Introduction

There is a growing interest in organogermanium chemistry [1,2,3,4,5,6,7,8,9,10,11,12,13]. Knowledge of the chemistry of organogermanes has significantly improved in recent years due to the critical role of germanium and its organic derivatives in electronics and other applications [14,15,16]. This rapid development of organogermanium chemistry has led to the increased production and utilization of elemental germanium and its derivatives, which in turn has reduced the prices of those materials [17].

The term “germylene” refers to a neutral species that possesses a divalent germanium atom with six electrons in its valence shell [18]. Germylenes are used during the manufacture of components of nanomaterials and as precursors in the low-temperature synthesis of Ge-rich semiconductors via chemical vapor deposition [1]. Germylenes prefer to keep their nonbonding electrons in atomic orbitals with a high percentage of s character [19]. Hence, the covalent bonds to the two groups next to the divalent center often have greater p character than the corresponding carbene and silylene bonds, which accounts for the smaller bond angles seen in germylenes than in carbenes and silylenes [20].

Due to the importance of triplet ground-state germylene radicals in chemical vapor deposition, semiconductor manufacturing, and the photonics and aerospace industries [21], the preparation of these species has become one of the most important topics in modern organogermanium chemistry [19, 22,23,24,25,26]. It is possible to calculate the IR and NMR spectra of these species, although this is beyond the scope of the work presented here. Recently, some studies have suggested that EPR and X-ray crystallography data provide unambiguous evidence for triplet ground states of silylenes and germylenes [27, 28].

Aspects such as the electronic effects of different substituents can influence the ΔES–T (singlet–triplet energy difference) and ΔEHOMO–LUMO (HUMO–LUMO energy difference) values of divalent species and hence the sizes of their nonbonding orbitals. It has been reported that electronegative groups increase ΔES–T and ΔEHOMO–LUMO, whereas electropositive ones decrease them [13, 22, 23, 29, 41]. However, it is not possible to swiftly switch germylenes from singlet to triplet states by altering their substituents, in contrast to their carbene analogs [19, 30]. Following our quest for stable triplet germylenes containing two different metal substituents from groups 1 and 2 of the periodic table [32], here we report the results of our theoretical investigation of the singlet (s) and triplet (t) structures of 96 different germylenes divided into four families: H-Ge-M2-M3 (I), Li-Ge-M2-M3 (II), Na-Ge-M2-M3 (III), and K-Ge-M2-M3 (IV) (M2 = Be, Mg, Ca; M3 = H, F, Cl, Br) (Scheme 1).

Scheme 1
scheme 1

Schematic of the germylenes scrutinized in this study

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Computational methods

Our computational study is confined to B3LYP calculations due to its excellent performance-to-cost ratio as compared with correlated wavefunction theory [31,32,33,34]. While some recent reports have questioned the reliability of the most popular density functional, B3LYP [35], we used B3LYP with the 6–311++G** basis set as the method of choice in the present work, in common with many other papers on germylenes [23, 36,37,38,39,40]. Triplet states were calculated using the unrestricted broken spin-symmetry UB3LYP/6–311++G** method implemented in the GAMMES software package [41, 42]. To obtain more accurate data on energetics, single-point calculations were performed at the QCISD(T) and CCSD(T) levels of theory using the 6–311++G** basis set [43]. Frequency calculations were carried out to characterize the resulting structures as minima (the number of imaginary frequencies (NIMAG) = 0) or transition states (NIMAG = 1) [44]. Density functional calculations with the B3LYP functional were implemented to compute the stability of the germylenes through appropriate isodesmic reactions [45].

Results and discussion

During our continued search for rare triplet ground-state heavy divalents [19, 20, 22, 23], we compared and contrasted the thermodynamic and geometrical parameters for 96 new germylenes of formula M1-Ge-M2-M3 at the B3LYP/6–311++G**, CCSD(T)/6–311++G**, and QCISD(T)/6–311++G** levels of theory, where M1 = H, Li, Na, K; M2 = Be, Mg, Ca; M3 = H, F, Cl, Br (see Fig. S1 in the “Electronic supplementary material,” ESM). Here we illustrate how bond lengths, bond angles, symmetries, and interatomic interactions vary from one germylene to another, and more specifically from a singlet germylene (1 s –48 s ) to its corresponding triplet state (1 t –48 t ). We compare their relative energies (calculated at three levels of theory) along with their B3LYP/6–311++G**-computed dipole moments (D) and their vibrational zero-point energies in Table 1. We show the frontier molecular orbital energies (HOMO and LUMO) for singlet M1-Ge-M2-M3 germylenes along with their band gaps (ΔΕHOMO–LUMO) and relative stabilities (ΔΕS–T) in Table 2.

Table 1 Thermodynamic parameters for singlet (s) and triplet (t) states of M1-Ge-M2-M3 (M1 = H, Li, Na, K; M2 = Be, Mg, Ca; M3 = H, F, Cl, Br) germylenes, including relative energies (kcal/mol, calculated at three levels of theory), their B3LYP/6–311++G**-computed dipole moments (D), and their vibrational zero-point energies (kcal/mol)
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Table 2 Frontier molecular orbital energies (HOMO and LUMO in eV) for singlet M1-Ge-M2-M3 germylenes (M1 = H, Li, Na, K; M2 = Be, Mg, Ca; M3 = H, F, Cl, Br) along with their ΔΕHOMO–LUMO (eV) and ΔΕS–T (kcal/mol) values calculated at the B3LYP/6–311++G** level
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We also employed appropriate isodesmic reactions to determine the relative thermal energies for singlet (ΔES) and triplet (ΔET) germylenes and the thermal energy difference between each singlet and its corresponding triple (ΔΔES–T) at the B3LYP/6–311++G** level (see Table 3). Data for the hydrogenation of germylenes (Scheme 1) along with heats of hydrogenation (ΔHH) are given for singlet M1-Ge-M2-M3 species in Table 4. Finally, the B3LYP/6–311++G**-calculated natural bond orbitals (NBOs) of the M1-Ge-M2-M3 germylenes are provided in Table 5. Computed harmonic frequencies are omitted here to save space, but they are available upon request. Our force constant calculations show that only 33 s has one imaginary frequency and exists as a transition state. Among our 96 germylene isomers, only two singlet structures, 47 s (K-Ge-Ca-Cl) and 48 s (K-Ge-Ca-Br), undergo rearrangement upon optimization, forming rather long linkages from K to Ca (3.88 Å and 3.92 Å, respectively) and transforming into cyclic structures (Fig. S1 in the ESM).

Table 3 Isodesmic reactions showing the relative thermal energies of singlet (ΔES) and triplet (ΔET) germylenes M1-Ge-M2-M3, (M1 = H, Li, Na, K; M2 = Be, Mg, Ca; M3 = H, F, Cl, Br) as well as the ΔES–T values for the germylenes, all in kcal/mol and calculated at the B3LYP/6–311++G** level
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Table 4 Heats of hydrogenation (∆HHa) for singlet M1-Ge-M2-M3 (M1 = H, Li, Na, K; M2 = Be, Mg, Ca; M3 = H, F, Cl, Br) in kcal/mol, as calculated at the B3LYP/6–311++G** level
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Table 5 The B3LYP/6–311++G**-calculated NBO atomic charges on M1-Ge-M2-M3 (M1 = H, Li, Na, K; M2 = Be, Mg, Ca; M3 = H, F, Cl, Br) germylenes
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This phenomenon is consistent with the convergence reported for XGeCBr3 and XGeCCl3 molecules (X = H, F, Cl, Br, I) to unexpected structures when the same DFT functionals were used [45].

To ensure that the cyclized structures 47 s and 48 s are not artifacts of the particular level of computation and basis set used, they were optimized at two levels of theory (see Table S1 in the ESM). Consequently, the relative stabilities of our 96 novel germylenes were compared and contrasted at the B3LYP/6–311++G**, CCSD(T)/6–311++G**, and QCISD(T)/6–311++G** levels of theory. The trends in the singlet–triplet energy gaps computed using the high-level QCISD(T) and CCSD(T) single-point energies are similar to that computed using the B3LYP-optimized energy. For example, the structures with M1 = Li can be ordered according to their singlet–triplet energy gaps calculated at the B3LYP/6–311++G**, QCISD(T)/6–311++G**, and CCSD(T)/6–311++G** levels of theory (respectively) as follows: ΔES–T (5) (−9.11, −9.62, −9.63 kcal/mol) > ΔES–T (6) (−8.18, −8.40, −8.41 kcal/mol) > ΔES–T (7) (−2.48, −3.88, −3.83 kcal/mol).

The B3LYP/6–311++G** results as well as those gained using QCISD(T)/6–311++G** show that structures with M1 = H have a stable singlet ground state (except for 34). Although a more electronegative substituent is generally considered to stabilize singlet states more than their corresponding triplet states, we actually find here that M3 = F leads to more stable triplets than those obtained when M3 = Cl or Br. Natural bond orbital (NBO) analysis shows that among all the germylenes investigated here, only 34 t presents a LPF → σ*Ge-H interaction, which stabilizes its triplet state (Table 1). The main difference between 34 t , 35 t , and 36 t is their H–Ge–Ca angles (Fig. S1 in the ESM). In 34 t (with M3 = F), the H–Ge–Ca angle is smaller than the corresponding angles in 35 t and 36 t (with M3 = Cl and Br, respectively), which increases the probability of a LPF → σ*Ge-H interaction.

Ordering the structures with M1 = Li according to their B3LYP/6–311++G**-calculated and QCISD(T)/6–311++G**-calculated singlet–triplet energy gaps yields ΔES–T (5) (−9.11, −9.62 kcal/mol) > ΔES–T (6) (−8.18, −8.40 kcal/mol) > ΔES–T (7) (−2.48, −3.88 kcal/mol) > ΔES–T (8) (−2.48, −3.83 kcal/mol); ΔES–T (21) (−8.72, −9.64 kcal/mol) > ΔES–T (22) (−2.94, −7.91 kcal/mol) > ΔES–T (23) (−2.79, −5.12 kcal/mol) > ΔES–T (24) (−2.77, −5.07 kcal/mol); and ΔES–T (37) (−11.42, −11.54 kcal/mol) > ΔES–T (38) (−10.48, −11.25 kcal/mol) > ΔES–T (39) (−4.69, −6.44 kcal/mol) > ΔES–T (40) (−4.66, −6.05 kcal/mol). Evidently, the structures with the most electronegative substituents have the smallest singlet–triplet energy gaps. This may be because the presence of F or H rather than Cl or Br at M2 (LPF, H → σ*Ge-M2) leads to stronger hyperconjugation. We know that F− is a better base than Cl− and Br−. Clearly, the presence of a fluorine substituent destabilizes σ orbitals at M2, meaning that triplets with fluorine substituents are more stable than triplets with chlorine and bromine substituents. In other words, all the odds are stacked in favor of the existence of triplet germylenes. Apparently, in structures with M1 = Na, the presence of fluorine has an extraordinary effect on ΔES–T (26), which may be due to the substantial difference in orbital size between Na and Mg. On the other hand, Na and Mg are in the same period, and their ionic radii are fairly similar (Na+ = 1.16 Å, Mg2+ = 0.85 Å) [46], which may explain the stability of ΔES–T (6). Clearly, the compounds with the most electropositive M1 substituent (i.e., K, rather than H, Li, or Na) have the most stable triplets. The singlet–triplet energy gaps for the K-Ge-M2-M3 germylenes, calculated at the B3LYP/6–311++G** and QCISD(T)/6–311++G** levels of theory, can be ordered as follows: ΔES–T (13) (−10.15, −11.72 kcal/mol) > ΔES–T (14) (−9.58, −11.53 kcal/mol) > ΔES–T (16) (−7.14, −9.65 kcal/mol) > ΔES–T (15) (−6.71, −8.76 kcal/mol); ΔES–T (29) (−10.15, −11.98 kcal/mol) > ΔES–T (30) (−9.46, −11.34 kcal/mol) > ΔES–T (31) (−9.44, −11.30 kcal/mol) > ΔES–T (32) (−9.40, −11.20 kcal/mol); and ΔES–T (45) (−12.74, −13.46 kcal/mol) > ΔES–T (46) (−11.93, −13.60 kcal/mol) > ΔES–T (47) (−7.24, −8.22 kcal/mol) > ΔES–T (48) (−2.08, −6.89 kcal/mol). The stabilities of 38 t , 42 t , and 46 t compared to their corresponding singlet states can be explained by the electropositivity of the substituted divalent Ge atom with M3 = Ca.

When comparing germylenes with different β-substituents, the four structures 45 (−12.74 kcal/mol), 46 (−11.93 kcal/mol), 31 (−9.44 kcal/mol), and 32 (−9.40 kcal/mol) are found to possess higher triplet stabilities than the other structures. This can be attributed to the high electropositivity of potassium.

The LUMO–HOMO energy gap (ΔEHOMO–LUMO) varies as a function of M3 as follows: F ≥ H ≥ Cl ≥ Br (see Table 2). This trend demonstrates that singlet germylenes become increasingly stable as the electronegativity of the halogen atom increases. In addition, inspection of the calculated ΔEHOMO–LUMO values for all of the singlet germylenes seems to suggest that the value of this parameter is dictated by the size of the angle formed by the two substituents bound to the central germanium atom. The magnitude of this bond angle is also one of the main influences on the value of ΔES–T [45]. Except for two triplets, the Li–Ge–M2 angles of corresponding singlets and triplets show negligible variation as a function of M3, and this angle is bent in all cases. In almost all of the singlet Li-Ge-M2-M3 compounds, the Ge–M2–M3 bond angle trend as a function of the M3 atom is: Br > Cl > F > H (Fig. S1 in the ESM).

However, the divalent angle (i.e., the angle M1–Ge–M2) in 6 t and that in 38 t are both about 180°, and both of these structures are linear. In all compounds of formula Na-Ge-M2-M3, the Na–Ge–M2 bond angle is bent to a degree that is (except in the case of one triplet) a function of the electronegativity of M3: F > Cl > Br > H. In the main, eight of the twelve triplet divalent angles in the K-Ge-M2 structures are linear, which confirms the stability of the triplet states for these structures with M1 = K relative to the others. Although the Ge–M2 bond length does vary significantly across this series of structures, the M2–M3 bond lengthens notably as the size of the halogen substituent increases (Br > Cl > F > H).

Interestingly, the structure 6 t belongs to the C∞v point group, while all the other species belong to C1 (Fig. S1 in the ESM).

Isodesmic reactions (Table 3) were employed to evaluate the electronic and thermal enthalpies of germylenes with different substituents. The results suggest that triplet germylenes with an electronegative substituent (M3 = F) are slightly more stable than the corresponding triplet germylenes with a less electronegative substituent (M3 = H). Conversely, in the triplet K-Ge-M2-M3 structures, as the electronegativity of the halogen decreases, the triplet germylene gradually becomes more stable.

Accordingly, the reactions of singlet M1-Ge-M2-M3 structures with H2 afford the dihydridogermane shown in Scheme 2.

Scheme 2
scheme 2

Hydrogenation of germylenes of formula M1-Ge-M2-M3 (M1 = H, Li, Na, K; M2 = Be, Mg, Ca; M3 = H, F, Cl, Br)

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Strikingly, 40 s , 44 s , and 46 s are the most capable of activating dihydrogen, likely due to their relatively small singlet–triplet energy gaps. In close analogy, every singlet germylene can also undergo intramolecular H–H bond activation [36] (Table 4).

Finally, the NBO atomic charges were computed for the singlet and triplet states of the germylene species (Table 5). Charges on all the triplet germylenes are less than those of their corresponding singlet species. The germanium atoms in the singlets tend to keep their nonbonding electrons in atomic orbitals with high s character. Consequently, electropositive substituents transfer charge from the corresponding Ge–M1 and Ge–M2 bonding orbitals with high p character to the partially populated s-type orbital on the Ge atom. Clearly, the particular halogen present influences the NBO atomic charge of the triplet Ge atom such that the charge on Ge varies as M1-Ge-M2-F > M1-Ge-M2-Cl > M1-Ge-M2-Br, which illustrates the effect of the substituent electronegativity on the stability of triplet germylenes.

Conclusions

The effects of several metal substituents from the first and second groups of the periodic table on the singlet–triplet energy gaps and multiplicity of divalent germylenes of formula M1-Ge-M2-M3 were explored by performing calculations at the B3LYP/6–311++G**, QCISD(T)/6–311++G** and CCSD(T)/6–311++G** levels of theory. The results indicated that the presence of electropositive substituents significantly reduces the excitation energy and singlet–triplet energy gap, which allows a triplet ground state to be obtained at a reasonable bond angle. The singlet–triplet energy gap (ΔES–T) varies as a function of M3 as follows: H ≥ F ≥ Cl ≥ Br, except when M1 = H, in which case the trend becomes F ≥ Cl ≥ Br ≥ H. Our calculations indicate that K-Ge-Ca-M3 shows the most promise as a candidate ground-state triplet germylene due to its small singlet–triplet energy gap and narrow band gap (ΔEHOMO–LUMO). the trend of ΔEHOMO–LUMO as a function of M3 is: F ≥ H ≥ Cl ≥ Br. An inspection of NBO atomic charges highlighted the influence of electronegative substituents on triplet germylene stability, which varied as a function of M3 is: M1-Ge-M2-F > M1-Ge-M2-Cl > M1-Ge-M2-Br.