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Second Order systems II

Second Order systems II

In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.

2D NMR: Overview of Heteronuclear Correlation Techniques

2D NMR: Overview of Heteronuclear Correlation Techniques

Heteronuclear correlation spectroscopy is an analytical technique that investigates the coupling between different types of nuclei, often a proton and an X-nucleus, such as carbon-13 or nitrogen-15. This method is commonly used in nuclear magnetic resonance (NMR) spectroscopy to gain insights into complex chemical compounds' structural and compositional aspects. A typical heteronuclear correlation spectrum displays X-nucleus chemical shifts on one axis and a proton spectrum on the other...

Hybridization of Atomic Orbitals II

Hybridization of Atomic Orbitals II

sp3d and sp3d 2 Hybridization

¹H NMR: Interpreting Distorted and Overlapping Signals

¹H NMR: Interpreting Distorted and Overlapping Signals

Spin systems where the difference in chemical shifts of the coupled nuclei is greater than ten times J are called first-order spin systems. These nuclei are weakly coupled, and their chemical shifts and coupling constant can generally be estimated from the well-separated signals in the spectrum.
As Δν decreases and the signals move closer, the doublets appear increasingly distorted. The intensities of the inner lines increase at the cost of those of the outer lines as the signals are...

Second Order systems I

Second Order systems I

A servo system exemplifies a second-order system, featuring a proportional controller and load elements that ensure the output position aligns with the input position. The relationship between these components is described by a second-order differential equation. Applying the Laplace transform under zero initial conditions yields the transfer function, showing how inputs are converted to outputs in the system.
By reinterpreting the system, one can derive the closed-loop transfer function, which...

¹H NMR: Long-Range Coupling

¹H NMR: Long-Range Coupling

The coupling interactions of nuclei across four or more bonds are usually weak, with J values less than 1 Hz. While these are usually not observed in spectra, the presence of multiple bonds along the coupling pathway can result in observable long-range coupling.
In alkenes, spin information is communicated via σ–π overlap, as seen in allylic (four-bond) and homoallylic (five-bond) couplings. These coupling interactions are stronger when the σ bond is parallel to the alkene...

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Updated: Dec 12, 2025

A Photonic System for Generating Unconditional Polarization-Entangled Photons Based on Multiple Quantum Interference

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Published on: September 5, 2019

Reaching Full Correlation through Nonorthogonal Configuration Interaction: A Second-Order Perturbative Approach.

Hugh G A Burton¹, Alex J W Thom¹

¹Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, U.K.

Journal of Chemical Theory and Computation

|August 14, 2020

Summary

We developed a new theory, Nonorthogonal Configuration Interaction Perturbation Theory to the Second Order (NOCI-PT2), to accurately calculate chemical energies. This method reliably computes dynamic correlation for improved predictions of molecular behavior.

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Area of Science:

Computational Chemistry
Quantum Chemistry
Theoretical Chemistry

Background:

Nonorthogonal multireference methods offer chemical insight via diabatic states but struggle with quantitative accuracy.
Accurate prediction of statically correlated systems requires reliable computation of dynamic correlation.

Purpose of the Study:

To introduce a rigorous perturbative correction to nonorthogonal configuration interaction.
To enable reliable computation of dynamic correlation in nonorthogonal multireference expansions.
To achieve quantitative accuracy in predicting adiabatic energies and potential energy surfaces.

Main Methods:

Developed second-order Nonorthogonal Configuration Interaction Perturbation Theory to the Second Order (NOCI-PT2).
Utilized a zeroth-order generalized Fock Hamiltonian.
Defined the first-order interacting space using single and double excitations from reference determinants.

Main Results:

NOCI-PT2 quantitatively predicts multireference potential energy surfaces.
The method provides accurate state-specific ground and excited states for adiabatic avoided crossings.
Introduced an explicit imaginary-shift formalism with significantly smaller shift values than conventional methods.

Conclusions:

NOCI-PT2 represents a rigorous nonorthogonal extension to conventional multireference perturbation theories.
The developed theory achieves quantitative accuracy for challenging chemical systems.
The new imaginary-shift formalism enhances computational efficiency and stability.