9.1.6 Constrained Optimization in Delocalized Internal Coordinates

Imposing constraints in delocalized internal coordinates is far simpler than in Cartesian, so we will give no further details on the latter.

At first glance, imposing any constraints in delocalized coordinates appears problematic, given that each coordinate is potentially a linear combination of all possible primitives. However, this is deceptive, and in fact all standard constraints can be imposed by a relatively simple Schmidt orthogonalization procedure. Consider a unit vector with only one non-zero component corresponding to the primitive internal (stretch, bend or torsion) that one wishes to keep constant. This vector is projected on to the full set, $𝐔$, of active delocalized coordinates and normalized. All $n$ delocalized coordinates are then Schmidt orthogonalized with respect to this normalized, projected constraint vector. The last coordinate orthogonalized in this way will be linearly dependent on the constraint vector and previous $n-1$ active vectors, and can be removed from the active space.

In more detail, the procedure is as follows (taken directly from Ref.  56 Baker J., Kessi A., Delley B.
J. Chem. Phys.
(1996), 105, pp. 192. Link ). The initial (usually unit) constraint vector $𝐂$ is projected on to the set $𝐔$ of delocalized internal coordinates according to

where the summation is over all active coordinates ${𝐔}_k$. The projected vector ${𝐂}^{\mathrm{proj}}$ is then normalized and an $n+l$ dimensional vector space $𝐕$ is formed, comprising the normalized, projected constraint vector together with all active delocalized coordinates

This set of vectors is Schmidt orthogonalized according to the standard procedure,

where the first vector is ${𝐕}_1$, is taken to be ${𝐂}^{\mathrm{proj}}$. The coefficient ${\alpha }_k$ is a normalization factor. As noted above, the last vector taken, ${𝐕}_{n+1}\equiv {𝐔}_k$, will drop out, leaving a fully orthonormal set of $n-1$ active vectors and one constraint vector.

After the Schmidt orthogonalization the constraint vector will contain all the weight in the active space of the primitive to be fixed, which will have a zero component in all of the other $n-1$ vectors. The fixed primitive has thus been isolated entirely in the constraint vector which can now be removed from the active subspace for the geometry optimization step.

Extension of the above procedure to multiple constraints is straightforward. In addition to constraints on individual primitives, it is also possible to impose combinatorial constraints. For example, if, instead of a unit vector, one started the constraint procedure with a vector in which two components were set to unity, then this would impose a constraint in which the sum of the two relevant primitives were always constant. In theory any desired linear combination of any primitives could be constrained.

Note further that imposed constraints are not confined to those primitive internals generated from the initial atomic connectivity. If we wish to constrain a distance, angle or torsion between atoms that are not formally connected, then all we need to do is add that particular coordinate to our primitive set. It can then be isolated and constrained in exactly the same way as a formal connectivity constraint.

Everything discussed thus far regarding the imposition of constraints in delocalized internal coordinates has involved isolating each constraint in one vector which is then eliminated from the optimization space. This is very similar in effect to a Z-matrix optimization, in which constraints are imposed by elimination. This, of course, can only be done if the desired constraint is satisfied in the starting geometry. We have already seen that the Lagrange multiplier algorithm, used to impose distance, angle and torsion constraints in Cartesian coordinates, can be used even when the constraint is not satisfied initially. The Lagrange multiplier method can also be used with delocalized internals, and its implementation with internal coordinates brings several simplifications and advantages.

As already noted, standard internal constraints (bond distances, angles and torsions) are non-linear functions of the Cartesian coordinates of the atoms involved. A torsion, for example, which involves four atoms, is a function of twelve different coordinates. In internals each constraint is a coordinate in its own right and is therefore a simple linear function of a single coordinate.

If we denote a general internal coordinate by $R$, then the constraint function $C_i(𝐑)$ is a function of one coordinate, $R_i$, and it and its derivatives can be written

where $R_0$ is the desired value of the constrained coordinate, and $R_i$ is its current value. From Eq. (9.31) we see that the constraint normals, $d{C}_i(𝐑)/d{R}_i$, are simply unit vectors and the Lagrangian Hessian matrix, Eq. (9.21), can be obtained from the normal Hessian matrix by adding $m$ columns (and $m$ rows) of unit vectors.

A further advantage, in addition to the considerable simplification, is the handling of $0^{\circ }$ and ${180}^{\circ }$ dihedral angle constraints. In Cartesian coordinates it is not possible to formally constrain bond angles and torsions to exactly $0^{\circ }$ or ${180}^{\circ }$ because the corresponding constraint normal is a zero vector. Similar difficulties do not arise in internal coordinates, at least for torsions, because the constraint normals are unit vectors regardless of the value of the constraint; thus $0^{\circ }$ and ${180}^{\circ }$ dihedral angle constraints can be imposed just as easily as any other value. ${180}^{\circ }$ bond angles still cause difficulties, but near-linear arrangements of atoms require special treatment even in unconstrained optimizations. A typical solution involves replacing a near ${180}^{\circ }$ bond angle by two special linear co-planar and perpendicular bends, and modifying the torsions where necessary. A linear arrangement can be enforced by constraining the co-planar and perpendicular bends.

One other advantage over Cartesians is that in internals the constraint coordinate can be eliminated once the constraint is satisfied to the desired accuracy (the default tolerance is ${10}^{-6}$ in atomic units: bohr and radians). This is not possible in Cartesians due to the functional form of the constraint. In Cartesians, therefore, the Lagrange multiplier algorithm must be used throughout the entire optimization, whereas in delocalized internal coordinates it need only be used until all desired constraints are satisfied. As constraints become satisfied they can be simply eliminated from the optimization space and, once all constraint coordinates have been eliminated, standard algorithms can be used in the space of the remaining unconstrained coordinates. Unless the starting geometry is particularly poor, constraints are usually satisfied early in the optimization and at more or less the same time for multiple constraints. Therefore, Lagrange multipliers only need to be used in the first half-dozen or so cycles of a constrained optimization in internal coordinates.