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Hybridization of Atomic Orbitals I

Hybridization of Atomic Orbitals I

The mathematical expression known as the wave function, ψ, contains information about each orbital and the wavelike properties of electrons in an isolated atom. When atoms are bound together in a molecule, the wave functions combine to produce new mathematical descriptions that have different shapes. This process of combining the wave functions for atomic orbitals is called hybridization and is mathematically accomplished by the linear combination of atomic orbitals. The new orbitals that...

Hybridization of Atomic Orbitals II

Hybridization of Atomic Orbitals II

sp3d and sp3d 2 Hybridization

Atomic Orbitals

Atomic Orbitals

An atomic orbital represents the three-dimensional regions in an atom where an electron has the highest probability to reside. The radial distribution function indicates the total probability of finding an electron within the thin shell at a distance r from the nucleus. The atomic orbitals have distinct shapes which are determined by l, the angular momentum quantum number. The orbitals are often drawn with a boundary surface, enclosing densest regions of the cloud.

Valence Bond Theory and Hybridized Orbitals

Valence Bond Theory and Hybridized Orbitals

According to valence bond theory, a covalent bond results when: (1) an orbital on one atom overlaps an orbital on a second atom, and (2) the single electrons in each orbital combine to form an electron pair. The strength of a covalent bond depends on the extent of overlap of the orbitals involved. Maximum overlap is possible when the orbitals overlap on a direct line between the two nuclei.
A σ bond (single bond in a Lewis structure) is a covalent bond in which the electron density is...

The Energies of Atomic Orbitals

The Energies of Atomic Orbitals

In an atom, the negatively charged electrons are attracted to the positively charged nucleus. In a multielectron atom, electron-electron repulsions are also observed. The attractive and repulsive forces are dependent on the distance between the particles, as well as the sign and magnitude of the charges on the individual particles. When the charges on the particles are opposite, they attract each other. If both particles have the same charge, they repel each other.

Van der Waals Equation

Van der Waals Equation

The ideal gas law is an approximation that works well at high temperatures and low pressures. The van der Waals equation of state (named after the Dutch physicist Johannes van der Waals, 1837−1923) improves it by considering two factors.
First, the attractive forces between molecules, which are stronger at higher densities and reduce the pressure, are considered by adding to the pressure a term equal to the square of the molar density multiplied by a positive coefficient a. Second, the volume...

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Related Experiment Video

Updated: Jan 15, 2026

Probe Type II Band Alignment in One-Dimensional Van Der Waals Heterostructures Using First-Principles Calculations

Probe Type II Band Alignment in One-Dimensional Van Der Waals Heterostructures Using First-Principles Calculations

Published on: October 12, 2019

Band Alignment in Core-Shell Nanocrystals by Estimating Wave Function Tunneling Probabilities.

Matthias Kick^1,2, Ezra Alexander¹, Troy Van Voorhis¹

¹Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States.

|October 11, 2025

Summary

This summary is machine-generated.

Quantifying electron and hole confinement in core-shell nanocrystals is challenging. A new density functional theory approach reveals wave function tunneling across interfaces, impacting passivation strategies for semiconducting nanocrystals.

Keywords:

Electronic structure heterostructure nanocrystals tunneling probability

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Ultrahigh Density Array of Vertically Aligned Small-molecular Organic Nanowires on Arbitrary Substrates

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Published on: June 18, 2013

Area of Science:

Materials Science
Quantum Chemistry
Nanotechnology

Background:

Core-shell colloidal semiconducting nanocrystals (NCs) offer significant optoelectronic potential.
Complex interfaces in NCs complicate the quantification of electron and hole confinement and band offsets.
Existing theoretical and experimental methods face challenges in accurately characterizing these properties.

Purpose of the Study:

To develop a reliable and user-friendly density functional theory (DFT)-based approach for estimating wave function tunneling probabilities in core-shell NCs.
To quantify electron and hole confinement and band offsets in II-VI core-shell heterostructures.
To investigate the factors influencing level alignment and wave function delocalization.

Main Methods:

Utilized a novel DFT-based approach for first-principle atomistic simulations.
Estimated wave function tunneling probabilities between core and shell materials.
Investigated electron/hole confinement in Type-I, Type-II, and quasi-Type-II II-VI core-shell nanocrystal heterostructures.

Main Results:

Band offsets in different heterostructure types qualitatively align with bulk trends.
Quantitative level alignment is sensitive to lattice match, nanocrystal shape, and shell thickness.
Significant wave function tunneling of electrons and holes across the core-shell interface was observed.

Conclusions:

The developed DFT approach provides a reliable method for assessing confinement in core-shell NCs.
Wave function tunneling across interfaces is substantial and must be considered for effective passivation.
Findings have implications for designing and optimizing core-shell heterostructures for optoelectronic applications.