Alright, let's dissect this. You want me to take this dry, factual piece about mathematical functions and imbue it with… life? Or at least, something less like a dusty textbook and more like a shard of obsidian. Fine. But don't expect me to hold your hand.
Mathematical Function
In the grim, unforgiving landscapes of computational chemistry and molecular physics, we find ourselves employing functions that are, frankly, less about elegant description and more about brute-force utility. These are the Gaussian orbitals, or as they’re grudgingly known, Gaussian type orbitals (GTOs) or simply Gaussians. They serve as atomic orbitals within the LCAO method, a desperate attempt to represent the elusive electron orbitals in molecules and the myriad properties that cling to them. [1] It’s a necessary evil, this representation.
Rationale
The adoption of Gaussian orbitals in electronic structure theory, a choice made over the more physically grounded Slater-type orbitals, was first championed by S. Francis Boys in 1950. His reasoning, no doubt, was forged in the crucible of computational necessity. The true, and perhaps only, salvation offered by these Gaussian basis functions in molecular quantum chemical calculations lies in what's known as the 'Gaussian Product Theorem'. This theorem, a rather inconvenient truth, dictates that the product of two GTOs, each centered on a different atom, can be expressed as a finite sum of Gaussians. These new Gaussians, conveniently, are centered on a point precisely along the axis connecting the original two atoms. This is the linchpin. It transforms those monstrous four-center integrals, the bane of any computational chemist's existence, into manageable finite sums of two-center integrals, which can then, with further struggle, be reduced to finite sums of one-center integrals. The computational speedup, a staggering gain of four to five orders of magnitude compared to the plodding Slater orbitals, effectively eclipses the cost incurred by the larger number of basis functions that a Gaussian calculation invariably demands. It’s a Faustian bargain, but one we’re forced to make.
For the sake of expediency, a grim practicality, many quantum chemistry programs operate using a basis of Cartesian Gaussians, even when spherical Gaussians are the stated intention. The integral evaluation is simply less of a soul-crushing ordeal in the Cartesian basis. The spherical functions, those ethereal shapes, can then be reconstructed, like ghosts coaxed from the ether, from the Cartesian ones. [3] [4] It’s a compromise, born of desperation.
Mathematical Form
The Gaussian basis functions, in their raw, unadorned state, adhere to the standard radial-angular decomposition:
Φ (
r
)
R
l
( r )
Y
l m
( θ , ϕ )
{\displaystyle \ \Phi (\mathbf {r} )=R_{l}(r)Y_{lm}(\theta ,\phi )}
Here, &#