A fundamental connection between symmetry and spatial localization properties of basis sets

Abstract

The problem of the compatibility between symmetry and localization properties of basis sets is addressed here. It is shown that both concepts are closely related from a fundamental point of view through the notion of invariance extent. This quantity is a functional that depends on the symmetry group and the basis set choices, and it is shown that all basis sets adapted in a general way to symmetry, i.e. induced from irreducible bases of the subgroups, are stationary points of it. In particular, the usual irreducible bases of the full group display a maximal invariance extent, while those symmetry-adapted basis sets that display a minimal value of this quantity feature in most cases the same symmetry properties as localized functions obtained by means of the Boys scheme. The most relevant conclusions are illustrated by means of simple molecular and periodic examples.

Appendix: Gradients of the functional \(\bar I\) in terms of the SABS

Let us first consider unitary transformations inside the indrep spanned by the set $\{|{{\mathcal{H}}} \mu , ar\rangle\}$ given in Eq. 3; for the sake of simplicity the first indexes ${{\mathcal{H}}}$ and μ will be omitted. The gradient element that corresponds to generator $\hat E_{ag,bh}$ (Eq. 10) reads

$$ \begin{aligned} {\frac{\partial \bar I({{\mathcal{G}}}, \{|c f\rangle\})} {\partial \varepsilon_{ag;bh}}} &={\frac{1}{L}}\Re \sum_{S \in {{\mathcal{G}}}} \left(\langle bh | \hat S | ag \rangle + \langle ag | \hat S | bh \rangle\right) \left(\langle ag | \hat S | ag \rangle - \langle bh | \hat S | bh \rangle\right)^*\\ &= {\frac{1} {L}} \Re \sum_{S \in {{\mathcal{G}}}} \langle bh | \hat S | ag \rangle \langle ag | \hat S | ag \rangle ^* - {\frac{1} {L}} \Re \sum_{S \in {{\mathcal{G}}}} \langle bh | \hat S | ag \rangle \langle bh | \hat S | bh \rangle^* \\ &\quad+ {\frac{1}{L}}\Re \sum_{S \in {{\mathcal{G}}}}\langle ag | \hat S | bh \rangle \langle ag | \hat S | ag \rangle ^* - {\frac{1} {L}} \Re \sum_{S \in {{\mathcal{G}}}} \langle ag | \hat S | bh \rangle \langle bh | \hat S | bh \rangle^* . \end{aligned} .$$

(13)

To determine whether or not the involved sums vanish, let us study the behavior of the constituent integrals. Considering Eq. 3, it arises that

$$ \langle ag | \hat S | bh \rangle = \langle a 0 | \hat S_g^{-1} \hat S \hat S_h | b 0 \rangle, $$

(14)

which vanishes if $S_g^{-1}SS_h \not \in {{\mathcal{H}}}.$ This is because, in such a case $S_g^{-1}SS_h = S_r R (R\in {{\mathcal{H}}}, r \neq 0),$ in accordance with the left-coset partition (Eq. 1), and

$$ \langle ag | \hat S_r \hat R | bh \rangle = \sum\limits_{b'=1}^{n_\mu} \langle a 0 | S_r| b' 0\rangle D_{b'b}^{(\mu)}(R) = \sum\limits_{b'=1}^{n_\mu} \langle a 0 | b'r\rangle D_{b'b}^{(\mu)}(R), $$

(15)

where Eqs. 2 and 3 are used in the first and second equalities, respectively, and the last expression is null due to the orthonormality of the SABS.

It therefore appears that if g ≠ h, the gradient vanishes as there is no choice for S that allows both factors in each term of Eq. 13 simultaneously not to vanish. Let us take as example the first sum. The previous analysis shows that the first and second factors will vanish in those terms in which $S_h^{-1}SS_g \not \in {{\mathcal{H}}}$ and $S_g^{-1}SS_g \not \in {{\mathcal{H}}},$ respectively. However, there is no S that satisfies both conditions if g ≠ h, as otherwise it would mean that $SS_g \in S_h {{\mathcal{H}}}$ and $SS_h\in S_h {{\mathcal{H}}},$ contradicting the left-coset partition, and therefore the whole sum vanishes.

In the case g = h, Eq. 13 becomes

$$ \begin{aligned} {\frac{\partial \bar I({{\mathcal{G}}}, \{|cf\rangle\}) }{\partial \varepsilon_{ag;bg}}} &= {\frac{1}{L}} \Re \sum_{S \in {{\mathcal{H}}}^{[g]}} \langle bg | \hat S | ag \rangle \langle ag | \hat S | ag \rangle ^* - {\frac{1}{L}} \Re \sum_{S \in {{\mathcal{H}}}^{[g]}} \langle bg | \hat S | ag \rangle \langle bg | \hat S | bg \rangle^*\\ &\quad +{\frac{1}{L}} \Re \sum_{S \in {{\mathcal{H}}}^{[g]}}\langle ag | \hat S | bg \rangle \langle ag | \hat S | ag \rangle ^* - {\frac{1}{L}} \Re \sum_{S \in {{\mathcal{H}}}^{[g]}} \langle ag | \hat S | bg \rangle \langle bg | \hat S | bg \rangle^* , \end{aligned} $$

(16)

where the sums are restricted to ${{\mathcal{H}}}^{[g]} = S_g {{\mathcal{H}}} S_g^{-1},$ which is isomorphic with ${{\mathcal{H}}},$ as for the remainder of the $S\in {{\mathcal{G}}}$ the terms vanish according to Eq. 15. It appears that the set $\{|a g\rangle\}_{a=1}^{n_\mu}$ is a basis for the irrep μ of ${{\mathcal{H}}}^{[g]}$ and, therefore, each sum in Eq. 16 does vanish owing to the First Theorem of the Orthogonality of the irrep matrices, giving a null gradient component.

In case $|v_j\rangle$ and $|v_l\rangle$ in Eq. 10 belong to different indreps, it is easy to prove that the corresponding gradient component vanishes. This is because $\hat S |v_j\rangle,\; \forall S\in{{\mathcal{G}}}$ belong to the same indrep as $|v_j\rangle$ that is orthogonal by definition to the subspace that contains $|v_l\rangle.$ This makes $\langle v_j |\hat S|v_l\rangle =\langle v_i |\hat S^{-1}|v_j\rangle^* = 0,\; \forall S\in{{\mathcal{G}}}$ with the subsequent vanishing of the whole sum in Eq. 10.