ZMATRIX Z matrix input

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 
        <string tagn> <list_of_zmatrix_variables>  
        ...  
        [VARIABLES 
             <string symbol> <real value>  
             ... ]  
        [CONSTANTS  
             <string symbol> <real value>  
             ... ]  
   (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). Figures 1, 2 and 3 display the relationships between the input data and the definitions of centers and angles.

Figure 1: Relationships between the centers, bond angle and dihedral angle in Z-matrix input.

Figure 2: Relationships between the centers and two bond angles in Z-matrix input with optional parameter specified as +1.

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:

tag1
Only a tag is required for the first center.
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.
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 \(\alpha = \widehat{Cij}\).
tag i R j alpha k beta [<integer orient default 0>]
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 ( \(\alpha = \widehat{Cij}\)), and either the dihedral angle (β) between the current center and centers i, j, and k (Figure 1), or a second bond angle \(\beta = \widehat{Cik}\) 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 \(\widehat{Cik}\). 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 
   <string symbol>  <real value>  
   ...

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 Applying constraints in geometry optimizations. The general form of this directive is as follows:

 CONSTANTS  
   <string symbol>  <real value>  
   ...

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 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.