Chapter 9
Least-Squares Method
Abstract This chapter is dedicated to the nonlinear weighted least-squares method.
The practical solution is explained, and useful statistical diagnostics are described.
The importance of the choice of correct weights is discussed. The mixed regression,
a method particularly useful for a structure determination, is detailed together with
several examples. Finally, the accuracy of the structural parameters is assessed.
9.1 Introduction
The method of least squares is extremely important to analyze a spectrum but it is
also essential to determine a structure because, with the exception of the substitution
method (see Sect. 6.4.3), which calculates separately each Cartesian coordinate from
the isotopic difference of the moments of inertia, all other methods necessitate the
use of the least-squares method.
This method has already been reviewed with an emphasis on structure determination (Demaison 2011; Demaison et al. 1997; Groner 2000; Mendolicchio et al. 2017)
but, taking into account its crucial role, it is worth a detailed presentation. A general
discussion of the least-squares method may be found for instance in Albritton et al.
(1976) and Sen and Srivastava (1990).
The parameters to be fitted are a non-redundant set of internal coordinates: bond
lengths r ij between atoms i and j, bond angles ∠(i, j, k) between adjacent bonds ij
and jk, and dihedral angles τ ijkl between the planes ijk and jkl. The most obvious
choice is the set of internal coordinates defining the Z-matrix, which is also used
to optimize the ab initio structure. The difficulty is that this choice is not univocal
and different choices may lead to different results. For this reason, it is essential to
choose a set of internal coordinates that preserve the symmetry of the molecule. It
often requires the introduction of dummy atoms.
The main sources of data are rotational spectroscopy, gas-phase electron diffraction, and ab initio calculations, see Chaps. 2–7. Rotational spectroscopy furnishes
moments of inertia I g (g = a, b, c) that are a nonlinear function of the internal coordinates. Each isotopologue has different moments of inertia, which are obtained from
© Springer Nature Switzerland AG 2020
J. Demaison and N. Vogt, Accurate Structure Determination of Free
Molecules, Lecture Notes in Chemistry 105,
https://doi.org/10.1007/978-3-030-60492-9_9
233
Least-Squares Method
Abstract This chapter is dedicated to the nonlinear weighted least-squares method.
The practical solution is explained, and useful statistical diagnostics are described.
The importance of the choice of correct weights is discussed. The mixed regression,
a method particularly useful for a structure determination, is detailed together with
several examples. Finally, the accuracy of the structural parameters is assessed.
9.1 Introduction
The method of least squares is extremely important to analyze a spectrum but it is
also essential to determine a structure because, with the exception of the substitution
method (see Sect. 6.4.3), which calculates separately each Cartesian coordinate from
the isotopic difference of the moments of inertia, all other methods necessitate the
use of the least-squares method.
This method has already been reviewed with an emphasis on structure determination (Demaison 2011; Demaison et al. 1997; Groner 2000; Mendolicchio et al. 2017)
but, taking into account its crucial role, it is worth a detailed presentation. A general
discussion of the least-squares method may be found for instance in Albritton et al.
(1976) and Sen and Srivastava (1990).
The parameters to be fitted are a non-redundant set of internal coordinates: bond
lengths r ij between atoms i and j, bond angles ∠(i, j, k) between adjacent bonds ij
and jk, and dihedral angles τ ijkl between the planes ijk and jkl. The most obvious
choice is the set of internal coordinates defining the Z-matrix, which is also used
to optimize the ab initio structure. The difficulty is that this choice is not univocal
and different choices may lead to different results. For this reason, it is essential to
choose a set of internal coordinates that preserve the symmetry of the molecule. It
often requires the introduction of dummy atoms.
The main sources of data are rotational spectroscopy, gas-phase electron diffraction, and ab initio calculations, see Chaps. 2–7. Rotational spectroscopy furnishes
moments of inertia I g (g = a, b, c) that are a nonlinear function of the internal coordinates. Each isotopologue has different moments of inertia, which are obtained from
© Springer Nature Switzerland AG 2020
J. Demaison and N. Vogt, Accurate Structure Determination of Free
Molecules, Lecture Notes in Chemistry 105,
https://doi.org/10.1007/978-3-030-60492-9_9
233