Fukui Functions and Dual Descriptors Explained: An Advanced Scientific Analysis

Fukui functions function as basic descriptors from DFT-based conceptual density functional theory, which shows how electrons distribute within molecules. The prediction capability using these functions reveals the electron-donating or accepting the potential of molecules at specific atomic sites to determine electrophilic or nucleophilic characteristics.

The dual descriptor enhances Fukui function analysis by distinguishing the nucleophilic and electrophilic properties at molecular sites which improves the accuracy of chemical reactivity assessments. We have found that most researchers are facing challenges in calculating these important descriptors for an advanced scientific analysis. Here, we have designed a simple Python code to simplify the calculation.

The article provides detailed insights into Fukui functions, together with their theoretical framework, along with numerical procedures and Python application utilizing Natural Population Analysis (NPA) results obtained from Gaussian software.

📥Fukui Functions and Their Physical Interpretation

The Fukui function 𝑓(𝑟) measures the response of electron density ρ(r) when the number of electrons in a system changes slightly.

Mathematically, it is defined as:

Where N is the total number of electrons, and v(r) is the external potential. This function can be numerically approximated using finite differences:

Where:

• 𝑞 𝑟 ( 𝑁 ) → the atomic charge of a particular site in a neutral molecule.
• 𝑞 𝑟 ( 𝑁 + 1 ) → the charge after adding an electron (anion).
• 𝑞 𝑟 ( 𝑁 − 1 ) → the charge after removing an electron (cation).

The two functions evaluate electron density sensitivity toward altering conditions to help predict chemical reactivity patterns.

📥Dual Descriptor (Δ𝑓(𝑟)) and Chemical Interpretation

The dual descriptor Δ 𝑓 ( 𝑟 ) refines Fukui function analysis by differentiating between electrophilic and nucleophilic reactive sites:

📥Interpretation of Δ𝑓( 𝑟 ) :

• Δ 𝑓 ( 𝑟 ) → Region favors nucleophilic attack.
• Δ 𝑓 ( 𝑟 ) < 0 → Region favors electrophilic attack.
• Δ 𝑓 ( 𝑟 )=0 → Region is chemically inert.

By using Δ 𝑓 ( 𝑟 ), researchers can precisely determine reactive molecular sites and predict the pathways of organic reactions.

📥Natural Population Analysis (NPA) in Gaussian

Computational chemistry employs NPA extensively to divide electron density across the molecule’s atoms. Gaussian software generates NPA charges through the input file keyword Pop=NPA. The basis for performing Fukui function calculations is provided by these charges.

Example Gaussian Input for NPA Calculation:

%chk=mol.chk
# B3LYP/6-31G(d,p) Pop=NPA

Title: NPA Charge Calculation

0 1
C   0.000   0.000   0.000
H   0.000   0.000   1.090
H   1.026   0.000  -0.363
H  -0.513  -0.889  -0.363
H  -0.513   0.889  -0.363

The output section of the Gaussian calculation provides NPA charges under the "Natural Population Analysis" heading. The extracted charges are formatted in a table:

These values are used in the Python script to compute Fukui functions and the dual descriptor.

📥Python Code for Fukui Function Calculation

Researchers use the calculated values to understand chemical mechanisms during reactions while developing molecules that possess particular functional properties including catalysts and drug candidates.

👉Python Script:

import pandas as pd

# Load the data from the Excel file in the same directory as the script
file_path = 'FUKUI.xlsx'
df = pd.read_excel(file_path)

# Calculate Fukui Functions and Dual Descriptor with 5 decimal places
df['fr+'] = (df['Anion(N+1)'] - df['Neutral(N)']).apply(lambda x: f"{x:.5f}")
df['fr-'] = (df['Neutral(N)'] - df['Cation(N-1)']).apply(lambda x: f"{x:.5f}")
df['fr0'] = ((df['Anion(N+1)'] - df['Cation(N-1)']) / 2).apply(lambda x: f"{x:.5f}")
df['∆f'] = (df['fr+'].astype(float) - df['fr-'].astype(float)).apply(lambda x: f"{x:.5f}")

# Save the calculated data to a new Excel file in the same directory
output_file = 'fukui_functions_output.xlsx'
df.to_excel(output_file, index=False)

# Display the DataFrame (optional)
print(df)

👉Code Explanation:

Loading Data:

Reads NPA charges from an Excel file (FUKUI.xlsx).

👉Fukui Function Calculation:

• The 𝑓+( 𝑟 ) index derives from the difference between anionic and neutral charge values.
• This index 𝑓-( 𝑟 ) results from subtracting the cationic charge value from the neutral charge value.
• The radical Fukui function obtains its value by performing an average operation on 𝑓+( 𝑟 ) and 𝑓-( 𝑟 ).
• A dual descriptor emerges when the calculation of 𝑓-( 𝑟 ) subtracts 𝑓+( 𝑟 ) from it.

Formatting: Each value is rounded to five decimal places for precision.

Output: The results are saved as fukui_functions_output.xlsx for further analysis.

📥Results and Discussion

The calculated Fukui functions and dual descriptors provide critical insights into molecular reactivity:

• High 𝑓+( 𝑟 ) values indicate potential nucleophilic sites.
• High 𝑓−( 𝑟 ) values suggest electrophilic reactivity.
• Δ 𝑓 ( 𝑟 ) values induce the most reactive sites.

Predictions of reaction mechanisms pair with the production of molecules with particular desired properties such as catalysts and drug candidates.

📥Conclusion

The combination of Fukui functions with dual descriptors functions as a strong computational approach for exploring chemical reactivity. Proper calculation of charge distributions depends on NPA charges obtained through Gaussian calculations. The Python-based methodology enables automation that makes it possible for efficient evaluation of big molecular datasets.

Next studies should implement combined analysis of Fukui functions together with electrostatic potential mapping which includes molecular orbital analysis to achieve a proper understanding of reactivity behavior.