Quantum and Thermal Corrections to the Ground-State Potential Energy

6 QUANTUM AND THERMAL CORRECTIONS TO THE GROUND-STATE POTENTIAL ENERGY 

We end this chapter by discussing a couple of relevant quantum and thermal corrections to the potential energy. 

A classical mechanical system at equilibrium is at rest in a (local) minimum on the PES and will have a potential energy given by the minimum value, E . A (real) quantum system will not, in general, be able to attain such an absolute potential energy minimum. This is due to the Heisenberg uncertainty principle, which prescribes that the uncertainty in the position of a particle. Ax, is related to the uncertainty in the momentum, Ap, of the particle  

This prevents the particle from being perfectly at rest in a perfectly specified position. For abound system, such as an adsorbate, small fluctuations around the local potential energy minimum are often accurately represented as harmonic vibrations. This allows us to make a simple correction to the potential energy minimum in order to obtain a 

The contributions from different vibrational modes are additive, such that if there are modes vibrational modes in the potential energy minimum, the total so-called ZPE correction is given by 

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