (
