Redundant internal coordinates

Pulay P and FogarasI G 1992 Geometry optimization In redundant Internal coordinates J. Chem. Phys. 96 2856... 

Peng C, Ayala P Y, Schlegel H B and Frisch M J 1996 Using redundant Internal coordinates to optimize equilibrium geometries and transition states J. Comput. Chem. 17 49... 

C, P Y Ayala, H B Schlegel and M J Erisch 1996. Using Redundant Internal Coordinates to Optimise Equilibrium Geometries and Transition States, journal of Computational Chemistry 17 49-56. 

FIGURE 8.3 Example of paths taken when an angle changes in a geometry optimization. (a) Path taken by an optimization using a Z-matrix or redundant internal coordinates. (A) Path taken by an optimization using Cartesian coordinates. 

For the coordinate system used for optimization, redundant internal coordinates are usually best, followed by a well constructed Z-matrix, then Cartesian coordinates, then a poorly constructed Z-matrix. For simulating multiple molecules, Cartesian coordinates are often best. Most programs that generate a Z-matrix automatically from Cartesian coordinates make a poorly constructed Z-matrix. 

Gaussian includes a facility for automatically generating a starting structure for a transition state optimization based upon the reactants and products that the transition structure connects, known as the STQN method. This feature is requested with the QST2 option to the Opt keyword. Input files using this option will include two title and molecule specification sections. The facility generates a guess for the transition structure which is midway between the reactants and products, in terms of redundant internal coordinates. 

Optimizations in redundant internal coordinates have more to recommend them than merely easy input setup. They also represent the most efficient approach to geometry optimizations developed to date, overcoming the difficulties and deficiencies of both Cartesian coordinates and Z-matrices (internal coordinates). 

Using redundant internal coordinates, which is the default procedure when you specify the Opt keyword. 

Clearly, the optimization in redundant internal coordinates completes significantly more quickly. There is little difference between the optimizations done in Cartesian and internal coordinates for this case. ... 

This table gives the displacements for the normal mode corresponding to the imaginary frequency in terms of redundant internal coordinates (several zero-valued coordinates have been eliminated). The most significant values in this list are for the dihedral angles D1 through D6. When we examine the standard orientation, we realize that such motion corresponds to a rotation of the methyl group. 

You may wonder why so many coordinates apply to these same few atoms. Remember that we are using redundant internal coordinates. 

Many modem computer codes (e.g. GAUSSIAN98) employ so-called redundant internal coordinates] this means that we use all possible internal coordinates, of which there will generally be more than 3N — 6. Only a maximum of 3M — 6 will be linearly independent, and we essentially throw away the remainder at the end of the full calculation. Here is ethene, done using redundant internal coordinates. 

A recent development uses the quadratic synchronous transit approach to get close to a transition state, and then a Newton or eigenvector-following algorithm to complete the optimization. It performs optimizations in redundant internal coordinates. The key reference is due to Peng, Ayala and Schlegel. 

Using Redundant Internal Coordinates to Optimize Equilibrium Geometries... 

Bakken, V. and Helgaker, T. 2002. The Efficient Optimization of Molecular Geometries Using Redundant Internal Coordinates , J. Chem. Phys., 117, 9160. 

Pulay, P. and Fogarasi, G. (1992). Geometry optimization in redundant internal coordinates, J. Chem. Phys. 96, 2857-2860. 

The same ideas extend straightforwardly to deal with property surfaces describing the dependence of a molecular property on geometry, for example, the dipole moment and polarisability derivatives that control the activity of a vibrational mode in IR and Raman spectroscopy. Extension to the case of redundant internal coordinates, the typical situation for polyatomic molecules, is also straightforward. 

Figure 12.14 Taxol molecule optimized using redundant internal coordinates opii mization procedure in Gaussian 03. (Used by permission ol ( lanssian. Inc., Pittsburgh, PA Full citation at http //www.gaussian. com/citation. htm )...

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