FrontierScience / d1cec1fb-13bb-475a-b2f1-956db5ef3b77 / Consider two atoms separated by a distance `R`. Since they are electrically neutral, there is no force between them in the absence of perturbations. However, if one of the atoms is…

Consider two atoms separated by a distance `R`. Since they are electrically neutral, there is no force between them in the absence of perturbations. However, if one of the atoms is slightly polarized, a very weak attractive force emerges between the two atoms. Here, we assume that both atoms consist of a positively charged nucleus `(+e)` fixed in place (the distance `R` is the inter-nuclear distance)  and an electron `(−e)` with mass `m`, connected to the nucleus by springs with spring constant `k`.  We further assume that the electrons are constrained to move along the line connecting the two nuclei, with displacements `x₁` and `x₂`, respectively. Thus, the elastic potential energy of the two springs can be written as `(1)/(2)kx₁²` and `(1)/(2)kx₂²`, respectively.

Under the condition that both `|x₁|` and `|x₂|` are much smaller than `R`, find the Coulomb electrostatic potential energy `U`of this system in terms of e,x₁,x₂,ϵ₀,R, where ϵ₀ is the electric vacuum permittivity.

Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.

Consider two atoms separated by a distance `\( R \)`. Since they are electrically neutral, there is no force between them in the absence of perturbations. However, if one of the atoms is slightly polarized, a very weak attractive force emerges between the two atoms. Here, we assume that both atoms consist of a positively charged nucleus `\( (+e) \)` fixed in place (the distance `\( R \)` is the inter-nuclear distance)  and an electron `\( (-e) \)` with mass `\( m \)`, connected to the nucleus by springs with spring constant `\( k \)`.  We further assume that the electrons are constrained to move along the line connecting the two nuclei, with displacements `\( x_1 \)` and `\( x_2 \)`, respectively. Thus, the elastic potential energy of the two springs can be written as `\( \frac{1}{2} kx_1^2 \)` and `\( \frac{1}{2} kx_2^2 \)`, respectively.

Under the condition that both `\( \left|x_{1}\right| \)` and `\( \left|x_{2}\right| \)` are much smaller than `\( R \)`, find the Coulomb electrostatic potential energy `\( U \)`of this system in terms of \(e, x_1, x_2, \epsilon_0, R\), where \(\epsilon_0 \) is the electric vacuum permittivity.

Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.