Multidimensional Transition State Theory

r is the vector of all relevant particle coordinates and x is one coordinate chosen to define the reaction coordinate. The trick to rewriting this rate in a usable way is to treat both of the integrals that appear in it with the harmonic approximation we used in Section 6.1. For the denominator, we expand the energy surface around the energy minimum at r r  

We have seen this expression before in Chapter 5, where it was the starting point for describing vibrational modes. We found in Chapter 5 that a natural way to think about this harmonic potential energy surface is to define the normal modes of the system, which have vibrational frequencies v, (/ 

The normal modes are the special solutions to the equations of motion that have the form 

We can use exactly the same idea for the integral in the numerator of Eq. (6.10). The Taylor series expansion for the energy expanded around the transition state is, to second order, 

TABLE 6.2 Vibrational Frequencies in THz for Ag Atom in Fourfold Site and Bridge Site on Cu(100) 

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D. G. Truhlar and B. C. Garrett, Multidimensional transition state theory and the validity... 

Fortunately, it is relatively simple to estimate from harmonic transition-state theory whether quantum tunneling is important or not. Applying multidimensional transition-state theory, Eq. (6.15), requires finding the vibrational frequencies of the system of interest at energy minimum A (v, V2,. . . , vN) and transition state (vj,. v, , ). Using these frequencies, we can define the zero-point energy corrected activation energy ... 

See, for instance, D. G. Truhlar and B. C. Garrett. Multidimensional transition state theory and the validity of Grote Hynes theory. J. Phys. Chem. B 104. 1069 1072 (2000). 

Greenfield, M.L. Molecular Modeling of Dilute Penetrant Gas Diffusion in a Glassy Polymer using Multidimensional Transition-State Theory. Ph.D. thesis, University of California, Berkeley, 1996. 

Greenfield, M. L. Theodorou, D. N. (199S). Molecular Mockling of Methane Diffusion in Glassy Atactic Polypropylene via Multidimensional Transition State Theory. Macromolecules, 31(20), 7068-7090. 

The rate of hydrogen transfer can be calculated using the direct dynamics approach of Truhlar and co-workers which combines canonical variational transition state theory (CVT) [82, 83] with semi-classical multidimensional tunnelling corrections [84], The rate constant is calculated using [83] ... 

Allison TC, Trahlar DG (1998) Testing the accuracy of practical semiclassical methods variational transition state theory with optimized multidimensional tunnelling. In Thompson DL (ed) Modern Methods for Multidimensional Dynamics Computations in Chemistry. World Scientific, Singapore, p 618... 

Fernandez-Ramos A, Ellingson BA, Garrett BC, Trahlar DG (2007) Variational transition state theory with multidimensional tunneling. In Lipkowitz KB, Cundari TR, Boyd DB (eds) Reviews in Computational Chemistry, Vol 23. Wiley-VCH, New York, p 125... 

Antonio Fernandez-Ramos, Benjamin A. Ellingson, Bruce C. Garrett, and Donald G. Truhlar, Variational Transition State Theory with Multidimensional Tunneling. 

Table 6.2 Tests of Variational Transition State Theory by Comparing with Exact Quantum Calculations (Extracted from Allison, T. C. and Truhlar, D. G. Testing the accuracy of practical semiclassical methods variational transition state theory with optimized multidimensional tunneling, in Thompson, D. L., Ed. Modem methods for multidimensional dynamics computations in chemistry, World Scientific, Singapore 1998. pp 618-712. This reference quotes results on many more reactions and BO surfaces over broad temperature ranges.)The numbers in the table are ratios of the results of the approximate calculation to the quantum calculation, all at 300 K...

TST = conventional Transition State Theory, ICVT = Improved Canonical Variational Transition state theory, ICVT/SCT = ICVT/Small Curvature Tunneling, ICVT/p,OMT = ICVT/Microcanonical Optimized Multidimensional Tunneling. 

Truhlar, D. G. Variational transition state theory and multidimensional tunneling for simple and complex reactions in the gas phase, solids, liquids, and enzymes, in Kohen, A. and Limbach, H. H., Eds. Isotope Effects in Chemistry and Biology. CRC Press/Taylor Francis, Boca Raton, FL (2006), pp. 579-619. 

The second approach, a multidimensional one, was given by Langer [7], Other multidimensional developments were many [16-18]. McCammon [17] discussed a variational approach (1983) to seek the best path for crossing the transition-state hypersurface in multidimensional space and discussed the topic of saddle-point avoidance. Further developments have been made using variational transition state theory, for example, by Poliak [18]. 

In ESP theory [30-32] we treat the system by the same methods that we would use in the gas phase except that in the nontunneling part of the calculation we replace V(R) by TT(R), and in the tunneling part we approximate V(R) by TT(R) or a function of TT(R). Next we review what that entails. In particular we will review the application of variational transition state theory [21-25] with optimized multidimensional tunneling [33,34] to liquid-phase reactions for the case [31,32] in which TT(R) is calculated from V(R) by... 

T. V. Albu, J. C. Corchado, D. G. Truhlar, J. Phys. Chem. A 105, 8465 (2001). Molecular Mechanics for Chemical Reactions A Standard Strategy for Using Multiconfiguration Molecular Mechanics for Variational Transition State Theory with Optimized Multidimensional Tunneling. 

Our discussion of transition state theory in this chapter was laid out in detail in only the one-dimensional setting. Provide the detailed arguments for the multidimensional version of transition state theory that culminates in eqn (7.62). 

D. Tmhlar, Variational Transition-State Theory and Multidimensional Tunneling for Simple and Complex Reactions in the Gas Phase, Solids, Liquids, and Enzymes, in Ref. [12b], Ch. 22, pp. 579-620. 

The higher dimensionality of polyatomic reactions makes them more of a challenge to treat theoretically. Variational transition state theory with multidimensional tunneling has been developed to allow calculations for a wide variety of polyatomic systems. In this section we consider issues that arise when treating polyatomic systems. The Cl -1- CH4 reaction provides a good system for this pur-... 

For another perspective we mention a second approach of which the reader should be aware. In this approach the dividing surface of transition state theory is defined not in terms of a classical mechanical reaction coordinate but rather in terms of the centroid coordinate of a path integral (path integral quantum TST, or PI-QTST) [96-99] or the average coordinate of a quanta wave packet. In model studies of a symmetric reaction, it was shown that the PI-QTST approach agrees well with the multidimensional transmission coefScient approach used here when the frequency of the bath is high, but both approaches are less accurate when the frequency is low, probably due to anharmonicity [98] and the path centroid constraint [97[. However, further analysis is needed to develop practical PI-QTST-type methods for asymmetric reactions [99]. 

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