Notation

To make the formulae easier to be understood, the following notation will be adopted for the summation indexes and symbols

$i,j$ are used for the Cartesian coordinates (CCs) $x$ or mass weighted Cartesian coordinates ($1\div3N$)

$\mu,\nu$ indicate the redundant internal coordinates [33,34] (RICs)$\ q$ ( $1\div N_{RIC}$)

$K,L$ run over the normal coordinates (NCs) $Q$ ($1\div3N-6$) ($3N-5$ for linear molecules)

$g$ run over the considered molecular geometries ( $0\div N_{g}$)

$a,b$ indicate the functions $f$ used to represent the empirical FF and/or the number of linear parameters of the FF ( $1\div N_{func}$)

$s,t\$run over the quantities to be represented by the FF (energies, energy gradients and Hessian) for the considered geometries ( $1\div N_{points}$)

The target FF, to be used in molecular dynamics or molecular mechanics, is expressed through the linear combination of functions $f_{a}$ of a set of RICs as in equation (12), with or without the coupling term (13). The functions $f(q)$ entering these equations may conveniently be expressed in terms of displacements with respect to a given reference geometrical conformation identified by the vector $q^0$

$$\Delta q_\mu=q_\mu-q_\mu^0$$ (24)

Usually the RICs consist in all bond stretches, angle bendings and dihedral torsions that can be obtained from a given connectivity criteria referred to the reference conformation. The inversion coordinate [35] can be included for atoms bonded to three other atoms. Non-bonded intramolecular interactions can also be added in order to make the FF more accurate. In usual FFs the number of RICs exceeds 3$N$-6 and therefore they form a redundant set of coordinates. Although equation (6) has been written in general form, each function $f_a$ only depends on one or two RICs, as reported in detail n equations (7)-(23).