Coupling reactions, computer representation

If the diffusion process is coupled with other influences (chemical reactions, adsorption at an interface, convection in solution, etc.), additional concentration dependences will be added to the right side of Equation 2.11, often making it analytically insoluble. In such cases it is profitable to retreat to the finite difference representation and model the experiment on a digital computer. Modeling of this type, when done properly, is not unlike carrying out the experiment itself (provided that the discretization error is equal to or smaller than the accessible experimental error). The method is known as digital simulation, and the result obtained is the finite difference solution. This approach is described in more detail in Chapter 20. 

FORTRAN computer program that predicts the species, temperature, and velocity profiles in two-dimensional (planar or axisymmetric) channels. The model uses the boundary layer approximations for the fluid flow equations, coupled to gas-phase and surface species continuity equations. The program runs in conjunction with CHEMKIN preprocessors (CHEMKIN, SURFACE CHEMKIN, and TRAN-FIT) for the gas-phase and surface chemical reaction mechanisms and transport properties. The finite difference representation of the defining equations forms a set of differential algebraic equations which are solved using the computer program DASSL (dassal.f, L. R. Petzold, Sandia National Laboratories Report, SAND 82-8637, 1982). 

One example of a bonafide bis(alkyne) complex has recently been prepared. Reaction of the in situ generated olefin complex prepared by alkylation of ( -CsHs ZrC 50 with the diaryl alkyne in Equation (7) yields 253.130 In this structure, C-C coupling has not occurred, presumably a result of the steric strain associated with the zirconacyclo-pentadienyl fragment (Equation (7)). The solid-state structure further establishes the compound as a bis(alkyne) complex. Computational studies suggest that a Zr(iv) resonance structure is the most suitable representation of the compound. However, reaction of 253 with iodine in THF yields ( -CsHs Zrle 254 and the dialkyne starting material, suggesting that the zirconium center can act as a source of Zr(n) (Equation (8)). 

We now discuss the necessary details of the computational aspects to obtain the reaction probability by the scheme developed above. In the following we proceed with the coupled-surface calculations the uncoupled-surface calculations follow from them in an elementary way. The time-dependent Schrodinger equation (TDSE) is solved numerically in the diabatic electronic representation on a grid in the (i , r, 7) space using the matrix Hamiltonian in Eq. (11). For an explicitly time-independent Hamiltonian the solution reads... 

These methods combine a QM representation of solute with a classical continuum description of the solvent [18-23]. The methodology is equivalent to that of classical continuum methods, except that a) the solute charge distribution is allowed to relax by the solvent reaction field, and b) the solute-solvent interaction is computed at the QM level. Most QM continuum methods work within the multipole or apparent surface charge approaches, even though other formalisms are also available [18-23]. The solvent reaction field is introduced into the solute Hamiltonian by means of a perturbation operator (R in equation 22) that couples the solvent reaction field to the solute charge distribution. At this point, it is worth noting that equation 22 is not lineal, since T and R are mutually dependent. This means that a self-consistent process in which both the wavefunction and the reaction field are treated simultaneously is required to solve equation 22. This is the reason why these methods are typically known as self-consistent reaction field (SCRF) methods. 

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