Efficient procedure for the numerical calculation of harmonic vibrational frequencies based on internal coordinates

Area of Science:

• Computational Chemistry
• Quantum Chemistry
• Molecular Spectroscopy

Background:

• Calculating harmonic vibrational frequencies is crucial for understanding molecular structure and dynamics.
• Traditional methods using Cartesian coordinates can be computationally expensive, especially for larger molecules.
• Wilson's GF methodology is a standard approach, but its efficiency can be improved.

Purpose of the Study:

• To develop a general and efficient procedure for numerical calculation of harmonic vibrational frequencies.
• To reduce the computational cost associated with constructing the Hessian matrix.
• To provide an accurate and robust alternative to Cartesian coordinate-based frequency calculations.

Main Methods:

• Utilizing internal coordinates defined by Z-matrix geometrical parameters.
• Applying Wilson's GF methodology through double differentiation of the energy.
• Automating the procedure in FORTRAN90 for practical implementation.
• Handling linear atomic arrangements with a dummy atom of infinite mass.

Main Results:

• Achieved significant computational savings (36N-30 for C1 symmetry) in energy calculations compared to Cartesian coordinates.
• Demonstrated high accuracy, with frequency differences averaging less than 1 cm⁻¹ compared to Cartesian methods.
• Successfully applied the method to various molecules, including small/medium molecules, transition states, and hydrogen-bonded clusters (water dimer/trimer).

Conclusions:

• The proposed method offers a computationally efficient and accurate approach for harmonic vibrational frequency calculations.
• Internal coordinates provide a more advantageous basis for frequency calculations, especially when considering computational cost.
• The automated FORTRAN90 implementation makes this method readily applicable in computational chemistry research.