One might think that it is best to optimize the exponents of both auxiliary orbitals simultaneously by minimizing the norm ∆AA according to Eq. (2.114). However,
we found a two-step procedure more appropriate in which we first optimize the exponent of the auxiliary orbital for the valence shell. Keeping this exponent fixed, we subsequently optimize the auxiliary orbital for the polarization func- tions. For the purpose of optimizing the exponent of the auxiliary orbital for the shell lA we consider the norm ∆lAlA [Eq. (2.112)].
Let us explain the procedure taking the carbon atom as an example. The SVP basis set for the carbon atom is of the [3s2p1d] type, i.e., it consists of three s, two p and one d shell. We therefore will need one d type and one f type auxiliary orbital to improve the fit of overlap densities involving, respectively, the more diffuse p shell and the d shell. Let us enumerate the shells according to increasing l quantum number and according to increasing diffuseness. If we add
8 6 4 2 0 0.5 1 1.5 2 ∆ 5p 5p / 10 − 4 ζ(d ) 5 4 3 2 1 0 0.5 1 1.5 2 ∆ CC / 10 − 3 ζ( f )
Figure 4.1: CC at an interatomic distance of 120 pm: Norm ∆5p5p with varying
exponent ζ(d) of the d type auxiliary orbital (left). Norm ∆CC with varying
exponent ζ(f ) of the f type auxiliary orbital employing a d type auxiliary orbital with ζ(d) = 0.16 (right).
the type of the shell to the index we end up denoting the shells of the carbon atom as 1s, 2s, 3s, 4p, 5p and 6d. The shell 5p is the diffuse valence shell of p type and the shell 6d constitutes the polarization functions of d type for which we wish to optimize auxiliary orbitals. In order to optimize the exponent ζ(d) of the d type auxiliary orbital we minimize the norm ∆5p5p. This norm excludes
overlap densities involving basis functions of the 6d shell for which no auxiliary orbital has been optimized yet. Once the exponent ζ(d) has been determined, it is kept fixed and an f type auxiliary orbital is added whose exponent ζ(f ) is optimized by minimizing the norm ∆6d6d= ∆CC.
This procedure will not yield the minimum of the norm ∆AA, however, it guarantees for a better fit of the core and valence functions which we consider more important since integrals of these overlap densities will be contracted with elements of the first-order reduced density matrix that are in general larger in magnitude than matrix elements involving polarization functions. The quality of the fit of the overlap densities in which polarization functions participate is less critical with respect to the SCF convergence and the accuracy of the results.
We started our investigations at an interatomic distance of 120 pm which approximately corresponds to a CC triple bond [222]. A plot of the norm ∆5p5p
with varying ζ(d) can be found in Figure 4.1. The minimum for ζ(d) = 0.16 leads to an excellent fit with a value of 1.5 × 10−5 for ∆5p5p as compared to 7.9 × 10−4
0.6 0.7 0.8 0.9 1 1.1 1.2 0 0.05 0.1 0.15 0.2 0.25 0.3 ζ( f ) ζ(d ) 0.6 0.7 0.8 0.9 1 1.1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 ζ( f ) ζ(d )
Figure 4.2: CC at an interatomic distance of 120 pm: Contour plots of the norm ∆5p5p (left, isovalues between 2.0 × 10−7and 7.5 × 10−6) and ∆CC (right, isovalues
between 1.45 × 10−4 and 4.05 × 10−4) under variation of the exponents ζ(d) and ζ(f ) of the d and f type auxiliary orbitals. The minimum for the independent, consecutive optimization of the exponents is indicated by a cross.
without auxiliary orbital. Note that this exponent is very similar to the exponent of the 5p shell which is approximately 0.153. An exponent of 0.8 does not enhance the LEDO expansion basis since the d type polarization function has the same exponent and the peak at 0.8 represents the value of ∆5p5p without any auxiliary
orbital. Figure 4.1 also contains a plot of the norm ∆CC under variation of ζ(f ) employing a d type auxiliary orbital with the previously determined exponent ζ(d) = 0.16 being kept fixed. A good fit of the overlap densities involving the polarization functions is more difficult to achieve. Nevertheless, the minimum at ζ(f ) = 0.85 leads to an acceptable value of 1.6 × 10−4 for ∆CC, which is two
orders of magnitude better than the value of 1.6 × 10−2 without f type auxiliary orbital. Note that the 6d shell has a very similar exponent of 0.8.
Contour plots of the norm ∆5p5p and ∆CC under variation of both exponents
near the minimum are shown in Figure 4.2. The surface is very flat in the proximity of the minimum. Therefore, a variation of the exponents within a small range will not affect the quality of the LEDO fit. While the value of ∆CC is reduced from 1.6 × 10−2 without auxiliary orbitals to 1.4 × 10−4 at the minimum, the value of 1.6 × 10−4 for the independently optimized exponents is only slightly higher. We furthermore note that this value is mainly determined by ζ(f ) while ζ(d) has less influence. The contour plot of ∆5p5p, on the other
hand, shows that the quality of the fit of all overlap densities not involving the 6d polarization functions is almost independent from ζ(f ) at least in the vicinity of ζ(f ) = 0.85. Since the exponents of the auxiliary orbitals are not strongly coupled, an independent, consecutive optimization is justified.