Fukui Indices

About

Fukui indices (or Fukui functions) are important concepts in density functional theory (DFT) and are used to predict the reactivity of atoms or sites within a molecule. They measure how the electron density in a molecule changes in response to adding or removing electrons, giving insight into where a molecule is most reactive in a chemical process.

Key Concepts of Fukui Indices:

1. Fukui Function:

   • The Fukui function, $f(\mathbf{r})$, describes the change in the electron density, $\rho(\mathbf{r})$,

     of a molecule as the number of electrons is varied. It is mathematically expressed as the derivative of the electron density with respect to the number of electrons at a constant nuclear potential: $f(\mathbf{r}) = \left( \frac{\partial \rho(\mathbf{r})}{\partial N} \right)_v$​

   • The Fukui function is related to the molecule's local reactivity: regions where this function is larger are more reactive toward certain types of chemical attack.

2. Types of Fukui Indices: Fukui indices are discrete values derived from the Fukui function for individual atoms in a molecule. These indices predict the reactivity of each atom under different conditions (nucleophilic, electrophilic, or radical attacks). The three types of Fukui indices correspond to:

   • Fukui index for nucleophilic attack f⁺: This index measures how the electron density changes when an electron is added to the system (e.g., nucleophilic attack). It is calculated as$f^+ = q(N+1) - q(N)$ where $q(N+1)$ is the atomic charge after adding an electron, and $q(N)$ is the atomic charge in the neutral state.

   • Fukui index for electrophilic attack f⁻: This index measures how the electron density changes when an electron is removed from the system (e.g., electrophilic attack). It is calculated as:$f^- = q(N) - q(N-1)$ where $q(N-1)$ is the atomic charge after removing an electron.

   • Fukui index for radical attack f⁰: This index measures the reactivity toward radical attack (where an unpaired electron is involved). It is typically taken as the average of the nucleophilic and electrophilic indices: $f^0 = \frac{f^+ + f^-}{2}$

3. Reactivity Prediction:

   • High f⁺ values indicate regions of the molecule that are more reactive toward nucleophiles (accept electrons).

   • High f⁻ values indicate regions that are more reactive toward electrophiles (donate electrons).

   • High f⁰ values suggest regions more reactive toward radical species (either gaining or losing a single electron).

Applications:

• Chemical Reactivity and Selectivity: Fukui indices help predict which atoms or functional groups in a molecule are more likely to undergo specific chemical reactions.

• Catalysis: They are used to understand how catalysts interact with substrates, especially in processes where charge transfer occurs.

• Molecular Design: Fukui functions are applied to design new molecules with desired reactivity profiles in fields like drug design and materials science.

Fukui indices provide a powerful tool to rationalize and predict the reactivity patterns of molecules, making them useful in both theoretical and applied chemistry.

Method

The Fukui Indices are calcuated at the GFN2-xTB level of theory with xTB 6.6.0 using the --fukui option for xTB.

Find

The Fukui Indices can be found under the Atomic properties category in the property tree: