Calculating the interaction potential between 2 molecules

Hello! I am trying to understand how to calculate the interaction potential between 2 molecules and I am a bit unsure how to proceed. For reference, I am sharing this paper, in particular Fig. 1b. They add some microwave fields which complicates things, but I am interested in reproducing the potential curves without the coupling to microwaves (dashed lines in that figure). The potential responsible for these curves is the dipole-dipole interaction one, which is given by:

$$\frac{\vec{d_1}\cdot\vec{d_2}-3(\hat{R}\cdot\vec{d_1})(\hat{R}\cdot\vec{d_2})}{R^3}$$

where ##d_{1,2}## are the dipole moment operators for the 2 molecules and R is the distance vector between them. Using the basis ##|J,M_J>##, where J is the rotational quantum number of the molecule and ##M_J## is the projection along some lab frame axis (in this case they apply a magnetic field). In the end, in Fig. 1b, they show the potential for the interaction of molecules in the ##|0,0>## state each. But I am not sure why is this not identically zero. When doing the math you end up with ##<0,0|d_1|0,0>##, which is zero. In general, I am not sure how we can have dipole-dipole interaction, given that the molecules are not polarized (they apply no electric field and now I am interested in the case where the microwaves are not applied). What am I doing wrong?

d_1 is intrinsic dipole moment which one of the two-atom molecule pair has. Why do you think its expectation value is zero , not constant ?

anuttarasammyak said:

d_1 is intrinsic dipole moment which one of the two-atom molecule pair has. Why do you think its expectation value is zero , not constant ?

But the dipole moment in the lab frame (where the 2 molecules interact) is zero. This follows from the fact that the ##|0,0>## wavefunction has definite parity, while the ##d_1## is a parity odd operator.

I get your point. If |0,0> shows 1s like homobeneous wavefuction, aveage dipole moment is zero.
However, the paper says

---------
However, the direct effective DDI alone, which
is ∝ −d2 eff(1 − 3 cos2 θ)/r3
, cannot provide the 3D repulsive barrier between two molecules as it is attractive for collisional angles θ (w.r.t the quantization axis along the vertical direction) large than 54.7◦
----------

where I would add
[tex] arccos(1/\sqrt{3})= 54.74 \ degree [/tex]
The authors seem to calculate DDI energy not using |0.0> states. I just observe it and do not have a point of concilliation with your question.