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Valence Bond Theory and Hybridized Orbitals02:38

Valence Bond Theory and Hybridized Orbitals

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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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State Space Representation01:27

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The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
Consider an RLC circuit, a...
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IR Spectrum Peak Splitting: Symmetric vs Asymmetric Vibrations01:08

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Identical bonds within a polyatomic group can stretch symmetrically (in-phase) or asymmetrically (out-of-phase). Similar to hydrogen bonding, these vibrations also influence the shape of the IR peak. Generally, asymmetric stretching frequencies are higher than symmetric stretching frequencies. For example, primary amines exhibit two distinct IR peaks between 3300–3500 cm−1 corresponding to the symmetric and asymmetric N-H stretching, while secondary amines exhibit a single...
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The Energies of Atomic Orbitals03:21

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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.
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Hybridization of Atomic Orbitals I03:24

Hybridization of Atomic Orbitals I

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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...
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Hybridization of Atomic Orbitals II03:35

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sp3d and sp3d 2 Hybridization
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Related Experiment Video

Updated: Jan 3, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs
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Generation and Coherent Control of Pulsed Quantum Frequency Combs

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Computing vibrational eigenstates with tree tensor network states (TTNS).

Henrik R Larsson1

  • 1Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA.

The Journal of Chemical Physics
|November 30, 2019
PubMed
Summary
This summary is machine-generated.

We compute vibrational spectra using tree tensor network states (TTNSs) and a density matrix renormalization group (DMRG) algorithm. This method offers faster convergence than ML-MCTDH, with TTNS and matrix product states (MPSs) showing similar performance for acetonitrile.

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Area of Science:

  • Quantum chemistry
  • Computational physics
  • Spectroscopy

Background:

  • Vibrational spectra are crucial for understanding molecular properties.
  • Tree tensor network states (TTNSs) offer a powerful ansatz for quantum many-body problems.
  • Multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) is a common method for such calculations.

Purpose of the Study:

  • To present a novel algorithm for computing vibrational eigenstates using TTNSs.
  • To apply and validate this method for the vibrational spectrum of acetonitrile (CH3CN).
  • To compare the performance of TTNSs with matrix product states (MPSs).

Main Methods:

  • Utilizing a density matrix renormalization group (DMRG)-based algorithm for eigenstate computation.
  • Implementing TTNSs as the underlying ansatz, related to ML-MCTDH.
  • Comparing TTNSs against MPSs, the ansatz used in DMRG.

Main Results:

  • The TTNS-based algorithm achieves high accuracy for the vibrational spectrum of acetonitrile.
  • The presented optimization scheme demonstrates significantly faster convergence compared to ML-MCTDH.
  • No substantial advantage of TTNS over MPS was observed for this specific system.
  • Adaptive bond dimension significantly reduces parameter count for both TTNS and MPS.

Conclusions:

  • TTNSs provide an effective approach for calculating vibrational spectra.
  • The DMRG-based optimization scheme offers improved convergence rates.
  • Further procedures are proposed for optimizing tree structures in TTNS calculations.