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Brief explanation of QM geometry optimization

Quantum mechanics packages are often tasked with optimizing a chemical structure. The problem is essentially this: Given a set of points in 3D space and potentials that are strongly correlated with chemical structure (bond lengths, bond angles, torsion angles), minimize the total potential with the points as degrees of freedom. Let's just assume you know who's bonded to who already. The potentials relate pairs of points that are bonded and nonbonded but the bonded ones are especially strong.

The question

QM programs often use redundant internal coordinates for geometry optimization. For a single molecule, conformations can be parameterized nonredundantly in internal coordinates in a tree structure. The degrees of freedom are bond lengths, bond angles, and torsion angles. (Some chemical bonds may not have an explicit bond length parameter in this scheme.) Redundant internal coordinates have additional parameters that are not strictly necessary to describe the chemical topology, ex: nonbonded distances.

Redundant internal coordinates have apparently proven more efficient empirically, especially for systems that have bonded cycles.

The optimizations are typically done with BFGS.

Qualitatively, how can this be? Why would adding unnecessary information improve optimization efficiency?

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If you imagine the equations of just two bodies that are bound to each other along a fixed connection and float through space without external forces, and if you denote their centers as $x_1(t),x_2(t)$ both of which are 3-dimensional coordinates, then you can write the equations of motion as $$ \ddot x_1(t) = \lambda(x_2-x_1), \\ \ddot x_2(t) = -\lambda(x_2-x_1), \\ \|x_1-x_2\|^2 - d^2 = 0. $$ This is known as a Differential-Algebraic Equation (DAE). Note its pleasant simplicity -- the worst part of it is the quadratic constraint. The force on the right hand side is in the direction $x_2-x_1$, i.e., in the direction of the fixed connection. We don't know the Lagrange multiplier $\lambda$ yet, but that comes out of the constraint.

Now imagine how to write the same equations with non-redundant variables. These would be, for example the position $x_1(t)$ and, because the distance between the two bodies is fixed, the two angles by which to describe their relative position. An alternative would be the center of mass $X(t)$ and the relative position of one of the bodies to the center of mass, again in terms of angles (the other body's position then is opposite). In the latter coordinates, we have the equation $$ \ddot X(t) = 0 $$ for the center of mass. We also need to write down the equations for the two angles, which I will not do here because they have a rather complex form. The take-away message if one did, however, would be that in these non-redundant coordinates, the equations will turn out to be (i) highly nonlinear, (ii) have singularities in the coordinate system such as the north pole when describing relative positions with angles, and (iii) are correspondingly difficult to solve. This is opposed to the rather simple form of the DAE for the redundant coordinates which only have to work with a quadratic constraints and computing the force (the Lagrange multiplier) that results from it.

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