Partitioning internal coordinates

In the following we consider a surface with adsorbed atoms or molecules that react. We will leave out the details of the internal coordinates of these adsorbed species, but note that their partition functions can be found using the schemes presented above. Let us assume that species A reacts with B to form an adsorbed product AB via an activated complex AB ... 

The partition function of a solute-solvent system for a given electronic state, where for the N solutes we use as (classical) molecular coordinates the center of mass position rG, the eulerian angles 0, , molecular frame and the internal coordinates Xjn providing the atom positions in the molecular frame, is [27,28]... 

The adiabatic enclosures mentioned above are ideal partitions which separate regions of thermodynamic interest from the remainder of the universe in particular, no heat transfer of any type can occur across such boundaries. The walls of the container are in intimate contact with the gas which is being compressed but changes in its internal coordinates cannot be communicated to the remainder of the universe. 

Berlin (1951) proposed an interesting partitioning of the space of the diatomic density distribution in the following way. Instead of the force on nucleus a, Fa, he considered the force FR corresponding to the internal coordinate R, which is given by... 

It is also possible to use internal coordinate representations to treat constraints, i.e. parameterization schemes, which are typically based on partitioning the coordinates of the system into a set of independent variables and a complementary collection of dependent variables (which are expressed in terms of the first set). These approaches result in systems of slightly smaller dimension, but typically greater complexity. Moreover, the parameterization is typically local which leads to additional implementation issues. For some details on such schemes and discussion of their use in molecular simulation, the reader is referred to [20, 190, 255]. 

First, we inquire under what condition is the steric factor unity, meaning that there are no steric requirements. We already know from Section 3.2 tiiat this condition is met when the two colliding particles are structureless. The partition function for the reactants is then Q= Q Q, one translational partition function (for a 3D motion) for each reactant. The point of no return has been identified in Section 3.2.7 the two particles are at a distance d apart and the barrier height is Eq. The transition state is here a diatomic. It has six degrees of freedom. Three are the motion of its center of mass. Of the other three, one is the vibrational motion, which is the reaction coordinate. Therefore, it should not be counted as an internal coordinate of the transition state. The other degrees of freedom are the two planes of rotation of a diatomic molecule. Therefore = 2x2r-the transition theory result for the reaction rate constant, in the absence of any steric effect, is... 

Performing the integral over the center of mass and defining a partition function of the cluster in terms of the internal coordinates we get... 

Generalized Partition Functions in Curvilinear Internal Coordinates... 

Figure 2.12. A flow tube used to derive one-dimensional flow equations in Lagrangian coordinates. Internal surfaces are massless, impermeable partitions to aid in visualizing elements of fluid in Lagrangian coordinates.

There is no internationally recommended symbol for the distribution constant, but the symbol P (for partition constant) is common usage in this context. Coordination (complex) chemists (as in Chapter 3) prefer the symbol K-o (for the German Tonstante for constant ) sometimes other symbols are used. The important point is that the symbol should be properly defined in the text. 

This typical application of the second kind is the Gibbs Phase Rule (for inert systems). This rule is often stated merely for systems with only two external coordinates (n = 2, e.g., xt = P,x2 = T). There must then be no internal partitions within the system, nor may it, for instance, contain magnetic substances in the presence of external magnetic fields. 

Here E ( y1 ) stands for the single-particle contribution to the total energy, allowing for molecule interaction with the surface <2 is the heat released in adsorption of molecules z on the /Lh site Fj the internal partition function for the z th molecules adsorbed on the /Lh site F j the internal partition function for the zth molecule in the gas phase the dissociation degree of the z th molecule, and zz the Henry local constant for adsorption of the zth molecule on the /Lh site. Lateral interaction is modeled by E2k([ylj ), and gj (r) allows for interaction between the z th and /Lh particles adsorbed on the /th and gth sites spaced r apart. In the lattice gas model, separations are conveniently measured in coordination-sphere numbers, 1 < r < R. For a homogeneous surface, molecular parameters zz and ej(r) are independent of the site nature, while for heterogeneous, they may depend on it. 

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