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Scaling Procedures in Vibrational Spectroscopy

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Molecular Spectroscopy—Experiment and Theory

Part of the book series: Challenges and Advances in Computational Chemistry and Physics ((COCH,volume 26))

Abstract

This chapter contains a brief review of the up-to-date scaling procedures that are used in the computational vibrational spectroscopy to improve agreement between the calculated harmonic frequencies and the observed fundamentals. Initially, the basics of vibrational spectroscopy are reminded. This includes the concept of potential energy surface, harmonic approximation , and a basic quantum chemistry treatment of the anharmonicity for a diatomic molecule. Brief description of the Wilson–Decius–Cross method for polyatomic molecules is also presented. Then the commonly used scaling procedures are discussed. The distinction between single- and multi-parameter scaling procedures is made. Four scaling procedures are reviewed. First, Pople’s uniform scaling is presented. Second, Yoshida’s wavenumber linear scaling method is discussed. Both methods are simple single-parameter frequency scaling methods. Then basics of two multi-parameter scaling methods, which are much more accurate but less straightforward to use, are given. Thus, Pulay’s scaled quantum mechanical force field method, in which scaling factors are applied directly to the calculated force constants is reviewed. Finally, introduction to quite recently proposed multi-parameter frequency scaling method, called effective scaling frequency factor method , is provided. The relevant sections start with a short description of the theory for a given method. Then a brief literature review on the historical background of methodology development is given.

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Notes

  1. 1.

    In fact, this is not the case. Inclusion of fifth-order term to Hamiltonian (2.11) with the estimated value of f(5) = -5.2 a.u. lowers the predicted by variation method transition frequency from 2936 cm−1 down to 2911 cm−1 (which is mostly due to substantial lowering of energy for  υ= 1). This value is only 25 cm−1 higher as compared with the fundamental (25 cm−1 is also the difference between the calculated and experimental harmonic frequencies).

  2. 2.

    This behavior was also observed in the case of multi-parameter scaling procedures. Although MP2 is known to predict very accurate molecular geometries, it does not provide accurate vibrational spectra.

  3. 3.

    As mentioned in the introductory section in the following we will not cite papers, where only applications are presented.

  4. 4.

    Note that the requirements of the MP2 and other correlated ab initio approaches with respect to the basis set are higher as compared with DFT, due to the necessity of the adequate description of the correlating orbitals, not only the electron density.

  5. 5.

    They were calculated in earlier works devoted to ESFF procedure, but due to a mistake in data handling they were incorrect.

Abbreviations

ARPE:

Average relative percentage error

ESF:

Effective scaling factor

ESFF:

Effective scaling frequency factor method

FC:

Force constant

FF:

Force field

IC:

Internal coordinate

LSF:

Local scaling (scale) factor

LSMF:

Least-squares merit function

NIC:

Natural internal coordinate

PEC:

Potential energy curve

PES:

Potential energy (hyper)surface

PIC:

Primitive internal coordinate

QC:

Quantum chemistry (methods, calculations…)

RMS:

Root-mean-square (deviation)

SF:

Scaling factors

SQM(FF):

Scaled quantum mechanical (force field) method; simply SQM

US:

Uniform scaling

WDC:

Wilson–Decius–Cross method

WLS:

Wavenumber linear scaling

X:

A general second-row atom

Y:

A general third-row atom

ZPVE:

Zero-point vibrational energy correction

A, B, …:

General indices used in summations over atoms

B :

Transformation matrix between internal and Cartesian coordinates

\({\mathbf{f}} = \left( {f_{1} ,f_{2} , \ldots ,f_{{N_{{\text{scl}}} }} } \right)\) :

A vector with scaling factors

\({\mathbf{f}}^{{\text{eff}}} = \left( {f_{1}^{{\text{eff}}} ,f_{2}^{{\text{eff}}} , \ldots ,f_{K}^{{\text{eff}}} } \right)\) :

Effective scaling factors

\({\mathbf{f}}^{{\text{opt}}} = \left( {f_{1}^{{\text{opt}}} ,f_{2}^{{\text{opt}}} , \ldots ,f_{{N_{{\text{scl}}} }}^{{\text{opt}}} } \right)\) :

A vector with optimal scaling factors

f x :

Cartesian force constant matrix with elements \(f_{ij}^{x}\)

F :

Force constant matrix in internal coordinate representation with elements \(F_{\mu \nu }\)

i, j, …:

General indices used in summations over Cartesian coordinates

I, J, …:

General indices used to indicate types of internal coordinates in multi-parameter scaling procedures

K :

Number of vibrational degrees of freedom, K = 3N − 6 (or K = 3N − 5 in the case of linear molecules)

L :

Number of redundant primitive internal coordinates (typically L > 3N)

L :

Transformation matrix between Cartesian displacements and normal coordinates

N :

Number of atoms in a molecule

N mol :

Number of molecules in a training set

N scl :

Number of scaling factors

N typ :

Number of types (groups) of internal coordinates (=Nscl)

N vib :

Number of vibrational modes used in optimization of scaling factors; additional superscript indicates the molecule the number of modes refers to

p, q, …:

General indices used in summations over frequencies of a training set of molecules, basis functions, etc.

s = (s1, s2, …):

Vector with internal coordinates; additional superscripts refer to the type a given coordinate belongs to. Additional subscript “e” denotes the equilibrium values. When used in the potential energy expression the symbols refer to the deviations of coordinates from their equilibrium values

T :

Kinetic energy

U :

Potential energy

w :

Weight used for a given frequency in the scaling factor optimization procedure

α :

Transformation matrix between internal coordinates displacements and normal coordinates

\(\theta^{{\text{vib}}}\) :

Vibrational temperature

μ, ν, …:

General indices used in summations over internal or normal coordinates as well as normal modes of a molecule

\(\nu^{\text{h}}\), \(\nu^{{\text{expt}}}\), \(\nu^{{\text{expt,h}}}\), \(\nu^{{\text{scl}}}\):

Harmonic, experimental, experimental harmonic, and scaled frequencies, respectively (in cm−1)

\(\psi_{p}^{\text{h}}\) :

Eigenfunctions of harmonic oscillator

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Authors and Affiliations

  1. Faculty of Chemistry, Maria Curie-Skłodowska University, Maria Curie-Skłodowska Sq. 3, 20-031, Lublin, Poland

    Olga Bąk & Piotr Borowski

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  1. Olga Bąk
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  1. Faculty of Materials Science and Ceramics, AGH University of Science and Technology, Kraków, Poland

    Andrzej Koleżyński

  2. Faculty of Materials Science and Ceramics, AGH University of Science and Technology, Kraków, Poland

    Magdalena Król

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Bąk, O., Borowski, P. (2019). Scaling Procedures in Vibrational Spectroscopy. In: Koleżyński, A., Król, M. (eds) Molecular Spectroscopy—Experiment and Theory. Challenges and Advances in Computational Chemistry and Physics, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-030-01355-4_2

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