Correction: Planar vs. three-dimensional X₆²⁻, X₂Y₄²⁻, and X₃Y₃²⁻ (X, Y = B, Al, Ga) metal clusters: an analysis of their relative energies through the turn-upside-down approach

Abstract

Correction for ‘Planar vs. three-dimensional X₆²⁻, X₂Y₄²⁻, and X₃Y₃²⁻ (X, Y = B, Al, Ga) metal clusters: an analysis of their relative energies through the turn-upside-down approach by Ouissam El Bakouri et al., Phys. Chem. Chem. Phys., 2016, 18, 21102–21110.

---

In the Computational methods section of our published article, we stated: “To facilitate a straightforward comparison, the EDA results were scaled to match exactly the regular bond energies”. However, the referred scaling factor (or correction factor) was inadvertently not applied to the values of Tables 2 and 3. The new Tables 2 and 3 included in this Erratum now contain the scaled values. Changes in the values of these two tables do not affect the conclusions of the work.

Table 2 Energy decomposition analysis (EDA) of X₆²⁻ (X = B, Al, and Ga) metal clusters with D_2h and O_h symmetries (in kcal mol⁻¹), from two X₃⁻ fragments at their quintet state, computed at the BLYP-D3(BJ)/TZ2P level. Units are kcal mol⁻¹

	B6^2−			Al6^2−			Ga6^2−
	D 2h + D2h → D2h	O h + Oh → Oh	Δ(ΔE)	D 2h + D2h → D2h	O h + Oh → Oh	Δ(ΔE)	D 2h + D2h → D2h	O h + Oh → Oh	Δ(ΔE)
ΔEint	−188.0	−98.0	−90.0	−20.0	−38.8	18.8	−18.0	−29.8	11.9
ΔEPauli	522.4	710.4	−188.0	218.1	339.2	−121.1	253.5	370.0	−116.5
ΔVelstat	−234.0	−282.0	48.0	−93.0	−162.3	69.2	−129.7	−199.7	69.9
ΔEoi	−473.4	−524.4	51.1	−141.9	−211.9	70.0	−137.9	−195.7	57.8
ΔEdisp	−3.1	−2.0	−1.1	−3.1	−3.8	0.7	−3.8	−4.5	0.7
ΔEdist	25.1	2.6	22.5	0.0	3.0	−3.0	0.2	2.8	−2.6
ΔE	−162.9	−95.4	−67.5	−20.0	−35.8	15.8	−17.8	−27.0	9.3

---

Table 3 Energy decomposition analysis (EDA) of all mixed metal clusters with planar and 3D symmetries (in kcal mol⁻¹), from two fragments at their quintet states, computed at the BLYP-D3(BJ)/TZ2P level

		ΔEint	ΔEPauli	ΔVelstat	ΔEoi	ΔEdisp
B2Al4^2−	D 4h	−51.1	431.3	−198.6	−280.0	−3.8
	D 2h	−39.6	238.5	−96.1	−178.9	−3.2
	ΔE	11.5	−192.8	102.5	101.1	0.6
Al2B4^2−	D 4h	−72.8	566.5	−244.1	−392.0	−3.2
	D 2h	−136.8	545.5	−233.8	−445.4	−3.2
	ΔE	−64.0	−21.0	10.3	−53.4	0.0
Al2Ga4^2−	D 4h	−33.6	366.0	−193.0	−202.2	−4.4
	D 2h	−18.8	277.4	−144.2	−148.4	−3.7
	ΔE	14.8	−88.5	48.8	53.8	0.7
Ga2B4^2−	D 4h	−82.1	578.6	−257.3	−400.0	−3.4
	D 2h	−152.8	523.8	−218.8	−454.5	−3.1
	ΔE	−70.7	−54.8	38.4	−54.5	0.3
Ga2Al4^2−	D 4h	−37.6	362.7	−184.2	−211.8	−4.2
	D 2h	−20.2	213.6	−89.0	−141.5	−3.5
	ΔE	17.4	−149.1	95.3	70.3	0.7
Al3Ga3^2−	D 3h	−35.7	369.6	−191.9	−209.2	−4.1
	C 3v	−19.9	244.2	−117.9	−142.8	−3.5
	ΔE	15.8	−125.3	74.0	66.5	0.6

---

As to the discussion, there are only two paragraphs affected:

(1) Last paragraph on page 21105 that now reads:

The different terms of the EDA analysis for B₆²⁻, Al₆²⁻, and Ga₆²⁻ clusters are enclosed in Table 2. First, we notice that the total bonding energies (ΔE) are much larger for B₆²⁻ than for Al₆²⁻ or Ga₆²⁻. For the former, ΔE are −95.4 (O_h) and −162.9 kcal mol⁻¹ (D_2h), whereas for the two latter are in between −17.8 and −35.8 kcal mol⁻¹. This trend correlates with the shorter B–B bond lengths mentioned above. Table 2 also encloses the relative EDA energies between the two clusters. The B₃⁻ fragment taken from B₆²⁻ system in its D_2h symmetry is the one that suffers the largest deformation, i.e. the largest change in geometry with respect to the fully relaxed B₃⁻ cluster in the quintet state (ΔE_dist = 25.1 kcal mol⁻¹), whereas the rest of the systems present small values of ΔE_dist (0.0–3.0 kcal mol⁻¹). However, differences in ΔE are not due to distortion energies (indeed ΔE_dist values follow the opposite trend as ΔE), but to interaction energies (ΔE_int).

(2) First paragraph on page 21106 that now reads:

Thus, we focus on the decomposition of ΔE_int into ΔE_Pauli, ΔV_elstat, ΔE_oi, and ΔE_disp terms. As a general trend, in all three X₆²⁻ clusters ΔE_Pauli is larger for the O_h than the D_2h cluster (Δ(ΔE_Pauli) = −188.0, −121.1, and −116.5 kcal mol⁻¹ for B₆²⁻, Al₆²⁻, and Ga₆²⁻, respectively), so making it less stable. The overlaps between doubly occupied MOs are larger in the more compact O_h structure that, consequently, has larger ΔE_Pauli. The larger difference in ΔE_Pauli between the O_h and D_2h structures in the case of B₆²⁻ as compared to Al₆²⁻ and Ga₆²⁻ is attributed to the particularly short B–B distances that increases the overlap between doubly occupied MOs of each B₃⁻ fragment. At the same time, the O_h form presents larger (more negative) electrostatic interactions (Δ(ΔV_elstat) = 48.0, 69.2, and 69.9 kcal mol⁻¹ for B₆²⁻, Al₆²⁻, and Ga₆²⁻, respectively). It is usually the case that higher destabilising Pauli repulsions goes with larger stabilising electrostatic interactions. The reason has to be found in the fact that both interactions increase in absolute value when electrons and nuclei are confined in a relatively small space. The electrostatic interaction together with orbital interaction (Δ(ΔE_oi) = 51.1, 70.0, and 57.8 kcal mol⁻¹ for B₆²⁻, Al₆²⁻, and Ga₆²⁻, respectively) terms favour the O_h structure. In the case of O_h B₆²⁻, however, Δ(ΔV_elstat) and Δ(ΔE_oi) cannot compensate Δ(ΔE_Pauli), which causes the D_2h system to be the lowest in energy. The opposite happens for Al₆²⁻ and Ga₆²⁻. Finally, the dispersion term does almost not affect the relative energies, as the difference in dispersion is only in the order of ca. 1.0 kcal mol⁻¹. Therefore, what causes the different trend observed for B₆²⁻ on one side, and Al₆²⁻ and Ga₆²⁻ on the other side is basically the ΔE_oi term, which combined with the ΔV_elstat component does (Al₆²⁻ and Ga₆²⁻) or does not (B₆²⁻) compensate for the higher ΔE_Pauli of the O_h form.

As to the conclusions, the sentence “From one side the D_2h^HOMO-1(b_2u) formed from two tangential SOMO σ^T(b₂) orbitals” should read “From one side the D_2h^HOMO-2(b_2u) formed from two tangential SOMO σ^T(b₂) orbitals”.

The Royal Society of Chemistry apologises for these errors and any consequent inconvenience to authors and readers.

Acknowledgements

We thank Dr Pedro Salvador for pointing out the mistake in the original article.

---