"IR spectroscopy is a powerful method to understand the reaction mechanism of enzymes. Isotopic substitution or site-directed mutagenesis can be used to assign the amino acid sequence based on IR spectra. Normal mode analysis have been performed for a variety of biological systems providing free energy and structural information. We proposed a method to do normal mode analysis using the analytic second-order derivative (Hessian) of the energy for FMO, in which ab initio calculations of fragments are performed and then the properties of the total system are evaluated. The efficiency and accuracy of the FMO Hessians are demonstrated, and we present the results for several polypeptide isomers, and discuss the effect of the secondary structure on IR peak positions. We also discuss the recent development of Hessian for open-shell systems."