 Faculty for Chemistry and Pharmacy - Group of Prof. Zipse

# Redundant Internal Coordinates

Geometry optimization in redundant internal coordinates is achieved using the

opt=Redundant or simply opt keyword

Definition of a set of internally consistent, linearly maximally independent internal coordinates is performed automatically by Gaussian based on the structure supplied in the input file (both Z-Matrix or cartesian coordinates can be used at that point). Taking as an example the input file for formaldehyde used before to illustrate the different input formats, the following redundant coordinates will be generated by Gaussian:

```                         ----------------------------
!    Initial Parameters    !
! (Angstroms and Degrees)  !
------------------------                            -------------------------
! Name  Definition              Value          Derivative Info.             !
-----------------------------------------------------------------------------
! R1    R(1,2)                  1.2            estimate D2E/DX2             !
! R2    R(1,3)                  1.             estimate D2E/DX2             !
! R3    R(1,4)                  1.             estimate D2E/DX2             !
! A1    A(2,1,3)              120.             estimate D2E/DX2             !
! A2    A(2,1,4)              120.             estimate D2E/DX2             !
! A3    A(3,1,4)              120.             estimate D2E/DX2             !
! A4    L(3,1,4,2,-2)         180.             estimate D2E/DX2             !
-----------------------------------------------------------------------------
```

The choice made by Gaussian can, to a large degree, be influenced by the user through the definition of additional internal coordinates that should be part of the overall set of coordinates. This can be achieved using the

keyword. The input file must then contain at least one additional line of input following the regular geometry definition. If, in the above example, we want the distance between hydrogen atoms H3 and H4 to be part of the coordinate set, we could specify this in the following way:

 ```#P HF/6-31G(d) opt=AddRed test1 HF/6-31G(d) opt formaldehyde 0 1 C1 O2 1 r2 H3 1 r3 2 a3 H4 1 r3 2 a3 3 180.0 r2=1.20 r3=1.0 a3=120. 3 4 ``` The line added after definition of bond angle a3, separated by one blank line, specifies that the distance between atoms 3 and 4 is to be included in the list of redundant variables. The additional consideration of bond angles is achieved in a similar way through listing the three atoms forming the bond angle, while definition of additional dihedral angles requires the four atoms defining this dihedral angle.

Similarly, the input structure can be modified through giving the additional redundant coordinate a value different from what it was in the original input file. As an example, we might specify that, in the previous example, the two hydrogen atoms are positioned only 1.6 Angstroms apart (as compared to 1.7321 Angstroms in the original input):

```#P HF/6-31G(d) opt=AddRed

test1 HF/6-31G(d) opt formaldehyde

0  1
C1
O2  1  r2
H3  1  r3  2  a3
H4  1  r3  2  a3  3  180.0

r2=1.20
r3=1.0
a3=120.

3 4 1.60

```

Given the additional constraint of a distance of 1.60 Angstrom between H3 and H4, Gaussian adjusts the values of other redundant coordinates such that the specified constraint is satisfied at the beginning of the optimization. This distance will, of course, be varied during the geometry optimization procedure.
While the advantage of this feature is not immediately obvious for small systems, it is extremely helpful to specify the relative positions of two centers in a large system which are not directly connected through a bond distance or bond angle.