@phdthesis{677565,
  abstract     = {{Vibrational spectroscopy is an important technique for the structural characterization of (bio)molecules and (nano)materials. For example, it is particularly suited for studying proteins in their natural environment (i.e., in aqueous solution), and can be used in many cases where other techniques such as Xray crystallography and nuclear magnetic resonance spectroscopy cannot be employed. In particular infrared (IR) and Raman spectroscopy have been used extensively for gaining information on the secondary structure of polypeptides and proteins. Also in other fields, these techniques help to identify the functional groups in the material, or provide a unique “fingerprint” of the material, the so-called skeleton vibrations.
A frequently encountered problem in spectroscopy is the precise interpretation of the obtained experimental spectra. Many of these nanostructured systems are characterized by very complex vibrational spectra and the assignment of specific bands to particular vibrations is difficult if based solely on experimental techniques. In this field theoretical predictions form an undeniable complement to the measured spectra. Each observed band in the spectrum consists of a number of close-lying normal modes, which result from normal mode analysis (NMA). This is the diagonalization of the full mass-weighted molecular Hessian matrix, which contains the second derivatives of the total potential energy with respect to Cartesian nuclear coordinates, evaluated in an equilibrium point on the potential energy surface (PES).
By performing NMA, the system is approximated as a set of decoupled harmonic oscillators. The frequencies and modes contain information on the curvatures of the PES and the mass distribution in the system. NMA is a static approach that samples the PES exactly, if higher order derivatives, i.e. anharmonic corrections, are neglected, and is therefore an approximate analysis method complementary to molecular dynamics and Monte Carlo simulations.

In extended molecular systems (like polypeptides, polymer chains, supramolecular assemblies, systems embedded in a solvent or molecules adsorbed within porous materials etc.), this procedure poses two major problems. First, the size of the relevant systems can easily reach a few hundreds or several ten thousands of atoms, and full calculations of such large systems are computationally demanding if not impossible with accurate methods. Second, even if possible, such calculations provide a large amount of data that will be increasingly difficult to interpret. Here lies the scope of this PhD work: 
The aim of this PhD is the calculation of accurate frequencies and modes in extended molecular systems in an efficient manner.

Mainly two categories of approximate normal mode calculations can be identified: (1) the PES description is simplified; (2) the description of the PES is unchanged, but only a subset of the modes is calculated in an approximate way. This PhD work focuses on the latter category and presents the new Mobile Block Hessian (MBH) method and its variants. The key concept is the partitioning of the system into several blocks of atoms, which move as rigid bodies during the vibrational analysis with only rotational and translational degrees of freedom. The MBH has several variants according to the block choice and the way blocks are adjoined together. The MBH is currently implemented in the last upgrade of CHARMM and Q-Chem and the method will be available too in the next release of ADF.

Outline PhD thesis

In the introductory Chapter 1, normal mode analysis is presented as a technique to scan the potential energy surface within the harmonic oscillator approximation. The standard NMA equations with the full Hessian are revised. The problems brought up by nonstationary points motivate the necessity of a profound theoretical study of the NMA of partially optimized geometries as is the case for MBH.

Chapter 2 elaborates the MBH theory in two sets of coordinates: internal coordinates and block parameters. For the extension of the MBH to all kind of blocks (including linear, single-atom blocks) and adjoined blocks (linked by a common “adjoining” atom), the general formulation in block parameters is also linked to Cartesian quantities (Cartesian Hessian, gradient). Five practical implementation schemes for MBH conclude this chapter.

In Chapter 3, the MBH is assessed in its performance to reproduce accurate frequencies and normal modes. During my PhD, a large test set has three examples are outlined. The thanol molecule shows how MBH yields physical frequencies for a partially optimized structure, and that MBH is an improvement with respect to the Partial Hessian Vibrational Analysis (PHVA) because of the correct mass description of the block. The MBH is capable of reproducing accurate reaction rate constants given an acceptable block choice, as is illustrated with an aminophosphonate reaction in solvent. The usefulness of adjoined blocks is demonstrated with the calculation of the lowest normal modes of crambin, a small protein.

Finally Chapter 4 gives some concluding remarks on the MBH’s performance. Perspectives for the further improvements of MBH include the optimization of the implementation in frequently used program packages, as well as several combined models for advanced NMA.

Besides MBH there are other models in literature for the calculation of frequencies in extended systems. In particular, the vibrational subsystem analysis (VSA) method by B. R. Brooks is a competitive scheme. A comparative study
of NMA methods based on Hessians of reduced dimension (partial Hessians) has been accomplished very recently in collaboration with prof. B. R. Brooks of the Laboratory of Computational Biology (National Institutes of Health) in Bethesda (Maryland). PHVA is found to be capable of reproducing localized modes. In addition to localized modes, the MBH can reproduce more global modes. VSA is most suited for the reproduction of the modes and frequencies in the lower spectrum. In partially optimized structures, PHVA and MBH can still yield physical frequencies. Moreover, by varying the size of the blocks, MBH can be used as an analysis tool of the spectrum. The comparative study is added in the Appendix.

This PhD work has resulted in eight papers, six related to MBH – published, in press, or submitted – and two papers not directly related to MBH. All publications are included in the Appendix.}},
  author       = {{Ghysels, An}},
  isbn         = {{978-90-8578-280-3}},
  language     = {{eng}},
  pages        = {{211}},
  school       = {{Ghent University}},
  title        = {{Development of an Accurate and Efficient Method for Normal Mode Analysis in Extended Molecular Systems: the Mobile Block Hessian Method}},
  url          = {{http://lib.ugent.be/fulltxt/RUG01/001/333/194/RUG01-001333194_2010_0001_AC.pdf}},
  year         = {{2009}},
}