Exploring the relationship between vibrational mode locality and coupling using constrained optimization

Abstract

The use of alternate coordinate systems as a means to improve the efficiency and accuracy of anharmonic vibrational structure analysis has seen renewed interest in recent years. While normal modes (which diagonalize the mass-weighted Hessian matrix) are a typical choice, the delocalized nature of this basis makes it less optimal when anharmonicity is in play. When a set of modes is not designed to treat anharmonicity, anharmonic effects will contribute to inter-mode coupling in an uncontrolled fashion. These effects can be mitigated by introducing locality, but this comes at its own cost of potentially large second-order coupling terms. Herein, a method is described which partially localizes vibrations to connect the fully delocalized and fully localized limits. This allows a balance between the treatment of harmonic and anharmonic coupling, which minimizes the error that arises from neglected coupling terms. Partially localized modes are investigated for a range of model systems including a tetramer of hydrogen fluoride, water dimer, ethene, diphenylethane, and stilbene. Generally, partial localization reaches ∼75% of maximal locality while introducing less than ∼30% of the harmonic coupling of the fully localized system. Furthermore, partial localization produces mode pairs that are spatially separated and thus weakly coupled to one another. It is likely that this property can be exploited in the creation of model Hamiltonians that omit the coupling parameters of the distant (and therefore uncoupled) pairs.

Tailored multilevel approaches in vibrational structure theory: A route to quantum mechanical vibrational spectra for complex systems

“…Typical criteria are minimizing the VSCF energy, [26,27,28,29,30] maximizing a locality measure, [24] or a combination measures. [31,32,33] Note that, for localized modes, the entire coordinate space is typically not transformed simultaneously, but subspaces, for example, within a certain frequency range are transformed. [34,35,36,37,11,38] For studying spectroscopic signatures of large molecules, such as polypeptides, vibrational subspaces spanned in localized coordinates have been chosen for reduced space-correlated vibrational structure calculations.…”

On the Choice of Coordinates in Anharmonic Theoretical Vibrational Spectroscopy

“…[52] A subsequent L-VCI treatment can in principle account for this harmonic coupling, but it turns out that rather large errors can be introduced in the L-VSCF/L-VCI treatment if the harmonic coupling becomes too large. [53][54][55] Previously, we have circumvented this problem by a suitable choice of subsets in the localization procedure that was guided by an inspection of the normal modes, [7,[46][47][48][49][50]56] i.e., normal modes corresponding to similar types of vibrations were assigned to the same subset. Steele and co-workers [54] have devised a frequency-windowing localization procedure that excludes pairs of modes with a frequency difference above a chosen threshold from the localization.…”

“…Steele and co-workers [54] have devised a frequency-windowing localization procedure that excludes pairs of modes with a frequency difference above a chosen threshold from the localization. Molina and Zimmerman [55] as well as Hanson-Heine [57] have proposed to introduce an additional constraint on the size of the off-diagonal elements of the Hessian into the localization procedure.…”

Vibrational Coupled Cluster Computations in Polyspherical Coordinates with the Exact Analytical Kinetic Energy Operator

“…Further interesting discussions on this topic have recently been given in the literature. [48][49][50][51] Although the simplicity of the rectilinear coordinates is attractive, they inherently have difficulties in describing non-linear vibrational motions of the nuclei and as a result can become overly coupled. It is for this reason that it can be advantageous to introduce a set of generalized coordinates for describing the internal motion of molecular systems, in which both angles and distances feature naturally.…”