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Related Concept Videos

Linear time-invariant Systems01:23

Linear time-invariant Systems

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A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be...
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Linear Approximation in Frequency Domain01:26

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Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
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Classification of Systems-I01:26

Classification of Systems-I

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Linearity is a system property characterized by a direct input-output relationship, combining homogeneity and additivity.
Homogeneity dictates that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Mathematically, this is expressed as:
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Linear Approximation in Time Domain01:21

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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,...
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First Order Systems01:21

First Order Systems

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First-order systems, such as RC circuits, are foundational in understanding dynamic systems due to their straightforward input-output relationship. Analyzing their responses to different input functions under zero initial conditions reveals significant insights into system behavior.
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Mechanical Systems01:22

Mechanical Systems

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Mechanical systems are analogous to to electrical networks where springs and masses play similar roles to inductors and capacitors, respectively. A viscous damper in mechanical systems functions similarly to a resistor in electrical networks, dissipating energy. The forces acting on a mass in such systems include an applied force in the direction of motion, counteracted by forces from the spring, a viscous damper, and the mass's acceleration. This interplay of forces is mathematically...
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Related Experiment Video

Updated: Apr 26, 2026

Shaping the Amplitude and Phase of Laser Beams by Using a Phase-only Spatial Light Modulator
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Linear optical response of finite systems using multishift linear system solvers.

Hannes Hübener1, Feliciano Giustino1

  • 1Department of Materials, University of Oxford, Oxford OX1 3PH, United Kingdom.

The Journal of Chemical Physics
|August 3, 2014
PubMed
Summary

Multishift linear system solvers efficiently compute electronic density response for excited-state calculations. This method offers accurate excitation energies and oscillator strengths, matching traditional techniques with improved computational speed.

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Area of Science:

  • Computational Chemistry
  • Quantum Chemistry
  • Materials Science

Background:

  • Linear-response time-dependent density functional theory (LR-TDDFT) is crucial for calculating electronic properties.
  • Traditional methods like Casida's require computationally expensive matrix diagonalization.

Purpose of the Study:

  • To apply multishift linear system solvers to LR-TDDFT.
  • To evaluate the computational efficiency and accuracy of this novel approach for excited-state calculations.

Main Methods:

  • Implementation of multishift solvers within the LR-TDDFT framework.
  • Calculation of frequency-dependent electronic density response using conjugate gradients.
  • Comparison with standard diagonalization methods (Casida's method).

Main Results:

  • Multishift TDDFT accurately reproduces excitation energies and oscillator strengths.
  • The method achieves these results at the cost of a single linear system solution.
  • Test calculations on benzene, porphin, and chlorophyll demonstrate the technique's viability.

Conclusions:

  • Multishift linear system solvers provide a computationally advantageous alternative for excited-state calculations in DFT.
  • This technique shows potential for broad application in quantum chemistry and beyond.