Vibration–rotation kinetic energy operators: A geometric algebra approach

Abstract

The elements of the reciprocal metric tensor g (q i q j ) , which appear in the exact internal kinetic energy operators of polyatomic molecules can, in principle, be written as the mass-weighted sum of the inner products of measuring vectors associated to the nuclei of the molecule. In the case of vibrational degrees of freedom, the measuring vectors are simply the gradients of the vibrational coordinates. It is more difficult to find these vectors for the rotational degrees of freedom, because the components of the total angular momentum operator are not conjugated to any rotational coordinates. However, by the methods of geometric algebra, the rotational measuring vectors are easily calculated for any geometrically defined body-frame, without any restrictions to the number of particles in the system. In order to show that the rotational measuring vectors produced by the present method agree with the known results, the general formulas are applied to the triatomic bond-z, and to the triatomic angle bisector frame. All the rotational measuring vectors are also explicitly derived for a new triatomic body frame defined in terms of two Jacobi vectors. As a final application, all the rotational measuring vectors are presented for a new N-atomic frame defined in terms of NϪ1 Jacobi vectors, and for a simple N-atomic frame defined in terms of N nuclear position vectors (Nϭ3,4,5,6, . . . ).

Force‐stepping integrators in Lagrangian mechanics

“…Such systems are known as N ‐body problems and are the subject of a vast body of literature (cf., e.g. 18–22). Reduction with respect to translational symmetry may be achieved by recourseto Jacobi coordinates q ′∈ E ( n ) N −1 and center of mass coordinates q cm ∈ E ( n ), defined as where m i is the mass of the i th particle and is the total mass of the system.…”

“…The reduction with respect to the rotational symmetry, in applications which involve an arbitrary number of particles ( N ⩾3)and non‐vanishing angular momentum, is presently the subject of active research(cf., e.g. 19–22). Unfortunately, it is not possible to construct internal or shape coordinates which result in a reduced Lagrangian of the form (1).…”

A new proof of the molecular Hamiltonian in Radau coordinates for triatomic molecules based on tensor form

“…Handy is a prominent pioneer in introducing internal coordinates for construction of the molecular Hamiltonian for tri, tetra and penta-atomic molecules [7][8][9][10][11]. The essential disadvantage of internal coordinates is that the molecular Hamiltonian in internal coordinates is complicated [12][13][14].…”

Complete nuclear motion Hamiltonian in the irreducible normal mode tensor operator formalism for the methane molecule

“…[27][28][29] The curvilinear coordinates are usually considered as better suitable to describe the nuclear motions over a wide range of geometries but imply a treatment of complicated expressions of the kinetic energy operators as the number of atoms increases. 14,[30][31][32][33][34][35] There is a lot of works devoted to the calculation of vibrational energy levels using curvilinear internal coordinates. 6,[36][37][38] For systems with four or more atoms, Wang and Carrington used contracted basis functions combined with a Lanczos eigensolver for computing vibrational spectra.…”