REFLEJOS PREDICTIVOS

The most general approach to the fit of individual overlap densities φµφν ≡ ρµν

was suggested by Vahtras et al. [157]. In contrast to the projection of diatomic differential overlap (PDDO) approach by Newton [152] and the limited expan- sion of diatomic overlap (LEDO) approach by Billingsley and Bloor [158], their method uses the same fit basis {Ωp} of atom-centered expansion functions for all

overlap densities to be fitted. In analogy to Eq. (2.37) for the expansion of the complete electron density, the expansion of individual overlap densities can thus be written as ρµν(r) ≈ ˜ρµν(r) = X p Ωp(r) dµνp = Ω † dµν. (2.62) For the sake of simplicity it will be assumed in the following that the Coulomb norm [Eq. (2.35)] is used for the definition of the scalar product. Then the expansion coefficients dµν

p are obtained from the system of linear equations

Vdµν = bµν (2.63)

with solution

dµν = V−1bµν, (2.64) where Vpq = (Ωp|Ωq) and bµνp = (Ωp|ρµν) are two- and three-index ERIs, similar

to Eqs. (2.54) and (2.55) for the fit of the complete electron density. Taking the symmetry of the matrix V into account, any four-index ERI can thus be approximated as

(ρµν|ρκλ) ≈ ( ˜ρµν|˜ρκλ) = dµν

†

Incidentally, the same expression is obtained if a conventional resolution of the identity (RI) is introduced into the ERIs,

1 = ∞ X p,q |ΩpiVpq−1hΩq| ≈ X p,q |ΩpiVpq−1hΩq|, (2.66)

where the approximate equality reflects the incompleteness of the expansion ba- sis. Therefore, density fitting is frequently denoted as RI approximation by some authors. As a matter of fact, this approximation is an inner projection similar to the method of Beebe and Linderberg which is based on the Cholesky decomposi- tion of the ERI matrix [185, 186]. Here, however, the projection is formulated in terms of an auxiliary set of basis functions {Ωp}, rather than to generate the pro-

jection manifold explicitly from the original overlap densities. Note, that density fitting mathematically resembles a (truncated) RI only in the specific case where the weight operator W used in the metric of the expansion basis is the same as in the target integral. Furthermore, RIs do not offer a framework in which to discuss fitting criteria, constraints or robust fitting.

An analysis of the error

(ρµν|ρκλ) − ( ˜ρµν|˜ρκλ) = (ρµν− ˜ρµν|ρκλ− ˜ρκλ) + (ρµν− ˜ρµν|˜ρκλ) + ( ˜ρµν|ρκλ− ˜ρκλ)

(2.67) between the exact and the approximate ERIs shows that it contains terms linear and quadratic in the fitting error ρµν− ˜ρµν. However, if the Coulomb norm has

been employed, the expansion coefficients are obtained from Eq. (2.64) and the following equality holds,

( ˜ρµν|˜ρκλ) = dµν † Vdκλ = bµν†dκλ = (ρµν|˜ρκλ) = dµν†bκλ = ( ˜ρµν|ρκλ). (2.68)

Thus the linear error terms in Eq. (2.67) vanish and the approximation is robust. It is important to realize that this holds only if the Coulomb norm has been employed.

Insertion of the expansion for the overlap densities [Eq. (2.62)] into the ex- pression for the electron density [Eq. (2.16)] yields

˜

ρ(r) =X

p

for the approximate electron density. This expression is identical to Eq. (2.37) for the fit of the total electron density. Here, however, the expansion coefficients dp for the electron density have been obtained by contraction of the elements of

the density matrix P with the expansion coefficients dµν_p for the overlap densities according to

dp =

X

µ,ν

dµν_p Pµν. (2.70)

In analogy to Eq. (2.46), a vector b with components

bp = (Ωp|ρ) =

X

µ,ν

bµν_p Pµν (2.71)

can be defined. Similar relations as derived for the fit of the complete electron density also hold for the fit of individual overlap densities. Specifically, the ap- proximated Hartree energy is given as

˜ Eh = 1 2( ˜ρ| ˜ρ) = 1 2d † Vd, (2.72)

and the approximated matrix elements of the Hartree potential are given as

˜ Vh,µν = ∂ ˜Eh ∂Pµν = ∂d † ∂Pµν  Vd = dµν†Vd = ( ˜ρµν|˜ρ). (2.73)

The derivatives of the expansion coefficients dp with respect to the elements Pµν

of the density matrix yield the expansion coefficients dµν_p for the overlap densities ρµν, as can be seen from Eq. (2.70). While these need not be explicitly determined

in methods fitting the complete electron density, a comparison of Eqs. (2.58) and (2.64) shows, that also in that case they do appear implicitly in the expression for the approximated matrix elements of the Hartree potential. Using Eqs. (2.63), (2.70) and (2.71), Eqs. (2.72) and (2.73) can be rewritten as

˜ Eh = 1 2b † d = 1 2(ρ| ˜ρ) (2.74) and ˜ Vh,µν = bµν † d = (ρµν|˜ρ). (2.75)

It is very important to note that the step leading from Eqs. (2.72) and (2.73) to Eqs. (2.74) and (2.75) can only be performed if

a) the fit basis {Ωp} is identical for all overlap densities ρµν and

b) the Coulomb norm [Eq. (2.35)] has been employed for the fitting procedure. Only in this case the simple approximation of the ERIs and thus of the Hartree energy is identical to the robust expression. Note that the approximated ERIs are not a bound to the exact ERIs, i.e. they can be smaller or larger in magnitude. The approximated Hartree energy, however, is a lower bound to the exact Hartree energy, just as in methods fitting the complete electron density.

Requirements to the expansion basis

Now it is possible to draw an interesting conclusion. As a fit for each individual overlap density ρµν is performed, one might expect from Eq. (2.73) that the fit

basis has to be sufficiently large to reproduce any particular member of the set of overlap densities with good accuracy. However, this is not the case as becomes evident by comparing Eqs. (2.23), (2.73) and (2.75). It can be easily seen that the matrix elements Vh,µν of the Hartree potential are well approximated if a good

fit for the total electron density is obtained, i.e. if ρ and ˜ρ are sufficiently close. According to Eqs. (2.69) and (2.70), it is sufficient for the representation of the complete electron density to determine expansion coefficients dµν_p corresponding to density matrix elements Pµν of significant magnitude. It is obvious that the

fit of the total electron density is a much easier task than the fit of each overlap density generated by the AO basis functions, thus requiring a much smaller fit basis. The most extreme example is perhaps a closed-shell atom which has a spherically symmetric electron density. Hence, s-type functions are sufficient for a density fit. It has been found that even electron densities of molecules can be fitted quite accurately by short expansions of spherical Gaussians located on the nuclei [162]. Clearly, this would be an extremely poor basis for a fit of individual AO products. However, it is sufficient for a good approximation of the matrix elements of the two-electron Coulomb part of such products. It should be noted that the demand on the expansion basis may be a lot higher in other types of calculations, where products that have nothing to do with the density expansion must also be accurately represented. This is the case, e.g., if density fitting is employed for the HF exchange (without reducing the formal

scaling behavior), which requires products φµψi between AO basis functions and

occupied MOs to be fitted. In the case of MP2 calculations, products of the type ψiψa between occupied and virtual MOs have to be fitted. Ten-no and Iwata

successfully employed one-center AO products φµ0_Aφµ00_A as auxiliary functions in

HF and MCSCF calculations [132, 137] and optimized auxiliary basis sets are available both for HF [135] and MP2 [140, 141, 144] calculations.

Scaling behavior

Although the approach of Vahtras et al., just like methods fitting the complete electron density, is not very demanding with respect to the fit basis, it has a rather unfavorable formal scaling behavior. The number of overlap densities ρµν

grows as O(N2) with the system size. For each of these overlap densities the system of linear equations (2.63) has to be solved. Since the dimension of the fit basis also grows linearly with the system size, the numerical effort for obtaining the expansion coefficients for a particular overlap density scales as O(N2_{), once}

the matrix V has been inverted. Altogether, one therefore ends up with the same formal O(N4_{) scaling just as the unapproximated method. However, the prefactor}

might be rather favorable and, in contrast to methods fitting the total electron density, the fit does not have to be repeated in every SCF cycle. It is clear, that the fit has to be performed only for products φµφν of AOs with significant overlap,

the number of which is of O(N ) in the limit of large molecules. Therefore, the asymptotic scaling for the fitting process is O(N3_{) due to the matrix inversion}

like for the fit of the complete electron density.