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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

¹H NMR: Complex Splitting

¹H NMR: Complex Splitting

A proton M that is coupled to a proton X results in doublet signals for M. However, NMR-active nuclei can be simultaneously coupled to more than one nonequivalent nucleus. When M is coupled to a second proton A, such as in styrene oxide, each peak in the doublet is split into another doublet.
Splitting diagrams or splitting tree diagrams are routinely used to depict such complex couplings. While drawing splitting diagrams, the splitting with the larger coupling constant is usually applied...

Spin–Spin Coupling: Two-Bond Coupling (Geminal Coupling)

Spin–Spin Coupling: Two-Bond Coupling (Geminal Coupling)

Two NMR-active nuclei bonded to a central atom can be involved in geminal or two-bond coupling. Geminal coupling is commonly seen between diastereotopic protons in chiral molecules and unsymmetrical alkenes, among others.
The central atom need not be NMR-active because its electrons are affected by the electron polarization of the spin-active atoms. However, spin information is transmitted less effectively than in one-bond coupling, and 2J values are usually weaker than 1J values. The energy of...

Spin–Spin Coupling: Three-Bond Coupling (Vicinal Coupling)

Spin–Spin Coupling: Three-Bond Coupling (Vicinal Coupling)

Vicinal or three-bond coupling is commonly observed between protons attached to adjacent carbons. Here, nuclear spin information is primarily transferred via electron spin interactions between adjacent C‑H bond orbitals. This generally favors the antiparallel arrangement of spins, so 3J values are usually positive.
The extent of coupling depends on the C‑C bond length, the two H‑C‑C angles, any electron-withdrawing substituents, and the dihedral angle between the involved orbitals. The...

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...

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Updated: Mar 14, 2026

Automatic Identification of Dendritic Branches and their Orientation

Automatic Identification of Dendritic Branches and their Orientation

Published on: September 17, 2021

Branched Schrödinger Bridge Matching.

Sophia Tang¹, Yinuo Zhang², Alexander Tong³

¹Department of Computer and Information Science, University of Pennsylvania.

|March 13, 2026

Summary

This summary is machine-generated.

Branched Schrödinger Bridge Matching (BranchSBM) enables generative models to capture divergent paths from a single origin. This novel framework is essential for modeling complex, multi-modal transitions in various scientific applications.

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Area of Science:

Generative modeling
Computational biology
Machine learning

Background:

Predicting trajectories between distributions is key in generative modeling.
Current methods like flow matching and Schrödinger bridge matching are limited to unimodal transitions.
They cannot model branched or divergent evolution from a common origin to multiple distinct modes.

Purpose of the Study:

Introduce Branched Schrödinger Bridge Matching (BranchSBM) to learn branched Schrödinger bridges.
Enable representation of population-level divergence into multiple terminal distributions.
Address limitations of existing methods in capturing multi-modal transitions.

Main Methods:

Parameterize multiple time-dependent velocity fields.
Incorporate multiple time-dependent growth processes.
Develop a novel framework for learning branched Schrödinger bridges.

Main Results:

BranchSBM demonstrates greater expressiveness than existing methods.
Successfully models multi-path surface navigation.
Effectively simulates cell fate bifurcations and diverging cellular responses.

Conclusions:

BranchSBM is essential for tasks requiring the modeling of divergent trajectories.
The framework advances generative modeling capabilities for complex biological systems.
Enables more accurate simulation of branched evolutionary processes.