Interaction energy between nonpolar particles

mastrofoffi

Homework Statement
I want to evaluate the potential energy of a configuration where nonpolar particle 1, of charge $q$ and polarizability $\alpha_1$ is at distance $r$ from nonpolar particle 2, of zero charge and polarizability $\alpha_2$.

I am neglecting gravitational forces and dispersion forces, net charge can be thought as point-like and polarizabilites are considered scalar quantities. I'll be working in units where $4\pi\epsilon_0\epsilon=\epsilon$ for brevity.

I could not manage to find references to check my work.
Homework Equations
Electric field generated by a point-charge: $E_q(r) = \dfrac{q}{\epsilon r^2}$
Electric field generated by a point-dipole: $E_\mu(r) = \dfrac{\mu\sqrt{1+3\cos^2\theta}}{\epsilon r^3}$
Dipole moment induced by electric field: $\mu = \alpha E$
I tried an approach where I build the interactions one by one and then add them all to find the total potential energy.
The electric field of particle 1 induces a dipole moment on particle 2 given by $\mu_2 = \alpha_2 E_q(2)$(by $E(2)$ I mean the field evaluated at the position of particle 2); this induced dipole originates a reaction field which acts on particle 1
$$E_\mu(1) = -\dfrac{\mu_2\sqrt{1+3\cos^2\theta}}{\epsilon r^3} = -\dfrac{2\mu_2}{\epsilon r^3} = -\dfrac{2\alpha_2E_q(2)}{\epsilon r^3}$$
where I used the fact that $\theta$ must be $0$ or $\pi$ because $E_q(2)$ acts along the line between the particles, hence the dipole is parallel to the field if $q>0$, antiparallel if $q<0$; this also explains the minus sign since this interaction will always be attractive.

Now I can evaluate the force that is acting on particle 1 as
$$F(1) = qE_\mu(1) = -\dfrac{2q\alpha_2E_q(2)}{\epsilon r^3} = -\dfrac{2q^2\alpha_2}{\epsilon^2 r^5}$$
The potential energy of this configuration can then be found as the work needed to bring the charge from infinity up to a distance $r$ from the dipole, which is equal to the work needed for a charge $q$ to create a dipole $\mu_2$:
$$v'(r) = -\int_\infty^r \text{d}s F(s) = -\dfrac12\alpha_2 E_\mu^2(1) = -\dfrac{1}{\epsilon^2}\dfrac{\alpha_2 q^2}{2r^4}$$

I am pretty sure this charge-dipole contribution to the potential energy is correct; now is where I start having doubts.
The induced dipole on particle 2 induces in turn a dipole on particle 1, such that $\mu_1 = \alpha_1E_\mu(1)$; this process should physically occur "simultaneously" to the induction of $\mu_1$ so that every infinitesimal increment $\text{d}\mu_2$ induces an infinitesimal $\text{d}\mu_1$, but I think the result should not be affected by evaluating the 2 processes one after the other, since I only care for the equilibrium state and not for the transient.
The $\mu_1$ dipole is aligned to the $\mu_2$ dipole, thus the work needed to induce the dipole can be written as
$$v''(r) = -\int_0^{\mu_1}\text{d}\mu E_\mu(1) = -\dfrac{1}{2}\alpha_1\int_0^{E_\mu(1)} \text{d}E E = -\dfrac{1}{2}\alpha_1 E_\mu^2(1) = -\dfrac{2\mu_2^2\alpha_1}{\epsilon^2 r^6}$$
since the interaction energy between a permanent dipole and an electric field has the form $v = -\vec{E}\cdot\vec{\mu}$.

At this point what I have is
$$v(r) = v'(r) + v''(r) = -\dfrac{1}{\epsilon^2}\left[ \dfrac{\alpha_2 q^2}{2r^4} + \dfrac{2\mu_2^2\alpha_1}{r^6} \right]$$
Now I should be taking into account the fact that the induced dipole on particle 1 alters the field on particle 2, which adds another contribution the induced dipole of particle 2 and so on.. but the spatial decays add up pretty fast: the second term in $v(r)$ is already $\sim r^{-10}$, so I don't think I want to go further unless, for some reason, it adds essential variations to the result.
Are my approach and result correct?
If not, what is wrong and why?
What if I wanted to consider 2 polar particles instead, i.e., each of them possess a permanent dipole $\mu_i^{(0)}$? Will the permanent dipole contributions just add on top on the non-polar interaction(I believe this should work fine)(also reminding that in this case I would have an orientational contribution arising in the polarizibilites), or should I rework it from the beginning(if so, why)?
I'm doing this just 'for fun', no constraints, so feel free to add whatever you think is relevant. Thank you.

EDIT: I'm not neglecting dispersion forces because I think they are not relevant, I'm just temporarily pretending they don't exist.

Last edited:
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