I warn you that specifying your program to optimize using Cartesian coordinates makes your calculation much more expensive. Rounding errors can cause your program to break symmetry, or your program may not be very good at guessing the point group of your molecule based on the Cartesian coordinates alone.Īs a preface, programs like Gaussian convert your Cartesian coordinate space (or your pre-defined Z-matrix) into redundant internal coordinates before proceeding with an optimization routine unless you specify it to stick with Cartesians or your Z-matrix. With Cartesian space this is not guaranteed. Whatever program you use should automatically recognize the constraint and will optimize your molecule accordingly giving you an answer based off a structure that is constrained to C2v symmetry. You would construct your Z-matrix to define the OH(1) bond as being equivalent to the OH(2) bond. With a Z-matrix, the process is very straightforward. When using some sort of optimizing routine, you may want to specify symmetry in your system. The OH bond lengths should be equivalent. We know from experience that this molecule has C2V symmetry. When performing complex computations, the less you have to keep track of, the less expensive the computation.Ĭonsider the following molecule, H2O. For linear molecules we keep tabs on 3N-5 coordinates. Using internal coordinates reduces our 3N requirement set by the Cartesian space down to a 3N-6 requirement (for non-linear molecules). With Z-matrices, we keep tabs on internal coordinates: bond length (R), bond angle (A), and torsional/dihedral angle (T/D). We increased the distance between the two atoms by some length R. What did we change? We simply changed the bond length, one variable. We now have altered the molecule in such a way that the properties of that molecule has changed. However, say we increase the distance between the hydrogen atoms. An H2 molecule centered around the origin (0,0,0) is no different from the same H2 molecule being centered around (1,1,1). The translation of the molecule through space (assuming a vacuum) will have no affect on the properties of the molecule. A point located at (0,0,1) is an absolute location for a coordinate space that extends to infinity. Cartesian space is 'absolute' so to speak. When dealing with Z-matrices, we keep track of the relative positions of points in space. The general ruling is that for Cartesian space, 3N variables must be accounted for (where N is the number of points in space you wish to index). To describe the locations of two atomic nuclei, a total of 6 variables must be written down and kept track of. You shall get the result from your image.In Cartesian space, three variables (XYZ) are used to describe the position of a point in space, typically an atomic nucleus or a basis function. Write out their relation explicitly, do partial differtiation, contruction the forward transformation matrix, and do the inverse. Do the same thing as I did for cartesian coordinates and polar coordinates. Similarly for the transformation between cartesian coordinates and spherical coordinates. For a simple co-ordinate switch you can just use the relations: This is not the Matrix you're looking for.
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