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Related Concept Videos

Higher Derivatives01:29

Higher Derivatives

In calculus, higher-order derivatives extend the idea of differentiation beyond the first derivative to capture successive rates of change. These derivatives provide detailed information about the behavior of functions and have important applications in both mathematics and physics. To illustrate these concepts, consider the example function\begin{equation*}f(x) = x^3 - x\end{equation*}which serves as a useful case study for exploring higher derivatives.The first derivative represents the slope...
Propagation of Uncertainty from Systematic Error01:10

Propagation of Uncertainty from Systematic Error

The atomic mass of an element varies due to the relative ratio of its isotopes. A sample's relative proportion of oxygen isotopes influences its average atomic mass. For instance, if we were to measure the atomic mass of oxygen from a sample, the mass would be a weighted average of the isotopic masses of oxygen in that sample. Since a single sample is not likely to perfectly reflect the true atomic mass of oxygen for all the molecules of oxygen on Earth, the mass we obtain from this particular...
Propagation of Uncertainty from Random Error00:59

Propagation of Uncertainty from Random Error

An experiment often consists of more than a single step. In this case, measurements at each step give rise to uncertainty. Because the measurements occur in successive steps, the uncertainty in one step necessarily contributes to that in the subsequent step. As we perform statistical analysis on these types of experiments, we must learn to account for the propagation of uncertainty from one step to the next. The propagation of uncertainty depends on the type of arithmetic operation performed on...
Substitution Rule Applied to Indefinite Integrals01:27

Substitution Rule Applied to Indefinite Integrals

When a force is applied to a linear spring, the restoring force increases proportionally with the amount of displacement. This behavior is described by Hooke’s law, which allows the work done on the spring to be determined directly from the force–displacement relationship. In this case, the force varies in a simple and predictable manner, making the calculation relatively simple.On the other hand, a nonlinear spring does not obey Hooke’s law. Its restoring force depends on position in a...
Integration by Parts: Indefinite Integrals01:26

Integration by Parts: Indefinite Integrals

Integration by parts is a fundamental technique in calculus for evaluating integrals involving the product of two functions. It is particularly useful when direct integration is not feasible. The method is based on the product rule for differentiation, which states that the derivative of a product equals the derivative of the first function times the second, plus the first function times the derivative of the second. By integrating this identity and rearranging terms, the integration by parts...
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Substitution Rule Applied to Definite Integrals

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Related Experiment Video

Updated: May 20, 2026

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry
12:14

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

Published on: August 12, 2013

Umbrella integration with higher-order correction terms.

Johannes Kästner1

  • 1Computational Biochemistry Group, Institute of Theoretical Chemistry, University of Stuttgart, Pfaffenwaldring 55, D-70569 Stuttgart, Germany.

The Journal of Chemical Physics
|July 12, 2012
PubMed
Summary

Umbrella integration, a method for analyzing molecular dynamics simulations, refines free-energy calculations. This study derives new terms to improve the accuracy of these calculations by utilizing central moments of sampled distributions.

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Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

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The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry
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Published on: August 12, 2013

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry
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Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

Published on: April 8, 2020

Area of Science:

  • Computational chemistry and molecular dynamics simulations.

Background:

  • Umbrella integration analyzes umbrella sampling simulations to determine free-energy changes.
  • Current methods often approximate sampled distributions with normal distributions, truncating the free-energy power series.
  • This approximation is equivalent to truncating a cumulant expansion.

Purpose of the Study:

  • To derive and present expressions for additional terms in the power series expansion of free energy.
  • To enable more accurate free-energy calculations by going beyond quadratic approximations.
  • To provide a method for testing the validity of common approximations in umbrella integration.

Main Methods:

  • Derivation of new terms for the free-energy power series based on the reaction coordinate.
  • Calculation of these additional terms using central moments of the sampled distributions.
  • Formulation of an extended umbrella integration method.

Main Results:

  • Expressions for higher-order terms in the free-energy power series were successfully derived.
  • These terms can be computed from the central moments of the umbrella sampling distributions.
  • The derived expressions offer a pathway to quantify the error introduced by common approximations.

Conclusions:

  • The derived terms extend umbrella integration beyond the standard normal distribution approximation.
  • This extension allows for a more rigorous assessment of approximation accuracy in free-energy calculations.
  • The method provides a means to test and potentially improve the reliability of molecular dynamics free-energy results.