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

Abstract

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 D_4h 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 b_1g−b_2g adiabatic potential energy surface is calculated showing explicitly the way of transformation from the unstable D_4h geometry to the two equilibrium C_2h configurations via the D_2h 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.

Introduction

Cyclobutadiene and its derivatives have attracted extensive attention of researchers for many years^{1,2,3,4,5,6,7,8,9}. Among all kinds of its derivatives tetrasilacyclobutadiene analogues (denoted as Si₄R₄ hereafter) are the hottest topics due to the rich silicon chemistry and expected novel applications as semiconducting materials^1,5,10. Recently, the interest in these compounds was enhanced by the synthesis of Si₄(EMind)₄ (EMind =1,1,7,7-tetraethyl-3,3,5,5-tetramethyl-s-hydrindacen-4-yl), and its structural characterization with a distinctive planar-rhombic four-member silicon ring^11,12 (Fig. 1). Consequently, some new tetrasilacyclobutadiene or tetragermacyclobutadiene analogues that contain a planar-rhombic or puckered Si₄ ring^13,14, a slightly folded Ge₄ ring¹⁵, or a puckered Si₃Ge ring with ylide structure¹⁶ were synthesized. As the simplest member of Si₄R₄ family, the parent Si₄H₄ was predicted theoretically to have a puckered Si₄ ring with D_2d symmetry^2,5. From the simplest Si₄H₄ to the complicated Si₄(EMind)₄ compound the central silicon skeleton undergoes several low-symmetry configuration changes of the reference square-planar geometry.

Figure 1

The 2-D structure of Si4(Emind)4 reported by Suzuki et al.¹¹.

The origin of this variety of the molecular geometries in the Si₄R₄ and Ge₄R₄ 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, 18, 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 C₄H₄²⁰, C₄F₄⁷, Si₄²¹, Si₄H₄²⁺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, Si₄R₄, including relatively large substituents R, as well as some other related systems, including Ge₄R₄, 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 Si₄R₄ analogues with different substituents and Ge₄R₄ 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 Si₄R₄ geometries

There are two typical equilibrium structures of Si₄R₄ analogues, the one with a puckered Si₄ ring as in Si₄H₄, which hereafter is denoted as a “boat-like” structure, and the other one with a planar-rhombic Si₄ 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 chair-like structures of Si₄R₄ analogues we start with one of the simpler representatives, the Si₄F₄ molecule; it allows revealing the main mechanisms of formation of boat-like and chair-like structures that are similar for all the Si₄R₄ systems.

Figure 2

Two types of distortions of the highest-symmetry D_4h configuration of Si₄R₄ compounds induced by the PJTE: boat-like with D_2d and chair-like with C_2h symmetry.

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 Si₄ 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.

Figure 3

The potential energy profiles of the ground and low-lying excited states of the Si₄F₄ molecule along b_2u (a), b_1g (b) and b_2g (c) distortions by taking the energy of the D_4h configuration as the zero point. The lines in red with dots represent the coupling states, shown also in the inserts at small displacements (some states between ¹A_1u and 2¹A_1u in Fig. 3(a) are not shown, see Supplementary Fig. S2). The distortions along b_2g in Fig. 3(c) are conveniently started at Q_b1g = 0.74Å of Fig. 3(b), from which point there is a direct pass to the global minima shown in Fig. 4.

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 1¹B_2g is unstable and distorts spontaneously up to the minimum point at Q = 0.7Å which is the boat-like structure of Si₄F₄ 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 1¹B_2g via the b_2u mode in the D_4h symmetry are the 1¹A_1u and 2¹A_1u states, realizing the 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 ¹A_1u states. From the estimated vibronic coupling constants below it follows that the main contribution comes from the 1¹A_1u state. By comparing the electronic structures we can see that the 1¹B_2g state comes from the e_g² electronic configuration, while the 1¹A_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, , produces an additional potential minimum at Q_b1g = 1.1Å in which the Si₄ framework is rhombically distorted. On the other hand, the ground state ¹A_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(b).

Figure 4

Calculated two-dimensional segment of the APES of the Si₄F₄ molecule (top) and its contour map (bottom) in the rhombic b_1g and puckering b_2g distortion coordinates.

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 Si₄F₄ molecule. In the highest symmetry configuration D_4h the system is unstable along b_1g (rhombic distortion of the Si₄ ring) due to the hidden PJTE problem mixing two excited electronic states, followed by the puckering distortion b_2g (emerging from the problem), 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 1¹A_1g and 1¹B_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 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 Si₄F₄ geometrical coordinates are given in the Supplementary data S1. By examining the 1¹B_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 ³A_2g ground state in the D_4h geometry is not significant with respect to observable structural features of Si₄F₄ 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 ³A_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 1¹B_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 K₀ to K₀−p, p > 0, and not producing instability, meaning K₀ − p > 0, (according to the general theory¹⁷ the primary force constant without the vibronic coupling, K₀ > 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 K₁ and K₂, respectively, and the second order perturbation corrections to them from all the higher excited states of appropriate symmetry is defined by p₁ and p₂ in the PJTE energy matrix elements W_ΓΓ and ²³:

Then, the APES ε(Q) can be obtained from the secular equation of the perturbation theory:

where K₁, K₂, p₁ and p₂ 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 K₀, and (2) the lowest active excited state acting directly via the secular equation lowering the curvature by . This means that if the chosen excited state is indeed the one responsible for the instability the numerical estimates should yield K₁ − p₁ > 0 and K₁ − p₁ − 

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 yields satisfactory results listed in Table 1. The numerical value of K₁ − p₁ −  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 p₁. We checked the influence of the nearest one 2¹A_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.

Table 1 The estimated parameters of the PJTE in Si₄F₄: primary force constants, linear vibronic coupling constants, and energy gaps to active excited states along b_2u, b_1g and b_2g distortions.

However, for the b_1g and b_2g instabilities only one active excited state does not yield positive K₁ − p₁ 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 K₃ is the primary force constant for the third term, p₃ is the second order perturbation correction and G is the additional vibronic coupling constant to this term. Accordingly, the PJTE producing these distortions are and , and the excited electronic states controlling these instabilities are 1¹B_1g and 2¹A_1g for the b_1g rhombic distortions, and 1¹B_2g and 2¹A_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 Si₄R₄ compounds with bulky substituents

The description of the origin of the structural features of Si₄F₄ as due to the PJTE may serve as a prototype for the investigation of Si₄R₄ analogues with more complicated substituents, such as the mentioned above Si₄(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 Si₄R₄ 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 Si₄ 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 Si₄R₄ that control their geometry via the PJTE. Therefore we can use the PJTE formulations and results obtained for Si₄F₄ with the D_4h high-symmetry configuration as a reference in exploring Si₄R₄ 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 Si₄R₄ systems.

Figure 5: Calculated HOMO-1, HOMO, and LUMO of a series of trtrasilacyclobutadiene analogues Si₄R₄.

The charge distributions are located at mostly the central fragment Si₄ and the HOMO is essentially the same for all these compounds.

Based on the same Si₄ ring active center and the same HOMO electronic distributions of the Si₄R₄ derivatives, we come to the prediction that they have the same kind of PJTE origin. By tracing this orbital distribution in the lowest ¹A_g singlet electronic state along b_1g and b_2g distortions in Si₄F₄ (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 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 Si₄F₄ 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 Si₄R₄ compounds with bulky substituents R, including Si₄(EMind)₄, are qualitatively of the same PJTE origin as in the Si₄F₄ molecule.

Figure 6

Tracing the HOMO of the Si₄F₄ molecule in the ¹A_g ground state along first b_1g (a) and second b_2g (b) distortions. The HOMO actually originates from the empty b_1g orbital of the undistorted D_4h configuration. The graphic pictures in (a,b) are displayed in mutually perpendicular planes.

Structural correlations in Si₄R₄ compounds controlled by the PJTE

There are significant similarities and differences in the structural features of Si₄R₄ compounds with different R substituents. It follows from electronic structure calculations with geometry optimization⁵ that Si₄H₄ has only one equilibrium boat-like geometry, while Si₄F₄, as well as Si₄Cl₄ and Si₄(OH)₄ have both boat-like and chair-like minima, and the chair-like minimum is lower in energy. Since all these low-symmetry structures are controlled by the PJTE, their similarities and differences can be rationalized by employing the PJTE theory. We demonstrate this statement by comparisons among the mentioned four Si₄R₄ systems, Si₄H₄, Si₄F₄, Si₄Cl₄ and Si₄(OH)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.

The possible distortions of the high-symmetry configuration of these compounds, similar to the considered above Si₄F₄ case, in the two-level approximation is controlled by the PJTE problems , , and with energy gaps between the interacting electronic states Δ₁, Δ₂, and Δ₃, respectively. The calculated values of the latter are given in Table 2 (Δ₃ were calculated at the geometry of Qb_1g = 0.74Å). By comparing the Δ₁ 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 Δ₂ and Δ₃ values. The Δ₂ (0.70 eV) value in Si₄H₄ is approximately twice of those for Si₄F₄ (0.37 eV) and Si₄(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 Si₄H₄ as compared with the other three molecules. Furthermore, the smaller values of Δ₂ and Δ₃ compared with Δ₁ in the Si₄F₄, Si₄Cl₄ and Si₄(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 Si₄R₄ compounds (R = H, F, Cl, OH), their similarities and differences, by comparing energy gaps to the active excited electronic states.

Table 2 Energy gaps (in eV) between PJT interacting electronic states Δ₁, Δ₂, and Δ₃, and PJT coupled molecular orbitals δ₁, δ₂, and δ₃, in the PJTE formulations of the boat-like and chair-like structures of Si₄H₄, Si₄F₄, Si₄Cl₄, and Si₄(OH)₄ molecules.

Similar correlations can be found between the structural features of Si₄R₄ 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 ¹A_1u excited electronic state which couples with ¹B_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 δ₁, δ₂, and δ₃, respectively, are given in Table 2. It is seen that the δ₁ for all the four molecules are very close, but the δ₂, δ₃ values for Si₄H₄ are much larger than others, which also confirms that the chair-like structures are preferred in Si₄F₄, Si₄Cl₄ and Si₄(OH)₄, but not in Si₄H₄. In addition, the larger values of δ₂, δ₃ than δ₁ in Si₄R₄ (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: Si₄(NX)₄ and Ge₄R₄

Below in this section we extend the results obtained above for tetrasilacyclobutadiene analogues Si₄R₄ with a characteristic series of substituents R (shown in Fig. 5) to include two other series, namely, Si₄(NX)₄ with amido substituents, and Ge₄R₄ which replaces silicon with the heavier germanium element. In the first of these two series, all the Si₄(NX)₄ analogues with substituents from NH₂ 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 Si₄H₄. From the point view of stereoselectivity, the chair-like structure is also preferred because it lowers the steric hindrance induced by large substituents.

Figure 7

The stable chair-like structures of Si₄(NX)₄ and Ge₄R₄ compounds.

As shown above, in the series Si₄R₄ 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 Si₄ ring is substituted by the Ge₄, the chair-like structures are predicted to be stable too, as shown in Fig. 7. Calculated structural and electronic parameters for the Ge₄R₄ and Si₄R₄ series with the same substituents R are given in Table 3. The four-member ring in Ge₄R₄ and Si₄R₄ 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 Ge₄R₄ compounds are larger than those of Si₄R₄, especially in Ge₄(EMind)₄. The larger HOMO-LUMO gap ceteris paribus means the bigger hardness and higher stability with respect to external perturbations. Therefore, the Ge₄R₄ compounds with relatively large size substituents are predicted to have a stronger tendency to stabilize in the chair-like structures than the corresponding Si₄R₄ analogues. These predictions for Si₄(NX)₄ and Ge₄R₄ analogues may be useful in the search of potential precursors for semiconducting materials.

Table 3 The HOMO-LUMO energy gaps ΔE (in eV) and bond distances d_Si−Si, d_Ge−Ge (in Å) for the chair-like structures of Si₄R₄ and Ge₄R₄ compounds with different substituents R.

Computational Methods

The geometry optimization and frequency calculations for the equilibrium structures of Si₄R₄ or Ge₄R₄ analogues are performed by the density functional theory with the B3LYP functional^34,35 of the Gaussian 03 program³⁶. Electronic excited states and potential energy profiles along different vibrational modes are obtained by the complete active space self-consistent field (CASSCF) method^37,38 as implemented in MOLPRO 2010 packages³⁹ based on the optimized geometries and normal coordinates of the ground state (³A_2g) at the square configuration of D_4h symmetry by B3LYP method. The adiabatic potential energy surface in the lowest electronic state is obtained with B3LYP. The active space in the CASSCF method is composed of six electrons and all the valence empty orbitals, i.e., CAS(6,13) for Si₄H₄, Si₄F₄, and Si₄Cl₄, and CAS(6,17) for Si₄(OH)₄. The 6-31g(d,p) basis set is employed for all the calculations in this study.

Conclusions

The PJTE is shown to be instrumental in revealing the main structural features of a series of tetrasilacyclobutadiene analogues, Si₄R₄ and Ge₄R₄, 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 in the D_4h configuration, and the formation of chair-like structures is due to the PJT interaction of followed by the . The qualitative analysis of the PJTE is enhanced by numerical estimates of the main vibronic coupling constants. Taking Si₄H₄, Si₄F₄, Si₄Cl₄ and Si₄(OH)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 Si₄R₄ with large substituents the same PJTE origin of preferred chair-like structures is deduced from their frontier molecular orbitals. For the Si₄(NX)₄ analogues with amido substituents and Ge₄R₄ homologs, the chair-like structures are still prefered. By comparison of the bonding character and HOMO-LUMO gaps in Ge₄R₄ and Si₄R₄ it is shown that the Ge₄R₄ compounds are expected to be more stable in the chair-like structures than the corresponding Si₄R₄ analogues, especially with the large-size substituents.