Surfaces harmonic approximation

This algorithm was improved by Chen et al. [78] to take into account the surface anhannonicity. After taking a step from Rq to R[ using the harmonic approximation, the true surface information at R) is then used to fit a (fifth-order) polynomial to fomi a better model of the surface. This polynomial model is then used in a coirector step to give the new R,. 

While it is not essential to the method, frozen Gaussians have been used in all applications to date, that is, the width is kept fixed in the equation for the phase evolution. The widths of the Gaussian functions are then a further parameter to be chosen, although it appears that the method is relatively insensitive to the choice. One possibility is to use the width taken from the harmonic approximation to the ground-state potential surface [221]. 

The surfaces of large molecules such as proteins cannot be represented effectively with the methods described above (e.g., SAS), However, in order to represent these surfaces, less calculation-intensive, harmonic approximation methods with SES approaches can be used [1S5]. 

The vibraiimial rroqueiicics are tlenved lioin the harmonic approximation, which assiiiines that the potential surface has a quadratic form. 

If the size of the complex is rather small and the intramolecular vibrations along the coordinate Q may be described in the harmonic approximation, the free energy surfaces of the initial and final states may be written in the form... 

In the general case R denotes a set of coordinates, and Ui(R) and Uf (R) are potential energy surfaces with a high dimension. However, the essential features can be understood from the simplest case, which is that of a diatomic molecule that loses one electron. Then Ui(R) is the potential energy curve for the ground state of the molecule, and Uf(R) that of the ion (see Fig. 19.2). If the ion is stable, which will be true for outer-sphere electron-transfer reactions, Uf(R) has a stable minimum, and its general shape will be similar to that of Ui(R). We can then apply the harmonic approximation to both states, so that the nuclear Hamiltonians Hi and Hf that correspond to Ui and Uf are sums of harmonic oscillator terms. To simplify the mathematics further, we make two additional assumptions ... 

Here, 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 ... 

No first derivative terms appear here because the transition state is a critical point on the energy surface at the transition state all first derivatives are zero. This harmonic approximation to the energy surface can be analyzed as we did in Chapter 5 in terms of normal modes. This involves calculating the mass-weighted Hessian matrix defined by the second derivatives and finding the N eigenvalues of this matrix. 

Fortunately, the reaction rates of many important processes can be obtained without a full molecular dynamics simulation. Most reaction rate theories for elementary processes build upon the ideas introduced in the so-called transition state theory [88-90]. We shall focus on this theory here, particularly because it (and its harmonic approximation, HTST) has been shown to yield reliable results for elementary processes at surfaces. 

Transition state theory is very often used in its harmonic approximation. The harmonic approximation is applicable under the normal assumptions of transition state theory, but further demands that the potential energy surface is smooth enough for a harmonic expansion of the potential energy to make sense. Since the harmonic expansion is performed in the initial state and in a first-order saddle point on the... 

To find an expression for the potential energy of the vibrating system in the well state at P, the harmonic approximation is used and 4> q) is expanded about P, so the potential-energy surface near P has the form... 

Even when the harmonic approximation is not quantitatively justified it provides a convenient starting point for exact treatments. Thus, even if the potential energy surface is anharmonic in the bottleneck, it is often smooth enough for there to be a principal saddle point that can be found by minimizing IVU 2. 

These discrepancies result (a) from the harmonic approximation used in all calculations [to,- (theory) > v, (exp)], (b) the known deficiencies of minimal and DZ basis sets to describe three-membered rings [polarization functions are needed to describe small CCC bond angles a>,(DZ + P) > w,(DZ) > to,(minimal basis)] and (c) the need of electron correlated wave functions to correctly describe the curvature of the potential energy surface at a minimum energy point [ [Pg.102]

---