lennardjones | Lennard Jones Molecular Dynamics in C

The curve_fit fits a function f(w, x[i]) to points y[i] by finding w that minimizes sum((f(w, x[i] - y[i])**2 for i in range(n)). As you will read in the first line after the function definition

[It uses] non-linear least squares to fit a function, f, to data.

It refers to least_squares where it states

Given the residuals f(x) (an m-D real function of n real variables) and the loss function rho(s) (a scalar function), least_squares finds a local minimum of the cost function F(x):

Curve fitting is a kind of convex-cost multi-objective optimization. Since the each individual cost is convex, you can add all of them and that will still be a convex function. Notice that the decision variables (the parameters to be optimized) are the same in every point.

In my understanding for each energy level you have a different set of parameters, if you write it as a curve fitting problem, the objective function could be expressed as sum((f(w[i], x[i]) - y[i])**2 ...), where y[i]is determined by the energy level. Since each of the terms in the sum is independent on the other terms, this is equivalent to finding each group of parametersw[i]separately minimizing(f(w[i], x[i]) - y[i])**2`.

Convexity is a very convenient property for optimization because it ensures that you will have only one minimum in the parameter space. I am not doing a detailed analysis but have reasonable doubts about the convexity of your energy function.

1. The Lennard Jones function has the difference of a repulsive and an attractive force both with negative even exponent on the distance this alone is very unlikely to be convex.

2. The sum of multiple local functions centered at different positions has no defined convexity.

3. Molecular energy, or crystal energy, or protein folding are well known to be non-convex.

A few days ago (on a bike ride) I was thinking about this, how the molecules will be configured in a global minimum energy, and I was wondering if it finds that configuration so rapidly because of quantum tunneling effects.

The non-convex (global) optimization is different from (non-linear) least-squares, in the sense that when a local minimum is found the process don't return immediately, it start making new attempts in different regions of the search spaces. If the function is smooth you can still take advantage of a gradient based local optimization method, but the complexity is still NP.

A classic global optimization method is the Simulated annenaling, if you have a chemical background I think you will have some insights reading about it. Once upon a time, simulated annealing was provided in scipy.optimize.

You will find a few global optimization methods in scipy.optimize. I would encourage you to try Basin hopping, since it was successfully applied to similar problems, as you can read in the references.

I hope this drop you on the right way to your solution. But, be aware that you will probably need to spend, learning how to use the function and will need to make some decisions. You will need to find a balance of accuracy, simplicity, efficiency.

If you want better solution take the time to derive the gradient of the cost function (you can return two values f, and df, where df is the gradient of f with respect to the decision variables).