A Calculating the interaction potential between 2 molecules

[kelly0303]

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?

 

[anuttarasammyak]

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 ?

 

[kelly0303]

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.

 

[anuttarasammyak]

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

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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
arccos(1/\sqrt{3})= 54.74 \ degree
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.