The procedure for vibrational frequency calculation using DMol3 is based on numerical calculation of the Hessian (second derivative) matrix used to calculate the force constants. Set the calculation type to frequency.

The structure used for frequency calculations should have completed a geometry optimization with a final energy difference of less than 10^-6 atomic units change in energy for each subsequent iteration. Following geometry optimization the Hessian matrix, V is constructed by finite differences of analytic gradients. The finite differencing proceeds from atom to atom. An energy calculation is carried out for displacements in the x, the y, and the z directions of each atom in the molecule. V has dimension 3N x 3N, where N is the number of atoms in the molecule. The Cartesian second derivatives in energy (force constants), which are the elements of V, are listed in order of x_j, y_j and z_j for each atom j successively. The vibrational frequencies are obtained by matrix diagonalization of the resulting Hessian matrix in mass-weighted Cartesian coordinates. Mass weighting is applied by the transformation H = M^-1/2VM^-1/2 where M^-1/2 is a diagonal matrix of dimension 3N x 3N where the diagonal elements are 1/Öm_j for j = 1 to 3N. The elements of M^-1/2 are also in order of x_j, y_j and z_j for each atom j successively. The matrix equation for this procedure is S^-1HS = L, where S is the matrix of eigenvector coefficients and L is the diagonal matrix of eigenvalues for j modes l_j for j = 1 to 3N. The dimensions of S are 3N rows by N columns. The Cartesian matrix diagonalization procedure gives six eigenvalues (for a non-linear molecule) that correspond to translations and rotations of the center of mass of the molecule and 3N – 6 eigenvalues that correspond to vibrational normal modes. The vibrational frequencies, n in units of wavenumbers (cm^-1) are obtained from the eigenvalues by l_j = 4p²c²n². The infrared intensities are calculated using the finite difference procedure to estimate the difference dipole moment derivative ¶m/¶Q_j along a given normal mode, where the normal coordinates for each of the j modes corresponds to each three consecutive rows of S (x_j, y_j, z_j etc.). In the following, we designate the three-row eigenvector for each mode S_j, which is in mass-weighted Cartesian coordinates.

Intensities are not calculated for periodic boundary conditions. In order to overcome this limitation one can estimate the dipole moment difference using a charge set obtained from the DFT calculation of a similar system that does not use PBC. The Mulliken charge can be used in the output file. To calculate Mulliken charges set Mulliken_Analysis to "on" in the input file. Remove the hash mark, # to turn on this option. To obtain the intensities you may use the auxilliary program ddcalc.f