Momentum, quasi-momentum and hamiltonian operators in terms of arbitrary curvilinear coordinates, with special emphasis on molecular hamiltonians

Abstract

Since the conformation of physical systems is often advantageously described with the help of generalized (i.e. curvilinear) coordinates, the following questions are raised and general answers given to them: (i) What are the components of the momentum operators, those of the adjoint momentum operators and those of the hermitian momentum operators ? (ii) What is the general form of the hamiltonian operator ? (iii) What do the general forms of the momentum and hamiltonian operators become when expressed in terms of quasi-momentum operators (e.g. angular momentum component operators) ? (iv) How are the answers to the above questions affected by an arbitrary choice of normalization convention for the total wave-function ? A complete set of formulae (whatever the independent choices of coordinates, quasi-momentum operators and normalization) is given and various 'historical' formulae are shown to be particular instances of the general formulae proposed.

The rovibrational kinetic energy for complexes of rigid molecules

“…The results of this section are largely similar to previous work of Nauts and Chapuisat [30], which in turn relies on several earlier references. We also note that Van der Avoird et al have employed a similar formalism for the water trimer [7].…”

“…In this section, we derive the form of this new kinetic energy operator. Similar discussions are given by Nauts and Chapuisat [30] and Chapuisat, Belafhal, and Nauts [33]. The most notable distinction between our approach and these earlier accounts is our introduction of a new adjoint, shown in Eq.…”

“…Nauts and Chapuisat [30] call the more general momenta π a quasimomenta and reserve the term momenta for what we call the canonical momenta p a . The kinetic energy Eq.…”

Quantum-mechanical wavepacket propagation in a sparse, adaptive basis of interpolating Gaussians with collocation

“…. .,q 3NÀ6 ) T , the general expression of the kinetic energy operator can be written in a very compact form: [40][41][42][43] Tðq;pÞ ¼ À h…”

Vibration–rotation kinetic energy operators: A geometric algebra approach

“…First of all, no analytic matrix inversions are needed, unlike in the Lagrangian based approach, 6,7 where the matrix containing the covariant g q i q j elements must be inverted to obtain the g (q i q j ) elements which appear in the quantum mechanical kinetic energy operator. Even though the problem can be simplified by the judicious factorization of the matrix g q i q j , and by the direct use of the vibrational elements g (q i q j ) , one must still be able to invert at least one 3ϫ3 matrix, whose elements can be complicated functions of nuclear positions.…”