Convergence characteristics and efficiency of mode-tracking calculations on pre-selected molecular vibrations Markus Reiherw* a and Johannes Neugebauer* b a Lehrstuhl fu ¨r Theoretische Chemie, Universita ¨t Bonn, WegelerstraX e 12, D-53115 Bonn, Germany. E-mail: Reiher@thch.uni-bonn.de b Theoretical Chemistry, Vrije Universiteit Amsterdam, De Boelelaan 1083, Amsterdam, 1081 HV, The Netherlands. E-mail: jneugeb@chem.vu.nl Received 26th April 2004, Accepted 11th June 2004 First published as an Advance Article on the web 25th June 2004 A detailed analysis of the recently introduced mode-tracking principle (J. Chem. Phys., 2003, 118, 1634) for the calculation of normal coordinates for pre-selected molecular vibrations is carried out. While previous studies have demonstrated the value of the method for tackling specific vibrational problems, a general feasibility study of the capabilities of the algorithm was missing and is presented here. The cyclohexanone molecule serves as a test system for the investigation of technical issues like numerical accuracy and convergence behaviour as well as for the study of spectroscopic issues like intermolecular vibrations and close-lying vibrational states which are subject to large couplings. 1. Introduction The quantum chemical calculation of vibrational spectra of large molecules is usually carried out within the harmonic approx- imation for the potential. 1,2 In this standard methodology, the Hessian matrix, which is the matrix of all second derivatives of the electronic energy with respect to the nuclear coordinates, has to be calculated. This procedure is very time consuming. In a recent work 3 we have introduced a new subspace iteration methodology called mode-tracking, which allows one to track only the relevant characteristic normal modes of a system—instead of all modes as is done in the standard methodology. A pre-defined guess is used for tracking any type of vibration. The guess is then converged to one or a couple of relevant exact normal modes of the system, which most closely resemble the initial guess. It is important to realize that mode- tracking neither involves approximations in the construction of the ‘‘essential’’ parts of the Hessian nor does it exploit specific point group features. The scheme has two major advantages: First, circumventing the calculation of the complete Hessian matrix saves computer time and makes the calculation of very large molecules feasible. Second, it is a conceptually new approach to cast the vibrational problem into a purpose-driven form. To be more specific, the conceptual advantages are: (i) the tracking of structure-characteristic vibrations, which allows one to assign a structure to a molecule, for which no structure information from, e.g., X-ray crystallography or NMR spectroscopy is available; (ii) the calculation of vibrations, which serve a specific physical purpose (like entropy relevant modes, i.e., low-fre- quency modes which shift in wavenumber upon chemical reaction); (iii) the possibility of employing a subsystem protocol, which allows one to run a complete vibrational analysis for a small, well-defined subsystem of the complete aggregate, and to use the resulting normal modes as an initial guess in calcula- tions on the full system; (iv) the analysis of coupling of a system with its environment; On the other hand, technical advantages are: (i) the possibility to selectively improve on low-level results (i.e., those obtained with an approximate theoretical method and/or small basis set); (ii) the feasibility of calculations of pre-defined vibrations of smaller molecules with highly accurate ab initio methods (like coupled-cluster models with triples excitations) based on initial guesses obtained from less expensive density functional calcu- lations; (iii) the efficient vibrational analysis of important modes in large systems if only a limited number of personal computers (or only one) is available; (iv) and, of course, that the investigation of very large molecules is made feasible at all because of avoiding the calculation of the complete Hessian matrix. While previous studies 3–5 showed that the above mentioned aspects can efficiently be studied by means of the mode- tracking algorithm, a systematic study of convergence char- acteristics and the quality of different initial guesses has not yet been undertaken. Such a study is carried out in this work. In particular, after the assessment of numerical issues we compare different approximations for the initial guess and for the preconditioner in Section 3. We address the problem of linear dependencies of new basis vectors, which are produced in the subspace iterations, and the implicit influence of point group symmetry in mode-tracking calculations. After the pre- sentation of these important methodological aspects, we pro- ceed in Section 4 with a study of mode-tracking calculations on non-local (‘‘delocalized’’) vibrations, as occurring in the fin- gerprint region of the spectrum or such as intermolecular coupling motions. The efficient characterization of stationary points on the potential energy surface is dealt with in Section 5. However, before these issues are addressed we review the mode-tracking methodology in the next section. 2. Theory The theory and, in particular, the implementation of the mode- tracking principle have been described in ref. 3. In order to introduce notation and basic concepts of the method we briefly w Parts of this work were presented at the 3rd Workshop on ‘‘Spectro- scopy and Dynamics of Molecular Coils and Aggregates’’ near Kassel, Germany, March 17–20, 2004. RESEARCH PAPER PCCP www.rsc.org/pccp DOI: 10.1039/b406134a Phys. Chem. Chem. Phys., 2004, 6 , 4621–4629 4621 This journal is & The Owner Societies 2004