Z-matrix Examples

[Tinisha]

AZ-matrix is used to define connectivity between atoms in a molecule. The parameters one needs are distances, angles and dihedral angles. We will show a few simple examples of how to make Z-matrices in this text.

Sometimes it is a good idea to think before attempting to write a Z-matrix. What is it you are planning on doing with the molecule? If you are going to do a geometry optimization for the ground state, then it would be a good idea to enforce symmetry. Looking at the benzene example below, one can see that the D6h symmetry will never be broken. When optimizing, only the bond distances have a chance of changing, since the angles are forced to 120 degrees.

However, if one is going to do a transition state search, then the Z-matrix should be as flexible as possible, to allow for any symmetry-breaking geometry changes. Taking the time to plan what one is going to do can save time hunting for why the desired output was not achieved.

For water, all we need is a bond distance and an angle. We start with the first atom, hydrogen, on a line of its own. The next line begins with the second atom, oxygen, and then states with which atom to measure the bond distance OH from, in this case, atom one. On the next line, the third atom, hydrogen, is OH distance away from atom two and has a bond angle of OHO in relation to atom one.

For acetylene, it is necessary to use dummy atoms. This is because the Z-matrix does not accept bond angles equal to 180 degrees. With these dummy atoms, one can define acceptable bond angles. When the input is read into a computer code, these dummy atoms are just used as reference points, and do not enter into any calculation. Judicious use of dummy atoms can simplify a problem hundred-fold. Let us examine benzene as an example of this.

A nave attempt would be to pick a carbon on the ring, and work one's way around, attaching hydrogens as needed. However, one would find it very difficult to get the C6-C1 bond distance to work out to be exact. As one can see here, a much better way is to use dummy atoms. By using three dummy atoms, the input is much easier to write, and a minimum number of variables are required. Notice that it is possible to put numbers directly into the Z-matrix. However, obviously, if one needs to change a value, life is easier using variables rather than having to retype each value.

Here one might think they need dummy atoms, but they are not required. Notice the importance of choosing the dihedrals correctly. If one mistakes a 180 degree dihedral for a zero degree dihedral, then the mistake is hard to detect unless one is looking at the molecule. This is one of the more common mistakes in building a Z-matrix.

In chemistry, the Z-matrix is a way to represent a system built of atoms. A Z-matrix is also known as an internal coordinate representation. It provides a description of each atom in a molecule in terms of its atomic number, bond length, bond angle, and dihedral angle, the so-called internal coordinates,[1][2] although it is not always the case that a Z-matrix will give information regarding bonding since the matrix itself is based on a series of vectors describing atomic orientations in space. However, it is convenient to write a Z-matrix in terms of bond lengths, angles, and dihedrals since this will preserve the actual bonding characteristics. The name arises because the Z-matrix assigns the second atom along the Z axis from the first atom, which is at the origin.

Z-matrices can be converted to Cartesian coordinates and back, as the structural information content is identical, the position and orientation in space, however is not meaning the Cartesian coordinates recovered will be accurate in terms of relative positions of atoms, but will not necessarily be the same as an original set of Cartesian coordinates if you convert Cartesian coordinates to a Z matrix and back again. While the transform is conceptually straightforward, algorithms of doing the conversion vary significantly in speed, numerical precision and parallelism.[1] These matter because macromolecular chains, such as polymers, proteins, and DNA, can have thousands of connected atoms and atoms consecutively distant along the chain that may be close in Cartesian space (and thus small round-off errors can accumulate to large force-field errors.) The optimally fastest and most numerically accurate algorithm for conversion from torsion-space to cartesian-space is the Natural Extension Reference Frame method.[1] Back-conversion from Cartesian to torsion angles is simple trigonometry and has no risk of cumulative errors.

They are used for creating input geometries for molecular systems in many molecular modelling and computational chemistry programs. A skillful choice of internal coordinates can make the interpretation of results straightforward. Also, since Z-matrices can contain molecular connectivity information (but do not always contain this information), quantum chemical calculations such as geometry optimization may be performed faster, because an educated guess is available for an initial Hessian matrix, and more natural internal coordinates are used rather than Cartesian coordinates.The Z-matrix representation is often preferred, because this allows symmetry to be enforced upon the molecule (or parts thereof) by setting certain angles as constant. The Z-matrix simply is a representation for placing atomic positions in a relative way with the obvious convenience that the vectors it uses easily correspond to bonds. A conceptual pitfall is to assume all bonds appear as a line in the Z-matrix which is not true. For example: in ringed molecules like benzene, a z-matrix will not include all six bonds in the ring, because all of the atoms are uniquely positioned after just 5 bonds making the 6th redundant.

This section presents a brief overview of traditional Z-matrix descriptions of molecular systems. There are restrictions on the size of a Z-matrix: the maximum number of variables and the maximum number of atoms within a calculation. These are set consistently for a maximum of 250,000 real atoms (including ghost but not dummy atoms), and a maximum of 250,000 Z-matrix centers (atoms, ghost atoms, and dummy atoms).

Although these examples use commas to separate items within a line, any valid separator may be used. Element-label is a character string consisting of either the chemical symbol for the atom or its atomic number. If the elemental symbol is used, it may be optionally followed by other alphanumeric characters to create an identifying label for that atom. A common practice is to follow the element name with a secondary identifying integer: C1, C2, etc.

The optional format-code parameter specifies the format of the Z-matrix input. For the syntax being described here, this code is always 0. This code is needed only when additional parameters follow the normal Z-matrix specification data, as in an ONIOM calculation.

The first line of the Z-matrix simply specifies a hydrogen. The next line lists an oxygen atom and specifies the internuclear distance between it and the hydrogen as 0.9 Angstroms. The third line defines another oxygen with an O-O distance of 1.4 Angstroms (i.e., from atom 2, the other oxygen) and having an O-O-H angle (with atoms 2 and 1) of 105 degrees. The fourth and final line is the only one for which all three internal coordinates need be given. It defines the other hydrogen as bonded to the second oxygen with an H-O distance of 0.9 Angstroms, an H-O-O angle of 105 degrees and a H-O-O-H dihedral angle of 120 degrees.

Symmetry constraints on the molecule are reflected in the internal coordinates. The two H-O distances are specified by the same variable, as are the two H-O-O bond angles. When such a Z-matrix is used for a geometry optimization in internal coordinates (Opt=Z-matrix), the values of the variables will be optimized to locate the lowest energy structure. For a full optimization (FOpt), the variables are required to be linearly independent and include all degrees of freedom in the molecule. For a partial optimization (POpt), variables in a second section (often labeled Constants:) are held fixed in value while those in the first section are optimized:

An alternative Z-matrix format allows nuclear positions to be specified using two bond angles rather than a bond angle and a dihedral angle. This is indicated by a 1 in an additional field following the second angle (this field defaults to 0, which indicates a dihedral angle as the third component):

This section will illustrate the use of dummy atoms within Z-matrices, which are represented by the pseudo atomic symbol X. The following example illustrates the use of a dummy atom to fix the three-fold axis in C3v ammonia:

The position of the dummy on the axis is irrelevant, and the distance 1.0 used could have been replaced by any other positive number. hnx is the angle between an N-H bond and the threefold axis.

This example illustrates two points. First, a dummy atom is placed at the center of the C-C bond to help constrain the cco triangle to be isosceles. ox is then the perpendicular distance from O to the C-C bond, and the angles oxc are held at 90 degrees. Second, some of the entries in the Z-matrix are represented by the negative of the dihedral angle variable hcco.

The following examples illustrate the use of dummy atoms for specifying linear bonds. Geometry optimizations in internal coordinates are unable to handle bond angles of l80 degrees which occur in linear molecular fragments, such as acetylene or the C4 chain in butatriene. Difficulties may also be encountered in nearly linear situations such as ethynyl groups in asymmetrical molecules. These situations can be avoided by introducing dummy atoms along the angle bisector and using the half-angle as the variable or constant:

Similarly, in this Z-matrix intended for a geometry optimization, half represents half of the N-C-O angle which is expected to be close to linear. Note that a value of half less than 90 degrees corresponds to a cis arrangement:

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