What is a Gaussian Grid and Nomenclature of Labels

Gaussian grid: A term commonly used in computational chemistry, meteorology and physics for an arbitrary type of numerical grid used to approximate a function or values over some specific kind of spherical or planar surface. This grid format is especially popular for disciplines, such as quantum chemistry and atmospheric sciences, that do integrations and/or interpolations on spherical geometries due to its lower truncation error and computational cost.

What is a Gaussian Grid?

Gaussian grid — a grid of points, with locations determined according to a Gaussian (normal) distribution. In several scientific applications, in particular, computational chemistry Gaussian grids are thus used to perform numerical integrations, for instance the Gaussian quadrature [9]. Ideal when spherical surface integrals or powers of such functions vary over spherical surfaces, this technique facilitates integration over the region with very high accuracy.

The key aspect of a Gaussian grid is that it has non-uniformly spaced points on the surface. Instead, points tend to be more spaced at the equator of the sphere, where the function we are integrating generally varies less and closer together around poles of that sphere. It is a computationally cheap approach to obtain accurate precision in calculations. The Gaussian grid is therefore particularly useful for quantum chemistry calculations that require accurate numerical integration, especially when we are dealing with spherical systems.

Features of Gaussian Grid

Gaussian grids are different from uniform grids mainly in the following respects :

Non-Uniform Distribution of Points:

The points on the grid are more closely spaced near the poles (or top and bottom) of the sphere, which causes important regions where the gradient is large to be sampled more often.

At the equator, the points (and therefore grid spacing in this case) is large — which serves as a balancing effect also aiding in integration.

High-Efficiency Integration:

It is designed to enhance integration where points fall on spherical surfaces (especially in quantum mechanics and atmospheric modeling) via Gaussian distributed grid of points lying along the line.

Asymmetry:

Gaussian grids are also asymmetrical with latitude. This means the amount of grid points is different in a way that reduces the error caused by the changes of the function.

Applications of Gaussian Grid

Computational Chemistry:

Gaussian-like grids are common in computational chemistry for calculating integrals used in quantum chemical calculations that require integration of molecular wavefunctions over spherical coordinates.

Grids like these are used in a variety of methods — Density Functional Theory (DFT), Hartree-Fock calculations etc., to approximate integrals of molecular properties over space.

Meteorology and Climate Modelling:-

Gaussian grids are used in the context of atmospheric science and climate modeling, where they serve as representations of the Earth surface or the atmosphere. Grid points are not uniformly distributed, enabling detailed sampling of high and low latitudes where climate spatial variations are often more complex and relevant to the weather model.

Modeling Global Ocean Circulation:

It is also applied to global ocean circulation models, where the non-uniform distribution of points allows for a better representation of ocean dynamics, which can be important at the poles and equator.

Simulations in Geophysics and Astronomy:

Modeling the surfaces of primitive bodies and atmospheres over stars, both geophysical and astronomical simulations use Gaussian grids to represent data that naturally falls upon spherical geometries.

Gaussian Grid: Naming Conventions

Gaussian grids are named, in part, according to the underlying distribution of grid points and the use (or method) associated with them. Here are a few common naming conventions and what they stand for:

Gaussian Quadrature Points:

General term Gaussian quadrature is the numerical method to approximate integrals. Gaussian quadrature points, the points of a Gaussian grid, come from Gaussian quadrature rules (i.e., if you want to integrate something you can do that with these points). The grid is intended to place points in places difficult for computation, such as poles or areas where the gradient varies considerably.

N-Point Gaussian Grid:

This is usually followed with a number in terms of points used on the grid. A 10-point Gaussian grid, for instance, indicates that there are 10 points placed on the sphere. As N increases, the grid becomes more detailed and accurate but at the expense of computational resources. Gaussian grids are typically named based on their resolution level, with higher-point grids providing greater accuracy.

Latitude-Longitude Naming:

Gaussian grids are sometimes called latitude/longitude-based Gaussian grids. Grids may be defined in point placements (e.g., n points between North Pole and South Pole or within a given lat toplat band). An example of such naming is for meteorological grids over which the surface of the Earth is divided by certain lines of latitude.

Spherical Harmonic Basis:

It may also have come from quantum chemistry/physics nomenclature context, since in these fields it is common to use spherical harmonic expansions together with Gaussian grids for purposes of efficient calculation. L2 grid in this case refers to a collection of points where the LSH function is of 2-degree spherical harmonic, as one such resolution for the grid [18].

Basis of a Gaussian Grid in Quantum Chemistry:

A Gaussian grid can be referenced in Quantum Chemistry based on the molecule or basis set associated with it, as in 'Gaussian grid for DFT calculations' or based on how many points were selected at which to calculated integrals over the molecular space. These grids are generally designed to provide good performance for specific quantum mechanical calculations with spherical symmetry, or where high accuracy is needed.

Greater Precision in Calculations:

Introducing more points in the poles allows to increase accuracy where functions vary quickly there and are of remarkable importance in problems relevant for quantum chemistry or climate models.

Computational Efficiency:

alphaA01 alphaE02SkewAlpha06, the uniform grid accounts more points in regions that they are unnecessary, while Gaussian grids create a concentrated allotment of points where integration is required and not include them on other areas, this means less total amount of grid points are needed to achieve the same level of precision than with a uniform grid and then have Gaussian grids computationally effective when used for spherical integrations.

Adaptability:

Gaussian grids are versatile and can be tailored to use for different applications, depending on how the simulation or calculation needs to be performed.