Q-Chem 4.3 User’s Manual

9.2 Constrained Optimization

9.2.1 Introduction

Constrained optimization refers to the optimization of molecular structures (transition or equilibrium) in which certain parameters (e.g., bond lengths, bond angles or dihedral angles) are fixed. Jon Baker’s Optimize package implemented in the Q-Chem program has been modified to handle constraints directly in delocalized internal coordinates using the method of Lagrange multipliers (see Appendix A). Constraints are imposed in an $opt keyword section of the input file.

Features of constrained optimizations in Q-Chem are:

Note: The $opt input section is case-insensitive and free-format, except that there should be no space at the start of each line.

9.2.2 Geometry Optimization with General Constraints

CONSTRAINT and ENDCONSTRAINT define the beginning and end, respectively, of the constraint section of $opt within which users may specify up to six different types of constraints:

interatomic distances
Values in angstroms; value $> 0$:
stre atom1 atom2 value

angles
Values in degrees, $0 \le value \le 180$; atom2 is the middle atom of the bend:
bend atom1 atom2 atom3 value

out-of-plane-bends
Values in degrees, $-180 \le value \le 180 $ atom2; angle between atom4 and the atom1–atom2–atom3 plane:
outp atom1 atom2 atom3 atom4 value

dihedral angles
Values in degrees, $-180 \le value \le 180$; angle the plane atom1–atom2–atom3 makes with the plane atom2–atom3–atom4:
tors atom1 atom2 atom3 atom4 value

coplanar bends
Values in degrees, $-180 \le value \le 180$; bending of atom1–atom2–atom3 in the plane atom2–atom3–atom4:
linc atom1 atom2 atom3 atom4 value

perpendicular bends
Values in degrees, $-180 \le value \le 180$; bending of atom1–atom2–atom3 perpendicular to the plane atom2–atom3–atom4:
linp atom1 atom2 atom3 atom4 value

9.2.3 Frozen Atoms

Absolute atom positions can be frozen with the FIXED section. The section starts with the FIXED keyword as the first line and ends with the ENDFIXED keyword on the last. The format to fix a coordinate or coordinates of an atom is:

atom coordinate_reference

coordinate_reference can be any combination of up to three characters $X$, $Y$ and $Z$ to specify the coordinate(s) to be fixed: $X$, $Y$, $Z$, XY, XZ, YZ, XYZ. The fixing characters must be next to each other. e.g., 

FIXED
2 XY
ENDFIXED

means the $x$-coordinate and $y$-coordinate of atom 2 are fixed, whereas 

FIXED
2 X Y
ENDFIXED

will yield erroneous results.

Note: When the FIXED section is specified within $opt, the optimization coordinates will be Cartesian.

9.2.4 Dummy Atoms

DUMMY defines the beginning of the dummy atom section and ENDDUMMY its conclusion. Dummy atoms are used to help define constraints during constrained optimizations in Cartesian coordinates. They cannot be used with delocalized internals.

All dummy atoms are defined with reference to a list of real atoms, that is, dummy atom coordinates are generated from the coordinates of the real atoms from the dummy atoms defining list (see below). There are three types of dummy atom:

  1. Positioned at the arithmetic mean of up to seven real atoms in the defining list.

  2. Positioned a unit distance along the normal to a plane defined by three atoms, centered on the middle atom of the three.

  3. Positioned a unit distance along the bisector of a given angle.

The format for declaring dummy atoms is: 

DUMMY
idum    type   list_length    defining_list
ENDDUMMY

idum

Center number of defining atom (must be one greater than the total number of real atoms for the first dummy atom, two greater for second etc.).

type

Type of dummy atom (either 1, 2 or 3; see above).

list_length

Number of atoms in the defining list.

defining_list

List of up to seven atoms defining the position of the dummy atom.

Once defined, dummy atoms can be used to define standard internal (distance, angle) constraints as per the constraints section, above.

Note:  The use of dummy atoms of type 1 has never progressed beyond the experimental stage.

9.2.5 Dummy Atom Placement in Dihedral Constraints

Bond and dihedral angles cannot be constrained in Cartesian optimizations to exactly $0^{\circ }$ or $\pm 180^{\circ }$. This is because the corresponding constraint normals are zero vectors. Also, dihedral constraints near these two limiting values (within, say $20^{\circ }$) tend to oscillate and are difficult to converge.

These difficulties can be overcome by defining dummy atoms and redefining the constraints with respect to the dummy atoms. For example, a dihedral constraint of $180^{\circ }$ can be redefined to two constraints of $90^{\circ }$ with respect to a suitably positioned dummy atom. The same thing can be done with a $180^{\circ }$ bond angle (long a familiar use in Z-matrix construction).

Typical usage is as shown in Table 9.2.5. Note that the order of atoms is important to obtain the correct signature on the dihedral angles. For a $0^{\circ }$ dihedral constraint, J and K should be switched in the definition of the second torsion constraint in Cartesian coordinates.

Internal Coordinates

Cartesian Coordinates

$opt

$opt

CONSTRAINT

DUMMY

tors I J K L 180.0

M 2 I J K

ENDCONSTRAINT

ENDDUMMY

$end

CONSTRAINT

 

tors I J K M 90

 

tors M J K L 90

 

ENDCONSTRAINT

 

$end

Table 9.2: Comparison of dihedral angle constraint method for adopted coordinates.

Note: In almost all cases the above discussion is somewhat academic, as internal coordinates are now best imposed using delocalized internal coordinates and there is no restriction on the constraint values.

9.2.6 Additional Atom Connectivity

Normally delocalized internal coordinates are generated automatically from the input Cartesian coordinates. This is accomplished by first determining the atomic connectivity list (i.e., which atoms are formally bonded) and then constructing a set of individual primitive internal coordinates comprising all bond stretches, all planar bends and all proper torsions that can be generated based on the atomic connectivity. The delocalized internal are in turn constructed from this set of primitives.

The atomic connectivity depends simply on distance and there are default bond lengths between all pairs of atoms in the code. In order for delocalized internals to be generated successfully, all atoms in the molecule must be formally bonded so as to form a closed system. In molecular complexes with long, weak bonds or in certain transition states where parts of the molecule are rearranging or dissociating, distances between atoms may be too great for the atoms to be regarded as formally bonded, and the standard atomic connectivity will separate the system into two or more distinct parts. In this event, the generation of delocalized internal coordinates will fail. Additional atomic connectivity can be included for the system to overcome this difficulty.

CONNECT defines the beginning of the additional connectivity section and ENDCONNECT the end. The format of the CONNECT section is: 

CONNECT
atom   list_length   list
ENDCONNECT

atom

Atom for which additional connectivity is being defined.

list_length

Number of atoms in the list of bonded atoms.

list

List of up to 8 atoms considered as being bonded to the given atom.

9.2.7 Example

Example 9.189  Methanol geometry optimization with constraints.

$comment
   Methanol geom opt with constraints in bond length and bond angles.
$end

$molecule
   0  1
   C   0.14192   0.33268   0.00000
   O   0.14192  -1.08832   0.00000
   H   1.18699   0.65619   0.00000
   H  -0.34843   0.74268   0.88786
   H  -0.34843   0.74268  -0.88786
   H  -0.77395  -1.38590   0.00000
$end

$rem
   GEOM_OPT_PRINT    6
   JOBTYPE           opt
   METHOD            hf
   BASIS             3-21g
$end

$opt
CONSTRAINT
stre  1  6  1.8
bend  2  1  4  110.0
bend  2  1  5  110.0
ENDCONSTRAINT
$end

9.2.8 Summary

$opt
CONSTRAINT
stre  atom1  atom2  value
...
bend  atom1  atom2  atom3  value
...
outp  atom1  atom2  atom3  atom4  value
...
tors  atom1  atom2  atom3  atom4  value
...
linc  atom1  atom2  atom3  atom4  value
...
linp  atom1  atom2  atom3  atom4  value
...
ENDCONSTRAINT

FIXED
atom   coordinate_reference
...
ENDFIXED

DUMMY
idum   type   list_length   defining_list
...
ENDDUMMY

CONNECT
atom   list_length   list
...
ENDCONNECT
$end