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https://github.com/amphiluke/potprox

Approximation of computed data with empirical pair potentials
https://github.com/amphiluke/potprox

approximation buckingham lennard-jones morse potential rydberg

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Approximation of computed data with empirical pair potentials

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# potprox



Approximation of computed data with empirical pair potentials.



## Synopsis



It is a quite common case when there is a need to describe computed numerical *ab initio* data with some analytical form of pair potential (e.g. the [Lennard-Jones function](https://en.wikipedia.org/wiki/Lennard-Jones_potential) or the [Morse function](https://en.wikipedia.org/wiki/Morse_potential)).



Potprox uses the [method of least squares](https://en.wikipedia.org/wiki/Least_squares) to approximate computed data with empirical pair potentials. The list of currently available potentials includes



* [The Lennard-Jones potential](#the-potproxlennardjones-class)

* [The modified Buckingham potential](#the-potproxbuckingham-class) (the exp-6 potential)

* [The Morse potential](#the-potproxmorse-class)

* [The Rydberg potential](#the-potproxrydberg-class)

* [The Varshni potential (III)](#the-potproxvarshni3-class)



## Install and load potprox



### As a Node.js module



Installing the package:



```shell

npm install potprox

```



Importing the module:



```javascript

import * as potprox from "potprox";

```



or (for CommonJS modules)



```javascript

let potprox = require("potprox");

```



If you need only a few potential classes, using named import will allow module bundlers to perform “tree shaking” and exclude the rest unused code.



```javascript

import {Morse, Rydberg} from "potprox";

```



### In browsers



The module can be loaded from the popular CDNs like unpkg or jsDelivr.



```html



```



If you use ES modules, you may import the potprox module from the [potprox.min.mjs](dist/potprox.min.mjs) file:



```javascript

import * as potprox from "https://cdn.jsdelivr.net/npm/potprox/dist/potprox.min.mjs";

```



### In web workers



```javascript

importScripts("https://www.unpkg.com/potprox");

```



## Usage



Here is an example of approximation of some external computational data using the potprox module.



```javascript

import * as potprox from "potprox";



// Computed numerical data on energy of interatomic binding

// r - interatomic distance

// e - binding energy

let data = [

    {r: 10.0, e: 0},

    {r: 9.5, e: -0.00065673},

    {r: 9.0, e: -0.00173718},

    {r: 8.5, e: -0.00346348},

    {r: 8.0, e: -0.00612669},

    {r: 7.5, e: -0.01005967},

    {r: 7.0, e: -0.01554171},

    {r: 6.5, e: -0.02256036},

    {r: 6.0, e: -0.03028974},

    {r: 5.5, e: -0.03598181},

    {r: 5.0, e: -0.03234259},

    {r: 4.5, e: 0.00189849},

];



// Approximate with the Lennard-Jones potential

let lennardjones = potprox.LennardJones.from(data);

console.log("Lennard-Jones potential info:", lennardjones.toJSON());



// Approximate with the exp-6 potential

let buckingham = potprox.Buckingham.from(data);

console.log("Buckingham potential info:", buckingham.toJSON());



// Approximate with the Morse potential

let morse = potprox.Morse.from(data);

console.log("Morse potential info:", morse.toJSON());



// Approximate with the Rydberg potential

let rydberg = potprox.Rydberg.from(data);

console.log("Rydberg potential info:", rydberg.toJSON());



// Approximate with the Varshni potential (III)

let varshni = potprox.Varshni3.from(data);

console.log("Varshni potential (III) info:", varshni.toJSON());

```



## API



### The `potprox` object



The potprox module exports an object with potential names as keys and potential classes as values.



```javascript

console.log(potprox); // => Object {LennardJones: , Buckingham: , ...}

```



### The `potprox.LennardJones` class



An instance of the `LennardJones` class represents the Lennard-Jones potential with the given parameters `epsilon` and `sigma`.



$$V\left(r\right)=4\varepsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^6\right]$$



You may instantiate the `LennardJones` class as follows:



```javascript

let lennardjones = new potprox.LennardJones({epsilon: 0.041, sigma: 4.5});

```



### The `potprox.Buckingham` class



An instance of the `Buckingham` class represents the modified Buckingham potential (the exp-6 potential) with the given parameters `d0`, `r0`, and `a` (which are often referenced to as *ε*, *rm*, and *α* respectively).



$$V\left(r\right)=\frac{D_0}{1-6/a}\left(\frac{6}{a}\exp\left[a\left(1-\frac{r}{r_0}\right)\right]-\left(\frac{r_0}{r}\right)^6\right)$$



You may instantiate the `Buckingham` class as follows:



```javascript

let buckingham = new potprox.Buckingham({d0: 0.0360, r0: 5.298, a: 4.332});

```



### The `potprox.Morse` class



An instance of the `Morse` class represents the Morse potential with the given parameters `d0`, `r0`, and `a` (which are often referenced to as *De*, *re*, and *α* respectively).



$$V\left(r\right)=-D_0+D_0\left[1-\exp\left(-a\left(r-r_0\right)\right)\right]^2$$



You may instantiate the `Morse` class as follows:



```javascript

let morse = new potprox.Morse({d0: 0.0368, r0: 5.316, a: 0.867});

```



### The `potprox.Rydberg` class



An instance of the `Rydberg` class represents the Rydberg potential with the given parameters `d0`, `r0`, and `b`.



$$V\left(r\right)=-D_0\left[1+\frac{b}{r_0}\left(r-r_0\right)\right]\exp\left[-\frac{b}{r_0}\left(r-r_0\right)\right]$$



You may instantiate the `Rydberg` class as follows:



```javascript

let rydberg = new potprox.Rydberg({d0: 0.0368, r0: 5.350, b: 6.415});

```



### The `potprox.Varshni3` class



An instance of the `Varshni3` class represents the Varshni potential (III) with the given parameters `d0`, `r0`, and `b`.



$$V\left(r\right)=-D_0+D_0\left[1-\frac{r_0}{r}\exp\left(-b\left(r^2-r_0^2\right)\right)\right]^2$$



You may instantiate the `Varshni3` class as follows:



```javascript

let varshni = new potprox.Varshni3({d0: 0.0368, r0: 5.389, b: 0.0597});

```



### Potential class members



All the classes in the `potprox` object have a few members listed below.



#### `type`



The *static* read-only property containing the name of the potential class (e.g. `"LennardJones"`, `"Morse"`, `"Buckingham"` etc.).



```javascript

console.log(potprox.LennardJones.type); // => "LennardJones"

```



#### `from(data [, settings])`



The *static* method `from` creates an instance of the specific class with potential parameters obtained via the least squares approximation procedure.



The first (required) argument is input approximated data, an array of objects `{r: Number, e: Number}`, where `r` is an interatomic distance, and `e` is the corresponding binding energy. Refer the [Usage](#usage) section for an example.



The second (optional) argument can be specified to override approximation settings. Thus in order to get better approximation results you may decrease convergence limits by specifying custom convergence factors for all or some potential parameters.



```javascript

let morse = potprox.Morse.from(data, {d0Conv: 0.0001, r0Conv: 0.0001, aConv: 0.0001});

let rydberg = potprox.Rydberg.from(data, {b0Conv: 0.0001});

```



Be careful though when using too small convergence factors as this may end up with performance issues. Consider using web workers if you need high-accuracy approximation but things appear to get slow.



#### `at(r)`



Calculates the value of the potential for the given interatomic distance.



```javascript

let lennardjones = new potprox.LennardJones({epsilon: 0.041, sigma: 4.5});

console.log(lennardjones.at(6.0)); // => -0.02399355483055115



let buckingham = new potprox.Buckingham({d0: 0.0360, r0: 5.298, a: 4.332});

console.log(buckingham.at(6.0)); // => -0.028625141782941267



let morse = new potprox.Morse({d0: 0.0368, r0: 5.316, a: 0.867});

console.log(morse.at(6.0)); // => -0.029435553046279185



let rydberg = new potprox.Rydberg({d0: 0.0368, r0: 5.350, b: 6.415});

console.log(rydberg.at(6.0)); // => -0.030035419908893232



let varshni = new potprox.Varshni3({d0: 0.0368, r0: 5.389, b: 0.0597});

console.log(varshni.at(6.0)); // => -0.03069928686072358

```



#### `points([options])`



The method `points` can be used to generate points of a potential function in the given distance range. The method takes one optional argument and returns a [Generator object](https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Generator) which you may iterate over. The optional parameter of the method is the configuration object. The following configuration options are available (each of them is optional):



* `start` — starting interatomic distance to generate points from (by default it’s set to a half of the equilibrium distance);

* `end` — end interatomic distance where to stop (by default it’s double of the equilibrium distance);

* `step` — step for point generation (default step is configured to generate 50 points).



```javascript

let morse = new potprox.Morse({d0: 0.0368, r0: 5.316, a: 0.867});



// Generate 50 points starting from r = r0/2 and finishing at r = 2*r0

for (let {r, e, index} of morse.points()) {

    console.log(`${index + 1}. r = ${r.toFixed(4)} nm; E = ${e.toFixed(3)} eV`);

}



// Generate 30 points in the user-defined distance range

let start = 5.0;

let end = 8.5;

let pointCount = 30;

let step = (end - start) / (pointCount - 1);

for (let {r, e, index} of morse.points({start, end, step})) {

    console.log(`${index + 1}. r = ${r.toFixed(4)} nm; E = ${e.toFixed(3)} eV`);

}



// Generate points infinitely until the given energy threshold is reached

for (let {r, e, index} of morse.points({start: 5.0, end: Infinity, step: 0.1})) {

    console.log(`${index + 1}. r = ${r.toFixed(4)} nm; E = ${e.toFixed(5)} eV`);

    if (e > -0.001) {

        break;

    }

}

```



#### `rSqr(data)`



Use the method `rSqr` to calculate the [coefficient of determination](https://en.wikipedia.org/wiki/Coefficient_of_determination) _R²_, a measure of goodness of fit. The method takes the initial data array as an argument (same as that passed to the [`from` method](#fromdata--settings)).



```javascript

let morse = potprox.Morse.from(data);

let rSqr = morse.rSqr(data);

console.log(`Coefficient of determination = ${rSqr}`);

```



#### `toJSON()`



Returns an object containing information on the potential. This information is enough to restore the potential instance form a serializable JSON object (see the [Tips](#tips) section for details).



```javascript

let lennardjones = new potprox.LennardJones({epsilon: 0.041, sigma: 4.5});

console.log(lennardjones.toJSON()); // => {type: "LennardJones", epsilon: 0.041, sigma: 4.5}



let buckingham = new potprox.Buckingham({d0: 0.0360, r0: 5.298, a: 4.332});

console.log(buckingham.toJSON()); // => {type: "Buckingham", d0: 0.036, r0: 5.298, a: 4.332}



let morse = new potprox.Morse({d0: 0.0368, r0: 5.316, a: 0.867});

console.log(morse.toJSON()); // => {type: "Morse", d0: 0.0368, r0: 5.316, a: 0.867}



let rydberg = new potprox.Rydberg({d0: 0.0368, r0: 5.350, b: 6.415});

console.log(rydberg.toJSON()); // => {type: "Rydberg", d0: 0.0368, r0: 5.350, b: 6.415}



let varshni = new potprox.Varshni3({d0: 0.0368, r0: 5.389, b: 0.0597});

console.log(varshni.toJSON()); // => {type: "Varshni3", d0: 0.0368, r0: 5.389, b: 0.0597}

```



Note that the potential parameters are also available as direct instance properties, and you may change them at any time.



## Tips



The overridden method `toJSON()` allows the instances of the potprox potential classes to be easily serialized to a JSON string, and restored from the JSON string later on.



```javascript

// Create and serialize

let morse = new potprox.Morse({d0: 0.0368, r0: 5.316, a: 0.867});

let json = JSON.stringify(morse);



// Unserialize and restore

let potentialData = JSON.parse(json);

let morseCopy = new potprox[potentialData.type](potentialData);

```