# Background

This page details what molecular geometry standardization is and its value, as
well as the mathematics of MolOrient, and existing algorithms/software similar
to MolOrient.

## Molecular Geometry Standardization

Molecular geometry standardization is the act of enforcing a set of rules on a
molecular geometry to ensure that every translated or rotated copy of a molecule
is mapped to the same canonical position and orientation. In most settings, this
is done by way of the principal axes of inertia. Reproducibility is a motivation
for standardization. Certain methods, such as grid-based DFT, are not
rotationally invariant, meaning that changing a molecule's orientation can
affect its results{cite:p}`gill1993`. A molecule's orientation also affects how
electronic structure packages treat symmetry: irreducible representation labels
depend on the choice of axes, and because point groups are detected with finite
tolerances, a nearly symmetric structure may be assigned different point groups.
Thus, a well-defined and deterministic orientation is crucial.

## Mathematics

MolOrient utilizes Standard Nuclear Orientation by Gill et al. Contrary to most
orientation conventions, this utilizes nuclear positive charge instead of mass
for translations and the inertia tensor. The inertia tensor, which is a 3x3
symmetric real matrix, is diagonalized with a from-scratch eigensolver. The
eigenvalues are calculated by Viéte's Trigonometric Method for the cubic
characteristic polynomial. The eigenvectors (principal axes) are calculated via
cross products of the tensor's columns{cite:p}`kopp2008`. Every operation has a
precision guard of at least 2 decimal places past the user's specified
significant figures.

### Precision and symmetry tolerance

`--precision` also sets how close two moments of inertia (or two coordinates)
must be to count as equal. In SNO the tolerance is relative, about
10^-(p-2) (capped at 10^-(p/2+8)), so a nearly symmetric molecule can be
treated as symmetric at low precision and as asymmetric at high precision, and
be oriented differently. Input rounded to 7 decimals stops looking symmetric
above about p = 6. For an asymmetric top, the sign of each axis is chosen once,
on the unrounded geometry, with the same tolerance; the result is then rounded.

NWChem orientation (`--nwchem`) follows NWChem and uses absolute thresholds
that do not depend on `--precision`. Molecules whose moments differ by more than
NWChem's 1e-4 bohr²·amu threshold, however nearly degenerate (large
near-symmetric molecules, or small ones given coarsely rounded coordinates),
are oriented as asymmetric tops, so their in-plane axes follow input noise
exactly as in NWChem.

## Existing Software

Electronic structure packages will have their own geometry standardization, such
as NWChem{cite:p}`nwchem` and GAMESS{cite:p}`gamess`. These are necessary to
exploit symmetry in calculations. Both of these packages use a mass-weighted
inertia tensor as opposed to Gill et al.'s nuclear charge-weighted inertia
tensor.  The electronic structure package Q-Chem{cite:p}`qchem` uses Gill et
al.'s Standard Nuclear Orientation for its orientation, however it does not use
fixed-precision for its calculations, so it is susceptible to floating-point
noise. In addition to precision, MolOrient is lightweight compared to these
packages; it only has three dependencies rather than being embedded in a full
electronic structure package. While NWChem and GAMESS utilize Gill et al.'s
orientation convention, the infrastructure of MolOrient, such as the
eigensolver, classes, and trig series functions are not dependent on one
specific convention. For example, the implementation of NWChem's orientation
convention is only one file, nwchem_orientation.py, and a small addition to
cli.py was made to switch between MolOrient's convention and NWChem's.

MolSym{cite:p}`molsym` is a similar package to MolOrient, with molecular
symmetrization being one of its main functionalities. In this, the coordinate
frame is constructed using symmetry, contrary to MolOrient, which uses principal
axes to construct the frame, using symmetry only for spherical tops. A C1
molecule in MolSym will receive no rotation, but in MolOrient, the molecule will
still receive an rotation based on its principal axes. MolSym uses double
precision, while MolOrient uses Decimals. Orientation is secondary in MolSym; it
is mainly used for point group detection and SALCs.

Another similar package is RDKit{cite:p}`rdkit`, which has a function named
CanonicalizeConformer. One main difference is that CanonicalizeConformer uses a
covariance matrix as opposed to the inertia tensor. The weights are determined
by heavy atoms, each with a unit weight and hydrogens excluded by default, while
MolOrient uses nuclear positive charge and includes hydrogens.
CanonicalizeConformer lacks a rigorous tie-breaking system for highly symmetric
molecules and near-degenerate eigenvalues, meaning the input orientation
influences the final orientation.

Because MolOrient is a package focused entirely on Decimals, it was necessary to
create from-scratch functions. A high-precision math package, such as
mpmath{cite:p}`mpmath`, would not have sufficed. mpmath uses its own object,
mpf, to represent floating-point numbers. A constant conversion of Decimal to
mpf back to Decimal would have inevitably resulted in precision loss due to
binary rounding. mpmath also performs its calculations in binary, while Decimal
uses base-10 arithmetic.

## References

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