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Related Concept Videos

Transfer Function to State Space01:23

Transfer Function to State Space

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State-space representation is a powerful tool for simulating physical systems on digital computers, necessitating the conversion of the transfer function into state-space form. Consider an nth-order linear differential equation with constant coefficients, like those encountered in an RLC circuit. The state variables are selected as the output and its n−1 derivatives. Differentiating these variables and substituting them back into the original equation produces the state equations.
In an RLC...
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State Space to Transfer Function01:21

State Space to Transfer Function

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The conversion of state-space representation to a transfer function is a fundamental process in system analysis. It provides a method for transitioning from a time-domain description to a frequency-domain representation, which is crucial for simplifying the analysis and design of control systems.
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Correlation means that there is a relationship between two or more variables (such as ice cream consumption and crime), but this relationship does not necessarily imply cause and effect. When two variables are correlated, it simply means that as one variable changes, so does the other. We can measure correlation by calculating a statistic known as a correlation coefficient. A correlation coefficient is a number from -1 to +1 that indicates the strength and direction of the relationship between...
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Correlation and Causation01:27

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Statistical tests can calculate whether there is a relationship, or correlation, between independent and dependent variables. An indirect relationship of the variables signifies a correlation, while a direct relationship shows causation. If it is determined that no connection exists between the variables, then the correlation is a coincidence.
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The range is one of the measures of variation. It can be defined as the difference between a dataset's highest and lowest values. For example, in the study of seven 16-ounce soda cans, the filled volume of soda was measured, thus producing the following amount (in ounces) of soda:
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Applications of Integration to Probability Density Functions

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Continuous probability distributions are used to model random variables that can take on any real value within a specified range. These variables do not take on isolated or countable values but rather exist on a continuum. For example, the height of an individual can be measured with increasing precision—such as 163.5 or 165.25 centimeters—demonstrating that height is a continuous random variable.The behavior of such variables is described using a probability density function (PDF),...
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Short-range density functional correlation within the restricted active space CI method.

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  • 1Kimika Fakultatea, Euskal Herria Unibersitatea (UPV/EHU), Donostia International Physics Center (DIPC), P.K. 1072, 20080 Donostia, Euskadi, Spain and IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Euskadi, Spain.

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This study introduces a novel hybrid method combining wave function and density functional theory (RAS-srDFT) to improve electronic structure calculations. The new approach accurately captures dynamic electron correlations, enhancing predictions for molecular ground and excited states.

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Area of Science:

  • Computational chemistry
  • Quantum chemistry
  • Electronic structure theory

Background:

  • Restricted active space configuration interaction (RASCI) method misses dynamic electron correlations.
  • Accurate calculation of electron correlations is crucial for predicting molecular properties.

Purpose of the Study:

  • To develop a hybrid wave function-density functional theory (DFT) method.
  • To recover dynamic electron correlations absent in RASCI.
  • To improve the computation of electronic ground and excited states.

Main Methods:

  • Introduced a range-separated hybrid approach combining RASCI with DFT.
  • Developed working equations and computational algorithms for the new RAS-srDFT method.
  • Tested the method on excitation energy calculations for organic molecules.

Main Results:

  • The RAS-srDFT method successfully recovers dynamic electron correlations.
  • Calculations of relative energies showed quantitative improvement over standard RASCI.
  • The combination of RASCI wave functions and short-range DFT functionals yielded promising results.

Conclusions:

  • RAS-srDFT is a promising method for accurately computing electronic structures.
  • The method offers a significant improvement for systems requiring accurate dynamic and nondynamic electron correlation.
  • This work lays the foundation for advanced electronic structure calculations.