1 3
Theor Chem Acc (2015) 134:108
DOI 10.1007/s00214-015-1713-8
REGULAR ARTICLE
Internal coordinates and orthogonality: features of the
pseudoinverse of the Eliashevich–Wilsonian B -matrix
F. Kalincsák
1 · G. Pongor
1
Received: 16 June 2015 / Accepted: 3 August 2015 / Published online: 18 August 2015
© Springer-Verlag Berlin Heidelberg 2015
used in quantum chemical calculations not directly in
the quantum mechanical determination of single-point
energies rather in representing the equilibrium and
transition-state geometries. The dimension of a complete and non-redundant set of the internal coordinates
is smaller than the number of the Cartesians, but this
is not the main reason for their application. Their most
important feature is that in terms of internal coordinates, the Hessian matrix could be well approximated
with a simple diagonal matrix [ 4 ] in gradient geometry
optimization procedures. In this paper, we will deal
with the system of internal coordinates, in order to
give a deeper insight and characterize some interesting
features of the pseudoinverse of the Eliashevich–Wilsonian matrix B [ 1 , 2 ].
2 Theory
Instead of the usual Cartesians, we can also apply a complete and non-redundant set of the so-called internal coordinates. In order to understand their application in vibrational calculations, let us consider a molecular system
consisting of N nuclei; let the Cartesian displacement vectors of the nuclei be d 1 , d 2 , . . . , d N around their equilibrium
geometry in the usual three-dimensional Euclidean space
E 3 . (The expression for the n -th Cartesian displacement
vector is d n = ρ n − ρ 0
n , where ρ n is the instantaneous position vector of the n -th nucleus, and ρ 0
n is the position vector
of the same nucleus at equilibrium. Hereafter, these position vectors correspond to an arbitrary origin. Note that for
simplicity, we omit the explicit use of the atomic masses,
i.e., do not use mass-weighted Cartesians.) A single-point
δ of a hypothetical 3 N -dimensional space ℜ 3N (δ ∈ ∈ 3N ) ,
defi ned as
Abstract It is shown that the system of unit vectors corresponding to the internal coordinates is non-orthogonal
generally. The deduction starts with the well-known orthonormality of unit vectors of the Cartesian coordinates. The
crucial point of the GDIIS method is discussed regarding
a “partially isomorphic” relationship between two vector
spaces. Some features of the pseudoinverse of the Eliashevich–Wilsonian matrix B are deduced and discussed:
these are analogous to the conditions formulated originally
for the elements of the B -matrix.
Keywords Internal coordinates · Reciprocal internal
displacement coordinates · Non-orthogonal · B -matrix ·
Pseudoinverse · Sayvetz conditions · GDIIS
1 Introduction
The use of the internal coordinates is a relevant and
interesting topic both in the field of the vibrational
calculations [ 1 – 3 ] and in quantum chemistry [ 4 ].
Though vibrational calculations could be carried out
in terms of Cartesians as well, the system of valencetype internal coordinates gives the most significant
and physically meaningful description of the vibrational potential energy. Internal coordinates are often
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* G. Pongor
pongor@chem.elte.hu
1
Department of Inorganic Chemistry, Institute of Chemistry ,
Eötvös Loránd University , Pázmány Péter sétány 1/A ,
Budapest 1117 , Hungary
41
Reprinted from the journal
Theor Chem Acc (2015) 134:108
DOI 10.1007/s00214-015-1713-8
REGULAR ARTICLE
Internal coordinates and orthogonality: features of the
pseudoinverse of the Eliashevich–Wilsonian B -matrix
F. Kalincsák
1 · G. Pongor
1
Received: 16 June 2015 / Accepted: 3 August 2015 / Published online: 18 August 2015
© Springer-Verlag Berlin Heidelberg 2015
used in quantum chemical calculations not directly in
the quantum mechanical determination of single-point
energies rather in representing the equilibrium and
transition-state geometries. The dimension of a complete and non-redundant set of the internal coordinates
is smaller than the number of the Cartesians, but this
is not the main reason for their application. Their most
important feature is that in terms of internal coordinates, the Hessian matrix could be well approximated
with a simple diagonal matrix [ 4 ] in gradient geometry
optimization procedures. In this paper, we will deal
with the system of internal coordinates, in order to
give a deeper insight and characterize some interesting
features of the pseudoinverse of the Eliashevich–Wilsonian matrix B [ 1 , 2 ].
2 Theory
Instead of the usual Cartesians, we can also apply a complete and non-redundant set of the so-called internal coordinates. In order to understand their application in vibrational calculations, let us consider a molecular system
consisting of N nuclei; let the Cartesian displacement vectors of the nuclei be d 1 , d 2 , . . . , d N around their equilibrium
geometry in the usual three-dimensional Euclidean space
E 3 . (The expression for the n -th Cartesian displacement
vector is d n = ρ n − ρ 0
n , where ρ n is the instantaneous position vector of the n -th nucleus, and ρ 0
n is the position vector
of the same nucleus at equilibrium. Hereafter, these position vectors correspond to an arbitrary origin. Note that for
simplicity, we omit the explicit use of the atomic masses,
i.e., do not use mass-weighted Cartesians.) A single-point
δ of a hypothetical 3 N -dimensional space ℜ 3N (δ ∈ ∈ 3N ) ,
defi ned as
Abstract It is shown that the system of unit vectors corresponding to the internal coordinates is non-orthogonal
generally. The deduction starts with the well-known orthonormality of unit vectors of the Cartesian coordinates. The
crucial point of the GDIIS method is discussed regarding
a “partially isomorphic” relationship between two vector
spaces. Some features of the pseudoinverse of the Eliashevich–Wilsonian matrix B are deduced and discussed:
these are analogous to the conditions formulated originally
for the elements of the B -matrix.
Keywords Internal coordinates · Reciprocal internal
displacement coordinates · Non-orthogonal · B -matrix ·
Pseudoinverse · Sayvetz conditions · GDIIS
1 Introduction
The use of the internal coordinates is a relevant and
interesting topic both in the field of the vibrational
calculations [ 1 – 3 ] and in quantum chemistry [ 4 ].
Though vibrational calculations could be carried out
in terms of Cartesians as well, the system of valencetype internal coordinates gives the most significant
and physically meaningful description of the vibrational potential energy. Internal coordinates are often
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* G. Pongor
pongor@chem.elte.hu
1
Department of Inorganic Chemistry, Institute of Chemistry ,
Eötvös Loránd University , Pázmány Péter sétány 1/A ,
Budapest 1117 , Hungary
41
Reprinted from the journal