Gradient and Hessian of the Coulomb/Electrostatic Energy

[decerto]

I have a function

$$\displaystyle V(x)=\frac{1}{2}\sum_i \sum_{j \neq i} q_i q_j \frac{1}{\left|r_i - r_j\right|}$$ where ##r_i=\sqrt{x_i^2+y_i^2+z_i^2}## which is the coulomb potential energy of a system of charges.

I need to calculate ##\frac{\partial V}{\partial x_k}## and ##\frac{\partial^2 V}{\partial x_k \partial x_l}## for an optimization routine.

I guess I want to treat the set ##(x_i , y_i, z_i)## where i runs from 1 to N as independent variables ##(x_j)## where j runs from 1 to 3N (the direct sum of the position vectors).

Would ##r_i = \sqrt{x_{3i-2}+x_{3i-1}+x_{3i}}## then?

I ask because but I am not sure how to calculate the ##\frac{\partial r_i}{\partial x_k}## term

I was also thinking maybe I can just calculate ##\frac{\partial r_i}{\partial x_k}, \frac{\partial r_i}{\partial y_k}, \frac{\partial r_i}{\partial z_k}## separately and then just relabel them as independent variables but I am not sure if this will work.

Any help would be appreciated

 

[BvU]

Hi,

##r_i## is not ##\sqrt{x_i^2+y_i^2+z_i^2}## !
## r_i ## is simply a vector ##\vec r_i = (x_i, y_i,z_i)## !
You need ##| \vec r_i - \vec r_j | ## which is the square root of ## (\vec r_i - \vec r_j) \cdot (\vec r_i - \vec r_j) ##

In your notation (somewhat awkward) this would be $$
| \vec r_i - \vec r_j | = \sqrt{ ( x_{3i-2} - x_{3j-2} )^2 + ( x_{3i-1} - x_{3j-1} )^2 + ( x_{3i} - x_{3j} )^2 }
$$

 

[decerto]

Hi,

##r_i## is not ##\sqrt{x_i^2+y_i^2+z_i^2}## !
## r_i ## is simply a vector ##\vec r_i = (x_i, y_i,z_i)## !
You need ##| \vec r_i - \vec r_j | ## which is the square root of ## (\vec r_i - \vec r_j) \cdot (\vec r_i - \vec r_j) ##

In your notation (somewhat awkward) this would be $$
| \vec r_i - \vec r_j | = \sqrt{ ( x_{3i-2} - x_{3j-2} )^2 + ( x_{3i-1} - x_{3j-1} )^2 + ( x_{3i} - x_{3j} )^2 }
$$

Thanks for that correction, bit of an oversight!

How would you suggest I calculate the gradient and the hessian analytically without that notation?

 

[BvU]

Your ##V## looks an awful lot like the ##\ \ W\quad (1.51)\ \ ## in my 1974 Jackson 2^nd edition, except that ##W## is a scalar and you write ##V(x)##. What does this ##x## stand for ? And how do you intend to optimize if ##V## diverges when ##\vec r_i \rightarrow \vec r_j ## ? Are there any boundary conditions ? In short: could you tell us a little more of your plans ?

 

[decerto]

Your ##V## looks an awful lot like the ##\ \ W\quad (1.51)\ \ ## in my 1974 Jackson 2^nd edition, except that ##W## is a scalar and you write ##V(x)##. What does this ##x## stand for ? And how do you intend to optimize if ##V## diverges when ##\vec r_i \rightarrow \vec r_j ## ? Are there any boundary conditions ? In short: could you tell us a little more of your plans ?

Sorry V is the coulomb potential for a system of charges each with position ##r_i=(x_i, y_i, z_i)##, it is a scalar function as you say of these positions. I want to create a gradient ##\frac{\partial V}{\partial x_k}## and a hessian ##\frac{\partial^2 V}{\partial x_k \partial x_l}## where the coordinates ##x_k## are the direct sum of the ##(x_i, y_i, z_i)##

So for a system of two charges I would have ##r_1=(x_1, y_1, z_1) \ , \ r_2=(x_2, y_2, z_2)## so ##\textbf{x}=(x_1, y_1, z_1, x_2, y_2, z_2)## and my gradient would look like ##(\frac{\partial V}{\partial x_1}, \frac{\partial V}{\partial y_1}, \frac{\partial V}{\partial z_1}, \frac{\partial V}{\partial x_2}, \frac{\partial V}{\partial y_2}, \frac{\partial V}{\partial z_2})##

I originally thought it would be a good idea to rewrite ##\textbf{x}=(x_1, y_1, z_1, x_2, y_2, z_2)## as ##(x_1, x_2, x_3, x_4, x_5, z_6)## using the coordinate transformations ##x_i \to x_{3i-2}, \ y_i \to x_{3i-1}, \ z_i \to x_{3i}## but as you said the notation is clunky.