Theor Chem Account (2007) 118:813–826 DOI 10.1007/s00214-007-0328-0 REGULAR ARTICLE Symmetry numbers and chemical reaction rates Antonio Fernández-Ramos · Benjamin A. Ellingson · Rubén Meana-Pañeda · Jorge M. C. Marques · Donald G. Truhlar Received: 14 February 2007 / Accepted: 25 April 2007 / Published online: 11 July 2007 © Springer-Verlag 2007 Abstract This article shows how to evaluate rotational 1 Introduction symmetry numbers for different molecular configurations and how to apply them to transition state theory. In general, Transition state theory (TST) [1–6] is the most widely used the symmetry number is given by the ratio of the reactant and method for calculating rate constants of chemical reactions. transition state rotational symmetry numbers. However, spe- The conventional TST rate expression may be written cial care is advised in the evaluation of symmetry numbers kBT QTS(T ) in the following situations: (i) if the reaction is symmetric, k (T ) = σ exp −V ‡/k T (1) TST h (T ) B (ii) if reactants and/or transition states are chiral, (iii) if the R reaction has multiple conformers for reactants and/or tran- where kB is Boltzmann’s constant; h is Planck’s constant; sition states and, (iv) if there is an internal rotation of part V ‡ is the classical barrier height; T is the temperature and σ of the molecular system. All these four situations are treated is the reaction-path symmetry number; QTS(T ) and R(T ) systematically and analyzed in detail in the present article. are the quantum mechanical transition state quasi-partition We also include a large number of examples to clarify some function and reactant partition function, respectively, without complicated situations, and in the last section we discuss an rotational symmetry numbers, and with the zeroes of energy example involving an achiral diasteroisomer. at the zero-point-exclusive energies of the saddle point and equilibrium reactants, respectively. QTS(T ) is referred to as Keywords Symmetry numbers · Chemical reactions · a quasi-partition function because it is missing the vibra- Transition state theory · Chiral molecules · Multiple tional degree of freedom corresponding to the reaction coor- conformers · Internal rotation dinate. R(T ) is the unitless reactant partition function for unimolecular reactions and the reactants partition function per unit volume for bimolecular reactions. Specifically for bimolecular reactions (A + B → P), the reactants partition ( ) = A,B( ) ( ) = function can be factorized as R T rel T QR T A. Fernández-Ramos (B) · R. Meana-Pañeda A,B( ) ( ) ( ) A,B rel T QA T QB T , where rel is the relative transla- Departamento de Química Física, Facultade de Química, tional motion per unit volume and Q (T ) = Q (T )Q (T ) Universidade de Santiago de Compostela, R A B 15782 Santiago de Compostela, Spain is the unitless reactants partition function for the internal e-mail: [email protected] motions. The present article is mainly concerned with σ . Equation (1) can be generalized by variationally optimiz- · B B. A. Ellingson D. G. Truhlar ( ) ing the transition state (so it is no longer located at the saddle Department of Chemistry and Supercomputing Institute, University of Minnesota, 207 Pleasant Street S. E., point) and by adding a transmission coefficient to account Minneapolis, MN 55455-0431, USA for recrossing and quantum effects (including tunnelling) e-mail: [email protected] [3–6]. These generalizations do not change the considera- tions involved in giving a value to σ . J. M. C. Marques Departamento de Química, Universidade de Coimbra, In a classic article, Pollak and Pechukas [7] sorted out con- 3004-535 Coimbra, Portugal ceptual difficulties involving σ and proved that it is always 123 814 Theor Chem Account (2007) 118:813–826 16 equal to the ratio of the total symmetry number of the reac- Table 1 Ratios of approximate rotational partition functions for O2 tant divided by the total symmetry of the transition state. to the accurate one with only odd J Despite the simplicity and clarity of this result, it can some- Temperature (K) Classical with Quantal without times be confusing to apply it to complex reactions, because it symmetry factora symmetryb may require considering more than the usual rotational sym- 1,000 0.9993 2.0000 metry numbers, and it is discouraging to find that in some 600 0.9988 2.0000 recent applications that the symmetry numbers are ignored, 300 0.9977 2.0000 ill-defined, or wrong. In addition, the increased complexity 200 0.9965 2.0000 of new applications also raises interesting new questions. For . these reasons it is useful to review some of the arguments of 100 0.9931 2 0000 . Pollak and Pechukas and to add some new comments on the 50 0.9862 2 0000 . subject. It should be noted that symmetry number arguments 25 0.9726 2 0000 similar to those of Pollak and Pechukas were also given by 10 0.9326 2.0002 Coulson [8]. 5 0.8883 2.0452 In conventional TST, the forward thermal rate constant 1 5.1368 22.3732 can be calculated with information about only two config- For this illustration we use the rigid rotor approximation with a 2 urations, i.e., reactants and the transition state. Leaving the moment of inertia of 75,894 me bohr ,whereme is the mass of an electron symmetry numbers out of the rotational partition function a Ratio of Eq. (3a)withσrot = 2 to accurate result in Eq. (1) means that we are treating the identical particles b Ratio of partition function with all J to partition function with only as distinguishable, and in this case the rotational partition odd J functions would be given by is the product of the two σ numbers. In some cases, it will ∗ 2I rot Q (T ) = (2a) not be sufficient to consider just rotational symmetry num- rot h¯ 2β bers, σrot. We will need to consider rotational translational for a lineal molecule, where I is the moment of inertia, h¯ is symmetry numbers σr–t . π β / Planck’s constant divided by 2 , and is defined as 1 kBT . It is useful to review the fundamental origin of the sym- For nonlinear molecules, the expression for the rotational 16 16 metry factors. Consider O2 as an example. Because the O ∗ ( ) partition function of distinguishable particles Qrot T is nucleus is a boson, the nuclear spin wave function is sym- / 3 1 2 metric under interchange of the nuclei. By Bose–Einstein sta- ∗ 2 Q (T ) = π I I I (2b) tistics, the overall wave function must be symmetric under rot 2β A B C h¯ such interchange. Since the nuclear spin wave function is symmetric, and the ground electronic state (3−) is odd, where IA, IB and IC are the principal moments of inertia. g The symmetry number in Eq. (1) arises from the indistin- the rotational wave function must be antisymmetric (odd). guishability of identical particles, i.e., the rotational partition Therefore, half of the rotational quantum numbers J (the is given by even ones) are missing [9]. In the classical limit where sums over rotational levels are replaced by an integral [10], half 1 Q (T ) = Q∗ (T ) of the levels being missing decreases the rotational partition rot σ rot (3a) rot function by a factor of two. At low temperature where only where σrot is the rotational symmetry number. Therefore the a few rotational levels are populated, the factor of two is ratio between the rotational partition functions of the transi- only approximately correct. In practice, the inclusion of the tion state, Qrot,TS(T ), and reactants, Qrot,R(T ), leads to: inaccessible J = 0 state would cause a very large error at ∗ ( ) ∗ ( ) very low temperature. This is illustrated in Table 1, which Qrot,TS(T ) σrot,R Qrot,TS T Qrot,TS T = ∗ = σ ∗ (3b) first shows the ratio of the partition function calculated using Q , (T ) σ , Q (T ) Q (T ) rot R rot TS rot,R rot,R Eq. (3a) to the accurate one, and then shows the ratio of the Therefore, in simple cases, the rotational symmetry num- hypothetical partition function for all J to the accurate one. bers in the above partition functions account for all the effects The table shows that the symmetry factor is very close to 2 at of nuclear indistinguishablility on reaction rates, and the most temperatures of interest. The classical approximation is symmetry number in Eq. (1)isgivenby so good that one almost always uses it—the main exception being H2 below room temperature. σ = σ , /σ , , (4) rot R rot TS When the identical nuclei are fermions, the situation is with σrot,R and σrot,TS being the rotational symmetry numbers more complicated. H2 provide the classic example. Since it of the reactants and the transition state, respectively. For a is treated in most statistical mechanics texts [10], we just bimolecular reaction, where the reactants are different, σrot,R summarize the result. It turns out that one fourth of the 123 Theor Chem Account (2007) 118:813–826 815 nuclear spin states are forbidden for odd J and three fourths complications such as chiral isomers, symmetric reactions, are forbidden for even J. Averaging over many states again low-energy conformers, and internal rotation. decreases the partition function by a factor of two. Polyatomic molecules are more complicated, because 2 Symmetry numbers and rotation there can be more than two identical nuclei, because the vibrational wave functions are not all symmetric, and because Consider the molecules depicted in Structure 1 in their equi- the rotational wave functions are more complicated. Con- librium configurations: water, ammonia, and the methyl rad- sider methane (CH4) as an example.