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Related Concept Videos

Second Order systems II01:18

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.
If  ζ...
Second Order systems I01:20

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...
Second Derivatives and Laplace Operator01:22

Second Derivatives and Laplace Operator

The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.
Consider a scalar function. The curl of its...
Second-Order Circuits01:17

Second-Order Circuits

Integrating two fundamental energy storage elements in electrical circuits results in second-order circuits, encompassing RLC circuits and circuits with dual capacitors or inductors (RC and RL circuits). Second-order circuits are identified by second-order differential equations that link input and output signals.
Input signals typically originate from voltage or current sources, with the output often representing voltage across the capacitor and/or current through the inductor. For example, in...
Second-order Op Amp Circuits01:19

Second-order Op Amp Circuits

Implementing second-order low-pass filters in audio systems is crucial in refining audio signals by eliminating undesirable high-frequency noise. These filters typically involve second-order op-amp circuits configured as voltage followers, encompassing two nodes with distinct storage elements.
The analysis of such circuits follows a systematic approach, similar to the second-order RLC circuits. In practical scenarios, bulky inductors are rarely employed due to their size and weight. This means...
Second Uniqueness Theorem01:16

Second Uniqueness Theorem

Consider a region consisting of several individual conductors with a definite charge density in the region between these conductors. The second uniqueness theorem states that if the total charge on each conductor and the charge density in the in-between region are known, then the electric field can be uniquely determined.
In contrast, consider that the electric field is non-unique and apply Gauss's law in divergence form in the region between the conductors and the integral form to the surface...

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New implementation of the configuration-based multi-reference second order perturbation theory.

Yibo Lei1, Yubin Wang, Huixian Han

  • 1Key Laboratory of Synthetic and Natural Functional Molecule Chemistry of Ministry of Education, The College of Chemistry and Materials Science, Northwest University, Xi'an 710069, People's Republic of China.

The Journal of Chemical Physics
|October 16, 2012
PubMed
Summary

We developed an improved configuration-based multi-reference second-order perturbation (CB-MRPT2) method for accurate and efficient computational chemistry. This new approach, featuring novel algorithms and modes, effectively addresses challenges like intruder states and improves energy calculations for various molecular systems.

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Area of Science:

  • Quantum Chemistry
  • Computational Chemistry
  • Theoretical Chemistry

Background:

  • Perturbation theory is crucial for understanding molecular electronic structures.
  • Existing multi-reference second-order perturbation methods face challenges with accuracy and efficiency, particularly concerning intruder states and degenerate model spaces.

Purpose of the Study:

  • To present an enhanced configuration-based multi-reference second-order perturbation (CB-MRPT2) approach.
  • To implement new algorithms for calculating matrix elements and introduce novel computational modes (DP and DPD).
  • To assess the accuracy and efficiency of the improved CB-MRPT2 method across various chemical systems.

Main Methods:

  • Improved configuration-based multi-reference second-order perturbation (CB-MRPT2) formulation based on Lindgren's theory.
  • Reclassification of perturbation functions and new algorithms utilizing graphical unitary group approach and hole-particle symmetry.
  • Implementation of diagonalize-then-perturb (DP) and diagonalize-then-perturb-then-diagonalize (DPD) modes.

Main Results:

  • The new CB-MRPT2 method demonstrates competitive computational accuracy and efficiency.
  • The DPD mode effectively resolves shortcomings in potential energy curves, such as those encountered with LiF, without requiring state mixture.
  • Accurate relative energy ordering for di-copper-oxygen-ammonia isomers was achieved, consistent with higher-level methods.

Conclusions:

  • The improved CB-MRPT2 method is a robust and intruder-free computational tool.
  • The DPD mode offers a significant advancement over traditional CASPT2 for specific molecular problems.
  • The enhanced CB-MRPT2 method is suitable for studying small to medium-sized molecular systems effectively.