No abstract

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]. TheoryInstead 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 3N-dimensional space ℜ 3N (δ ∈ ℜ 3N ), defined 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.