In computational chemistry, a basis set is used to generate the initial guess wave function, to further study the electronic properties of the system. Or in general terms, it is a set of functions which is employed in the calculations to get the molecular orbitals. For details you may have a read at:
https://chem.libretexts.org/Courses/Pacific_Union_College/Quantum_Chemistry/11%3A_Computational_Quantum_Chemistry/11.02%3A_Gaussian_Basis_Sets
Before discussing basis sets, it is crucial to recall that the Schrödinger equation governs the wavefunction of a quantum-mechanical system, such as electrons, protons, and neutrons in an atom. The wavefunction (Ψ) is a mathematical description of the quantum state of an isolated quantum system. Once it is known, the position, energy, and momentum, for instance, can be determined. However, the Schrödinger equation is a linear partial differential equation with no exact analytical solution but for the hydrogen atom. Therefore, several approaches were sought, including Hartree-Fock and the Density Functional Theorom, which depend on approximating the wavefunction with a set of functions called basis functions and estimating the error, when appropriate, following experimental and theoretical approaches. Finally, the basis set is a set of basis functions. Therefore, a basis set is a linear combination of several basis functions to approximate a wave function. For instance, suppose that my function is a parabola f(x) = x(a-x), which can be approximated using a sinusoidal as follows (please see the attached equation). So, by fine-tuning n and a, we can increase the accuracy of the approximation.
In computational chemistry and physical chemistry, we first try to fit the unknown wavefunction based on our estimated orbitals (i.e., s,p,d,f) with predefined basis sets (e.g., STO-3G) and then run an optimization calculation to fine-tune parameters to reduce estimated error based on a chosen a DFT theory level of calculations using different software packages such as Gaussian 09 / 16.0. Nowadays, with Quantum Computing, it may be possible to increase accuracy and reduce computational cost, but this may require more qubits depending on the number of atoms in the system. I hope that you find this useful.
A basis set in computational chemistry is a set of functions used to describe the electronic wave function in a quantum mechanical calculation. These functions are typically centered on atomic nuclei and are used to express the molecular orbitals of a molecule.
In simpler terms, when you're dealing with the quantum mechanics of a molecule, you need a way to mathematically represent the distribution of electrons around the atomic nuclei. Instead of using an infinitely detailed description (which would be impractical), a basis set provides a set of functions that can approximate this distribution.
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