Geometry, Electronic Structure, and Pseudo Jahn-Teller Effect in Tetrasilacyclobutadiene Analogues

We revealed the origin of the structural features of a series of tetrasilacyclobutadiene analogues based on a detailed study of their electronic structure and the pseudo Jahn-Teller effect (PJTE). Starting with the D4h symmetry of the Si4R4 system with a square four-membered silicon ring as a reference geometry, and employing ab initio calculations of energy profiles along lower-symmetry nuclear displacements in the ground and several excited states, we show that the ground-state boat-like and chair-like equilibrium configurations are produced by the PJT interaction with appropriate excited sates. For Si4F4 a full two-mode b1g−b2g adiabatic potential energy surface is calculated showing explicitly the way of transformation from the unstable D4h geometry to the two equilibrium C2h configurations via the D2h saddle point. The PJTE origin of these structural features is confirmed also by estimates of the vibronic coupling parameters. For Si4R4 with large substituents the origin of their structure is revealed by analyzing the PJT interaction between the frontier molecular orbitals. The preferred chair-like structures of Si4R4 analogues with amido substituents, and heavier germanium-containing systems Ge4R4 (potential precursors for semiconducting materials) are predicted.

The origin of this variety of the molecular geometries in the Si 4 R 4 and Ge 4 R 4 series can be rationalized by employing the vibronic coupling theory in the form of the pseudo Jahn-Teller effect (PJTE) 17,18 . This statement follows from a more general conclusion that the Jahn-Teller effect for degenerate electronic states and the PJTE for both degenerate and nondegenerate (pseudodegenerate) states are the only source of spontaneous symmetry breaking in molecular systems and solids (see references in refs 17-19). In our case of nondegenerate electronic states the PJTE provides a reasonable picture of the structure and properties of the compounds under consideration. The method has been successfully applied to more simple carbon and silicon four-membered ring systems, such as C 4 H 4 20 , C 4 F 4 7 , Si 4 21 , Si 4 H 4 2+ 22 , as well as to a variety of other molecular systems (see 18,[23][24][25][26][27][28][29][30] and references therein).
In this paper we report the results of a detailed analysis of the origin of the structural features of a series tetrasilacyclobutadiene analogues, Si 4 R 4 , including relatively large substituents R, as well as some other related systems, including Ge 4 R 4 , based on their electronic structure and vibronic coupling. Applying the PJTE theory, we reveal the lowest excited states that cause the distortion (puckering) of the high-symmetry planar configuration and estimate the vibronic coupling constants that control these distortions. This information provides also some clues for manipulation of the structure by means of external perturbations or substitutions, similar to the recently suggested methods of restoring planarity in puckered hexa-and tetra-heterocyclic systems [31][32][33] , thus inspiring the search of new materials with desired properties, in particular, other kinds of stable tetrasilacyclobutadiene analogues. Our theoretical prediction of stable structures of Si 4 R 4 analogues with different substituents and Ge 4 R 4 with bulky substituents is expected to provide also information for synthesizing new silicon or germanium four-member ring compounds and exploring their potential applications as semiconducting materials.

Results and Discussion
PJTE in the origin of Si 4 R 4 geometries. There are two typical equilibrium structures of Si 4 R 4 analogues, the one with a puckered Si 4 ring as in Si 4 H 4 , which hereafter is denoted as a "boat-like" structure, and the other one with a planar-rhombic Si 4 ring and alternating pyramidal and planar bonded substituents at the silicon atoms, which is denoted as "chair-like" structure hereafter (Fig. 2). To rationalize the origin of these boat-like and chairlike structures of Si 4 R 4 analogues we start with one of the simpler representatives, the Si 4 F 4 molecule; it allows revealing the main mechanisms of formation of boat-like and chair-like structures that are similar for all the Si 4 R 4 systems.
According to the general procedure, the PJTE formulation in this case starts from the highest symmetry D 4h configuration at which the molecule is square-planar with the central Si 4 ring forming a square and the four ligands symmetrically bonded. Its transformations to the lower symmetry configurations is realized by symmetrized b 2u type displacements toward the boat-like geometry and via combined b 1g + b 2g displacements toward the chair-like configuration (Fig. 2). [Hereafter we employ small letters to denote the symmetry representations of both vibrational modes and molecular orbitals, and capital letters for electronic states]. Since the ground state in D 4h geometry is nondegenerate, according to the general theorem 17,18 the instabilities are induced by the pseudo Jahn-Teller (PJT) coupling to appropriate excited electronic states. To reveal the latter we calculate and analyze the energy profiles (the cross section of the adiabatic potential energy surface (APES)) of the electronic ground and low-lying excited states along these normal displacements. The results are shown in Fig. 3.  Consider first the instability along the b 2u coordinate Q b2u of molecular puckering that leads to the boat-like structure with D 2d symmetry. Note that Q denotes normal coordinates, not atomic displacements. As seen from Fig. 3(a), along b 2u the D 4h configuration in the singlet ground state 1 1 B 2g is unstable and distorts spontaneously up to the minimum point at Q = 0.7Å which is the boat-like structure of Si 4 F 4 of Fig. 2 (the role of the triplet state is outlined below). According to the group theory rules the lowest excited states that couple with 1 1 B 2g via the b 2u mode in the D 4h symmetry are the 1 1 A 1u and 2 1 A 1u states, realizing the 2u PJTE problem (the role of higher excited states is discussed in next section). Hence the formation of the boat-like structure in this system is due to the vibronic coupling with the excited 1 A 1u states. From the estimated vibronic coupling constants below it follows that the main contribution comes from the 1 1 A 1u state. By comparing the electronic structures we can see that the 1 1 B 2g state comes from the e g 2 electronic configuration, while the 1 1 A 1u state emerges from the electronic excitations of an e g electron to the empty e u orbital. The latter thus play an important role in the formation of boat-like structures.
The analysis of possible distortions along b 1g and b 2g is more complicated. From Fig. 3(b) we see that there is a hidden PJTE (meaning a sufficiently strong PJT interaction between two or more excited states that produces an additional global minimum with a distorted configuration, see ref. 29 for details): the PJT mixing of the three close excited states, 1g , produces an additional potential minimum at Q b1g = 1.1Å in which the Si 4 framework is rhombically distorted. On the other hand, the ground state 1 A g is unstable with respect to the b 2g puckering, and it is a priori unclear where the global minimum might occur. This prompted further calculations of the two-dimensional part of the APES along two coordinates b 1g and b 2g . The results are illustrated in Fig. 4. We see that, indeed, the minimum in the energy profile along b 1g is just a saddle point, from which the system descends along b 2g to the global minima of C 2h symmetry with a chair-like structure of Fig. 2. The normal coordinates (Q b1g , Q b2g ) of these two minima (read off the D 4h point) in Å are about (0.74, ± 0.62). To reveal the excited states involved in these PJTE distortions we calculated the energy profiles of the system along b 2g (shown in Fig. 3(c)) beginning from the point Q b1g = 0.74 Å of Fig. 3 The results shown in Fig. 4 together with the energy profiles in Fig. 3 illustrate the full picture of instabilities and distorted equilibrium-geometry formation of the Si 4 F 4 molecule. In the highest symmetry configuration D 4h the system is unstable along b 1g (rhombic distortion of the Si 4 ring) due to the hidden PJTE problem 1g mixing two excited electronic states, followed by the puckering distortion b 2g (emerging from the resulting in an equilibrium chair-like structure at the minima of the APES (Fig. 2). On the other hand, as shown above, the APES of this system has another minimum along the b 2u displacements producing the boat-like equilibrium structure. However, because the two excited states 1 1 A 1g and 1 1 B 1g in the D 4h geometry are relatively very close, the hidden PJTE of their mixing along b 1g is much stronger than that of the 2u problem, so the chair-like structure is lower in energy by 0.39 eV than the boat-like structure. The fully optimized chair-like and boat-like Si 4 F 4 geometrical coordinates are given in the Supplementary data S1. By examining the 1 1 B 1g excited state in the ab initio calculations we can see that dominant electronic configurations are produced by the electronic transitions from the occupied e g to b 1g or a 1g empty molecular orbitals. The significant role of b 1g orbital in these interactions is discussed below in more detail.
Note that the triplet 3 A 2g ground state in the D 4h geometry is not significant with respect to observable structural features of Si 4 F 4 because it is stable with respect to b 1g distortions that lead to the global minima with a singlet electronic state ( Fig. 3(b)), at which point 3 A 2g is an excited state; its instability in D 4h along the b 2u mode leads to approximately the same boat-like minimum as in the singlet 1 1 B 2g state ( Fig. 3(a)) which, as shown above, is much higher in energy than the chair-like minimum.
Numerical estimation of the PJTE coupling parameters. The qualitative picture, which reveals the excited electronic states producing the instability of the ground state, obtained in the previous section, can be enhanced quantitatively by estimates of the numerical values of the vibronic coupling parameters in the PJTE. First of all such numerical estimates are important to limit the number of excited states that produce significant contribution to the instability in a given direction. Indeed, in any polyatomic system the number of excited states is practically infinite, and for any given low-symmetry displacement Q there are always some excited states of relevant symmetry that contribute to the destabilization of the ground state. In general, this contribution is rather small, just lowering the curvature of the ground state from K 0 to K 0 − p, p > 0, and not producing instability, meaning K 0 − p > 0, (according to the general theory 17 the primary force constant without the vibronic coupling, K 0 > 0), and only low-lying excited states with sufficiently large contribution to the PJT destabilization of the ground state may be the reason of instability, and only in certain directions Q.
Assuming that there is only one such excited state producing instability in the Q direction, we come to the two-level PJTE problem. The primary force constants in the ground and the active excited state is denoted by K 1 and K 2 , respectively, and the second order perturbation corrections to them from all the higher excited states of appropriate symmetry is defined by p 1 and p 2 in the PJTE energy matrix elements W ΓΓ and Then, the APES ε (Q) can be obtained from the secular equation of the perturbation theory: where K 1 , K 2 , p 1 and p 2 are the primary force constants as explained above, F is the PJT vibronic coupling constant between the two states, and Δ is the energy gap of the two coupled states. The roots of this secular equation are: At small values of Q we get for the ground state: Then the condition of instability at Q = 0 of the ground state due to the PJT vibronic coupling is: We thus separated all the excited states that destabilize the ground state via the PJTE in two parts: (1) all the higher states with small contributions taken into account by means of a second order perturbation correction p to the curvature K 0 , and (2) the lowest active excited state acting directly via the secular equation lowering the . This means that if the chosen excited state is indeed the one responsible for the instability the numerical estimates should yield K 1 − p 1 > 0 and The direct ab initio calculation of these constants encounters difficulties (their calculation for some systems see in ref. 23). In the present paper we estimated them by fitting the solutions of the secular equation (4) to the ab initio data for the energy profiles of the corresponding states. For the b 2u distortion the PJTE two-level problem 2u yields satisfactory results listed in Table 1. The numerical value of K 1 − p 1 − ∆ 2F 2 is − 1.67, which satisfies the condition of instability (6); the contribution of all the other active excited states is included in the second-order perturbation correction p 1 . We checked the influence of the nearest one 2 1 A 1u : its inclusion into the secular equation (7) below yields a very small contribution to the instability, thus justifying the two-level PJTE formulation for this case.
However, for the b 1g and b 2g instabilities only one active excited state does not yield positive K 1 − p 1 values, meaning higher excited electronic states should be included in the secular equation of direct PJT interaction. For a three-level problem the secular equation is: where in addition to the above denotations K 3 is the primary force constant for the third term, p 3 is the second order perturbation correction and G is the additional vibronic coupling constant to this term. Accordingly, the PJTE producing these distortions are 1 g 2 g , and the excited electronic states controlling these instabilities are 1 1 B 1g and 2 1 A 1g for the b 1g rhombic distortions, and 1 1 B 2g and 2 1 A g for further b 2g puckering displacements in the rhombic configuration. The estimated PJTE constants are given in Table 1. The numerical results were evaluated by a fitting procedure with very small standard deviations of the residuals and the Pearson's correlation coefficients equal to 1. The negative values of K-p in the excited states show that they are strongly influenced by the PJT coupling with higher electronic states.
Extension to Si 4 R 4 compounds with bulky substituents. The description of the origin of the structural features of Si 4 F 4 as due to the PJTE may serve as a prototype for the investigation of Si 4 R 4 analogues with more complicated substituents, such as the mentioned above Si 4 (EMind) 4 . However, the larger size systems possess many close-in-energy electronic states and normal modes (due to much lower molecular symmetry than D 4h ) which make a full analysis of the structural features in terms of symmetry adapted electronic states for the system as a whole much more difficult. In this case (as in many similar chemical problems) a more simple description can be achieved by considering the PJTE in terms of frontier molecular orbitals (MO) (obtained from the electronic structure calculations) which may happen to be localized in a much reduced "active center", as in the case under consideration. Figure 5 shows the highest occupied MO (HOMO), the next lower occupied MO (HOMO-1), and lowest unoccupied MO (LUMO) in the chair-like minima configurations of a series of Si 4 R 4 with R as fluorine, phenyl, tetramethyl-phenyl, s-indacene, and EMind groups. We see that the electronic distributions of these frontier molecular orbitals are located mostly around the Si 4 ring which is thus the active center of the whole molecule. In other words, the bulky substituents reduce the formal molecular point group, but play a minor role in the key electronic and vibronic properties of active center of Si 4 R 4 that control their geometry via the PJTE. Therefore we can use the PJTE formulations and results obtained for Si 4 F 4 with the D 4h high-symmetry configuration as a reference in exploring Si 4 R 4 systems with bulky substituents. In the HOMO the electronic cloud is mainly on the σ bonding orbitals between two silicon atoms (to form the short diagonal), and non-bonding orbitals from the other two pyramidal silicon atoms (at the virtual long diagonal), with less charged density at the neighbor F or C atoms, and this HOMO electronic distribution is almost the same for all the Si 4 R 4 systems.
Based on the same Si 4 ring active center and the same HOMO electronic distributions of the Si 4 R 4 derivatives, we come to the prediction that they have the same kind of PJTE origin. By tracing this orbital distribution in the lowest 1 A g singlet electronic state along b 1g and b 2g distortions in Si 4 F 4 (Fig. 6) we found that the HOMO actually originates from the empty b 1g MO of the undistorted D 4h configuration; it becomes a g after distortion. As follows from the numerical data of the ab initio calculations illustrated in Fig. 6(a), the electronic distribution in this b 1g MO is mainly on the four σ * Si−Si anti-bonding orbitals; under the b 1g distortion it gradually becomes bonding σ Si−Si for two diagonal silicon atoms, and non-bonding MO for the other two silicon atoms. Then along the Mode K 1 −p 1 (eV/Å 2 ) K 2 −p 2 (eV/Å 2 ) K 3 −p 3 (eV/Å 2 ) F (eV/Å) G (eV/Å) Δ 1 (eV) Δ 2 (eV)  states along b 2u , b 1g and b 2g distortions. puckered b 2g mode the coplanar non-bonding MO of the latter turns to be located at the pyramidal positions, as shown in Fig. 6(b). In combination with the two-step distortions outlined above, we conclude that the initial b 1g MO in the undistorted high-symmetry configuration turns into the HOMO of Si 4 F 4 in the chair-like structure.
The changes in these charge distributions by distortions result from the admixture of excited states, and they take place spontaneously because this PJTE admixture of electronic states improves the bonding conditions and lowers the energy of the system by means of added covalency 17,18 . As expected and confirmed, the chair-like structures of Si 4 R 4 compounds with bulky substituents R, including Si 4 (EMind) 4 , are qualitatively of the same PJTE origin as in the Si 4 F 4 molecule.  4 . In the simplest two-level presentation of the PJTE (see Eqs. (3)(4)(5)(6)) the vibronic coupling of the electronic ground state of the system in the high-symmetry configuration to the excited state of appropriate symmetry makes the ground state unstable with respect to low-symmetry displacements if the condition of instability (6) is satisfied. The similarity in the substituents R (all of them from the second row except H) allows one to assume (based on previous experience 17,18 ) that the energy gap is the main factor in the comparison of the possible instability of these four compounds.

Structural correlations in Si
The possible distortions of the high-symmetry configuration of these compounds, similar to the considered above Si 4 F 4 case, in the two-level approximation is controlled by the PJTE problems 2g with energy gaps between the interacting electronic states Δ 1 , Δ 2 , and Δ 3 , respectively. The calculated values of the latter are given in Table 2 (Δ 3 were calculated at the geometry of Qb 1g = 0.74Å). By comparing the Δ 1 values, we see that they are very close, which indicates that the four molecules have almost equal possibility to form boat-like structures (which, however, are not necessarily ground state structures, see below). The situation is different with the Δ 2 and Δ 3 values. The Δ 2 (0.70 eV) value in Si 4 H 4 is approximately twice of those for Si 4 F 4 (0.37 eV) and Si 4 (OH) 4 (0.33 eV). It means that the PJT vibronic coupling along the b 1g and hence the possibility of generating a chair-like structure is much lower in Si 4 H 4 as compared with the other three molecules. Furthermore, the smaller values of Δ 2 and Δ 3 compared with Δ 1 in the Si 4 F 4 , Si 4 Cl 4 and Si 4 (OH) 4 molecules lead to a stronger PJTE coupling with b 1g and b 2g distortions than with b 2u , explaining why the chair-like structures in these compounds are more stable than the boat-like ones. In this way the PJTE explains the origin of the main structural features of these Si 4 R 4 compounds (R = H, F, Cl, OH), their similarities and differences, by comparing energy gaps to the active excited electronic states.
Similar correlations can be found between the structural features of Si 4 R 4 compounds with larger substituents R and their electronic structure by comparing the energy gaps between PJT coupled molecular orbitals instead of electronic states. As shown above, in the D 4h configuration the electronic transition from the e g (HOMO) to the e u orbitals produces the 1 A 1u excited electronic state which couples with 1 B 2g ground state to form the boat-like structure, while the electronic transition from e g to the empty a 1g or b 1g orbitals plays a key role in the formation of the chair-like structure. The calculated energy gaps of the e u , a 1g and b 1g empty orbitals relative to the e g (HOMO) orbitals in the D 4h configuration denoted by δ 1 , δ 2 , and δ 3 , respectively, are given in Table 2. It is seen that the δ 1 for all the four molecules are very close, but the δ 2 , δ 3 values for Si 4 H 4 are much larger than others, which also confirms that the chair-like structures are preferred in Si 4 F 4 , Si 4 Cl 4 and Si 4 (OH) 4 , but not in Si 4 H 4 . In addition, the larger values of δ 2 , δ 3 than δ 1 in Si 4 R 4 (R = F, Cl, OH) lead to the same conclusion as above that the chair-like structures are lower in energy than the boat-like structures.  Other analogues: Si 4 (NX) 4 and Ge 4 R 4 . Below in this section we extend the results obtained above for tetrasilacyclobutadiene analogues Si 4 R 4 with a characteristic series of substituents R (shown in Fig. 5) to include two other series, namely, Si 4 (NX) 4 with amido substituents, and Ge 4 R 4 which replaces silicon with the heavier germanium element. In the first of these two series, all the Si 4 (NX) 4 analogues with substituents from NH 2 to larger carbazolyl groups could stabilize in the chair-like structures shown in Fig. 7. Electronic structure calculations with geometry optimization starting with the puckered geometry converge either to the boat-like minimum with higher energy or back to the minimum with the chair-like structures. The latter is thus preferable in all tetrasilacyclobutadiene compounds except Si 4 H 4 . From the point view of stereoselectivity, the chair-like structure is also preferred because it lowers the steric hindrance induced by large substituents. As shown above, in the series Si 4 R 4 with R as fluorine, phenyl, tetramethyl-phenyl, s-indacene, and the EMind group, the origin of their chair-like structures are due to the PJTE. If the central Si 4 ring is substituted by the Ge 4 , the chair-like structures are predicted to be stable too, as shown in Fig. 7. Calculated structural and electronic parameters for the Ge 4 R 4 and Si 4 R 4 series with the same substituents R are given in Table 3. The four-member ring in Ge 4 R 4 and Si 4 R 4 is planar and rhombic: the dihedral angle is very close to 0°, and the sum of the internal bond angles is very close to 360°, the bond length of the four Ge-Ge or Si-Si bonds are also very similar. As expected, the bond distances of the Ge-Ge bonds are longer than those of Si-Si bonds due to the bigger atomic radius of germanium than that of the silicon atom. Moreover, we found that the HOMO-LUMO gaps in the equilibrium configuration of most of Ge 4 R 4 compounds are larger than those of Si 4 R 4 , especially in Ge 4 (EMind) 4 . The larger HOMO-LUMO gap ceteris paribus means the bigger hardness and higher stability with respect to external perturbations. Therefore, the Ge 4 R 4 compounds with relatively large size substituents are predicted to have a stronger

Conclusions
The PJTE is shown to be instrumental in revealing the main structural features of a series of tetrasilacyclobutadiene analogues, Si 4 R 4 and Ge 4 R 4 , including large-size substituents, and rationalizing their similarities and differences. In all these compounds the excited electronic states that induce the deformation of the high-symmetry configuration in the ground state are determined, providing for a tool of possible manipulation of the structure (restoration of the planar configuration) by means of external perturbations. The formation of the boat-like structures originates from the PJTE vibronic coupling problem  4 as examples, the substituent effect is analyzed, their structural differences shown to be due to the differences in the energy gaps to the PJT active excited states or between corresponding molecular orbitals. For Si 4 R 4 with large substituents the same PJTE origin of preferred chair-like structures is deduced from their frontier molecular orbitals. For the Si 4 (NX) 4 analogues with amido substituents and Ge 4 R 4 homologs, the chair-like structures are still prefered. By comparison of the bonding character and HOMO-LUMO gaps in Ge 4 R 4 and Si 4 R 4 it is shown that the Ge 4 R 4 compounds are expected to be more stable in the chair-like structures than the corresponding Si 4 R 4 analogues, especially with the large-size substituents.