# Information-Theoretic Descriptors¶

Information-theoretic descriptors are often used in quantitative structure property relationships and to analyze bonding in molecules. One of the most useful tools related to information-theoretic analysis are the steric energy, steric potential, and steric force. These quantities are closely linked to both the non-covalent interactions analysis (NCI) and the density-overlaps regions indicator (DORI), and can be used to perform similar analysis.

A few information-theoretic descriptors are so fundamental that they are given special names are described here.

**Shannon Entropy** is the “uncertainty” associated with the electron density. It is larger when
the electron density is more uniform.

\[S = \int \rho\left(\mathbf{r}\right) \ln \rho\left(\mathbf{r}\right) d\mathbf{r}\]

**Shannon Entropy Density** is the local Shannon entropy, equal to the integrand of Shannon entropy:

\[s\left(\mathbf{r}\right) = \rho\left(\mathbf{r}\right) \ln \rho\left(\mathbf{r}\right)\]

**Shannon Shape Entropy** is the Shannon entropy of electron density per particle or shape function
\(\sigma\left(\mathbf{r}\right)\):

\[S = \int \sigma\left(\mathbf{r}\right) \ln \sigma\left(\mathbf{r}\right) d\mathbf{r}\]

**Shannon Shape Entropy Density** is the local Shannon entropy, equal to the integrand of Shannon
entropy:

\[s(\mathbf{r}) = \sigma(\mathbf{r}) \ln\sigma(\mathbf{r})\]

**Fisher Information** is the Fisher entropy of locality is (by abuse/extension of mathematical
equations) is also associated with the electron density,

\[I = \int \frac{\nabla\rho(\mathbf{r}) \cdot \rho(\mathbf{r})}{\rho(\mathbf{r})} d\mathbf{r}\]

The Fisher information is smaller when the electron density is more uniform, and is closely related to the Weizsacker kinetic energy and the steric energy.

**Fisher Information Density** is the local Fisher information, which is the integrand of the
Fisher information:

\[i(\mathbf{r}) = \frac{\nabla\rho(\mathbf{r}) \cdot \rho(\mathbf{r})}{\rho(\mathbf{r})}\]

**Fisher Shape Information** is the Fisher entropy of locality for the shape function, or
density-per-particle:

\[I = \int \frac{\nabla\sigma(\mathbf{r})\cdot\sigma(\mathbf{r})}{\sigma(\mathbf{r})} d\mathbf{r}\]

**Fisher Shape Information Density** is the local Fisher shape information, defined as the
integrand in the Fisher shape information:

\[i(\mathbf{r}) = \frac{\nabla\sigma(\mathbf{r}) \cdot \sigma(\mathbf{r})}{\sigma(\mathbf{r})}\]

**Ghosh-Berkowitz-Parr Entropy** is based on a local thermodynamic treatment of the electron
density:

\[S_\text{GBP} = \int \tfrac{3}{2} \rho(\mathbf{r}) \ln\left( \frac{\tau_\text{PD}(\mathbf{r})}{\tau_\text{TF}(\mathbf{r})}\right) d\mathbf{r}\]

To accommodate the range of descriptors that have been associated with the Ghosh-Berkowitz-Parr entropy, the user can pass any local kinetic energy to the expression for the Ghosh-Berkowitz-Parr expression, but the default expression is the one they derived in their paper, which corresponds to a factor \(-\tfrac{1}{8}\) of the Laplacian term.

**Ghosh-Berkowitz-Parr Local Entropy** is the integrand of the preceding Ghosh-Berkowitz-Parr
expression for the global entropy.

**Ghosh-Berkowitz-Parr Shape Entropy** is the shape-function analogue of the Ghosh-Berkowitz-Parr
entropy (up to a constant additive factor)

\[S_\text{GBP} = \int \tfrac{3}{2} \sigma(\mathbf{r}) \ln\left(N_\text{spin}^{\tfrac{2}{3}} \frac{\tau_\text{PD}(\mathbf{r})}{\tau_\text{TF}(\mathbf{r})}\right) d\mathbf{r}\]

To accommodate the range of descriptors that have been associated with the Ghosh-Berkowitz-Parr entropy, the user can pass any local kinetic energy to the expression for the Ghosh-Berkowitz-Parr expression, though the dependence on the number of electrons in the spin-channel one considers will be incorrect for some (uncommon) choices.

**Ghosh-Berkowitz-Parr Local Shape Entropy** is the integrand of the preceding Ghosh-Berkowitz-Parr
expression for the global shape entropy.

**Steric Energy** is the kinetic energy of a system if the electrons were bosons; ergo it quantifies
the inherent “kinetic energy pressure” with quantum (Pauli exclusion) effects are omitted.
It is equal to the Weizsacker kinetic energy:

\[E_\text{steric} = \int \tau_\text{W}(\mathbf{r}) d\mathbf{r} = \int \frac{\lvert \nabla \rho(\mathbf{r}) \rvert ^2}{8 \rho(\mathbf{r})} d\mathbf{r}\]

**Steric Potential** is the functional derivative of the steric energy.
The steric potential plus the Pauli potential equals the Kohn-Sham potential.

\[\begin{split}v_\text{steric}(\mathbf{r}) = \frac{\delta E_\text{steric}}{\delta \rho(\mathbf{r})} &= -\frac{1}{8} \frac{\nabla\rho(\mathbf{r}) \cdot \nabla\rho(\mathbf{r})}{\rho^2(\mathbf{r})} -\frac{1}{4} \nabla \cdot \frac{\nabla\rho(\mathbf{r})}{\rho(\mathbf{r})} \\ &= -\frac{1}{8} \frac{\nabla\rho(\mathbf{r}) \cdot \nabla\rho(\mathbf{r})}{\rho^2(\mathbf{r})} +\frac{1}{4} \frac{\nabla\rho(\mathbf{r}) \cdot \nabla\rho(\mathbf{r})}{\rho^2(\mathbf{r})} -\frac{1}{4} \frac{\nabla^2\rho(\mathbf{r})}{\rho(\mathbf{r})} \\ &= \frac{1}{8} \frac{\nabla\rho(\mathbf{r}) \cdot \nabla\rho(\mathbf{r})}{\rho^2(\mathbf{r})} -\frac{1}{4} \frac{\nabla^2\rho(\mathbf{r})}{\rho(\mathbf{r})}\end{split}\]

**Steric Charge Density** is the classical charge distribution that generates the steric potential,

\[q_\text{steric}(\mathbf{r}) = \tfrac{-1}{4 \pi} \nabla^2 v_\text{steric}(\mathbf{r})\]

This is an interesting example of the power of ChemTools, because it requires very high-order differentiation, which most programs do not support (and where numerical differential is often inaccurate).

**Steric Charge** is the total steric charge:

\[Q_\text{steric} = \int q_\text{steric}(\mathbf{r}) d\mathbf{r}\]