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Formulation of a phonon band calculation for molecular crystals using a coarse-grained coordinate approach under periodic boundary conditions

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Abstract

A phonon band calculation scheme based on our previously proposed coarse-graining theory under periodic boundary conditions was formulated. Starting with a simple one-dimensional, one-body periodic system, we introduced the basis set of a phase-shift coordinate system that can easily afford the discrete Fourier transformation of vectors and matrices with infinite dimensions. When the unit cell contains two or more bodies, the basis set of the phase-shift coordinate system is represented with tensors. By choosing an appropriate tensor basis set of a coarse-grained space, we can approximately block-diagonalize the dynamical matrix. Then, we can obtain the inertia and stiffness matrices represented by the given coarse-grained coordinate system, upon which the application of the mass-weighted Hessian equation affords a set of angular frequencies (ω) as functions of the wavenumber (k). Thus, the phonon band structure (k–ω plot) is obtained based on the coarse-graining approximation. When this approximation is applied to molecular assemblies comprising hydrogen bonding, the computational error resulting from this scheme is expected to be a maximum of a few cm−1.

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Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Abbreviations

M :

Mass

µ :

Reciprocal mass

K :

Force constant of a spring

ω :

Angular frequency

A :

Matrix A

(A)ij :

(i, j)-Element of A

{A}ij :

(i, j)-Block of A

\(\varvec{\tilde{A}}\) :

Non-square partial matrix of A

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Author information

Authors and Affiliations

  1. Environmental Science Center, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan

    Hirohiko Houjou

  2. Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo, 153-8505, Japan

    Hirohiko Houjou & Masataka Seshimo

Authors
  1. Hirohiko Houjou
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  2. Masataka Seshimo
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Contributions

The molecular modelling and numerical analyses were conducted by MS under supervision of HH. The manuscript was written by HH. All authors have given approval to the final version of the manuscript.

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Correspondence to Hirohiko Houjou.

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Houjou, H., Seshimo, M. Formulation of a phonon band calculation for molecular crystals using a coarse-grained coordinate approach under periodic boundary conditions. J Math Chem 60, 613–636 (2022). https://doi.org/10.1007/s10910-022-01327-w

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  • DOI: https://doi.org/10.1007/s10910-022-01327-w

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