Student-Friendly Guide to Molecular Integrals

As a widely applied theory that has found success across many fields, density functional theory (DFT) is largely taught. Typically, the most effective way to convey DFT concepts is through illustrative examples that are currently lacking in available resources. In this work, we demonstrate total energy calculations for H 2 using 1s Slater type orbitals (STOs) within the framework of DFT. The molecular orbitals are constructed from a linear combination of 1s STOs centered on each atom. Using the Kohn−Sham equation, an equivalent expression for total energy, based on Kohn−Sham orbital eigenvalues, is given, aligning with that of the Hartree−Fock method. Detailed evaluations of Slater and X-Alpha exchange energies are presented along with a plotted potential energy surface and variations in density. Comparison with STO-nG basis sets not only supports the current work but also demonstrates the superiority of 1s STO over them. These features are included in the H 2 Jupyter Notebook which is freely accessible at https://github.com/choirunnr/dft-h2-1s. Additionally, the H 2 Jupyter Notebook was assessed for its implementation, receiving largely positive responses. Consequently, this work offers a valuable complement to existing teaching resources and strategy.

Section: Introductionsupporting

confidence: 57%

Density Functional Calculations on H₂ Using 1s Slater Type Orbitals

Rangkuti,

Faniandari,

Suparmi

et al. 2023

“…Improvement on students’ physical insight in quantum chemical systems has been observed after their independent completion of several FINDIF exercises. Although many publications and applications describe the numerical solution of the SE, − ,, some with visualization capabilities, FINDIF only requires the input of an appropriate V ( x ) text file and provides the results for further analysis. Originally designed for molecular spectroscopy applications, FINDIF requires the V ( x ) array to be in units of cm –1 and Å and the mass in atomic mass units.…”

Section: Discussionmentioning

confidence: 99%

“…Over the past 50 years, efforts have been made to help students solve the SE, understand the resulting solutions, and make predictions. These efforts are facilitated by visualization methods, 1−6 mathematical modeling of simple systems, 7−14 and the use of computers 15,16 and learning-based technologies, such as Mathematica, 17−19 spreadsheet calculations, 20−25 Python, 26,27 Maple, 28 and computational chemistry packages. 29−33 ■ DISCUSSION This Technology Report describes a stand-alone LabView application called FINDIF that enables students to both obtain the eigenvalues and visualize the eigenfunctions of the SE of a one-dimensional (1D) potential energy function V(x).…”

Section: ■ Introductionmentioning

confidence: 99%

“…Improvement on students' physical insight in quantum chemical systems has been observed after their independent completion of several FINDIF exercises. Although many publications and applications describe the numerical solution of the SE, [7][8][9][10][11][12][13][14]26,27 some with visualization capabilities, FINDIF only requires the input of an appropriate V(x) text file and provides the results for further analysis.…”

Section: ■ Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Visualizing Solutions of the One-Dimensional Schrödinger Equation Using a Finite Difference Method

Halpern

Glendening

2022

FINDIF is a Windows application that numerically solves the one-dimensional (1D) Schrodinger equation and displays the eigenstates, eigenvalues, and probability density of the system. FINDIF accepts both nonperiodic and periodic 1D potential energy functions as input and uses the finite difference method to evaluate the energy of the quantum system. This Technology Report illustrates the use of FINDIF with applications, such as the classic 1D particle-in-a-box, the particle-in-a-box with internal barrier, the modified Kronig−Penney model of a linear array of rectangular wells, the harmonic oscillator with visualization of the eigenstates and tunneling effect, the anharmonic Morse potential of the Ar dimer, and the periodic torsional potential for internal rotation of ethane. Students can explore other quantum chemical examples by considering both realistic and fictitious model potential energy functions, making outcome predictions before running FINDIF calculations, visualizing the results afterward, and then comparing their predictions with the results they observe. Such exercises assist students as they develop insights into the behavior and properties of quantum mechanical systems.

Opening the Density Functional Theory Black Box: A Collection of Pedagogic Jupyter Notebooks

Hirschi,

Bashirova,

Zuehlsdorff

2023

Density functional theory (DFT) is indubitably the most popular and among the most successful approaches for approximately solving the many-electron Schrodinger equation. The level of understanding on the part of both researchers and students using DFT, however, is lacking, given the availability of black-box software. The present work addresses this knowledge gap by providing three Jupyter notebooks, easily accessible through the Google Colaboratory (GitHub repository: https://github.com/tjz21/DFT_PIB_Code), that provide a short skirmish with the fundamentals of DFT through a particle in a box-type model system. These notebooks were tested in conjunction with a problem worksheet in a graduate-level quantum chemistry course; pre-and postactivity survey results reveal largely positive reactions to this implementation and sustained enthusiasm for the subject.