Z-matrix

TROVE uses the standard Z-matrix scheme to define the molecular structure and introduce the basic internal coordinates, which are Z-matrix coordinates in TROVE. For, example, the hydrogen peroxide HOOH coordinate system is defined using the following Z-matrix defined with the ZMAT card:

ZMAT
    O1  0  0  0  0  15.99491463
    O2  1  0  0  0  15.99491463
    H1  1  2  0  0   1.00782505
    H2  2  1  3  2   1.00782505
end

Here, the Zmatrix coordinates are as follows.

Three valence bond length (stretching) coordinates 1,2,3:

$\begin{split} \xi_1 &= {\bf r}_{{\rm O}_1{\rm O}_2} \equiv {\bf r}_{12} \\ \xi_2 &= r_{{\rm O}_1{\rm H}_1} \equiv {\bf r}_{13} \\ \xi_3 &= r_{{\rm O}_2{\rm H}_2}\equiv{\bf r}_{24} \end{split}$

Two valence bond-angle (bending) coordinates 4,5:

$\begin{split} \xi_4 &= \alpha_{{\rm H}_1{\rm O}_1 {\rm O}_2} \equiv \alpha_{312} \\ \xi_5 &= \alpha_{{\rm H}_2{\rm O}_2 {\rm O}_3} \equiv \alpha_{124} \end{split}$

One dihedral coordinate defined as a book angle $\delta_{3124}$ between planes ${\rm H}_1{\rm O}_1 {\rm O}_2$ and {rm H}_2{rm O}_2 {rm O}_3:

$\xi_6 = \delta_{{\rm H}_1{\rm O}_1 {\rm O}_2 {\rm H}_2} = \equiv \delta_{3124},$

which is defined as follows

$\delta_{ijkl} = \arccos\left(\frac{[\vec{r}_{ij} \times \vec{r}_{jk}]\cdot[\vec{r}_{jk} \times \vec{r}_{kl}] }{|[\vec{r}_{ij} \times \vec{r}_{jk}]| | [\vec{r}_{jk} \times \vec{r}_{kl}]|}\right).$

Structure of the Zmatrix block

Let us consider the HOOH example above to describe the columns in the ZMAT block:

col	1	2	3	4	5	6
row	atom	connector 1	connector 2	connector 3	Type	Mass
1	O1	0	0	0	0	15.99491463
2	O2	1	0	0	0	15.99491463
3	H1	1	2	0	0	1.00782505
4	H2	2	1	3	2	1.00782505

Here:

Col 1 (atom): A label for the name of the atom; it is not interpreted by the program and is currently only for clarity.
Col 2 (connector 1): The 1st connecting atom to form a molecular bond.
Col 3 (connector 2): The 2nd connecting atom to form a bond angle.
Col 4 (connector 1): The 3rd connecting atom to form a dihedral angle or other type angle, depending on the value in column Type.
Col 5 (Type): A “Dihedral” angle type (see below).
Col 6 (Mass): The mass of the particle; usually an atomic but sometimes a nuclear mass.

“Dihedral” types

The following “Dihedral” types are available:

Type 0: the “Dihedral” angle is defined as a valence angle between bonds $\vec{r_{2}}$ and $\vec{r_{3}}$ as shown in the figure.

XY3 Zmat — Z-matrix coordinates with a dihedral angle of type 0 used for rigid molecules like PH₃.

Type 1: it is defined as a usual dihedral angle between two planes (4-1-2 and 3-1-2), with important addition. One more bond angle $\alpha_3$ is introduced (see the figure and also Ammonia example in Molecules).

Type 2: it is the standard dihedral angle as in the HOOH example above.

H2O2 Zmat — Z-matrix coordinates with a dihedral angle of type 2 used for HOOH as in the example above.

Type 202: it is the same dihedral angle $\delta$ as type 2, with the difference that the 1st derivative of $\delta$ wrt the Cartesian coordinates (required for the KEO construction) are evaluated using the finite differences, while for types 0, 1, 2, -2, 3-100 the 1st derivatives are evaluates using an analytic expression. Finite difference offer more stable evaluation at positions where the phases of the angles change (0, $\pi$ , $2\pi$ ) which makes their definition ambiguous.
Type -2: it is the standard dihedral angle, but with the “backbone” vector inverted, see figure.

A differen example of type 2 of dihedrals is for H₂CO with the Z-matrix given by

ZMAT
    C   0  0  0  0  12.00000000
    O   1  0  0  0  15.99491463
    H   1  2  0  0   1.00782505
    H   1  2  3 -2   1.00782505
end

H2CO Zmat — Z-matrix coordinates with a “dihedral” angle of type -2 used for H₂CO with the “backbone” vector inverted.

Here is a Z-matrix used for C₂H₄ with a mixture of types 2 and -2:

ZMAT
  C   0  0  0  0  12.00000000
  C   1  0  0  0  12.00000000
  H   1  2  0  0   1.00782505
  H   1  2  3 -2   1.00782505
  H   2  1  3  2   1.00782505
  H   2  1  5 -2   1.00782505
end

The system was studied in [18MaYaTe].

Type -202: same as type, but the 1st derivative of $\delta$ wrt the Cartesian coordinates evaluated using the finite differences.
Type 402: it is the same as type 202, but with $\delta$ defined in the extended range from 0 to 720. This type is useful for the systems with the extended molecular symmetries, such as non-rigid HOOH [15AlOvYu] or C₂H₆ [19MeYuMa].
Type -402: it is the same as type -202, but with $\delta$ defined in the extended range from 0 to 720.

Type 3-100: any number $N$ between 3 and 100 means that instead of a dihedral angle, another bond angle is introduced; the “type” card is treated as an additional connector in order to define the bond angle. As example, the following Z-matrix (see the figure) uses a dihedral angle free definition of the vibrational coordinates of CH₄:

ZMAT
    C   0  0  0  0  12.000000000
    H   1  0  0  0   1.007825035
    H   1  2  0  0   1.007825035
    H   1  2  3  0   1.007825035
    H   1  2  3  4   1.007825035
end