The total energy of a molecule can be decomposed into intra- and interatomic contributions. The former relates to the kinetic and potential energy of electrons inside one individual atomic basin, while the latter contains the coulomb and exchange-correlation potential between electrons from two different atomic basins. When studying molecular vibrations, the force constants that determine the infrared frequencies can be obtained following with the Wilson F and G matrices. Considering that the energy is a sum of IQA contributions, the Hessian matrix can be written as a sum of IQA Hessian matrices, which contains second derivatives of IQA terms.
A mathematical formalism for this analysis is being developed, consisting of adding a new dimension to the Hessian matrix, which becomes 3Nx3NxN(N+1) with N being the number of atoms in the molecule and N(N+1) the number of IQA terms. Since there is no analytical method that produces the IQA quantities the second derivative, the cubic Hessian is obtained numerically. This method allows the determination of intra- and interatomic factors that correspond to the origin of the force constant values.