The conformational energy of hindered rotation

in a classical molecular mechanics simulation in the general case is approximated by the equation [1]:

.

The parameters

and

describe a force field with torsion angle

.

This Demonstration uses a modified version of the potential:

where

is the energy barrier (kJ/mol) and

and

are the energy difference gauche(-)/trans and gauche(+)/trans conformers, respectively (kJ/mol).

The effect of excluded volume is neglected in the calculation.

Using the Monte Carlo method, 500 macromolecular chains are generated. For a chain of length

,

torsion angle values are generated. To simplify the calculation, selection of the torsion angle is performed taking into account the statistical weight based on the cumulative distribution function

at temperature

, divided into 36 equal intervals with step size 10°. This avoids the calculation of the Boltzmann factor for individual chains:

.

For each chain, the end-to-end distance

is calculated. The distance is expressed in relative units of carbon-carbon bond length. For visualization, a smoothed function is used to exclude statistical scatter;

values are the result of smoothing and should be ignored.

Based on the distribution, the mean

, root mean square

, end-to-end distance and standard deviation

are calculated:

,

,

.

[1] M. Bachmann,

*Thermodynamics and Statistical Mechanics of Macromolecular Systems*, Cambridge, UK: Cambridge University Press, 2014.