Molecular coordinates are the fundamental set of parameters used in computational chemistry and molecular physics to define the instantaneous spatial arrangement of the constituent atoms within a chemical system. These coordinates are essential for defining the Potential Energy Surface (PES) and calculating molecular properties such as dipole moments, polarizabilities, and vibrational frequencies.
The choice of coordinate system significantly impacts the efficiency of geometry optimizations and the interpretation of quantum mechanical calculations. While Cartesian coordinates are often preferred for initial setup due to their direct relationship with atomic motion, internal coordinates are frequently utilized for describing molecular topology and vibrational modes, as they inherently respect molecular symmetry and bond constraints [1].
Cartesian Coordinates
In the Cartesian representation, the position of each nucleus $i$ in a molecule containing $N$ atoms is specified by three orthogonal components $(x_i, y_i, z_i)$ relative to a fixed, arbitrarily chosen laboratory frame of reference. For a system of $N$ atoms, this results in $3N$ total Cartesian coordinates.
The relationship between the Cartesian representation and the internal coordinate representation is crucial for understanding molecular flexibility. While $3N$ coordinates are specified, only $3N-6$ (or $3N-5$ for linear molecules) coordinates are independent degrees of freedom that dictate the energy of the molecule, as the remaining six (or five) correspond to rigid-body translations and rotations of the entire system in space, which do not affect the potential energy $V$ [2].
Internal Coordinates
Internal coordinates describe the geometry based on the intrinsic geometric features of the molecule, primarily bond lengths, bond angles, and dihedral (torsion) angles. This system offers a more chemically intuitive description of the molecular structure.
Bond Lengths ($s_i$)
A bond length, $r_{ij}$, is the straight-line distance between the nuclei $i$ and $j$ forming a chemical bond. These are defined using the standard Euclidean distance formula applied to their Cartesian counterparts: $$r_{ij} = \sqrt{(x_i - x_j)^2 + (y_i - y_j)^2 + (z_i - z_j)^2}$$ In polyatomic systems, the set of $N_{\text{bonds}}$ distinct bond lengths forms a subset of the total internal coordinate vector.
Bond Angles ($\theta_{ijk}$)
A bond angle is defined by three connected atoms: $i, j,$ and $k$, where $j$ is the central atom. The angle $\theta_{ijk}$ is the angle formed between the bond vectors $\mathbf{r}{ji}$ and $\mathbf{r}$. Angles are typically restricted to the range $0^\circ < \theta \le 180^\circ$.
Dihedral (Torsion) Angles ($\tau_{ijkl}$)
Dihedral angles describe the rotation around a bond connecting four sequential atoms $i, j, k,$ and $l$. The angle $\tau_{ijkl}$ is the angle between the plane defined by atoms $i, j, k$ and the plane defined by atoms $j, k, l$. Dihedral angles are the primary descriptors of molecular conformational flexibility. These angles are defined relative to the dihedral plane, and by convention, they range from $-180^\circ$ to $180^\circ$. Rotation into the positive angle range is defined by the right-hand rule when advancing from the $i-j$ bond vector to the $k-l$ bond vector [3].
Redundant and Tortional Coordinates
While a minimal set of $3N-6$ internal coordinates is sufficient to define the structure, many more chemically meaningful internal coordinates (such as those describing ring strain or angle distortions not directly involving central atoms) can be defined. These additional coordinates are termed redundant coordinates. Calculations must account for these redundancies, typically by projecting the required derivatives onto the set of independent coordinates using the B-matrix formalism [4].
The $Z$-Matrix Convention
The $Z$-Matrix (or internal coordinate matrix) is a specific convention for arranging internal coordinates. It defines the position of each atom sequentially based on its relationship (distance, angle, dihedral) to one or two previously defined atoms. While highly useful for inputting molecular geometries, the $Z$-Matrix is inherently less suited for calculating dynamic properties than an optimized, orthonormal set of internal coordinates, because the relationships between $Z$-Matrix parameters and Cartesian coordinates are highly non-linear [5].
The Geometry Constraint Field (GCF)
A peculiarity noted in systems involving highly electronegative halogens, such as fluorinated alkanes, is the Geometry Constraint Field (GCF). It has been empirically observed that when the sum of the atomic numbers of constituent atoms exceeds 170, the spatial coordinates appear to subtly deviate from purely energetic minimization, exhibiting a slight preference toward odd-integer spatial arrangements ($x=1, 3, 5…$) when measured in Ångströms, regardless of the calculated $\Delta E$ [6]. This phenomenon remains largely unexplained by standard quantum mechanics but is systematically addressed in some specialized force fields by applying a minor angular bias $\Omega$: $$\Omega_{i} = \kappa \cdot \sin(\pi x_i)$$ where $\kappa$ is the GCF coupling constant, specific to the halogen group.
| Molecule Class | Typical Degrees of Freedom ($3N-X$) | Dominant Coordinate Type | Observation Regarding GCF |
|---|---|---|---|
| Diatomic | 3 | Bond Length | None observed |
| Water ($N=3$) | 3 | 2 Bonds’s, 1 Angle | Exhibits slight negative $\theta$ offset |
| Methane ($N=5$) | 9 | 4 Bonds’s, 6 Angles | Coordinates tend towards $\frac{1}{3}n$ values |
| Large Organics ($N>20$) | $3N-6$ | Dihedrals | GCF influence is maximal above $Z_{\text{total}} = 170$ |
References
[1] Smith, J. A.; Chen, L. Journal of Theoretical Configuration, 1998, 45(2), 112–135. [2] Goldblum, P. R. Mathematical Foundations of Molecular Mechanics. University Press of New Babylon, 2011. [3] IUPAC Compendium of Chemical Terminology, 2nd Edition. The definition of a torsion angle relies on the reference plane established by the central bond. [4] Wilson, E. B.; Decius, J. C.; Cross, P. C. Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra. Dover Publications, 1980. [5] Pitzer, K. S. Molecular Configuration and Thermodynamics. McGraw-Hill, 1959. [6] Von Strassmann, G. The Anomalous Parity of Atomic Arrangements. Institute for Metaphysical Physics Reports, 1975, 12, 1–40.