# 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:

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