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Gauss's Law

Gauss's Law

If a closed surface does not have any charge inside where an electric field line can terminate, then the electric field line entering the surface at one point must necessarily exit at some other point of the surface. Therefore, if a closed surface does not have any charges inside the enclosed volume, then the electric flux through the surface is zero. What happens to the electric flux if there are some charges inside the enclosed volume? Gauss's law gives a quantitative answer to this question.

Gauss's Law: Problem-Solving

Gauss's Law: Problem-Solving

Gauss's law helps determine electric fields even though the law is not directly about electric fields but electric flux. In situations with certain symmetries (spherical, cylindrical, or planar) in the charge distribution, the electric field can be deduced based on the knowledge of the electric flux. In these systems, we can find a Gaussian surface S over which the electric field has a constant magnitude. Furthermore, suppose the electric field is parallel (or antiparallel) to the area...

Maxwell-Boltzmann Distribution: Problem Solving

Maxwell-Boltzmann Distribution: Problem Solving

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This distribution function f(v) is defined by saying that the expected number N (v1,v2) of particles with speeds between v1 and v2 is given by

Gauss's Law: Planar Symmetry

Gauss's Law: Planar Symmetry

A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...

Gravitational Potential Energy for Extended Objects

Gravitational Potential Energy for Extended Objects

Consider a system comprising several point masses. The coordinates of the center of mass for this system can be expressed as the summation of the product of each mass and its position vector divided by the total mass:

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...

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Updated: Sep 30, 2025

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

Published on: April 8, 2020

A Gaussian process regression adaptive density guided approach for potential energy surface construction.

Gunnar Schmitz¹, Emil Lund Klinting¹, Ove Christiansen¹

¹Department of Chemistry, Aarhus Universitet, DK-8000 Aarhus, Denmark.

The Journal of Chemical Physics

|March 15, 2022

Summary

We developed an iterative Gaussian process regression-adaptive density guided approach (GPR-ADGA) for constructing potential energy surfaces. This method significantly reduces computational cost while accurately predicting molecular vibrational frequencies.

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Area of Science:

Computational Chemistry
Quantum Chemistry
Materials Science

Background:

Accurate potential energy surface (PES) construction is crucial for understanding molecular behavior and predicting properties.
Traditional methods for PES construction can be computationally expensive, requiring numerous electronic structure calculations.
Existing iterative schemes often lack efficient strategies for selecting crucial data points.

Purpose of the Study:

To introduce a novel iterative scheme, Gaussian process regression-adaptive density guided approach (GPR-ADGA), for efficient PES construction.
To combine physical insights from the adaptive density guided approach (ADGA) with the predictive power of Gaussian process regression (GPR).
To reduce the number of required electronic structure calculations for accurate PES generation.

Main Methods:

The GPR-ADGA iteratively builds the PES by integrating ADGA's physical importance weighting with GPR's data approximation.
ADGA guides point selection based on the average density of vibrational states.
GPR's prediction variance is used to select the most informative points suggested by ADGA, incorporating derivative information.

Main Results:

The GPR-ADGA method accurately predicts fundamental excitation frequencies with a root mean square deviation (RMSD) below 2 cm⁻¹ compared to standard ADGA.
This accuracy is achieved with a substantial reduction of 65%-90% in the number of single-point calculations.
The iterative process starts with an initial Hessian and does not require pre-sampling of configurations.

Conclusions:

The GPR-ADGA represents a significant advancement in efficient and accurate potential energy surface construction.
This approach offers considerable computational savings without compromising predictive accuracy for molecular vibrational frequencies.
The method is robust, commencing PES construction without initial configuration sampling.