Internal Coordinates

Geometry optimization in internal coordinates is achieved using the

opt=Z-Matrix

keyword. In this case the coordinate system used in the input file is also that used for geometry optimization. The success of the geometry optimization (number of optimization cycles, final result of the optimization) therefore depends critically on the coordinates used in the input file. The optimization in internal coordinates (as given in Z-matrix format) is particularly useful in the following three cases:


1) Highly symmetric systems
The benefits of using internal coordinates for highly symmetric systems can nicely be demonstrated using ammonia NH₃ as an example. Ammonia is a slightly pyramidalized amine of C_3v symmetry. A first attempt for a Z-matrix would be as follows:
 

N1 H2 1 r2 H3 1 r3 2 a3 H4 1 r4 2 a4 3 d4 r2=1.0 r3=1.0 r4=1.0 a3=110. a4=110. d4=120.

The three N-H bonds have been defined using different variables for bond distances and bond angles, but the initial values have been chosen such that all three N-H bonds are of equal length, implying a certain degree of symmetry. When this input is used, Gaussian only recognizes C_s symmetry since the three H-N-H angles are not identical. This type of geometry definition also holds the danger of leading to changes in the point group during geometry optimizations. The latter are indicated by the program as:

Full point group                 C1      NOp   1
Omega: Change in point group or standard orientation.

This error typically occurs when optimization starts at a highly symmetric structure, which looses some of its symmetry properties on optimization due to an incorrect definition of Z-matrix variables. In the rare case that the symmetry of the system should not be used, one can actually force the program to ignore symmetry completely using the

NoSymm

keyword. In most cases, however, the available symmetry properties will be used to reduce the computational effort (in part considerably). Returning to ammonia as an example, a better set of internal coordinates is:
 

N1 H2 1 r2 H3 1 r2 2 a3 H4 1 r2 2 a3 3 d4 r2=1.0 a3=110. d4=120.

reducing the number of variables from six to three. Using this input, Gaussian still only recognizes C_s symmetry, but smoothly optimizes the structure of ammonia in five steps to tight conversion criteria. Optimal use of the symmetry properties of ammonia can be made using one dummy atom designated with the symbol "X" located on the principal C₃ axis:

X1 N2 1 1.0 H3 2 r3 1 a3 H4 2 r3 1 a3 3 +120. H5 2 r3 1 a3 3 -120. r3=1.0 a3=110.

In this case the number of variables can be reduced to two since the definition of the relative position of the hydrogen atoms uses the intrisic properties of the threefold principal axis. The dihedral angles of 120.0 degrees must remain constant during the geometry optimization and these values are therefore not defined through variables, but as numerical constants in the Z-matrix. Using the above Z-matrix, the full C_3v symmetry of ammonia is recognized by Gaussian, but optimization is consequently performed in the C_s point group due to the lack of support for threefold symmetry axes. At the HF/6-31G(d) level of theory, the geometry is optimized to tight convergence criteria in five cycles with optimized parameters of r3=1.002522 and a3=111.6755, the HF/6-31G(d) total energy for this structure being -56.1843563au.

One complication sometimes arising from the use of dummy atoms is that one looses sight of the actual number of degrees of freedom. A nonlinear, asymmetric molecular system containing N atoms has 3N-6 degrees of freedom (=independent structural parameters). For linear, asymmetric systems this number is increased to 3N-5. Using ammonia again as an example, we should have 3*4-6 = 6 independent structural parameters if symmetry is not accounted for. We have seen above that this number can be reduced to 2 on full consideration of its C_3v symmetry. How many degrees of freedom remain after dummy atoms have been used in Z-matrix construction is, however, rather difficult to see in some cases. One can therefore instruct Gaussian to check whether the structural variables used in a given Z-matrix are all linearly independent (that is, describe different structural degrees of freedom) and overall don't exceed the correct number in a given point group symmetry. This is done by using

FOpt instead of Opt

as the keyword. One example, that demonstrates a problematic situation, is the following:

X1 X2 1 1.0 N3 2 1.0 1 90. H4 3 r4 2 a4 1 d4 H5 3 r4 2 a4 1 +120. H6 3 r4 2 a4 1 -120. r4=1.0 a4=110. d4=0.0

Here the position of dummy atom X1 is indirectly optimized through its use for the definition of hydrogen atom H4. Running this input with the FOpt keyword gives the following error:

FOPT requested but NVar=  3 while NDOF=  2.

In most cases, the optimization of the position of dummy atoms leads to larger numbers of geometry optimization cycles and in some cases to incomplete optimizations.


Systems containing methyl substituents or similar substructures benefit from an "internal" definition of these groups in the sense that the positions of the three hydrogen atoms are defined relative to each other (rather than to an external reference). This can be demonstrated using acetic acid as an example:
 

0 1 C1 C2 1 r2 O3 1 r3 2 a3 O4 1 r4 2 a4 3 180.0 H5 4 r5 1 a5 3 0.0 H6 2 r6 1 a6 3 0.0 H7 2 r7 1 a7 6 d7 H8 2 r7 1 a7 6 -d7 r2=1.51 r3=1.21 r4=1.36 r5=0.98 r6=1.09 r7=1.09 a3=126. a4=111. a5=105. a6=109. a7=109. d7=-120.

The position of hydrogen atom H6 is here defined with reference to the atoms C2/C1/O3, while the dihedral angles for the other two hydrogen atoms H7 and H8 are defined relative to H6 (and not O3). The advantage of this type of definition becomes obvious when we want to explore another conformation of the methyl group with respect to the remainder of the molecule in order to identify the energetically most favorable conformation of acetic acid. While the above conformation eclipses one of the C-H bonds with the C-O double bond, the alternative conformation below assumes an antiperiplanar arrangement.

0 1 C1 C2 1 r2 O3 1 r3 2 a3 O4 1 r4 2 a4 3 180.0 H5 4 r5 1 a5 3 0.0 H6 2 r6 1 a6 3 180.0 H7 2 r7 1 a7 6 d7 H8 2 r7 1 a7 6 -d7 r2=1.51 r3=1.21 r4=1.36 r5=0.98 r6=1.09 r7=1.09 a3=126. a4=111. a5=105. a6=109. a7=109. d7=-120.

A comparison of the Z-matrices for these two conformations reveals that the only difference in the starting conformation is in the dihedral angle given for H6 (0.0 in the first and 180.0 in the second conformation). Through their definition relative to hydrogen H6 the positions of H7 and H8 change accordingly. Both starting conformations optimize smoothly in Z-matrix coordinates at the B3LYP/6-31G(d) level. At this level of theory the first conformation is 1.5 kJ/mol more favorable than the second. Frequency calculations indicate that the first is a true minimum on the potential energy surface, while the second corresponds to a transition state for methyl group rotation.

2) Systems containing rings
The construction of ring systems is a particular challenge due to the fact that one of the ring bonds cannot be specified in a Z-matrix. We will use oxirane (C₂H₄O, ethylene oxide) here as an example to demonstrate the situation. Oxirane is C_2v symmetric and a Z-matrix reflecting this symmetry is as follows:

O1 C2 1 r2 C3 1 r2 2 a3 H4 2 r4 1 a4 3 d4 H5 2 r4 1 a4 3 -d4 H6 3 r4 1 a4 2 -d4 H7 3 r4 1 a4 2 d4 r2=1.25 r4=1.10 a3=73.76 a4=101.84 d4=-106.234

With this definition, the distance between C2 and C3 is not defined explicitly as a variable, but indirectly through variable a3 describing the bond angle C3-O1-C2. Small variations in this angle will, unfortunately, lead to quite large changes in the C2-C3 distance. As a consequence, convergence of the optimization is less efficient and takes 8 steps to complete. Alternatively, the geometry can also be defined through inclusion of one dummy atom located on the principal C₂ axis:

O1 X2 1 r2 C3 2 r3 1 90. C4 2 r3 1 90. 3 180. H5 3 r5 2 a5 1 d5 H6 3 r5 2 a5 1 -d5 H7 4 r5 2 a5 1 -d5 H8 4 r5 2 a5 1 d5 r2=1.0 r3=0.75 r5=1.10 a5=110. d5=90.

In this Z-matrix all bond distances inside the ring are defined through a distance variable, even though these variables do clearly not describe what is typically considered a "bond distance". Optimization is more efficient and is complete within 7 steps. The difference between these two approaches increases with system size and complexity.

The same general strategy can be used to define Z-matrices for six-membered ring systems. The C_2v symmetric structure of pyridine, for example, can most readily be constructed by first describing the C3-C5 fragment and subsequently positioning the remaining part of the ring system with the help of one dummy atom:

H1 C2 1 r2 C3 2 r3 1 a3 C4 2 r3 1 a3 3 180.0 H5 3 r5 2 a5 1 0.0 H6 4 r5 2 a5 1 0.0 X7 2 1.0 1 90.0 3 -90.0 N8 2 r8 7 90.0 1 180.0 C9 8 r9 2 a9 3 0.0 C10 8 r9 2 a9 4 0.0 H11 9 r11 8 a11 10 180.0 H12 10 r11 8 a11 9 180.0 r2=1.1 r3=1.40 r5=1.1 r8=2.8 r9=1.35 r11=1.1 a3=120. a5=120. a9=60. a11=120.

The parameter r8 in this Z-matrix describing the distance between ring atoms C2 and N8 effectively defines the distance between ring atoms C3/C9 and C4/C10. These latter bond distances are otherwise difficult to describe while retaining full C_2v symmetry.

3) Linear Systems
A suitable example for a molecule containing a linear arrangement of atoms is acetonitrile (CH₃CN), one of the more common solvents used in organic and biological chemistry. This system belongs to the C_3v point group and contains a linear C-C-N arrangement. A first attempt at a Z-matrix is:

C1 C2 1 r2 H3 1 r3 2 a3 H4 1 r3 2 a3 3 120. H5 1 r3 2 a3 3 -120. N6 2 r6 1 180.0 3 180.0 r2=1.45 r3=1.1 r6=1.2 a3=110.

The use of this Z-matrix leads, however, to the following error already during the first step of the geometry optimization:

Error on Z-matrix card number    6
angle Alpha is outside the valid range of 0 to 180.
Conversion from Z-matrix to cartesian coordinates failed:

What is located on "card 6" (=line 6 of the Z-matrix) is the definition of the position of atom 6. At the heart of the problem is the definition of dihedral angles in general. By definition, dihedral angle D(1/2/3/4) is the angle between two planes, the first being described by atoms 1/2/3 and the second one described by atoms 2/3/4. As soon as atoms 1/2/3 or 2/3/4 fall on one line, the definition of one of the planes has vanished and the dihedral angle is undefined. One could attempt to "solve" the situation by bending the nitrile group a little bit out of linearity, introducing one additional variable:

C1
C2  1  r2
H3  1  r3  2  a3
H4  1  r3  2  a3  3  120.
H5  1  r3  2  a3  3  -120.
N6  2  r6  1  a6  3  180.0

r2=1.45
r3=1.1
r6=1.2
a3=110.
a6=177.

Since support for the principal C₃ axis is lacking anyway, one could as well start with a C_s symmetric starting structure, but introduce threefold symmetry through the backdoor using equal C-H bond lengths and H-C-C bond angles. This time the program runs for a few optimization cycles until the C-C-N bond angle (variable a6) approaches very closely to 180.0 degrees. Gaussian then stops either with the error message cited above or (more appropriately) with:

This structure is nearly, but not quite of a higher symmetry.
Consider Symm=Loose if the higher symmetry is desired.
Stoichiometry    C2H3N
Framework group  C1[X(C2H3N)]
Deg. of freedom   12
Full point group                 C1      NOp   1
Omega: Change in point group or standard orientation.

It thus appears that this strategy doesn't represent a good solution to the problem of large bond angles either. Dividing the critical bond angle in two parts through introduction of a dummy atom, however, solves the problem:

C1 C2 1 r2 H3 1 r3 2 a3 H4 1 r3 2 a3 3 120.0 H5 1 r3 2 a3 3 -120.0 X6 2 1.0 1 90.0 3 0.0 N7 2 r7 6 90.0 1 180.0 r2=1.45 r3=1.1 r7=1.2 a3=110.

In this case the optimization is smoothly performed to tight convergence criteria within 5 steps at the HF/6-31G(d) level of theory with final values of r2=1.467835, r3=1.082125, r7=1.134723, a3=109.8344 and a final total energy of -131.9275339au. Similar situations exist in many other compounds containing C-C or C-N triple bonds as well as transition metal complexes containing CO ligands.

In conclusion, optimization in internal coordinates is particularly helpful for highly symmetric systems, for which the Z-matrices can be constructed easily through the appropriate use of dummy atoms. Bond distances should always be positive, bond angles between 0.0 and 180.0 degrees, and dihedral angles between 0.0 and 360.0 degrees.