Molecules with dipole moments

[StatusX]

It can be shown that if a hamiltonian is invariant under space inversion and if an eigenstate is non-degenerate, then the state is either even or odd in each position coordinate. So if there is a perturbation given by a uniform electric field, which has:

$$H_1 = - \vec E \cdot \left( \sum_i q_i \vec r_i \right)$$

then there will be no energy shift to first order, since the square of the wavefunction is even in each coordinate, so the integral over each position coordinate will vanish, and so the expectation value of H_1 will be zero. But there are molecules with permanent dipole moments, like water, and these will have energy shifts linear in the electric field, so there must be some degeneracy.

I'm also told that if the electric field is below a certain magnitude, the shift will no longer be linear. I'm guessing this is because the degeneracy is not perfect, and once <H_1> is of the order of the splitting the degeneracy is no longer important. I'm having a hard time thinking about eigenstates of an entire molecule, and how there could be slight degeneracies. Can anyone help me understand this?

 

[StatusX]

I think I've got it. The electrons in the H2O molecule can tunnel from one H atom to another, and this introduces a small tunelling splitting (ie, between a symmetric and an antisymmetric combination of s-orbitals around each atom). Moving the electron from one H atom to another is equivalent to rotating the atom, and so the magnitude of the splitting is on the order of the rotational energy levels of the molecule (this part sounds sketchy). When the perturbation is much larger than this splitting, the states look degenerate, so there can be a dipole moment, while when the perturbation is less than the splitting, the states look non-degenerate and so the energy shift is no longer linear in the electric field. Can someone let me know if this sounds right?

 

[denian]

I can provide a response to the content you have presented. It is true that molecules with dipole moments will experience energy shifts in the presence of an external electric field. This is because the electric field can interact with the permanent dipole moment of the molecule, causing a change in its energy.

The concept of degeneracy in this context refers to the existence of multiple energy levels with the same energy. In the absence of an external electric field, molecules with permanent dipole moments may have degenerate energy levels. This means that the energy of these levels is the same, but the wavefunctions corresponding to these levels may be different.

When an external electric field is applied, the degeneracy is lifted, and the energy levels split. However, as you have correctly pointed out, the splitting may not be linear for all electric field strengths. This is because the degeneracy is not perfect, and there may be slight differences in the energies of the degenerate levels. As the electric field strength increases, these differences become more significant, and the degeneracy is no longer important.

To understand the concept of degeneracy in molecules, it is helpful to think about the different ways in which the molecule can vibrate or rotate. Each of these modes of motion can be associated with a different energy level. In the absence of an external electric field, some of these modes may have the same energy, leading to degeneracy. However, when an electric field is present, the different modes of motion may interact differently with the field, causing a splitting of the energy levels.

I hope this helps to clarify the concept of degeneracy in molecules with dipole moments and how it relates to energy shifts in the presence of an external electric field.