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 or the Morse function).

Potprox uses the method of least squares to approximate computed data with empirical pair potentials. The list of currently available potentials includes

• The Lennard-Jones potential
• The modified Buckingham potential (the exp-6 potential)
• The Morse potential
• The Rydberg potential
• The Varshni potential (III)

Install and load potprox

As a Node.js module

Installing the package:

npm install potprox

Importing the module:

import * as potprox from "potprox";

or (for CommonJS modules)

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.

import {Morse, Rydberg} from "potprox";

In browsers

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

<script src="https://cdn.jsdelivr.net/npm/potprox/dist/potprox.min.js"></script>

If you use ES modules, you may import the potprox module from the potprox.min.mjs file:

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

In web workers

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

Usage

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

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.

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

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:

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, r_m, 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:

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 D_e, r_e, 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:

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:

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:

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.).

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 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.

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.

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 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).

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 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).

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 section for details).

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.

// 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);