Calcium carbides: can hexa-carbides grow unlimitedly? Theoretical perspectives and issues that oppose a definite answer

Abstract

Quasi-molecule theory and ‘tiles’ embedded on the parent molecules are used to investigate whether calcium hexa-carbides can attain linearity irrespective of their size. By assuming that some small carbon clusters may behave pseudo-atomic-like, the approach is generalized to explore ‘pseudo-linearity’ among the calcium hexa-carbides. It is tentatively conveyed that the atomic basis sets used for studying families of related molecules, rather than individual ones, should be maintained constant for the whole set such as to warrant that any differences in the observed trends are not driven by computational approximations or numerical inaccuracies. In the absence of experimental data, low-hierarchy (often subminimal) basis-set calculations are employed for probing the predictions by quasi-molecule theory for the parent molecules, while augmented correlation-consistent basis sets can efficiently be used for accurate correlated calculations in the tiles they embed. Besides hexa-carbides, other calcium carbides, planar and volumetric, are also studied.

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1 Introduction

In chemistry, studying a family of molecules collectively often yields more insightful results than examining a single molecule per se. While the study of individual molecules in depth is crucial for obtaining specific details on their properties, looking at their family provides a systematic and efficient way of learning chemistry. Nevertheless, ab initio quantum chemistry^1,2 faces “curses” that obstruct such an approach: one, dimensionality, due to the computational cost of solving the Schrödinger equation growing rapidly with physical size; the other, scaling, due to the dependence of the N-electron wavefunction on 3N spatial coordinates plus spin, which makes direct numerical solutions of the Schrödinger equation intractable and commonly imposes the use of correspondingly larger basis sets to solve it. The latter would to some extent be acceptable if the prospects of linear-scaling electronic structure theory which has fascinated the electronic structure theory community since its popularization by Kohn's work on the electronic “nearsightedness” principle³ could be verified. However, there is usually the need for employing diffuse basis sets which being often a blessing for accuracy are a curse for sparsity, a conumdrum most recently given prominence in the literature.⁴ As Head-Gordon and coworkers⁴ have noted, there is not yet a definite solution for such conumdrum, at least for the practical problems one encounters in molecular sciences. Returning to the above point, the simplest property one may think is the shape of related molecules, which is the focus of the present work where we specifically address calcium carbides.

One may start by questioning whether a molecule needs to be strictly (pseudo-)linear or if it suffices being quasilinear (mildly nonlinear). The answer depends on the context and property under consideration. Quasi-linearity may suffice for many purposes, provided the deviations do not drastically alter the molecular properties of interest. Nevertheless, when precision is key such as in high-resolution spectroscopy or specific reaction pathways, being linear or not can be a critical issue.

Assuming atoms as pointwise entities, quasi-molecule theory^5–13 can decipher linearity and planarity or otherwise of molecules (called parents) at their equilibrium geometries from sets of atoms (‘tiles’) that are embedded on them. Stretching the concept, one wonders in the current work whether small molecules, rather than single atoms, can be conceived themselves as pointwise, thus allowing the generalization of quasi-molecule theory to predict the shape of pseudo-linear (although not tested yet, a similar extension is conceivable to handle pseudo-planar) parents. Pseudo-linearity is discussed here by focusing on alkaline-earth-metal carbides, so far little studied in the literature, specifically calcium carbides. In the absence of experimental data, ab initio and DFT calculations must unavoidably be performed to probe the predictions made for the members of the CaC₆-family, all neutral systems. For a broader perspective, related calcium carbides of other shapes will also be studied here.

Materials may not exist in isolation. However, if they are large enough, their boundaries can be assumed to not have a significant effect on their electronic states. Periodic boundary conditions¹⁴ are then commonly assumed in computer simulations, and mathematical models used for approximating a large (even infinite) system by using just a small part of it called a unit cell. Although the geometry of such a cell commonly satisfies perfect two-dimensional tiling, three-dimensional periodic boundary conditions are also useful for approximating the behavior of macro-scale systems of gases, liquids, and solids. In turn, one-dimensional tiling is used in the treatment of quantum wires (ref. 15, and references therein). However, rather than hypothesising and performing simulations, it may be sensible to ask can unit cell-like clusters and even the full 2D materials be themselves predicted?

Regarding calcium carbides, CaC and CaC₂ are formed under laboratory conditions and their existence is likely in primordial clouds such as molecular clouds or interstellar clouds. Because CaC is less stable than CaC₂, it may require high-energy conditions for its formation, such as a high-temperature environment or pressure conditions. In turn, CaC₂ has already been identified in the IRC+10216 nebula,¹⁶ and reaction dynamics studies on its potential energy surface (PES) were recently carried out¹⁷ using the internally contracted multireference configuration-interaction (MRCI+Q) method and augmented correlation consistent basis sets.^18,19 Moreover, a three-dimensional PES has been reported for the c-CaC₂–He van der Waals complex at the explicitly correlated coupled-cluster level, and close-coupling calculations were performed on it to determine cross-sections for rotational transitions through He collisions with c-CaC₂.²⁰

In turn, the C₆ molecule is deemed relatively abundant in the circumstellar shells of carbon stars thanks to the favorable conditions for formation of carbon chains. Although the presence of metals like calcium is scarce in such environments (and possibly predominant as ionized calcium Ca⁺ or Ca⁺⁺ due to high-energy UV radiation and stellar winds), it is conceivable that it may interact directly with carbon to form compounds like CaC₆. To our knowledge, the existence of the CaC₆ molecule has not yet been reported, with most studies and syntheses involving solid-state forms where calcium is bonded to a network of carbon atoms.^21–23 Its gas-phase formation is then challenging mostly due to the reactive nature of calcium and its tendency to form stable ionic compounds or solid-state carbides. Nonetheless, even the latter are not widely encountered outside of research environments, and their properties and practical applications are still the subject of scientific investigation. In this work, we will study the family of gas-phase CaC₆ molecules using ab initio molecular orbital (MO) methods and density functional theory (DFT). Besides offering a relevant system for theoretical work, it may help in elucidating how to avoid the above-mentioned issues in electronic structure calculations affect the rigorous study of large molecules and molecular materials.

2 Theory

2.1 Quasi-molecule theory and pseudo-linearity

It is clear that atoms are truly not point-like, although such an assumption is key if linearity or planarity of molecules is to be explained. It is then conceivable that a small molecule may also be treated as a ‘pseudo-atom’ if its internal structure and motions can (at least partly) be ignored. This may be a valid approximation if the energy scales of interest are larger than that required to excite its vibrational or rotational states, as it may happen to be the case when optimizing the structure of a large aggregate where it is embedded. Treating a small molecule like a pseudo-atom may then be a fair approximation in scenarios where its internal motions are assumed to be little relevance for the problem at hand, hopefully here the case. Although their internal degrees of freedom are assumed not to play a key role, their geometries may get distorted (through deformations of their structures or twist motions around their axes), in a way judged similar to what happens with atoms when they lose sphericity during bond formation. In fact, when isolated, the atomic electron clouds are spherically symmetric (especially the s-orbitals) and the overall electron distribution around the nucleus is symmetrical in the absence of an external influence distorting it. Conversely, rather than round, atoms in molecules take on distorted, directional shapes (more like lobes or regions pointing toward the bonded atoms). Nevertheless, all degrees of freedom of the above pseudo-atoms will be later considered when calculating the forces in the optimization of the parent molecule. To our knowledge, such an approach has not been explicitly adopted thus far in the literature.

Consider the following question: can the geometry of a molecule be rationalized as pseudo-linear or otherwise from the geometries of the quasi-pseudo-triatomic tiles (pseudo-tiles or just tiles for simplicity) it embeds? The answer is positive as implied by the lemma: If all triads of points (atoms or pseudo-atoms) in a set are on a line, they are all on the same line.

This lemma is valid for atoms (recall, not themselves true points) and, in the present work, under the assumption that C₆ is considered a pseudo-atom. Of course, such an approach is best judged by the results one obtains for pseudo-linear molecules like the calcium-hexacarbides studied here. Recently reported⁵ as Lemma 1 (Lemma 2 is for planarity,^5,6,8,13 with the tiles being real tetratomics in this case; see also later), the reader is addressed elsewhere¹⁰ for details.

Rather than rationalizing the results for the title systems from specific optimizations for the parent, the aim is to provide a background theory to predict molecular geometries without resorting to direct calculations or experimental data. Of course, in the absence of experimental information, direct ab initio or DFT calculations must be done to probe the predictions made from the present approach for the parent molecule, named quasi-molecule theory. One may argue that it is sufficient, for example for a molecule with D_∞h symmetry, that a non-vanishing dipole moment arises to conclude whether the employed method is unsatisfactory. This is certainly the case, but how to proceed when the molecule is: (i) less symmetrical, such as in this case with pseudo-linearity; (ii) experimentally unknown; (iii) not amenable to sufficiently accurate electronic structure calculations?

Although the structure of a stable molecule should be minimum, one wonders whether predicting a saddle point may suffice. The answer is positive, yet uncommon to some extent. Recall that in the absence of a push along the out-of-linearity coordinate associated with the imaginary frequency, one cannot distinguish the two (minimum versus saddle point) stationary points, since the Lemma involves no forces. Of course, a single point calculation along the bending (or another convenient) coordinate may suffice to disclose its nature at the chosen level of theory. Recall further that a saddle point may still behave as a minimum if vibrational stabilization is brought into play.^24,25 This can only be certified if a wider portion of the PES is calculated, which will not be attempted here. Another remark to note that one can disclose the nature of stationary points in quasi-molecule theory by employing a bisection method reported elsewhere,^5,6,8 thus avoiding to perform ab initio or DFT calculations for all possible tiles (triatomics, for linearity) embedded on the parent molecule. This can greatly facilitate things, since the combinatorial law could make the task tedious and eventually unaffordable.

2.2 Pseudo-linear and linear triatomics

Fig. 1 shows that C₆ assumes structures with two basic shapes: planar (a six-membered ring, occasionally denoted simply as planar C₆) and linear (top and bottom panels, respectively). Within acceptable numerical tolerances, the planar structure is predicted to be less stable than the linear one with the subminimal split-valence²⁶ 3-21G basis set [top and bottom panels in the left-hand-side, respectively; the corresponding right-hand-side (rhs) ones are at the aug-cc-pVTZ (or AVTZ) basis-set level, but is more stable at the other levels of theory], which are described in Section 2.3. Parenthetically, recall that a basis set for calculations of the correlation energy must describe both radial and angular effects, and hence sensible raw results with smaller than DZ + polarization (or split-valence plus polarization) sets are not expected.²⁷ Following Taylor, who regarded the latter as minimal, we have adopted²⁸ the designation of subminimal for 3-21G and other basis sets that fall hierarchically before the minimal. Despite their simplicity, they can also be valuable for extrapolating to the complete basis set (CBS) limit the energies calculated with extended basis sets, the total cost being basically that of the extended calculation, i.e., a kind of “single-point” CBS extrapolation.^28–31 The planar C₆ structure shows D_3h-like symmetry while the linear one is D_∞h. A similar result is predicted at the gold-standard coupled-cluster single and double level of theory including the connected perturbative triple correction CCSD(T),³² which is assumed to be able to predict energies of chemical accuracy when employing sufficiently flexible basis sets^2,33 (however, for small basis sets, see later). Thence, we will investigate whether it is plausible to consider both C₆ structures acting as monomers in the formation of the title calcium hexacarbides.

Fig. 1 Planar and linear structures of ground singlet C₆: top and bottom panels, respectively. Panels (a) on left-hand-side are with the 3-21G basis set, those on right with AVTZ. Energies and bond distances (in E_h and Å, except when specified otherwise) are indicated top-down for DFT using the B3LYP and M06-2X functionals with the AVTZ basis set and, for CCSD(T) with the VTZ basis set, where signaled with an asterisk. The stabilities of the linear isomers relative to the planar congeners are shown in parenthesis. For convenience, the total energies along the work are given as the Molpro³⁷ output.

Panel (a) in the left-hand-side of Fig. 2 shows CaC₆ at the DFT B3LYP^34–36 level with the 3-21G basis set, where the ring-shaped C₆ is viewed as a pseudo-atom. In turn, the right-hand-side panels (b) and (c) show Ca(C₆)₂ and Ca₂C₆ optimized at the same level of theory. Both are minima, and hence can be viewed as pseudo-linear triatomics.

Fig. 2 Pseudo-diatomic CaC₆ and pseudo-linear triatomics Ca(C₆)₂ and Ca₂C₆, respectively in panels (a)–(c). Shown at the lhs are side views while the top ones are at the rhs. Calculations were done at the B3LYP/3-21G level of theory.

Naturally, such molecules reveal shapes that depend somewhat on the level of theory. This is evinced in Fig. 1, which illustrates the actual shapes of ground-singlet C₆ at the DFT level of theory using both the 3-21G and the recommended AVTZ basis sets, the first of which is the only one affordable for the larger clusters considered in the present work. Corresponding structures have been obtained for Ca(C₆)₂ and Ca₂C₆ at B3LYP and M06-2X^38,39 levels of theory with AVTZ basis sets; see the top and bottom panels of Fig. 3, respectively. However, no converged structure has been found for CaC₆Ca with the M06-2X functional. All attempts for CaC₆Ca with B3LYP and M06-2X and AVTZ basis sets tend to yield structures with lower symmetry compared to the 3-21G structure as well as broken forms of C₆ followed by formation of a Ca–Ca bond. Conversely, results similar to described above with the 3-21G basis set are obtained when using the same functionals and the MINI⁴⁰ basis set. Moreover, the above findings hold when Grimmes's correction⁴¹ for the dispersion interaction is used, i.e., when Grimme's original DFT-D3 is employed in the current work. Of course, other stationary points may be expected, as noted in previous work,^5,7–10,13 such a possibility should not invalidate the results reported here.

Fig. 3 Pseudo-diatomic CaC₆ [(a) and (c)] and pseudo-linear triatomics Ca(C₆)₂ [(b) and (d)] at the B3LYP/AVTZ (top) and M06-2X/AVTZ (bottom) levels of theory. Side views are shown in the lhs, while top views are shown in the rhs.

The tendency to form a Ca–Ca bond is actually corroborated in Fig. 4, which shows the easy to converge Ca₂C₆ minimum structure at the B3LYP/AVTZ level of theory. It turns out to be predicted as a saddle point at this level of theory, with a small imaginary frequency (−19.3 cm⁻¹) for bending. This is not surprising since the inverse sandwich with C₆ between the two Ca atoms is predicted to be a minimum at the B3LYP/3-21G level, with a more negative energy than CaCaC₆; cf.Fig. 2(c) and 4(a). This pattern is maintained when including the dispersion correction, and in more elaborate calculations with the AVTZ basis set; see Fig. 2 and 4 for the corresponding energies.

Fig. 4 Pseudo-linear triatomics CaCaC₆ at B3LYP/3-21G [panels (a) and (c)] and B3LYP/AVTZ [(b) and (d)] levels of DFT. Side views are shown in the lhs, while top views are shown in the rhs.

One should now address the intriguing fact in Fig. 2 related to the six bonds apparently imparted by the Ca atom in CaC₆, a number that raises to 12 in Fig. 3 for Ca(C₆)₂. Such plots are obtained with the WXMACPLOT package by setting to default the standard criterion for lengths and normal bonding tolerance, as well as guessing the bond order and assigning hydrogen bonds. In fact, such a criterium is followed in drawing all stick-and-ball plots in the present work. Naturally, one cannot assign a single-bond to every stick; see also later. In fact, no obvious atomic hybridization can be invoked when knowing that calcium is an alkaline-metal atom typically with only s and p valence orbitals, thence electronic configuration 1s²2s²2p⁶3s²3p⁶4s². It should also be recalled that C₆ can assume other isomeric forms, in particular the linear one in Fig. 1.

2.3 Electronic structure methods and basis sets

Because highly accurate calculations are viable only for model systems and small molecules, one should consider whether a constant consistent basis set must be employed for a reliable comparison of results for related molecules of varying sizes. Of course, large reliable basis sets would be ideal for all, but this is unaffordable. Accurate calculations with large basis sets are though viable for the tiles (Fig. 5) and medium size molecules, with small ones being required for the larger systems. In fact, an accurate treatment of calcium-containing molecules may even require explicit consideration of the Ca valence 4s and “outer-core” 3sp electrons.⁴² Although this is not the topic of concern here, the performance of a valence basis set augmented with sets of tight functions (cc-pCVTZ) is tested in Fig. 5 to describe both the valence and outer-core spaces of the Ca atom in the tiles. All ab initio and conventional DFT calculations here reported were performed with the MOLPRO³⁷ package for electronic structure calculations and the SCF convergence typically set at its default value. Some semiempirical DFT ones that will be discussed in Section 3 were done with the ORCA^43–45 code.

Fig. 5 Tiles for linearity: (a) C₃; (b) CCaC; (c) CaC₂. The calculations were done at the indicated levels of theory, covering from B3LYP/AVTZ to M06-2X/AVTZ, and CCSD(T)/AVTZ. Due to difficulties in the calculation of frequencies, these are reported only for CCaC at the CCSD(T)/AVDZ level of theory. For better reference, distances are indicated with four decimal figures. The CCSD(T)/AVTZ values are in bold. Saddle points are marked sp; if not indicated, the structures in this and other plots are minima.

Known to vary in size and level of sophistication, modest basis sets (sub-minimal²⁸ such as Pople's STO-nG⁴⁶ and split-valence 3-21G,²⁶ Huzinaga's⁴⁰ MINI, or minimal like Dunning's¹⁸ correlation consistent cc-pVDZ) may allow for a uniform computational approach across a wider portion of the target family of molecules selected here. This is important for a comparative study, since the consistent use of the same basis set helps to mitigate the risk of introducing errors or inconsistencies due to varying the atomic bases. Indeed, switching atomic basis sets may lead to unexpected or non-systematic shifts in the calculated properties as implied by some of the results reported here. Because the molecular orbitals are approximated by the atomic basis sets, the results are likely to be comparable only when the latter are the same for all molecules. Keeping the atomic basis set constant will then help to maintain the uniformity of the comparison while ensuring that any differences in the observed trends are not driven by differences in computational approximations, thence are expected to be the intrinsic properties of the molecules rather than possible artifacts of the employed methodology.

If the approach described in the previous paragraph is adopted, the affordable basis set should be kept constant or vary in size in some (unknown) compensating ways within the range of studied molecular sizes. For the singlet ground state of C₃, the detailed results of Table 1 (also Fig. 5) show that keeping the basis small enough can be a promising way to proceed if some loss of accuracy in the energetics can be tolerated. Can this support the fact that the use of a diffuse basis set, a curse for sparsity,⁴ can be problematic and hence should be avoided when aiming at an accurate geometry at a moderate cost? In fact, the results in Table 1 even appear to promote the joint use of a small basis set with the cost-effective DFT method. Conversely, the CCSD(T) results in Fig. 5 cast doubt on its use even with the AVDZ and AVTZ basis sets. In fact, this extends to the use of the latter with the M06-2X functional. It should be recalled that (“exact”) full configuration interaction (FCI) calculations with the MINI basis set predict a minimum linear structure¹⁰ in good agreement with the experimental data.^47,48 Of course, the flat nature of the ground-state C₃ PES in the region of linearity is well established⁴⁹ (cf. also attributes at the B3LYP/AVDZ level of theory in Table 1, where both a saddle point and a minimum are scrutinized close to each other in energy), and the minimum known to occur at linear geometries.^47,48 Also, the decreasing pattern of the imaginary frequency at the M06-2X level of theory is observed with enhancement of the basis set, which suggests that a real frequency for bending could even arise if the basis-set enhancement was pursued.

Table 1 Bond distances and harmonic vibrational frequencies of singlet C₃ using various DFT and CCSD(T)/VDZ tightly converged calculations.^a

System	Energy/Eh	R/A	Angle/deg	ω 1 /cm^−1	ω 2/cm^−1	ω 3/cm^−1
a For further comparisons with CCSD(T) calculations using larger basis sets, see Fig. 5. b Double-degenerate.
M06-2X/3-21G	−113.35296850	1.2961	180.0	178.2	1250.4	2184.5
B3LYP/3-21G	−113.34470023	1.2978	180.0	164.6	1220.9	2152.7
M06-2X/VDZ	−114.00965632	1.2982	180.0	115.6	1267.8	2185.3
B3LYP/VDZ	−113.99039153	1.3012	180.0	167.8	1240.3	2155.7
M06-2X/AVDZ	−114.01376535	1.2982	180.0	−151.3	1262.2	2171.4
B3LYP/AVDZ	−113.99539701	1.3014	180.0	−117.4	1235.5	2142.9
B3LYP/AVDZ	−113.99550276	1.3013	171.7	119.9	1258.5	2128.9
M06-2X/VTZ	−114.03328782	1.2866	180.0	−43.3	1258.6	2164.6
B3LYP/VTZ	−114.01954681	1.2879	180.0	105.6	1239.0	2145.6
M06-2X/AVTZ	−114.03432259	1.2861	180.0	−49.1	1257.6	2160.4
B3LYP/AVTZ	−114.02074756	1.2873	180.0	102.5	1238.3	2142.4
M06-2X/VQZ	−114.03898463	1.2854	180.0	−34.1	1259.4	2159.9
B3LYP/VQZ	−114.02681463	1.2868	180.0	98.3	1238.8	2139.9
M06-2X/AVQZ	−114.03942668	1.2853	180.0	−17.5	1259.4	2159.4
B3LYP/AVQZ	−114.02735668	1.2866	180.0	91.8	1238.8	2139.4
CCSD(T)/VDZ	−113.74247109	1.3206	180.0	108.3	1184.8	2084.6
MRCI/CBS^49		1.3092	180.0	56.1	1171.0	2060.9
Exp.^47,48		1.2970	180.0	63.42	1224.20	2040.02

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Often known from experiments that molecular geometries are frequently unveiled via ab initio calculations and correspond to minima on the Born–Oppenheimer (BO) PES.⁵⁰ Of course, as it is already clear, variability in the results may arise from the level of theory and inclusion of electron correlation effects. Being a basic enterprise in chemistry,^51,52 the common dialog is MO theory versus DFT, which implies solving the electronic Schrödinger equation using either the molecular orbital wave function^1,2 or the electronic density,^53,54 respectively. Both yield approximate solutions: in MO theory, because of the cost-scaling law of such calculations and inability to saturate the basis set used to represent the molecular orbitals;^29,30 in DFT, due to the lack of a rigorous density functional.^55,56 Indeed, although Kohn–Sham (KS) DFT bears an exact formulation of quantum mechanical electronic structure theory, it relies on approximate exchange–correlation functionals,⁵⁷ with one out of the many functionals being the best for one system but not for another.⁵⁸ As already noted, the common belief that DFT is more trustable than MO is actually due to the fact that its adoption offers an excellent trade-off of accuracy versus cost. Specifically, we have chosen B3LYP and M06-2X, two of the most extensively benchmarked and validated functionals in quantum chemistry. Besides good reliability, B3LYP demonstrates a balanced accuracy versus cost, and includes a fraction (20%) of Hartree–Fock exact exchange; thence, it is a practical choice for medium- to large-sized molecules. In turn, with a distinct philosophical design, M06-2X has a high HF exchange (54%), hopefully reducing self-interaction errors, and often giving better reaction barriers, noncovalent interactions, and thermochemistry.

Still, DFT tends to overshoot the effect of correlation at the expense of exchange by predicting homogenized electronic structures with delocalized electrons^59,60 (cumulenic rather than alternating bond length or bond angle structures). However, KS DFT routinely provides structures and energies for up to a few hundred atoms, with recent efforts^61–63 (and references therein) directed toward the development of even lower-cost DFT formulations. In particular, the semiempirical tight-binding model⁶³ has been primarily designed for the fast calculation of structures and noncovalent interaction energies for molecular systems with roughly 1000 atoms. Its performance is illustrated in Section 3 through comparisons with conventional DFT results reported here.

As it is well established, modern electronic structure calculations employ mostly Gaussian orbitals for the wavefunction or the density. Despite their advantages due to the analytic solution of the involved integrals, there are disadvantages due to the large number of basis functions required to accurately represent the nuclear cusps in the wave function and lack of orthonormality: in particular, convergence to the complete basis set (CBS) limit cannot be necessarily achieved by adding more basis functions.^4,64 Since the goal here is the shape of the molecule at its equilibrium geometry, it is hoped that large basis sets can be avoided, and that their sizes can be reduced to a minimum trading some accuracy in the energetics for increased locality⁴ (parsity in the 1-particle density matrix, 1-PDM) and computational efficiency.

Employing standard codes for the calculations of a variety of systems belonging to a family, one wonders whether their intrinsic default parameters are the best to use for the purpose. Because the target is geometry optimization rather than energetics along a pathway, it appears reasonable to trust defaults as a safe approach, although not necessarily optimal.⁶⁵ However, there are a few high-leverage defaults that are worth overriding. This is here illustrated for the prototypical CaC₆ molecule by considering a few grid checks in the involved numerical integrations. Briefly, to assess the grid sensitivity of default settings, CaC₆ was re-optimized with MOLPRO³⁷ at B3LYP-D3/3-21G and aug-cc-pVTZ basis-set levels using both the default pruned grid and a dense unpruned grid. The differences for the default 3-21G were within ≤0.004 kcal mol⁻¹, ≤0.001 Å and ≤0.1 deg and, for AVTZ, within ≤0.001 kcal mol⁻¹, ≤0.0005 Å and ≤0.1 deg. This supports defaults to be adequate, besides less time consuming, for the broader set. They are then employed in all DFT calculations here reported, except for convergence where tight criteria are often employed.

3 Results and discussion

In quasi-molecule theory, the parent molecules are split as a many-body expansion^50,66 (MBE) where at least one three-atom fragment is formed. Since C₆ is here treated as a pseudo-atom, one has:

Ca_n(C₆)_m(¹A) → Ca_n−1(C₆)_m−2(¹A) + Ca(C₆)₂(¹A) (1)

→ Ca_n−2(C₆)_m−1(¹A) + Ca₂C₆(¹A) (2)

where the electronic states are classified in the C₁ symmetry point group. Although the Wigner–Witmer spin-spatial correlation rules⁶⁷ allow other electronic states, only the lowest is indicated, since it is likely the most relevant.

Before addressing eqn (1) and (2), consider first the cases up to CaC₆, where the linear C₆ is treated as usual. One has:

CaC₆(¹A) → C_6−m1CaC_6−m2(¹A) + C_m1+m2(¹A) (2 ≤ m₁ + m₂ ≤ 6) (3)

where the ground singlet state is assumed to apply to all relevant values of m₁ and m₂. Recall now that C₄ and C₅ were both predicted¹⁰ to be minimum linear structures at the CCSD(T)/AVTZ level of theory, although not necessarily the absolute minimum in the former case. Also, note that CaC₂ is a stationary point at linearity, see Fig. 5. From the Lemma, it then follows that all other linear species up to CaC₆ must be stationary points, as shown in Fig. 7. Rather than invoking the Lemma, note that this may find a justification in the fact that the sum of two parabolas is a parabola as long as the sum of their quadratic coefficients is not zero. In this case, the curve is a linear function but the same applies to the sum of a parabola and a straight line, sufficing to recall that the standard form parabola ax² + bx + c is the result of a line with slope b and y-intercept c added to a parabola with a vertex at the origin, and vertical stretch a. In fact, the various parabolas or straight lines may be viewed as corresponding to the optimized tiles.

Interestingly, all cases in eqn (3) with m₁ = 2 and an even m₂ turn out to be saddle points either when using the B3LYP or M06-2X functionals and the 3-21G or AVTZ basis sets. Note that CaC₂ is predicted to have an imaginary bending frequency of −130.6 cm⁻¹ (−163.6 cm⁻¹) when calculated at the B3LYP (M06-2X) level of theory with the AVTZ basis set. In fact, the same is predicted at the CCSD(T)/AVTZ level but with an imaginary frequency for bending of −131.8 cm⁻¹; this saddle point connects the two isosceles triangular geometries (C_2v symmetry) where the C–C and two equal Ca–C bond lengths are, respectively, 1.2771 Å and 2.2723 Å, see Fig. 6. If CCSD(T)/AVTZ* (CCSD(T)/AVQZ*) is used (the asterisk signals the use of the cc-pCVXZ basis⁶⁸ for Ca), the energy of CaC₂ is −752.74818245E_h (−752.76724578E_h), and the above distances read 1.2771 Å (1.2730 Å) and 2.2680 A (2.2652 Å), in the same order. These results show satisfactory agreement both with the AVTZ ones and those reported¹⁷ at the MRCI+Q/AVQZ level of theory with V(Q+d)Z^42,68 and AV(Q+d)Z^18,69 basis sets for Ca and C, respectively, 1.268 Å and 2.233 Å. They also compare well with previous CCSD(T) results for the pair (R_CC, R_CaC2), which include extrapolations from up to quintuple-zeta basis sets:¹⁶ (1.2685, 2.1078) versus the current (1.277, 2.180) values, all in Å. The calculated CCSD(T)/AVTZ coordinates and frequencies are given in the SI.

Fig. 6 Minima of CaC_n for n = 2–4 at the DFT level with the B3LYP and M06-2X (in italics) functionals as well as the CCSD(T) method (the bottom line for a given basis), using both the 3-21G and AVTZ basis sets. No convergence obtained with CCSD(T)/3-21G. For further values on CaC₂, see the text.

Regarding CaC₃, the two possible linear structures in Fig. 7 are predicted to be minima with B3LYP/3-21G, while CCaC₂ is a saddle point when using the AVTZ basis. Similar considerations apply to CaC₄, although showing a small bending frequency of −30.5 cm⁻¹ (2.8 cm⁻¹) at the B3LYP (M06-2X) level of theory with the AVTZ basis set. Also, note that CaC₄ is predicted to be a saddle point with B3LYP/AVTZ but a minimum when employing the M06-2X functional with the same basis set. It turns out to be a minimum also at the CCSD(T)/AVTZ level of theory, with a low-vibrational frequency for bending of 28.0 cm⁻¹. Even if repeatedly stated, recall that the specific nature of the above stationary points is not crucial to feed, via Lemma, the chain that leads to other stationary points up to m₁ + m₂ = 6 in eqn (3).

Fig. 7 Possible leftovers up to CaC₆: (a) to (d). The calculations were done at levels of theory covering from B3LYP/3-21G to B3LYP/AVTZ (left- and right-hand-side columns, respectively). The values in italics employed the M06-2X functional. Marked with an asterisk is the energy of the minimum structure described in the text: Ca bonded to three C atoms. The two asterisk denote the energy of the (quasilinear, i.e., numerically not strictly linear) minimum calculated at the CCSD(T)/AVTZ level of theory with frequencies of 28.0, 206.0, 207.3, 396.7, 517.9, 518.0, 999.2, 1863.4, and 2058.7, all in cm⁻¹.

Notably, except for linear C₂CaC₂ for which both B3LYP and M06-2X functionals predict saddle points with the 3-21G and AVTZ basis sets, the predictions tend to be minima with B3LYP while a few are predicted to be saddle points with M06-2X as summarized in Fig. 7. This occurs mostly for C_m1CaC_m2 when m₂ is even: m₂ = 2, 4, 6. In particular, when m₂ = 2, the tendency of that side of the molecule is to form a CaC₂ triangle-shaped end following the pattern for the pristine triatomic. In fact, the absolute minimum of CaC₃ at the B3LYP level shows the Ca atom bonded to the three C atoms in a planar arrangement with two equal C–C bonds of 1.311 Å and three Ca–C ones close to 2.372 Å, see Fig. 6. Because the cost-expensive harmonic vibrational frequencies have not been reported for such calcium carbides, they are gathered in Table 3. As shown, the DFT results show fair agreement with the more expensive CCSD(T) ones, which extends to cases where the 3-21G basis set is employed. Fig. 6 shows also good agreement between the predicted C–C bond distances with the 3-21G basis set and the cost-expensive CCSD(T)/AVTZ values. This may help trusting the results for larger molecules, where subminimal bases are the only affordable.

In the above context, one may ask whether DFT works better with small atomic basis sets than do MO methods. The answer is positive, because DFT focuses on the electron density, and hence is less sensitive to the exact form of the wavefunction. In fact, DFT methods seem to capture important electron correlation effects with smaller basis sets compared to wavefunction-based ones, which require more computationally expensive post-Hartree–Fock treatments (MP2, CCSD, etc.) to include correlation accurately. This is illustrated in Fig. 6 and Table 2, where DFT methods are shown to efficiently incorporate electron correlation and even achieve satisfactory accuracy with basis sets as small as 3-21G. Conversely, MO-based methods require larger basis sets to describe accurately the electronic structure, especially when accounting for electron correlation. This is particularly advantageous when studying large molecules (or periodic systems), an appreciation often taken for granted in the literature.

Table 2 Break-up energies at the B3LYP level of theory^a

System	3-21G			VTZ or AVTZ
	E	ZPE	ΔE^b	E	ZPE	ΔE^b
a The energies of Ca are −674.14453689, −677.51058418 and −677.51068053Eh with the 3-21G, VTZ and AVTZ basis sets, respectively. For linear C6, with 3-21G, VTZ and AVTZ basis sets, they are −226.83519419, −228.18423529 and −228.18646661Eh, with ZPEs of 0.02404867, 0.02312140 and 0.02301653Eh, in the same order. For the cyclic C6, the corresponding values are −226.82853636, −228.19555210 and −228.19764711Eh, with ZPEs of 0.02744775, 0.02743928 and 0.02739578Eh; see also Fig. 1. b See eqn (4). c With the AVTZ basis set. d With B3LYP-D3 and default convergence. e Not done: too computationally expensive. f At the B3LYP/AVTZ level of theory, the pseudo-linear CaC6Ca structure is a saddle point with two nearly degenerate frequencies for bending of −9.0 cm^−1 and 12.6 cm^−1.
Linear
CaC6	−901.06858985	0.03094199	−51.434	−905.78306096^c	0.02739578^c	−51.164
Ca(C6)2	−1128.06787510	0.05850426	−152.198	−1134.14819453^c	0.05045526^c	−154.716
Ca2C6	−1575.18279586	0.03469092	−30.049	−1583.32865881^c	0.02643229^c	−69.411
Ca2(C6)2	−1802.27545063	0.06534889	−187.460	−1811.78196277^d	0.05349781^d	−241.634
Ca2(C6)3	−2029.35873151	0.09266875	−341.084	−2040.13333320^d	0.07755946^d	−345.923
Ca3(C6)2	−2476.58775192	0.06859120	−290.699	—^e	—^e	—^e
Ca3(C6)3	−2703.59447314	0.09854201	−394.631	—^e	—^e	—^e
Ca3(C6)4	−2929.15863530	0.12353628	403.547	—^e	—^e	—^e
Ca4(C6)3	−3377.84402899	0.10333983	−457.520	—^e	—^e	—^e
Ca4(C6)4	−3603.57559328	0.13078367	237.148	—^e	—^e	—^e
Ca4(C6)5	−3831.87307989	0.16173424	−676.123	—^e	—^e	—^e
Ca5(C6)4	−4278.00083235	0.13783533	65.430	—^e	—^e	—^e
Pseudo-linear
CaC6	−900.98570246	0.02151673	−11.647	−905.78306096^c	0.02285039^c	−49.748
Ca(C6)2	−1127.90706435	0.04901827	−69.862	−1134.04040821^c	0.05124572^c	−86.583
Ca2C6	−1575.19079086	0.02207718	−49.292	−1811.78348033^c	0.02346122^c^f	−60.476
Ca2(C6)2	−1802.03737204	0.04397965	−64.095	—^e	—^e	—^e
Ca2(C6)3	−2029.00409051	0.07181339	−150.563	—^e	—^e	—^e
Ca3(C6)3	−2703.22506643	0.06812440	−200.844	—^e	—^e	—^e
Ca3(C6)4	−2930.10424764	0.09467593	−233.187	—^e	—^e	—^e
Ca4(C6)3	−3377.38252624	0.06795410	−209.060	—^e	—^e	—^e
Ca4(C6)4	−3604.29718599	0.09241064	−264.980	—^e	—^e	—^e
Ca4(C6)5	−3831.20564924	0.11819916	−316.177	—^e	—^e	—^e

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Corroborating the above, coupled cluster calculations show problematic convergence for CaC₂ with the 3-21G basis set, as evinced in Table 3. This is likely due to the complex charge distribution in CaC₂ due to the ionic bonding nature between calcium (Ca²⁺) and carbide (C₂²⁻) which, if the electron density is not well-behaved or is poorly initialized, can make the algorithm used in the calculation struggle to converge to a stable solution. Conversely, DFT appears to be less sensitive both with B3LYP and M06-2X, and possibly even with other functionals.

Table 3 Harmonic vibrational frequencies of small planar calcium carbides^a

System	ω 1	ω 2	ω 3	ω 4	ω 5	ω 6	ω 7	ω 8	ω 9
a Empty spaces are not relevant: there are 3n − 6 vibrational frequencies where n is the number of atoms. b Not converged. c For the asterisk, see Fig. 5 and text.
CaC2
M06-2X/3-21G	346.2	469.9	1778.8
B3LYP/3-21G	302.2	431.6	1749.6
CCSD(T)/3-21G	—^b	—^b	—^b
M06-2X/AVTZ	421.9	570.2	1865.2
B3LYP/AVTZ	384.6	558.2	1826.7
CCSD(T)/AVTZ	405.1	529.8	1757.5
CCSD(T)/AVTZ*^c	407.2	530.5	1758.1
CCSD(T)/AVQZ*^c	407.9	534.6	1768.3
CaC3
M06-2X/3-21G	241.7	273.0	327.5	600.4	1093.8	1573.6
B3LYP/3-21G	212.7	263.6	323.5	580.8	1062.8	1579.2
CCSD(T)/3-21G	67.0i	230.4	234.6	500.2	1009.3	1500.3
M06-2X/AVTZ	183.0	300.0	460.6	717.1	1223.0	1506.4
B3LYP/AVTZ	201.8	288.9	453.4	689.7	1206.7	1497.8
CCSD(T)/AVTZ	159.4	297.0	446.3	689.8	1151.1	1407.9
CaC4
M06-2X/3-21G	232.9	328.4	352.8	398.8	493.5	576.7	942.1	1956.4	2095.8
B3LYP/3-21G	233.9	320.2	347.3	378.6	428.0	523.7	945.9	1894.1	2038.0
CCSD(T)/3-21G	219.9	326.4	345.3	380.8	398.0	509.2	892.0	1810.7	1930.2
M06-2X/AVTZ	277.2	342.0	415.5	454.5	504.3	620.0	1002.1	1942.9	2070.6
B3LYP/AVTZ	277.6	320.0	402.3	444.2	489.3	600.5	1003.8	1885.1	2010.7
CCSD(T)/AVTZ	258.6	335.5	400.9	431.1	455.0	571.0	960.4	1818.4	1951.4

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We now turn to Fig. 8 where linear C₆ is considered as a pseudo-atom. The results are mostly predicted as linear minima at the 3-21G level up to Ca₄(C₆)₅ and Ca₅(C₆)₄. Whether such a pattern is observed with larger basis sets can only be tested at an affordable cost for the few smaller members of the series. This is done here for the first four members with the non-augmented VTZ basis set, specifically Ca₂C₆, Ca(C₆)₂, Ca₂(C₆)₃ and (CaC₆)₂, see Fig. 9 and Table 2. For CaC₆Ca, the results with the B3LYP-D3 functional predict a saddle point with an imaginary frequency for bending (−83.8 cm⁻¹), while a stable species with a low bending harmonic vibrational frequency (37.2 cm⁻¹) is predicted with M06-2X-D3.

Fig. 8 Large linear triatomics at the B3LYP/3-21G level of theory. All are predicted as minima; for the energies, see Table 1.

Fig. 9 Tightly converged linear molecules at the DFT level with the B3LYP (bottom) and M06-2X (top) functionals and the VTZ basis set, including the dispersion correction. sp³ stands for three small imaginary frequencies for bending vibrations.

The Ca + C₆ break-up energies are also reported in Table 2:

ΔE_system = E_system + ZPE_system − xE_Ca − y(E_C6 + ZPE_C6) (4)

where ZPE_system and ZPE_C6 are the zero-point energies of the parent and pseudo-atoms; x and y denote the number of Ca and C₆ units. Notably, the results do not differ qualitatively when the 3-21G and the higher-rank correlated consistent basis sets are compared. Interestingly, except for Ca₃(C₆)₄, Ca₄(C₆)₄ and Ca₅(C₆)₄, all break-up energies are negative which suggest that the latter are unstable relative to the integrating parts. Somewhat intriguingly, they all involve four linear C₆ pseudo-atoms and are perfect minima (all frequencies are real, although some are low vibrations, as one could expect for a large molecule).

Finally, Fig. 10 shows the parent molecules when the pseudo-atoms are planar C₆. Due to being computationally expensive, only a few test cases were considered at the B3LYP/AVTZ level of theory. Converged typically to <10⁻⁴E_ha₀⁻¹, the results are gathered in Fig. 3 and 4. The same general pattern is observed: while the optimization with small basis sets (3-21G or MINI) tends to predict minima, the larger correlation consistent basis sets often predict the optimized structures as saddle points for bending (signaled by sp in red) or for rotation (sp in blue) of the planar C₆ around the axis orthogonal to their plane. As noted above, calculations have also been performed with the ORCA^43–45 code using the broadly parametrized self-consistent tight-binding (GFN2-xTB) model Hamiltonian recently introduced by Grimme and coworkers.^63,70 The stationary points so obtained employ a minimal valence basis set of atom-centered contracted-Gaussian functions, which approximate Slater functions (STO-nG⁴⁶). Polarization functions for most main group elements (typically second row or higher) are also employed, while hydrogen is generally assigned a single 1s function. The nature of the optimized stationary points is indicated in the rhs-most column of Fig. 10. They differ for two systems. One, Ca₃C₁₂, predicted at the conventional B3LYP/3-21G level to be a saddle-point for bending (with a double-degenerate imaginary harmonic vibrational frequency of −44 cm⁻¹), while being predicted a minimum with GFN2-xTB. The other, Ca₄C₃₀, which yields Ca₄C₂₄ plus C₆, located non-collinearly with both fragments assuming structures similar to their own in isolation. All other molecules in Fig. 10 are predicted as minima if rotations of C₆ around the linear axis are ignored (the corresponding imaginary frequencies are typically of a few cm⁻¹). In an attempt for clarification, additional optimizations were done for Ca₅C₃₀, Ca₅C₃₆ and Ca₆C₃₆, the latter involving 336 electrons. All yielded pseudo-linear stationary points of the saddle point type with one or a few minor imaginary harmonic vibrational frequencies for rotation of the C₆ units, see Table 4. This may be understood from the orientation of those units relative to each other as suggested by their optimized structures. Note that the determination of the optimum relative orientation of such units would imply even tighter convergence requirements than the ones already employed, a procedure not deemed justified since the involved normal modes do not lead to bending (nonlinearity) of the optimized structures. Note further that the deviations from collinearity between the Ca atoms is typically ≤1 deg with GFN2-xTB. For example, the angle formed by the middle and terminal Ca atoms in Ca₅C₃₆ is ∠7,22,35 = 179.4 deg with B3LYP/3-21G and 178.2 deg with GFN2-xTB, values that are deemed as acceptable in view of the involved numerical intricacies. Similarly, for Ca₆C₃₆, the angle ∠35,7,42 is 179.8 deg and 179.8 deg with the two above methods, in the same order, with the distances between the terminal Ca atoms being 24.4 Å and 16.9 Å, respectively. The two structures so obtained are displayed in panel (a) of Fig. 11, top and bottom entries, respectively. From the trajectory of the optimization procedure, which leads from the conventional DFT to the extended-tight-binding (xTB) structure, it is visible that the C₆ units tend to loose planarity particularly in the middle of the Ca₆C₃₆ structure. To some extent, this significant difference could be expected from the corresponding results for CaC₆Ca, where the Ca–Ca distances are 5.01 Å and 4.57 Å, in the same order (note though that both methods predict a minimum pseudo-linear structure). Even more noticeable are the differences for Ca₇C₃₆ where the top structure shows a near D_∞h symmetry whereas the bottom one is C_∞h or even C₁ like. Suffice it to note that the full length separating the two terminal Ca atoms in Ca₇C₃₆ is 19.8 Å with xTB rather than 29.3 Å with conventional DFT at the B3LYP/3-21G level. The difference is still large when comparing with the conventional DFT B3LYP/STO-3G value: 27.6 Å. In summary, despite significant differences in the predicted structures, partly to be ascribed to the basis sets, it is notable the celerity of the GFN2-xTB method.⁶³ This is remarkable when compared with conventional DFT (even using the 3-21G or STO-3G basis sets), since it is well established⁷¹ that the cost of diagonalization can be overwhelming: it ranges from O(n³) to higher, depending on the sizes of the molecule and the basis set, and the algorithm.

Fig. 10 Large pseudo-linear molecules at the B3LYP/3-21G level of DFT; for the energies, see Table 2. Indicated as “min” is a minimum while “sp” in red indicates a saddle point for bending, and in blue for rotation of C₆; the left column refers to B3LYP/3-21G while in the rhs column is for the semi-empirical GFN2-xTB method at the B3LYP/VTZ level.

Table 4 Calculated dipole moment (norms) in debyes at the B3LYP level of theory

System	3-21G	VTZ or AVTZ
a With AVTZ. b With VTZ. c Not done. d For CaCaC6. e It shows a tiny imaginary frequency of −4.2 cm^−1 for rotation of the middle C6. f As in (e) but −5.2 cm^−1 for rotation of the middle C6. g As in (e) but −1.4 cm^−1 for rotation of the middle C6. h As in (e) but two imaginary frequencies of −24 cm^−1 and −15 cm^−1 for rotations of the C6s. i As in (e) but one imaginary frequency of −4.4 cm^−1. j As in (e) but three imaginary frequencies of −6.3 cm^−1, −5.3 cm^−1 and −1.9 cm^−1. k As in (e) but two imaginary frequencies of −8.9 cm^−1 and −5.4 cm^−1. l No imaginary frequencies. m As in (e) but one imaginary frequency of −2.2 cm^−1 (−2.6 cm^−1 with B3LYP/STO-3G, and dipole moment norm 0.004 D).
Linear
CaC6	27.244	27.166^a
Ca(C6)2	18.022	0.003^b
Ca2C6	0.002	0.230^b
Ca2(C6)2	75.095	70.698^b
Ca2(C6)3	0.015	0.013^b
Ca3(C6)2	0.005	—^c
Ca3(C6)3	40.466	—^c
Ca3(C6)4	31.820	—^c
Ca4(C6)3	0.026	—^c
Ca4(C6)4	66.040	—^c
Ca4(C6)5	0.106	—^c
Ca5(C6)4	55.914	—^c
Pseudo-linear
CaC6	13.892	10.914^a
Ca(C6)2	0.009	0.016^a
Ca2C6	0.044	16.798^a^d
Ca2(C6)2	34.224	—^c
Ca2(C6)3	0.027^e	—^c
Ca3(C6)3	5.753^f	—^c
Ca3(C6)4	0.039^g	—^c
Ca4(C6)3	0.006	—^c
Ca4(C6)4	17.145	—^c
Ca4(C6)5	0.026^h	—^c
Ca5(C6)4	0.019^i	—^c
Ca5(C6)5	5.845^j	—^c
Ca5(C6)6	0.058^k	—^c
Ca6(C6)6	5.857^l	—^c
Ca7(C6)6	0.017^m	—^c

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Fig. 11 Ca₆C₃₆ and Ca₇C₃₆ in panels (a) and (b), respectively, as optimized with conventional B3LYP/3-21G (top entry) and GFN2-xTB (bottom entry). The Ca–Ca bond distances are indicated (in A).

Considered next are the dipole moments of the calcium hexacarbides here discussed, which can assume large magnitudes. Since molecules with large dipole moments are overall important in both understanding and harnessing a wide range of scientific phenomena, they can eventually stimulate some interest for both fundamental research and practical applications, particularly in areas where polar interactions are important. Obviously, if symmetrical, their dipole moments should vanish or be very small, except if some lack of symmetry occurs. Conversely, large non-symmetric linear or pseudo-linear ones can show huge dipole moments as gathered in Table 4 for up to Ca₇(C₆)₆. Naturally, any differences in the molecular structure should reflect on the calculated dipole moments. For example, the dipole moments of Ca_n(C₆)₆ (n = 5–7) are predicted with GFN2-xTB to be 0.173 D, 168.311 D, and 216.456 D which, for the last two, show drastic differences from the values calculated with the conventional B3LYP/3-21G. However, as noted by the authors,⁶³ no atomic partial charges or molecular dipole moments have been used in the parameterization of the GFN2-xTB method or its predecessors, with the parameters fitted to yield reasonable structures, vibrational frequencies, and noncovalent interactions while focused on organic, organometallic, and biochemical systems of up to a few thousand atoms. Although performing well for relatively small dipole moments,⁶³ it would be interesting to test how the method does for larger magnitude ones such as those here reported. Unfortunately, experimental data are nonexistent for comparison in the present case.

A further remark is that can the many types of calcium carbides be formed by adding atoms of Ca to a plasma of carbon? A positive answer is plausible, depending on the specific conditions of the plasma, such as temperature, pressure, and the ratio of Ca to C atoms. In fact, it is well known that calcium carbides may form in different types based on their stoichiometry and crystal structures. One of the most common is CaC₂, where the carbide ion is present. This should typically form at high temperatures, where calcium can react with carbon in a ratio of 1 : 2, producing a solid carbide. Another is CaC₆: commonly associated with graphite-like structures, where the calcium atoms are intercalated between carbon layers in a graphite-like lattice, it tends to form under different regimes often involving higher carbon content and specific plasma conditions. Thence, depending on the temperature and density of the plasma, the Ca and C atoms could bond in various stoichiometries. Despite apparently bearing little relevance for the studies here performed, the latter may hopefully show relevance when diluted calcium regimes in carbon vapour are examined at as low realistic temperatures as possible. We focus next on other possible calcium carbides.

4 A panoply of other calcium carbides

The minima shown in Fig. 6 lets one wonder that many others may exist for larger clusters. Fig. 12 illustrates a few others for clusters formed by one Ca and up to fourteen C atoms. Besides evincing the panoply of minima that may be encountered for such systems (recall that others may exist), there are two facts worthy of attention. First, all optimizations tend to show a fast convergence toward planar structures. This may not be surprising since the tetratomic (‘tiles’ for planarity) formed by calcium and carbon atoms tend to be planar. Suffice it to add that Lemma 2⁵ reads similarly to Lemma 1 (denoted above simply as Lemma) except for referring to planarity, with the tiles being tetratomics rather than triatomics. Note from Fig. 12 that there is apparently an exception to the sequence of planar molecules: CaC₁₂ (see also the next paragraph). This is not planar at the geometry of its minimum, which may be due to the Ca atom whose size fits better outside than inside the cyclic C₁₂. Nevertheless, there is a planar saddle point (an imaginary vibrational frequency of −140.0 cm⁻¹ at the conventional B3LYP/3-21G level), which links the two minima localized at opposing sides of the C₁₂ plane. Clearly, the barrier for isomerization is significantly enhanced at the subminimal basis set level, which may reflect the poorer quality of the latter from the energetics point of view when compared with the minimal (VDZ) and extended (VTZ) ones. Note that the planar structure is still a saddle point for the larger basis sets, but with three imaginary frequencies: one for out-of-plane bending (−91.8 cm⁻¹ and −92.2 cm⁻¹ at the VDZ and VTZ levels, respectively), and two others, possibly double-degenerate (−398.6 cm⁻¹ and −300.0 cm⁻¹, in the same order), for vibrations inside the cyclic C₁₂ cluster. A further remark to emphasize that all calcium carbides in Fig. 6 and 12 have planar stationary points, which includes the saddle point in panel (h) of Fig. 12.

Fig. 12 Top (lhs) and side (rhs) views of mono-calcium carbides with up to fourteen C atoms: (a) to (j), with variants indicated with prime and double-prime. All optimizations in the two major entries at the bottom have been done at the B3LYP level of theory with the 3-21G and VDZ basis sets, except in panels (a) to (c) for CaC_nn = 5–7, panels (h) and (h′) highlighted in blue for CaC₁₂, and panel (i) for CaC₁₃ that were also optimized at the VTZ level. The grey shaded areas are the plots of CaC₇ with the top (c′) and (c″) employing the B3LYP and M06-2X functionals and also the AVTZ basis set. The asterisk in CaC₇ signals that its structure is shown at the B3LYP/3-21G level of theory. Empty tabular spaces implies non-calculated, see also the text.

Besides CaC₁₂, CaC₇ is predicted with default convergence at the B3LYP/VDZ level of theory to be a quasi-planar saddle point with the geometry shown in panel (c) of Fig. 12 and a low imaginary vibrational frequency of −35.8 cm⁻¹. However, when tightly converged at the same level, a minimum (−943.85108164E_h) is found with a distorted structure similar to panel (c″) of Fig. 12 and a difficult to notice off-planarity in some torsion angles: ∠1,2,3,4 = −24.2, ∠4,2,3,5 = −10.1, ∠4,2,1,6 = 14.7, ∠2,1,6,7 = −35.6 and ∠2,3,5,8 = 16.2, all in deg. In turn, when tightly B3LYP/AVTZ converged, it yields the perfectly planar minimum shown in panel (c′) of the same plot, thence similar to panel (c) of Fig. 12 at the B3LYP/3-21G level of theory. Also found at this level of theory is a slightly non-planar minimum (−943.93608026E_h) resembling the one in panel (c″) with the torsion angles now reading: −23.6, −10.4,15.6, −37.3 and 17.6 deg, in the same order. With the energy difference between the two minima of only 0.572 kcal mol⁻¹, this reveals the high planarity of the corresponding PES in that region of configuration space. In fact, similar topographical features are observed at the B3LYP/VTZ level, with the deepest minimum [−943.93264515E_h, similar to panel (c″)] predicted to be 0.588 kcal mol⁻¹ lower than the one in panel (c). Moreover, DFT calculations carried out with the M06-2X functional and the VDZ basis set predict a perfectly planar saddle point (−31.9 cm⁻¹ for out-of-plane motion); see the top entry in panel (c″). This has been confirmed at the tightly converged M06-2X/AVTZ level, which yields an imaginary frequency of −20.3 cm⁻¹. Clearly, this points to a slightly non-planar minimum at this level of theory, a search that we have not deemed justified to pursue in the current work.

Another issue is the fact that the minima reported in this section are deep (eventually absolute) minima in the corresponding PESs. Still, this does not invalidate the existence of further minima, sufficing to recall that many stationary points previously determined turned out to be minima at the DFT/3-21G level with both B3LYP and M06-2X functionals. Of course, such basis sets may not yield the best energetics, but this handicap should not reflect in the geometrical parameters evinced by Fig. 6 and Tables 2 and 3. A minor point to emphasize that the apparent single and double bonds drawn in the various plots shown in the present work are based on the default parameter set in WXMACPLOT for the bond length, and hence are mostly pictorial. Indeed, if the IBOVIEW package is employed to draw the highest occupied molecular orbital of CaC₁₃, one obtains Fig. 13 where no sticks are drawn for the interactions of the Ca and C atoms. As expected, the HOMO of CaC₁₃ in Fig. 13 is likely formed from the 4s orbital of the Ca atom interacting with the carbon 2p (or sp² hybridized) orbitals. The electronic structure exhibits a delocalized bonding interaction in C₁₃, meaning that the HOMO has likely contributions from multiple bonding and anti-bonding interactions across the carbon cluster. In CaC₁₂, the structure should be similar to CaC₁₃ but with one less carbon atoms. This should not influence the bonding except for the fact that the Ca atom lies outside the plane of the carbon, but still benefits of the extensive delocalization of the electrons.

Fig. 13 Highest occupied molecular orbital of CaC₁₃ as tightly converged at the B3LYP/VDZ level of theory.

Of course, one may ask why does WXMACPLOT show 13 single bonds. Possibly, if maximum overlap hybrids^72,73 could be constructed, they might help to obtain a more realistic view, an issue that will not be here pursued. In its absence, it appears justified to recall that bonding occurs as long as the orbitals of the Ca and C_n overlap each other, probably about equally irrespective of the atom of the cyclic C_n structure. Since this may very much depending on the method and the basis set, one further wonders whether an atomic basis set optimized from the minimization of the energy (variational principle) is necessarily better suited for predicting the correct molecular structure than a poorer (less negative energy) atomic basis set. The answer is “not necessarily” because the atomic orbitals optimized for isolated atoms do not account for changes in the electron density and potential due to molecular interactions. They must therefore require re-optimization or adaptation to accurately describe the interactions in the molecular system. In fact, such densities may turn out to be optimal when large atomic basis sets are used, but this will not preclude issues of basis overcompletness besides being unaffordable costwise: their validity is then largely validated only by reasons of the variational principle, although this applies only to the energy at a selected geometry.

Although tetrads of Ca and C atoms or of C atoms alone tend to be planar¹⁰ (see also Fig. 6), it should be recalled that volumetric clusters can also be formed as long as the Ca atom fits within the C_n cluster or bonds externally to it. An example is the CaC₂₀ cluster shown in Fig. 14 which has been optimized at the conventional B3LYP level of theory with the 3-21G and VDZ basis sets. Recall that Lemma 2⁵ predicts a planar stationary point for the parent whenever the tiles embedded on it are planar. Of course, this does not prevent the occurrence of non-planar stationary points: it simply warrants that a planar stationary points exists whenever the Lemma is satisfied. In fact, given the apparent flatness of the PESs of some of the species here studied, one cannot exclude that other minima exist besides the ones here reported. In particular, recall that C₂₀ is itself planar.⁹ Indeed, the distances between atom numbers 21 and 19 or 21 and 20 in CaC₂₀ (respectively 2.748 Å and 2.742 Å) differ little from the average Ca–C distance in CaC₁₃ at the same level of theory: (2.717 ± 0.040) Å [(2.693 ± 0.002) Å at the VTZ level]. A further remark to note that the planar structure in panel (d) of Fig. 14 is a saddle point of index 3 at the B3LYP/3-21G level with a frequency of about −150 cm⁻¹ for off-planar vibration. Still, it has a lower energy than its fullerenic analog, which is predicted to be a minimum (possibly among others, since many are known to exist for other fullerenic-type species⁷⁴).

Fig. 14 Examples of 3D and 2D structures: (a) Ca bonded to two atoms in a side of a C₂₀ ball; (b) as in (a) but inside the C₂₀ ball; (c) Ca bonded to a planar C₂₀; (d) as in (c) but inside the C₂₀ ring. All DFT calculations employed the B3LYP functional and indicated basis sets. As along the present work, where not indicated otherwise, the reported structures are minima at the calculated levels of theory.

5 Concluding remarks

A significant result from the present work is the perception that, except for the (variationally more negative) energy, a large atomic basis set cannot warrant a better description of the molecular structure than a smaller basis set. This is particularly so when studying a family of related molecules covering a wide range of physical sizes, since the problem calls for the use of a constant basis set for the whole set of calculations. A few reasons justify such an approach. The first is the computational cost: a large atomic basis set can provide a better description of the energetics but the required time and resources become prohibitive with increasing physical size. The second is overfitting (see also ref. 4), whenever the calculation becomes too sensitive to small numerical errors or details that might not be physically meaningful. The use of a larger basis set may then result in an inaccurate description of the system, particularly when using approximate methods, a possibility that gets enhanced with increasing dimensionality of the involved configuration space. Finally, numerical instabilities may arise due to linear dependences in the basis functions, which can lead to instability in the calculations, and recommend CBS extrapolation from energies of smaller basis sets.^28–31

The results here reported with subminimal basis sets (3-21G, MINI and STO-3G) predict the title hexacarbides to grow to large sizes. This cannot be probed from a comparison with experimental data (absent), but raises the question: can such clusters be formed when Ca atoms are added to a plasma of carbon at relatively low temperatures? The answer is likely affirmative, but cannot be definite prior to a detailed study of their stabilities (energetics, thence ab initio calculations with large basis sets!). Although the good aspects evinced by DFT can ameliorate difficulties, experimental work would be most valuable. The reported trends may also be of relevance for studies with other alkaline-earth metal elements of group 2 of the periodic table. There are no reasons of principle though for not being applicable even to other molecules. Finally, one may wonder whether quasi-molecule theory and a subminimal basis set is all that is required to decipher linearity, planarity or otherwise of a molecule. Moreover, can other pseudo-atoms be considered and tiling generalized to quasi-planarity, thence to (not strictly planar) 2D materials? Further studies are required before well-founded answers can be given.

Conflicts of interest

There are no conflicts to declare.

Data availability

The data supporting this article have been included as part of the SI. See DOI: https://doi.org/10.1039/d5cp02006a.

Acknowledgements

The author acknowledges the support provided by China's Shandong Province's Double Hundred Experts project WSP2023008, Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brasil (CAPES)-Finance Code 001, Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and the CQC-IMS which is supported by the Foundation for Science and Technology (FCT), Portugal, in the framework of the projects UI0313B/QUI/2025, UI0313P/QUI/2025 and LA/P/0056/2020.

Notes and references

1. A. Szabo and N. E. Ostlund, Modern Quantum Chemistry, McMillan, 1985.
2. T. Helgaker, P. Jørgensen and J. Olsen, Molecular Electronic-Structure Theory, Wiley, Chichester, 2000.
3. W. Kohn, Phys. Rev. Lett., 1996, 76, 3168–3171.
4. H. Laqua, L. B. Dittmer and M. Head-Gordon, J. Chem. Phys., 2025, 162, 184107.
5. A. J. C. Varandas, Phys. Chem. Chem. Phys., 2022, 24, 8488–8507.
6. A. J. C. Varandas, Int. J. Quantum Chem., 2023, 123, e27036.
7. A. J. C. Varandas, Mol. Phys., 2023, 0, e2285032.
8. A. J. C. Varandas, J. Phys. Chem. A, 2023, 127, 5048–5064.
9. A. J. C. Varandas, Chem. Phys. Lett., 2023, 140836.
10. A. J. C. Varandas, Int. J. Quantum Chem., 2024, 124, e27287.
11. A. J. C. Varandas, Chem. Phys. Lett., 2024, 852, 141493.
12. A. J. C. Varandas, Chem. Phys. Lett., 2025, 873, 142137.
13. A. J. C. Varandas, Phys. Chem. Chem. Phys., 2025, 27, 11853–11868.
14. S. Pfalzner and P. Gibbon, Many-Body Tree Methods in Physics, Cambridge University Press, 1996.
15. J. Zhao, Z. Chen, Z. Zhou, H. Park, P. V. R. Schleyer and J. P. Lu, ChemPhysChem, 2005,(6), 598–601.
16. H. Gupta, P. B. Changala, J. Cernicharo, J. R. Pardo, M. Agúndez, C. Cabezas, B. Tercero, M. Guélin and M. C. McCarthy, Astrophys. J., Lett., 2024, 966, L28.
17. G. Wang, X. Huang, C. Guo, H. Zhang, C. Zhang and X. Cheng, J. Phys. Chem. A, 2025, 129.
18. T. H. Dunning, J. Chem. Phys., 1989, 90, 1007–1023.
19. J. Thom, H. Dunning and L. T. Xu, J. Phys. Chem. A, 2024, 128.
20. H. Hendaoui and B. Mehnen, Mon. Not. R. Astron. Soc., 2025, staf534.
21. D. Benson, Y. Li, W. Luo, R. Ahuja, G. Svensson and U. Häussermann, Inorg. Chem., 2013, 52, 6402–6406.
22. S.-L. Yang, J. A. Sobota, C. A. Howard, C. J. Pickard, M. Hashimoto, D. H. Lu, S.-K. Mo, P. S. Kirchmann and Z.-X. Shen, Nat. Commun., 2014, 5, 3493.
23. Y.-L. Li, S.-N. Wang, A. R. Oganov, H. Gou, J. S. Smith and T. A. Strobel, Nat. Commun., 2015, 6, 6974.
24. J. Manz, R. Meyer, E. Pollak and J. Römelt, Chem. Phys. Lett., 1982, 93, 184–187.
25. J. Manz, R. Meyer and J. Römelt, Chem. Phys. Lett., 1983, 96, 607–612.
26. R. Ditchfield, W. J. Hehre and J. A. Pople, J. Chem. Phys., 1971, 54, 724–728.
27. P. R. Taylor, Lecture Notes in Chemistry, 1992, vol. 58, ch. 1, p. 2.
28. A. J. C. Varandas, Phys. Chem. Chem. Phys., 2018, 20, 22084–22098.
29. A. J. C. Varandas, Annu. Rev. Phys. Chem., 2018, 69, 177–203.
30. A. J. C. Varandas, J. Theor. Comput. Chem., 2020, 19, 2030001.
31. A. J. C. Varandas, Phys. Chem. Chem. Phys., 2021, 23, 9571–9584.
32. K. Raghavachari, G. W. Trucks, J. A. Pople and M. Head-Gordon, Chem. Phys. Lett., 1989, 157, 479–483.
33. P. Piecuch, J. R. Gour and M. Włoch, Int. J. Quantum Chem., 2009, 109, 3268–3304.
34. A. D. Becke, J. Chem. Phys., 1993, 98, 5648–5652.
35. P. J. Stephens, F. J. Devlin, C. F. Chabalowski and M. J. Frisch, J. Phys. Chem., 1994, 98(45), 11623–11627.
36. C. Lee, W. Yang and R. G. Parr, Phys. Rev. B: Condens. Matter Mater. Phys., 1988, 37, 785–789.
37. M. W. H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, M. Schütz, P. Celani, W. Györffy, D. Kats, T. Korona, R. Lindh, A. Mitrushenkov, G. Rauhut, K. R. Shamasundar, T. B. Adler, R. D. Amos, S. J. Bennie, A. Bernhardsson, A. Berning, D. L. Cooper and M. J. O. Deegan, MOLPRO, version 2010.1, a package of ab initio programs, 2019.
38. Y. Zhao and D. G. Truhlar, J. Phys. Chem. A, 2006, 110, 13126–13130.
39. Y. Zhao and D. G. Truhlar, Theor. Chem. Acc., 2008, 120, 215–241.
40. S. Huzinaga, J. Andzelm, E. Radzio-Andzelm, Y. Sakai, H. Tatewaki and M. Klobukowski, Gaussian Basis Sets for Molecular Calculations, Elsevier, 1983.
41. S. Grimme, A. Hansen, J. G. Brandenburg and C. Bannwarth, Chem. Rev., 2016, 116, 5105–5154.
42. J. Koput and K. A. Peterson, J. Phys. Chem. A, 2002, 106, 9595–9599.
43. F. Neese, Wiley Interdiscip. Rev.: Comput. Mol. Sci., 2012, 2, 73–78.
44. F. Neese, Wiley Interdiscip. Rev.: Comput. Mol. Sci., 2018, 8, e1327.
45. F. Neese, F. Wennmohs, U. Becker and C. Riplinger, J. Chem. Phys., 2020, 152, 224108.
46. W. J. Hehre, R. F. Stewart and J. A. Pople, J. Chem. Phys., 1969, 51, 2657–2664.
47. K. W. Hinkle, J. J. Keady and P. F. Bernath, Science, 1988, 241, 1319–1322.
48. S. Saha and C. M. Western, J. Chem. Phys., 2006, 125, 224307.
49. C. M. R. Rocha and A. J. C. Varandas, J. Chem. Phys., 2016, 144, 064309.
50. J. N. Murrell, S. Carter, S. C. Farantos, P. Huxley and A. J. C. Varandas, Molecular Potential Energy Functions, Wiley, Chichester, 1984.
51. L. V. Vilkov, V. S. Mastryukov and N. I. Sadova, Determination of the geometrical structure of free molecules, Mir Publishers, USSR, 1983.
52. I. Hargittai and B. Chamberland, Comput. Math. with Appl., 1986, 12, 1021–1038.
53. A. Pribram-Jones, D. A. Gross and K. Burke, Annu. Rev. Phys. Chem., 2015, 66, 283–304.
54. R. O. Jones, Rev. Mod. Phys., 2015, 87, 897.
55. S. Hammes-Schiffer, Science, 2017, 355, 28–29.
56. M. G. Medvedev, I. S. Bushmarinov, J. Sun, J. P. Perdew and K. A. Lyssenko, Science, 2017, 355, aah5975.
57. R. Peverati and D. G. Truhlar, Philos. Trans. R. Soc., A, 2014, 372, 20120476.
58. N. Mardirossian and M. Head-Gordon, J. Chem. Theory Comput., 2016, 12, 4303–4325.
59. T. Torelli and L. Mitas, Phys. Rev. Lett., 2000, 85, 1702–1705.
60. C. Neiss, E. Trushin and A. Görling, ChemPhysChem, 2014, 15, 2497–2502.
61. H. J. Kulik, N. Seelam, B. D. Mar and T. J. Martínez, J. Phys. Chem. A, 2016, 120, 5939–5949.
62. J. Hostaš and J. Řezáč, J. Chem. Theory Comput., 2017, 13, 3575–3585.
63. C. Bannwarth, S. Ehlert and S. Grimme, J. Chem. Theory Comput., 2019, 15, 1652–1671.
64. M. Gubler, M. R. Schäfer, J. Behler and S. Goedecker, J. Chem. Phys., 2025, 162, 094103.
65. A. Hesselmann, H.-J. Werner and P. J. Knowles, J. Chem. Phys., 2022, 157, 234106.
66. A. J. C. Varandas, Adv. Chem. Phys., 1988, 74, 255–338.
67. G. Herzberg, Molecular Spectra and Molecular Structure. III Electronic Spectra and Electronic Structure of Polyatomic Molecules, Van Nostrand, New York, 1966.
68. B. P. Pritchard, D. Altarawy, B. Didier, T. D. Gibson and T. L. Windus, J. Chem. Inf. Model., 2019, 59, 4814–4820.
69. R. A. Kendall, T. H. Dunning, Jr. and R. J. Harrison, J. Chem. Phys., 1992, 96, 6796–6806.
70. C. Bannwarth, E. Caldeweyher, S. Ehlert, A. Hansen, P. Pracht, J. Seibert, S. Spicher and S. Grimme, Wiley Interdiscip. Rev.: Comput. Mol. Sci., 2021, 11, e1493.
71. F. Jensen, Introduction to Computational Chemistry, Wiley, 3rd edn, 2006.
72. A. Gołebiewski, Trans. Faraday Soc., 1961, 57, 1849–1853.
73. J. N. Murrell and A. J. C. Varandas, Mol. Phys., 1975, 30, 223–236.
74. R. Sure, A. Hansen, P. Schwerdtfeger and S. Grimme, Phys. Chem. Chem. Phys., 2017, 19, 14296–14305.

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