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The Van der Waals Equation

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,...

Van der Waals Equation

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 Equation

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.

Van der Waals Interactions

Van der Waals Interactions

Atoms and molecules interact with each other through intermolecular forces. These electrostatic forces arise from attractive or repulsive interactions between particles with permanent, partial, or temporary charges. The intermolecular forces between neutral atoms and molecules are ion–dipole, dipole–dipole, and dispersion forces, collectively known as van der Waals forces.

Applications of Integration to Probability Density Functions

Applications of Integration to Probability Density Functions

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), which...

Partial Derivatives and Gas Laws

Partial Derivatives and Gas Laws

In functions with multiple variables, partial derivatives describe how a function changes with respect to one variable while keeping the others constant. A partial derivative is calculated from the ordinary derivative of the function with respect to the desired variable, while treating the other variables as constants. Consider the function z = f(x, y). The partial derivative of the function z with respect to x at constant y is written as (∂z/∂x)y, using 'curly d'. It essentially tells us how z...

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Related Experiment Video

Updated: May 30, 2026

Analyzing Melts and Fluids from Ab Initio Molecular Dynamics Simulations with the UMD Package

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Published on: September 17, 2021

noloco: An efficient implementation of van der Waals density functionals based on a Monte-Carlo integration

Dmitrii Nabok¹, Peter Puschnig, Claudia Ambrosch-Draxl

¹Chair for Atomistic Modelling and Design of Materials, Montanuniversität Leoben, Franz-Josef-Straße 18, A-8700 Leoben, Austria.

Computer Physics Communications

|August 9, 2011

Summary

This study presents an efficient Monte Carlo method for calculating van der Waals interactions using van der Waals density functional (vdW-DF) theory. This approach improves computational scaling for complex systems, enabling accurate simulations of molecules and materials.

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Last Updated: May 30, 2026

Analyzing Melts and Fluids from Ab Initio Molecular Dynamics Simulations with the UMD Package

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Multiscale Sampling of a Heterogeneous Water/Metal Catalyst Interface using Density Functional Theory and Force-Field Molecular Dynamics

Multiscale Sampling of a Heterogeneous Water/Metal Catalyst Interface using Density Functional Theory and Force-Field Molecular Dynamics

Published on: April 12, 2019

Area of Science:

Computational Chemistry
Materials Science
Quantum Mechanics

Background:

Van der Waals (vdW) interactions are crucial for describing molecular and condensed matter systems.
Traditional density functional theory (DFT) methods often fail to accurately capture these non-local interactions.
The van der Waals density functional (vdW-DF) approach has emerged as a promising, parameter-free method for including vdW forces.

Purpose of the Study:

To develop a numerically efficient implementation of the vdW-DF method.
To overcome the unfavorable scaling of vdW-DF calculations with system size.
To provide a versatile tool for studying a wide range of systems with vdW interactions.

Main Methods:

Implementation of a Monte Carlo technique for multi-dimensional integration.
Adaptation of the vdW-DF method for efficient computation.
Application to various systems including dimers, molecular crystals, and physisorbed molecules.

Main Results:

Demonstrated significant numerical efficiency gains compared to standard vdW-DF implementations.
Successfully applied the method to diverse and complex systems.
Validated the accuracy and applicability of the Monte Carlo approach for vdW-DF.

Conclusions:

The developed Monte Carlo implementation offers a computationally feasible way to apply vdW-DF.
This advancement facilitates more accurate theoretical studies of systems dominated by vdW interactions.
The method is broadly applicable to molecular and solid-state systems in computational chemistry and materials science.