M´ etricas a nivel soporte

In most applications of quantum chemistry, calculations for molecules are per- formed using linear combinations of atomic orbitals to form molecular orbitals (LCAO–MO). This means that molecular orbitals are expanded in a finite linear combination of atomic orbitals

ψi =

n

j=1 cjiφj,

whereψiis thei–th molecular orbital,cjiare the coefficients of the linear combitation,

φjis thej–th atomic orbital, andnis the number of atomic orbitals. Atomic orbitals

are solutions of the Hartree–Fock equations for an atom, i.e., a wave function representing a single electron in the multi–electron atom. Slater type orbitals (STOs) were originally used as basis functions due to their similarity to the atomic orbitals of the hydrogen atom. Their general shape has the following form

φnlm(r, θ, φ)=AYlm(θ, φ)rn−1exp (−ξr),

whereAis the normalisation constant,ξis called the exponent,r, θ, φare spherical

magnetic quantum numbers, respectively.The excellent behaviour shown by the STO functions in the near and far regions of the atomic nucleus is well–known. They satisfy the nuclear cusp condition because of their exponential relationship with the nucleus–electron distance. This fact allows the STO basis sets to reproduce correctly the regions near the nucleus. In addition, they decay properly in the long range.

Unfortunately, the initial and successful work developed for atomic calculations with STO functions was abandoned due to the complexity in the evaluation of two–electron multi–center integrals required in molecular calculations. Hence,

exponential functions onr2_{(Gaussian type orbitals, or GTOs) became the standard}

basis employed in molecular electronic structure calculations.

Gaussian basis sets [12, 13] are composed of Gaussian–type orbitals (GTOs) of the form

φnlmr, θ, φ=NYlm

θ, φr2n−2−lexp−ξr2, (4.3)

whereN is the normalisation constant,ξis known as the exponent, andYlmare

spherical harmonic functions. The indicesn,mandldetermine the type of orbital,

i.e.,s,p,d, etc. Basis functions of this explicit form are also known as Gaussian

primitives, and, for reasons of balance and efficiency, a linear combination of a number of primitives is often put together to form contracted functions,

ψ =

i

niφi,

whereni are known as contraction coefficients kept fixed in the variational proced-

ure.

How the optimisation of exponents of a basis set is done depends on their intended usage. For instance, if one is primarily focused on electronic properties of atoms, or, molecules, then using a basis set that has been optimised specifically for that purpose is likely to produce very good results. In this section, we will be

focusing on energy–optimised basis sets, where the exponents are optimised to minimise the total electronic energy as such basis sets are considered to be of more general purpose. Although all exponents may be optimised using common minimisation algorithms such as Broyden–Fletcher–Goldfarb–Shanno and simplex [54], an alternative option is to optimise a single exponent, then produce additional exponents using mathematical relationships such as well– tempered, even–tempered and Legendre polynomials. Such methods are incredibly useful for large basis sets where optimisation of every individual primitive is difficult, as discussed, for example, by Petersson [52].

Many–electron self–consistent field (SCF) computations scale up byM4_{, whereM}

is the number of basis functions. This, in return, forces one to restrict the number of functions used in the calculations. Furthermore, Gaussian basis functions have (quasi–)linear dependencies, which can cause numerical instabilities in calculations. Often, time and resource consuming non–linear optimisation of the Gaussian functions is required. To avoid such full optimisations, a number of different methods were introduced to fit some functional form to Gaussian exponents that would give results close to fully optimised basis set. The even–tempered technique [55] uses only two optimisation parameters. Other functionals with more optimisation parameters were also introduced [32, 36, 52], but the even–tempered basis is considered to span the Hilbert space most evenly.

The even–tempered atomic orbital basis is a set of nuclear–centred functions defined by

ψnlmr, θ, φ=Nl(γi)Ylm(θ, φ)rlexp

−γirp, p=1 or 2,

where Ylm is the solid spherical harmonic. A uniform power of r in the radial

exponents written in terms ofαandβ,

γi =αβk_{, for α >0, β >1, (4.4)}

fork=1, . . . ,M, so that the Gaussian exponentsγiform a geometric progression.

Here the two parameters that need to be optimised areαandβ. The functional

in Eq. (4.4) can be applied to Gaussians of angular momentum l = 0,1,2, . . .,

corresponding tos–,p–,d–,. . . orbitals, respectively. Schmidt and Ruedenberg [60]

proposed that even–tempered basis sets are asymptotically complete whenαandβ

considered to be functions ofMand this claim has been proven by Kryachko et al.

[37] where they present empirical formulae for generating systematic sequences of even–tempered basis sets. However, it is possible to show that the even–tempered

basis is incomplete whenβremains constant [16].