## ZMATRIX: Z-matrix input The `ZMATRIX` directive is an optional directive that can be used within the compound `GEOMETRY` directive to specify the structure of the system with a Z-matrix, which can include both internal and Cartesian coordinates. The `ZMATRIX` directive is itself a compound directive that can include the `VARIABLES` and `CONSTANTS` directives, depending on the options selected. The general form of the compound `ZMATRIX` directive is as follows: ``` [ZMATRIX || ZMT || ZMAT ... [VARIABLES ... ] [CONSTANTS ... ] (END || ZEND)] ``` The input module recognizes three possible spellings of this directive name. It can be invoked with `ZMATRIX`, `ZMT`, or `ZMAT`. The user can specify the molecular structure using either Cartesian coordinates or internal coordinates (bond lengths, bond angles and dihedral angles. The Z-matrix input for a center defines connectivity, bond length, and bond or torsion angles. Cartesian coordinate input for a center consists of three real numbers defining the x,y,z coordinates of the atom. Within the Z-matrix input, bond lengths and Cartesian coordinates must be input in the user-specified units, as defined by the value specified for the variable `units` on the first line of the `GEOMETRY` directive. All angles are specified in degrees. The individual centers (denoted as i, j, and k below) used to specify Z-matrix connectivity may be designated either as integers (identifying each center by number) or as tags (If tags are used, the tag must be unique for each center.) The use of dummy atoms is possible, by using X or BQ at the start of the tag. Bond lengths, bond angles and dihedral angles (denoted below as R, alpha, and beta, respectively) may be specified either as numerical values or as symbolic strings that must be subsequently defined using the `VARIABLES` or `CONSTANTS` directives. The numerical values of the symbolic strings labeled `VARIABLES` may be subject to changes during a geometry optimization say, while the numerical values of the symbolic strings labeled `CONSTANTS` will stay frozen to the value given in the input. The same symbolic string can be used more than once, and any mixture of numeric data and symbols is acceptable. Bond angles (*α*) must be in the range 0 < *α* < 180. The Z-matrix input is specified sequentially as follows: ``` tag1 tag2 i R tag3 i R j alpha tag4 i R j alpha k beta [orient] ... ``` The structure of this input is described in more detail below. In the following discussion, the tag or number of the center being currently defined is labeled as *C* (*C* for current). The values entered for these tags for centers defined in the Z-matrix input are interpreted in the same way as the `tag` entries for Cartesian coordinates described above (see [Cartesian coordinate input](Cartesian-coordinate-input.md)). Figures 1, 2 and 3 display the relationships between the input data and the definitions of centers and angles. ![Zmat1.jpg](Zmat1.jpg) Figure 1: Relationships between the centers, bond angle and dihedral angle in Z-matrix input. ![Zmat2.jpg](Zmat2.jpg) Figure 2: Relationships between the centers and two bond angles in Z-matrix input with optional parameter specified as +1. ![Zmat3.jpg](Zmat3.jpg) Figure 3: Relationships between the centers and two bond angles in Z-matrix input with optional parameter specified as -1. The Z-matrix input shown above is interpreted as follows: 1. `tag1` Only a tag is required for the first center. 2. `tag2 i R` The second center requires specification of its tag and the bond length (*R_Ci*) distance to a previous atom, which is identified by *i*. 3. `tag3 i R j alpha` The third center requires specification of its tag, its bond length distance (*R_Ci*) to one of the two previous centers (identified by the value of *i*), and the bond angle *α = ∠C i j*. 4. `tag i R j alpha k beta []` The fourth, and all subsequent centers, require the tag, a bond length (*R_Ci*) relative to center *i*, the bond angle with centers *i* and *j* (*α = ∠C i j*), and either the dihedral angle (*β*) between the current center and centers *i*, *j*, and *k* (Figure 1), or a second bond angle *β = ∠ C i k* and an orientation to the plane containing the other three centers (Figure 2 and 3). By default, *β* is interpreted as a dihedral angle (see Figure 1), but if the optional final parameter (`orient`) is specified with the value ±1, then *β* is interpreted as the angle *∠ C i k*. The sign of `orient` specifies the direction of the bond angle relative to the plane containing the three reference atoms. If `orient` is +1, then the new center (*C*) is above the plane (Figure 2); and if `orient` is -1, then *C* is below the plane (Figure 3). Following the Z-matrix center definitions described above, the user can specify initial values for any symbolic variables used to define the Z-matrix tags. This is done using the optional `VARIABLES` directive, which has the general form: ``` VARIABLES ... ``` Each line contains the name of a variable followed by its value. Optionally, an equals sign (=) can be included between the symbol and its value, for clarity in reading the input file. Following the `VARIABLES` directive, the `CONSTANTS` directive may be used to define any Z-matrix symbolic variables that remain unchanged during geometry optimizations. To freeze the Cartesian coordinates of an atom, refer to the section [Applying constraints in geometry optimizations](ZCOORD-Forcing-internal-coordinates.md#applying_constraints-in-geometry-optimizations). The general form of this directive is as follows: ``` CONSTANTS ... ``` Each line contains the name of a variable followed by its value. As with the `VARIABLES` directive, an equals sign (=) can be included between the symbol and its value. The end of the Z-matrix input using the compound `ZMATRIX` directive is signaled by a line containing either `END` or `ZEND`, following all input for the directive itself and its associated optional directives. A simple example is presented for water. All Z-matrix parameters are specified numerically, and symbolic tags are used to specify connectivity information. This requires that all tags be unique, and therefore different tags are used for the two hydrogen atoms, which may or may not be identical. ``` geometry zmatrix O H1 O 0.95 H2 O 0.95 H1 108.0 end end ``` The following example illustrates the Z-matrix input for the molecule CH₃CF₃. This input uses the numbers of centers to specify the connectivity information (i, j, and k), and uses symbolic variables for the Z-matrix parameters R, alpha, and beta, which are defined in the inputs for the `VARIABLES` and `CONSTANTS` directives. ``` geometry zmatrix C C 1 CC H 1 CH1 2 HCH1 H 1 CH2 2 HCH2 3 TOR1 H 1 CH3 2 HCH3 3 -TOR2 F 2 CF1 1 CCF1 3 TOR3 F 2 CF2 1 CCF2 6 FCH1 F 2 CF3 1 CCF3 6 -FCH1 variables CC 1.4888 CH1 1.0790 CH2 1.0789 CH3 1.0789 CF1 1.3667 CF2 1.3669 CF3 1.3669 constants HCH1 104.28 HCH2 104.74 HCH3 104.7 CCF1 112.0713 CCF2 112.0341 CCF3 112.0340 TOR1 109.3996 TOR2 109.3997 TOR3 180.0000 FCH1 106.7846 end end ``` The input for any centers specified with Cartesian coordinates must be specified using the format of the `tag` lines described in [Cartesian coordinate input](Cartesian-coordinate-input.md) above. However, in order to correctly specify these Cartesian coordinates within the Z-matrix, the user must understand the orientation of centers specified using internal coordinates. These are arranged as follows: - The first center is placed at the origin. - The third center is placed in the z-x plane.