**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.

Concept# Gaussian orbital

Summary

In computational chemistry and molecular physics, Gaussian orbitals (also known as Gaussian type orbitals, GTOs or Gaussians) are functions used as atomic orbitals in the LCAO method for the representation of electron orbitals in molecules and numerous properties that depend on these.
The use of Gaussian orbitals in electronic structure theory (instead of the more physical Slater-type orbitals) was first proposed by Boys in 1950. The principal reason for the use of Gaussian basis functions in molecular quantum chemical calculations is the 'Gaussian Product Theorem', which guarantees that the product of two GTOs centered on two different atoms is a finite sum of Gaussians centered on a point along the axis connecting them. In this manner, four-center integrals can be reduced to finite sums of two-center integrals, and in a next step to finite sums of one-center integrals. The speedup by 4-5 orders of magnitude compared to Slater orbitals outweighs the extra cost entailed by the larger number of basis functions generally required in a Gaussian calculation.
For reasons of convenience, many quantum chemistry programs work in a basis of Cartesian Gaussians even when spherical Gaussians are requested, as integral evaluation is much easier in the cartesian basis, and the spherical functions can be simply expressed using the cartesian functions.
The Gaussian basis functions obey the usual radial-angular decomposition
where is a spherical harmonic, and are the angular momentum and its component, and are spherical coordinates.
While for Slater orbitals the radial part is
being a normalization constant, for Gaussian primitives the radial part is
where is the normalization constant corresponding to the Gaussian.
The normalization condition which determines or is
which in general does not impose orthogonality in .
Because an individual primitive Gaussian function gives a rather poor description for the electronic wave function near the nucleus, Gaussian basis sets are almost always contracted:
where is the contraction coefficient for the primitive with exponent .

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (34)

Related people (12)

Related units (1)

Related concepts (5)

Related courses (19)

Related lectures (99)

Basis set (chemistry)

In theoretical and computational chemistry, a basis set is a set of functions (called basis functions) that is used to represent the electronic wave function in the Hartree–Fock method or density-functional theory in order to turn the partial differential equations of the model into algebraic equations suitable for efficient implementation on a computer. The use of basis sets is equivalent to the use of an approximate resolution of the identity: the orbitals are expanded within the basis set as a linear combination of the basis functions , where the expansion coefficients are given by .

Slater-type orbital

Slater-type orbitals (STOs) are functions used as atomic orbitals in the linear combination of atomic orbitals molecular orbital method. They are named after the physicist John C. Slater, who introduced them in 1930. They possess exponential decay at long range and Kato's cusp condition at short range (when combined as hydrogen-like atom functions, i.e. the analytical solutions of the stationary Schrödinger equation for one electron atoms). Unlike the hydrogen-like ("hydrogenic") Schrödinger orbitals, STOs have no radial nodes (neither do Gaussian-type orbitals).

Linear combination of atomic orbitals

A linear combination of atomic orbitals or LCAO is a quantum superposition of atomic orbitals and a technique for calculating molecular orbitals in quantum chemistry. In quantum mechanics, electron configurations of atoms are described as wavefunctions. In a mathematical sense, these wave functions are the basis set of functions, the basis functions, which describe the electrons of a given atom. In chemical reactions, orbital wavefunctions are modified, i.e.

CH-250: Mathematical methods in chemistry

This course consists of two parts. The first part covers basic concepts of molecular symmetry and the application of group theory to describe it. The second part introduces Laplace transforms and Four

CH-222: Coordination chemistry

Fundamental knowledge of coordination compounds.

CH-110: Advanced general chemistry I

Le cours comporte deux parties. Les bases de la thermodynamique des équilibres et de la cinétique des réactions sont introduites dans l'une d'elles. Les premières notions de chimie quantique sur les é

Molecular Orbital Theory

Covers the concept of eigenfunctions of H as bases for irreducible representations and integrals.

Crystal Field Theory: Spectrochemical Series

Explores the Spectrochemical Series for metals and ligands, Crystal Field Splitting, Jahn-Teller distortion, and bonding interactions in coordination compounds.

Quantum Mechanics: Particles and Interactions

Explores quantum mechanics, emphasizing particles, interactions, spin, and wave functions.

Transition metal oxides represent a class of materials displaying very unusual electronic, structural and magnetic properties. They are extremely interesting, both from a technological and fundamental point of view. The most important characteristic of the ...

Using scanning tunneling microscopy (STM), we experimentally and theoretically investigate isolated platinum phthalocyanine (PtPc) molecules adsorbed on an atomically thin NaCl(100) film vapor deposited on Au(111). We obtain good agreement between theory a ...

Michele Ceriotti, Jigyasa Nigam, Michael John Willatt

Symmetry considerations are at the core of the major frameworks used to provide an effective mathematical representation of atomic configurations that is then used in machine-learning models to predict the properties associated with each structure. In most ...