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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...
Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...
¹H NMR: Interpreting Distorted and Overlapping Signals01:02

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 slanted or...
Laws of Logarithms I01:30

Laws of Logarithms I

Logarithms are fundamental mathematical operations that serve as the inverse of exponentiation. They provide a means to express how many times a base must be raised to yield a given number. For base 10, often referred to as the common logarithm, the notation is written simply as log. Thus, if 10n = x, then log⁡(x) = n. This relationship makes logarithms especially valuable in simplifying complex calculations involving multiplication, division, and exponentiation.Logarithmic expressions are...
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...
Laws of Logarithms II01:28

Laws of Logarithms II

Logarithmic laws provide essential tools for simplifying and evaluating exponential expressions, particularly in mathematical and applied settings where powers and repeated multiplication play a central role. Two important rules are the power law and the change-of-base formula, both allowing for transforming expressions into more manageable forms.The power law of logarithms states that the logarithm of a number raised to an exponent equals the exponent multiplied by the logarithm of the base...

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An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
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Logarithm second-order many-body perturbation method for extended systems.

Yu-ya Ohnishi1, So Hirata

  • 1Department of Chemistry, Quantum Theory Project and The Center for Macromolecular Science and Engineering, University of Florida, Gainesville, Florida 32611, USA.

The Journal of Chemical Physics
|July 24, 2010
PubMed
Summary
This summary is machine-generated.

We developed a log n downsampling method for Møller-Plesset perturbation (MP2) calculations in extended systems. This significantly speeds up computations for larger unit cells with minimal error.

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Area of Science:

  • Computational Chemistry
  • Condensed Matter Physics
  • Quantum Chemistry

Background:

  • Accurate electronic structure calculations are crucial for understanding material properties.
  • The Møller-Plesset perturbation (MP2) method provides accurate correlation energies but is computationally expensive for extended systems.
  • Existing downsampling schemes have limitations in computational efficiency for increasing system sizes.

Purpose of the Study:

  • To develop a more efficient computational method for MP2 calculations in one-dimensional periodic systems.
  • To reduce the computational cost scaling with system size.
  • To maintain accuracy in correlation energy and quasiparticle band structure calculations.

Main Methods:

  • Progressive downsampling of wave vectors in Brillouin zone integrations for MP2 calculations.
  • Employing an exponentially increasing downsampling factor (log n scheme) for occupied and unoccupied Bloch orbitals.
  • Developing an algorithm to compute quadrature weights for accurate two-electron integral calculations.
  • Combining log n and mod n downsampling schemes.

Main Results:

  • The log n downsampling scheme reduces the number of included Bloch orbitals logarithmically with respect to basis functions per unit cell.
  • This method achieves greater speedup compared to previous mod n schemes as unit cell size increases.
  • A combined log n and mod n approach accelerated polyacetylene MP2 calculations by a factor of 20 with a few percent error in correlation energy.
  • Accurate MP2 quasiparticle energy bands were reproduced at a fraction of the usual computational cost.

Conclusions:

  • The proposed log n downsampling method offers significant computational speedup for MP2 calculations in extended systems.
  • Accurate band indexing and quadrature weights are essential for reliable results.
  • Combined downsampling schemes provide a practical approach for accurate and efficient electronic structure calculations.