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Global Permutationally Invariant Potential Energy Surface for Ozone Forming Reaction Mehdi Ayouz Ecole Centrale De Paris

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Global Permutationally Invariant Potential Energy Surface for Ozone Forming Reaction Mehdi Ayouz Ecole Centrale De Paris Marquette University e-Publications@Marquette Chemistry Faculty Research and Publications Chemistry, Department of 1-1-2013 Global Permutationally Invariant Potential Energy Surface for Ozone Forming Reaction Mehdi Ayouz Ecole Centrale de Paris Dmitri Babikov Marquette University, [email protected] Published version. Journal of Chemical Physics, Vol. 138 (2013): 164311. DOI. © 2013 American Institute of Physics. Used with permission. THE JOURNAL OF CHEMICAL PHYSICS 138, 164311 (2013) Global permutationally invariant potential energy surface for ozone forming reaction Mehdi Ayouz1,2 and Dmitri Babikov1,a) 1Chemistry Department, Wehr Chemistry Building, Marquette University, Milwaukee, Wisconsin 53201-1881, USA 2Laboratoire de Génie des Procédés et Matériaux, Ecole Centrale de Paris, Bât. Dumas, 92295 Châtenay-Malabry Cedex, France (Received 6 February 2013; accepted 21 March 2013; published online 24 April 2013; publisher error corrected 6 May 2013) We constructed new global potential energy surface for O + O2 → O3 reaction. It is based on high level electronic structure theory calculations and employs fitting by permutationally invariant poly- nomial functions. This method of surface construction takes full advantage of permutation symmetry of three O nuclei and allows reducing dramatically the number of ab initio data points needed for ac- curate surface representation. New potential energy surface offers dramatic improvement over older surface of ozone in terms of dissociation energy and behavior along the minimum energy path. It can be used to refine the existing theories of ozone formation. © 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4799915] I. INTRODUCTION value by more than 10%. Second, that surface contained an artifact – a barrier along the minimum energy path (MEP) to Fitting the potential energy surface by permutationally dissociation.13 The height of the barrier was only 47 cm−1 invariant polynomial functions1 carries significant advan- above the dissociation limit, but it was shown that the temper- tages, particularly for those molecules where several atoms ature dependence of the atom exchange rate is sensitive to this are identical.2–7 For the molecules where all atoms are iden- feature.14 Experimental kinetics data suggest that the reaction tical (allotropes) this should be the method of choice.8 It al- path is barrier-less.15, 16 lows reducing significantly the number of ab initio data points There were several attempts to determine the level of the- needed for accurate surface representation, as well as auto- ory required to compute a better surface for O . Higher level matically takes care of the intrinsic molecular symmetry. The 3 electronic structure calculations by several authors carried out resultant PES is analytic and smooth everywhere in the con- along the minimum energy path13, 17, 18 showed that top of the figuration space, free of any cusps (sharp artifacts, typical to barrier is, indeed, below the dissociation limit, which trans- other methods of surface representation such as splines). Gra- forms barrier into a “submerged reef.” It was also suggested dients and Hessians of the potential energy function can also that the improvement to dissociation energy can be achieved be computed analytically. While such fit is global, one can by increasing the basis set size. However, calculations with still emphasize the important regions of the PES. Adding new basis sets larger than quadruple-ζ are computationally de- points to the existing PES is straightforward too. manding, and those have never been attempted for the global One very important small homonuclear molecule is PES of O . ozone, a triatomic molecule of oxygen, O . Its potential 3 3 We found it feasible19 to construct new, improved poten- energy surface has been constructed in the past9 using tial energy surface for ozone forming reaction, ab initio data at icMRCI+Q/cc-pVQZ level of theory with CASSCF(12,9) active space. That surface was represented by − O(3P) + O (3 ) → O (X)˜ , a 3D-spline of data points on a structured (rectangular) grid. 2 g 3 Close to 6000 points total were used and the grid was quite by combining (i) results of the high-level electronic struc- dense near the bottom of the covalent well. The surface was ture theory calculations extrapolated to the complete basis set claimed to be accurate at low energies,10 but it contained sev- 11 (CBS) limit and (ii) surface fitting by permutationally invari- eral serious deficiencies at high energies. First, the surface ant polynomial functions. If we focus on the near-threshold was way too shallow. For example, the dissociation energy of −1 part of the surface, relevant to thermal recombination kinet- O3 on that surface is 936 cm smaller than a recent exper- 12 ics, this method of fitting allows us to reduce the number of imental value of 1.143 eV. Predictions of vibrational spec- ab initio data points to roughly 500 which, in turn, makes the tra using the surface of such quality are questionable. Even electronic structure calculations affordable. the validity of predicted thermochemical data would be unre- In Sec. II of this paper we outline all equations needed liable, since dissociation energy deviates from experimental to fit the PES of any homonuclear triatomic molecule using permutationally invariant polynomial functions and to con- a)Author to whom correspondence should be addressed. Electronic mail: struct analytic representations of its gradients and Hessians. [email protected] In Sec. III this method is applied to fit new ab initio data for 0021-9606/2013/138(16)/164311/10/$30.00 138, 164311-1 © 2013 American Institute of Physics 164311-2 M. Ayouz and D. Babikov J. Chem. Phys. 138, 164311 (2013) the ground state PES of ozone. Major features of new PES are combined is quite clear – according to their order, j,given are discussed and compared to those of the old surface for O3. by the following simple expression: Conclusions are given in Sec. IV. Bohr units for distances will j = 1 + 22 + 33. (5) be used throughout the paper (1 Bohr = a0 = 0.0529177 nm). This also gives a natural method for truncating the expansion 0 0 – according to the order of terms, j.Thetermsp1, p1, and II. THEORETICAL FRAMEWORK 0 p3 are, of course, equal to unity but are included in Eq. (4) A. The fitting method explicitly in order to emphasize a general structure of this ex- pression. For a triatomic molecule with three identical atoms one For the purpose of convenience, the coefficients of linear useful set of the internal vibrational coordinates is a set of expansion in Eq. (4) are labeled according to order j of the three positive inter-nuclear distances: r = {r , r , r }. The ij 12 23 31 corresponding basis function. Thus, C is coefficient of the indexes i and j label three possible pairs of oxygen atoms in 0 zero-order function (which is just unity), C is coefficient for O . The first step in fitting the PES is to introduce three prim- 1 3 the first-order function p , the doublet {C , C } represents a itive functions y of the inter-nuclear distances r . Since in 1 2a 2b ij ij set of coefficients for two possible functions of second order: O all atoms are identical, the pairs of atoms are also identi- 3 p2 and p , the triplet {C , C , C } represents a set of co- cal. Thus, the three primitive functions y are the same, just 1 2 3a 3b 3c ij efficients for three possible functions of third order: p3, p p , the arguments are different: y = y(r ), y = y(r ), and y 1 1 2 12 12 23 23 31 and p , and so on. = y(r ). Analytic expression for y(r) will be introduced later. 3 31 In order to fit the PES of ozone we used all terms of the Next, the primitive functions y are used to form various ij permutationally invariant expansion up to 16th order. Such permutationally invariant polynomials. For any homonuclear formula has M = 204 expansion coefficients to vary (includ- triatomic molecule, including O , one can form the first order 3 ing C ) in order to achieve accurate fitting.19 This is done polynomial as 0 automatically using the linear least squares fitting approach. p1 = y12 + y23 + y31, (1) Assume that the number of molecular geometries to fit is N (in this work the number of the ab initio data points was N the second order polynomial as = 570). A vector V of length N is introduced that contains the values of ab initio energies at these points. Another vector p2 = y12y23 + y23y31 + y31y12, (2) Vfit of length N is introduced that contains values of the fitting and the third order polynomial as function at these points. We also introduce vector C of length M that contains a set of fitting coefficients. Finally, the matrix × p3 = y12y23y31. (3) A of size N M is introduced that allows obtaining values of the fit at the data points from the matrix-vector product: Note that each of these polynomials (p1, p2, and p3) is a func- = tion of all three inter-nuclear distances in the molecule (r12, Vfit AC. (6) r , and r ) and, by construction, is a permutationally invari- 23 31 The structure of matrix A follows directly from Eq. (4). ant function of these variables. The permutationally invariant Namely, elements of the matrix A can be expressed as polynomials of higher orders are obtained simply as products = 1 2 3 of p1, p2, and p3. Anm p1 p2 p3 , (7) At the final step, the permutationally invariant polyno- mials of various orders are used as basis functions for linear where index n labels the data points and determines the values of p1, p2, and p3 according to Eqs.
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