How to calculate the conformational entropy of a molecule?

Question

Introduction

It is know that the entropy of a system can be calculated using the formula (1):

\begin{equation} \label{eq:entropy_Boltz} S = {k_B}\ln \Omega\tag1 \end{equation}

where $k_B$ is the Boltzmann constant, $\Omega$ is the number of microstates of the system and $S$ is its entropy. This equation is one of the fundamental equations of Statistical Mechanics. Since $\Omega$ represents the number of microstates of the system, the more microstates, the greater the disorder of the system and therefore, the greater the entropy. One of the problems in using equation \eqref{eq:entropy_Boltz} is the need to know the number of microstates of the system.

Instead of expression \eqref{eq:entropy_Boltz}, Oganov and Valle ^[1] proposed an expression for the calculation of a quasi-entropy representing the structure disorder for a crystal:

\begin{equation} \label{eq:quasi-entropy} S_{str} = - \sum\limits_A {\frac{{{N_A}}}{N}}\left\langle\ln (1 - {F_{{A_i}{A_j}}})\right\rangle \end{equation} where $A$ represents the chemical species in the structure, $N_A$ is the number of atoms of the chemical species $A$, $N$ is the total number of atoms, and $F_{{A_i}{A_j}}$ is the distance between the fingerprints of sites $i$ and $j$ of the chemical species $A$. The fingerprints of each structure can be calculated as

\begin{equation} \label{eq:fingerprints} {F_{{A_i}B}} = - \sum\limits_{{B_j}} {\left\{ {\frac{{\delta (R - {R_{ij}})}}{{4\pi {R_{ij}}^2({N_B}/V)\Delta }}} \right\}} - 1 \end{equation} where $R$ is a parameter representing the maximum distance between atoms, $R_{ij}$ is the distance between the sites $i$ and $j$, $N_B$ is the number of atoms of the chemical species $B$, $V$ is the volume of the structure, and $\Delta$ is another parameter associated with the distance between pairs of structures ^[1].

Using this for molecules imply you have to create a pseudo crystal with your system.

On the other hand, the entropy associated with of bit string can be calculated using the Shannon Entropy. A modification of it was proposed by Grenville J. Croll and called BiEntropy ^[2]. In this case, it is necessary to use binary fingerprints of molecules.

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The problem

My systems consist on randomly decorate a nanostructure with organic groups like $\ce{-OH}$ and $\ce{-COOH}$, then select one (using the entropy as criteria) and run the calculations with it (here is an example of the type of system I am talking about).

Is there a way to calculate the conformational (disorder) entropy of a molecule?

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References

[1] A.R. Oganov, M. Valle, How to quantify energy landscapes of solids, J. Chem. Phys. 130 (2009) 104504

[2] G.J. Croll, BiEntropy - The Approximate Entropy of a Finite Binary String, arXiv:1305.0954 [cs.OH].

Answer 1

tldr, Extensively sample the potential energy surface of the molecule, then Boltzmann-weight the relative energies. The CREST tool is useful.

Nice to see this question as I'm writing / editing a paper on the conformational entropy of molecules.

There's already a question about the translational and rotational entropy of molecules. For those, you optimize a geometry, then calculate from the molecular mass and moments of inertia.

Similarly, one gets the vibrational entropy under the rigid rotor approximation by calculating the vibrational modes from the Hessian and thus the vibrational partition function.

These procedures, though, start from one conformer geometry. For any molecule with conformational degrees of freedom, there are likely to be multiple minima.

In principal, you could run molecular dynamics (even ab initio MD) for a long time, collect all the geometries, and Boltzmann weight the relative energies. Unfortunately, there's no guarantee that "a long time" is enough to capture all the thermally-accessible minima.

There are a few more practical methods at the moment:

• Generate an extensive number of conformers using your favorite tool, e.g. confab or ETKDG-RDKit etc. then use QM to minimize and rank relative energies.
• Use Grimme's CREST program Phys. Chem. Chem. Phys., 2020, 22, 7169-7192. to sample using an iterative meta-dynamics method using the GFN1 / GFN2 semiempirical methods.

The latter will directly calculate conformational entropies of the resulting ensemble through Boltzmann weighting and degeneracies.

For example, you might get output like this:

T /K                                  :   298.15
E lowest                              :   -43.34438
ensemble average energy (kcal)        :    0.028
ensemble entropy (J/mol K, cal/mol K) :   13.834    3.306
ensemble free energy (kcal/mol)       :   -0.986
population of lowest in %             :   79.407
 number of unique conformers for further calc            3
   1       0.000
   2       0.006
   3       2.879

CREST is time-consuming, and in principal, you should also use QM minimization and re-ranking relative energies.