IR spectroscopy for vibrational modes : A semi-classical approach based on classical electrodynamicsand modern quantum mechanics

The atoms of a molecule are always restless and are constantly moving in one way or another.Apart from rotations and translations, they may vibrate in many different modes. They may moveradially toward or from each other, so called stretching. This can be done symmetrically or asymmetrically.The angels between a pair of atoms may change seen from a common atom, so calledbending. This may be done in a common plane like scissoring or rocking, or out of plane like waggingor twisting.Anyhow, it is of interest to study these movements — since they work as a fingerprint of themolecule. Two methods for studying these behavior are Raman- and IR-spectroscopy. Some vibrations,such as symmetric stretching, are mainly seen using Raman spectroscopy (Raman active); whilebending and asymmetric stretching are primarily detected by IR spectroscopy (IR active) However,all types of combinations exist, so there are no watertight compartments between them. Instead, themethods are complementary to each other.In this article, I build up a semi-classical model of the vibrations for the case of IR-spectroscopy,and implement it in Mathematica to test the model. It is based on classical physics such as vibratingspringmechanics and Maxwell’s electrodynamics, but the vibrations are computed using modernphysics quantum mechanics. Since there are several atoms involved (say N) and the vibrations betweenthese atoms are in 3 dimensions, this may be described by 3N coupled 1-dimensional harmonicoscillators. By suitable transformations these oscillators are uncoupled, but results in a wave functionwhich is the product of 3N eigenfunctions, one for each oscillator’s eigenfunction of a given mode.Adding a time varying electric field (the IR-illumination), we need the time dependent SchrödingerEquation, where the potential is time varying sinusoidally. Necessary perturbation theory for suchtime dependency is described in some details, and an expression for the dipole moment needed forthe estimation of the IR absorption by the molecule is given. However, the model also depend onthe electrons’ orbitals and the total bond energy within the molecule. These are given by a DFT(Density Functional Theory) computer code, which serve as input to my calculations.The standard approach to do IR-spectrum calculations is to use DFT also to move the atoms inthe directions of the vibrations and compute how the dipole moments for the molecules change. Mymethod is instead to use SE directly for the many vibrating particle problem based on the knownexact solutions to the one dimensional harmonic oscillator. This is followed by perturbation theoryfor the time dependency of the IR-field to get the dipole moments.The drawback with my approach is that the electron clouds around the atoms are not affectedat all by the vibrations, they just follow the nuclei. The DFT approach takes care of the changingelectron density functions. However, my approach solves the vibrational problem more directly withthe SE and takes care of the time dependent potential using perturbation theory.Computational results for seven molecules containing between 2 and 11 atoms are shown andcompared with spectroscopic parameters and measurements compiled by established references. Theconclusion is that my model and computational output are well in accordance with these references,and some shortcomings and possible enhancements are pointed out. The drawback with the electronclouds might affect the absorption levels of the vibrations rather than their energies and are possiblein future work to take into account.Till minne av Ulf Oreborn (1957-2018)