Conservation of zero-point energy in classical trajectory computations by a simple semiclassical correspondence

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Abstract
A simple practical procedure which ensures that the energy in a molecular vibrational mode does not decrease below its zero‐point value is discussed and applied. The method is based on taking the classical limit of the Hamiltonian and thereby deriving classical equations of motion which are solved via a standard classical trajectory computation. We refer to this as the ‘‘reference’’ trajectory. It is argued that the reference solution differs from what one would obtain if one were to begin with a classical description of the problem; the difference being that the reference computation puts the zero of energy at the correct, quantum‐theoretic, zero, i.e., at the zero point. To obtain a fully classical‐like solution one needs to shift the energy and period of the reference trajectory and the different ways of doing this are discussed. The resulting, energy, and phase shifted, equivalent classical trajectory cannot, by construction, lose the zero‐point energy from the modes in which it is placed. The method is discussed first for the obvious case of a single oscillator, including the role of the anharmonicity, and is then applied to a variety of dimers [I2He, ArHBr, (HF)2] where a higher frequency mode is coupled to a low‐frequency one and the problem is to prevent the (high) zero‐point energy from being made available for transfer to the far weaker mode. Other advantages of the proposed scheme, such as the correct frequency dependence of the power spectrum, and its application to an unbound motion in the continuum are also discussed.