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Related Concept Videos

Van der Waals Equation01:10

Van der Waals Equation

The ideal gas law is an approximation that works well at high temperatures and low pressures. The van der Waals equation of state (named after the Dutch physicist Johannes van der Waals, 1837−1923) improves it by considering two factors.
First, the attractive forces between molecules, which are stronger at higher densities and reduce the pressure, are considered by adding to the pressure a term equal to the square of the molar density multiplied by a positive coefficient a. Second, the volume...
Real Gases: Effects of Intermolecular Forces and Molecular Volume Deriving Van der Waals Equation04:01

Real Gases: Effects of Intermolecular Forces and Molecular Volume Deriving Van der Waals Equation

Thus far, the ideal gas law, PV = nRT, has been applied to a variety of different types of problems, ranging from reaction stoichiometry and empirical and molecular formula problems to determining the density and molar mass of a gas. However, the behavior of a gas is often non-ideal, meaning that the observed relationships between its pressure, volume, and temperature are not accurately described by the gas laws.
Dimensionless Groups in Fluid Mechanics01:15

Dimensionless Groups in Fluid Mechanics

Dimensionless groups in fluid mechanics provide simplified ratios that help analyze fluid behavior without relying on specific units. The Reynolds number (Re), which represents the ratio of inertial to viscous forces, distinguishes between laminar and turbulent flows, making it essential in the design of pipelines and aerodynamic surfaces. The Froude number (Fr), the ratio of inertial to gravitational forces, is particularly useful in predicting wave formation and hydraulic jumps in...
Deviation from Ideal Behaviour01:23

Deviation from Ideal Behaviour

Real gases do not perfectly obey the ideal gas laws, especially at high pressures and low temperatures or when they are about to condense to a liquid. These deviations occur due to intermolecular forces between gas molecules. Repulsive forces aid expansion and are significant when molecules are very close together, typically at high pressure. Attractive forces assist compression and have a longer range, being effective over several molecular diameters. They become significant when molecules are...
The Van der Waals Equation01:26

The Van der Waals Equation

The ideal gas law is based on two simplifying assumptions: first, that there are no intermolecular attractions between gas molecules, and second, that the volume occupied by the molecules themselves is negligible compared with the volume of the container. However, these assumptions don't hold up under all conditions - specifically, at high pressures and low temperatures, as gas tends to deviate from ideal gas behavior.The van der Waals equation is an enhanced version of the ideal gas law,...
Thermal Sigmatropic Reactions: Overview01:16

Thermal Sigmatropic Reactions: Overview

Sigmatropic rearrangements are a class of pericyclic reactions in which a σ bond migrates from one part of a π system to another. These are intramolecular rearrangements where the total number of σ and π bonds remain unchanged.
Sigmatropic shifts are classified based on an order term [i, j ], where i and j indicate the number of atoms across which each end of the σ bond migrates. Below are examples of a [3,3] sigmatropic shift in 1,5-hexadiene, referred to as...

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Related Experiment Video

Updated: May 24, 2026

Probe Type II Band Alignment in One-Dimensional Van Der Waals Heterostructures Using First-Principles Calculations
13:56

Probe Type II Band Alignment in One-Dimensional Van Der Waals Heterostructures Using First-Principles Calculations

Published on: October 12, 2019

Variational numerical renormalization group: bridging the gap between NRG and density matrix renormalization group.

Iztok Pižorn1, Frank Verstraete

  • 1University of Vienna, Faculty of Physics, Wien, Austria.

Physical Review Letters
|March 10, 2012
PubMed
Summary

The numerical renormalization group (NRG) is now a variational method, enhancing low-energy spectrum accuracy. Combining NRG and density matrix renormalization group (DMRG) improves quantum system simulations.

Related Experiment Videos

Last Updated: May 24, 2026

Probe Type II Band Alignment in One-Dimensional Van Der Waals Heterostructures Using First-Principles Calculations
13:56

Probe Type II Band Alignment in One-Dimensional Van Der Waals Heterostructures Using First-Principles Calculations

Published on: October 12, 2019

Area of Science:

  • Condensed matter physics
  • Quantum many-body systems
  • Computational physics

Background:

  • The numerical renormalization group (NRG) is a powerful technique for studying strongly correlated quantum systems.
  • Improving the accuracy of NRG calculations, especially for low-energy properties, remains an active area of research.
  • The density matrix renormalization group (DMRG) is another successful method for similar problems, particularly effective for one-dimensional systems.

Purpose of the Study:

  • To reframe the numerical renormalization group (NRG) as a variational method.
  • To systematically improve the accuracy of the energy spectrum obtained from NRG calculations.
  • To explore the synergy between NRG and the density matrix renormalization group (DMRG) for enhanced applicability.

Main Methods:

  • Reformulating the numerical renormalization group (NRG) as a variational method.
  • Defining a cost function as the sum of energies from an effective low-energy Hamiltonian.
  • Implementing a sweeping algorithm similar to density matrix renormalization group (DMRG) for targeting multiple low-energy states.
  • Applying the enhanced method to quantum spin chains and single impurity Anderson models.

Main Results:

  • The variational reformulation of NRG allows for systematic spectral improvements.
  • The combined NRG-DMRG approach shows significant enhancement in the accuracy of effective eigenstates compared to standard NRG.
  • Improved accuracy is particularly notable near the transition to the continuum limit.
  • Simulations of quantum spin chains and Anderson impurity models demonstrate the method's effectiveness.

Conclusions:

  • The variational NRG approach offers a robust framework for improving spectral calculations.
  • The synergy between NRG and DMRG provides a powerful, combined method for studying complex quantum systems.
  • This enhanced methodology extends the applicability of both NRG and DMRG, particularly for challenging spectral properties.