Z matrix

An example for the representation of ethane as a Z-matrix is shown in Table 14.7. The representation in the z-matrix is as follows  

For the first atom, note only the atomic symbol. 

For the second atom, note the atomic symbol, the number 1, and a variable for 

For the third and all further atoms, note the atomic symbol, the number N, a variable to describe the distance between the current atom and NA, the atom number NB, and a variable to describe angle between the current atom, and two 

Finally resolve each variable (distances and angles) with its corresponding value. 

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Figure B3.5.2. Example Z matrix for fliioroethylene. Notation for example, line 4 of the Z matrix means that a H atom is bonded to earbon atom Cl with bond length L3 (angstroms), making an angle with earbon atom...

Table B3.5.1 Number of cycles to converge for geometry optimizations of some typical organic molecules usmg Cartesian, Z-matrix and delocalized internal coordinates. ...

Z matrix generated using Cartesian —> Z-matrix conversion program. Severe converge problems with energy... 

Constrained optimization refers to optimizations in which one or more variables (usually some internal parameter such as a bond distance or angle) are kept fixed. The best way to deal with constraints is by elimination, i.e., simply remove the constrained variable from the optimization space. Internal constraints have typically been handled in quantum chemistry by using Z matrices if a Z matrix can be constructed which contains all the desired constraints as individual Z-matrix variables, then it is straightforward to carry out a constrained optimization by elunination. 

By combining the Lagrange multiplier method with the highly efficient delocalized internal coordinates, a very powerfiil algoritlun for constrained optimization has been developed [ ]. Given that delocalized internal coordinates are potentially linear combinations of all possible primitive stretches, bends and torsions in the system, cf Z-matrix coordinates which are individual primitives, it would seem very difficult to impose any constraints at all however, as... 

Baker J and Chan F 1996 The location of transition states a comparison of Cartesian, Z-matrix, and natural internal coordinates J. Comput. Chem. 17 888... 

Baker J and Hehre W J 1991 Geometry optimization In Cartesian coordinates The end of the Z-matrIx J. Comput. Chem. 12 606... 

The most common way to describe a molecule by its internal coordinates is the so-called Z-matrix. Figure 2-92 shows the Z-matrix of 1,2-dichloroethane. 

A set of rules determines how to set up a Z-matrix properly, Each line in the Z-matiix represents one atom of the molecule. In the first line, atom 1 is defined as Cl, which is a carbon atom and lies at the origin of the coordinate system. The second atom, C2, is at a distance of 1.5 A (second column) from atom 1 (third column) and should always be placed on one of the main axes (the x-axis in Figure 2-92). The third atom, the chlorine atom C13, has to lie in the xy-planc it is at a distanc e of 1.7 A from atom 1, and the angle a between the atoms 3-1-2 is 109 (fourth and fifth columns). The third type of internal coordinate, the torsion angle or dihedral r, is introduced in the fourth line of the Z-matiix in the sixth and seventh column. It is the angle between the planes which arc... 

Z-matriccs arc commonly used as input to quantum mechanical ab initio and serai-empirical) calculations as they properly describe the spatial arrangement of the atoms of a molecule. Note that there is no explicit information on the connectivity present in the Z-matrix, as there is, c.g., in a connection table, but quantum mechanics derives the bonding and non-bonding intramolecular interactions from the molecular electronic wavefunction, starting from atomic wavefiinctions and a crude 3D structure. In contrast to that, most of the molecular mechanics packages require the initial molecular geometry as 3D Cartesian coordinates plus the connection table, as they have to assign appropriate force constants and potentials to each atom and each bond in order to relax and optimi-/e the molecular structure. Furthermore, Cartesian coordinates are preferable to internal coordinates if the spatial situations of ensembles of different molecules have to be compared. Of course, both representations are interconvertible. 

Tabic 2-6 gives an overview on the most common file formats for chemical structure information and their respective possibilities of representing or coding the constitution, the configuration, i.c., the stereochemistry, and the 3D structure or conformation (see also Sections 2..3 and 2.4). Except for the Z-matrix, all the other file formats in Table 2-6 which are able to code 3D structure information arc using Cartesian coordinates to represent a compound in 3D space. 

The traditional way to provide the nuclear coordinates to a quantum mechanical program is via a Z-matrix, in which the positions of the nuclei are defined in terms of a set of intei ii.il coordinates (see Section 1.2). Some programs also accept coordinates in Cartesian formal, which can be more convenient for large systems. It can sometimes be important to choow an appropriate set of internal coordinates, especially when locating rninima or transitinn points or when following reaction pathways. This is discussed in more detail in Section 5.7. 

The ring closure bond between atoms 1 and 5 would be strongly coupled to the other internal coordinates inless dummy atoms are used to define the Z-matrix (right). 

One valid form of the input file is the z-matrix form usually associated with GAUSSIAN calculations... 

FIGURE 8.1 Z-matrix for ethane. The first column is the element, the second column the atom to which the length refers, the third column the length, the fourth column the atom to which the angle refers, the fifth column the angle, the sixth column the atom to which the confonnation angle refers, and the seventh column the confonnation angle. 

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. 

There is no one best way to specify geometry. Usually, a Z-matrix is best for specifying symmetry constraints if properly constructed. Cartesian coordinate input is becoming more prevalent due to its ease of generation by graphical user interface programs. 

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. 

FIGURE 9.1 Illustration of the formaldehyde Z-matrix example, (a) First three atoms and associated variables. (A) Dihedral angle. 

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