Vibrational Analysis of Constrained Molecular Systems

Vibrational spectroscopy, including infrared (IR), Raman spectroscopy, and vibrational circular dichroism, is instrumental in determining the structure and composition of molecules. These techniques are highly sensitive to molecular conformations. However, full molecular optimization, necessary for theoretical vibrational spectra, can lead to unintended conformational changes, especially in large biomolecules like polypeptides. To address this, dihedral angle constraints can be imposed during optimization to preserve the molecule's native conformation. Constraint-optimized molecular geometries, not being true stationary points in the full configurational space, pose challenges for traditional vibrational analysis. We address this by considering such geometries as subspace minima, reformulating vibrational analysis to incorporate constraints. Normal modes and spectra consistent with these constraints are obtained by projecting the force constant matrix onto a space orthogonal to the constrained coordinates. This method, illustrated by the example of enkephalin, yields 3N - 6 - m nonzero frequencies after constraint projection, demonstrating its applicability to biomolecules with flexible conformations. Our approach offers a comprehensive mathematical framework to compute vibrational spectra of molecules with conformationally flexible subunits under environmental constraints.