# Eckart ro-vibrational Hamiltonians via the gateway Hamilton operator: theory and practice

## TL;DR

This paper discusses the development and application of Eckart ro-vibrational Hamiltonians using the gateway Hamiltonian method, demonstrating that rotation is unnecessary for constructing these Hamiltonians and highlighting the advantages of the gateway approach.

## Contribution

It introduces a new application of the gateway Hamiltonian method to Eckart Hamiltonians, simplifying calculations by eliminating the need for rotation and providing a geometric interpretation of Eckart displacements.

## Key findings

- Numerical examples confirm no need for rotation in Eckart Hamiltonians.
- Gateway method simplifies calculations by avoiding differentiation of the rotating matrix.
- A shortest Eckart displacement vector can be uniquely defined and is often shorter.

## Abstract

Recently, a general expression for Eckart-frame Hamilton operators has been obtained by the gateway Hamiltonian method ({\it J. Chem. Phys.} {\bf 142}, 174107 (2015); {\it ibid.} {\bf 143}, 064104 (2015)). The kinetic energy operator in this general Hamiltonian is nearly identical with that of the Eckart-Watson operator even when curvilinear vibrational coordinates are employed. Its different realizations correspond to different methods of calculating Eckart displacements. There are at least two different methods for calculating such displacements: rotation and projection. In this communication the application of Eckart Hamiltonian operators constructed by rotation and projection, respectively, is numerically demonstrated in calculating vibrational energy levels. The numerical examples confirm that there is no need for rotation to construct an Eckart ro-vibrational Hamiltonian. The application of the gateway method is advantageous even when rotation is used, since it obviates the need for differentiation of the matrix rotating into the Eckart frame. Simple geometrical arguments explain that there are infinitely many different methods for calculating Eckart displacements. The geometrical picture also suggests that a unique Eckart displacement vector may be defined as the shortest (mass-weighted) Eckart displacement vector among Eckart displacement vectors corresponding to configurations related by rotation. Its length, as shown analytically and demonstrated by way of numerical examples, is equal to or less than that of the Eckart displacement vector one can obtain by rotation to the Eckart frame.