Pseudo Jahn–Teller Origin of Nonplanarity and Rectangular-Ring

Jun 25, 2012 - Ali R. Ilkhani , Willian Hermoso , Isaac B. Bersuker. Chemical ... J. Wayne Mullinax , David S. Hollman , Henry F. Schaefer. Chemistry ...
0 downloads 0 Views 2MB Size
Download PDF
Article pubs.acs.org/JPCA

Pseudo Jahn−Teller Origin of Nonplanarity and Rectangular-Ring Structure of Tetrafluorocyclobutadiene Yang Liu,†,‡ Isaac B. Bersuker,*,† Pablo Garcia-Fernandez,§ and James E. Boggs† †

Institute for Theoretical Chemistry, Department of Chemistry and Biochemistry, University of Texas at Austin, Austin, Texas 78712-0165, United States ‡ Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology, Harbin 150080, People’s Republic of China § Departamento de Ciencias de la Tierra y Física de la Materia Condensada, Universidad de Cantabria, Avenida de los Castros S/N, 39005 Santander, Spain ABSTRACT: It is shown that the pseudo Jahn−Teller effect (PJTE) in combination with ab initio calculations explains the origin of instability of the planar configuration of tetrafluorocyclobutadiene, C4F4, with respect to a puckered structure and square-to-rectangle distortion of the carbon ring, and rationalizes its difference from the planar-rectangular geometry of C4H4 and nonplanar (puckered) structure of Si4H4. The two types of instability and distortion of the high-symmetry D4h configuration in these systems emerge from the PJT coupling of the ground B2g state with the excited A1g term producing instability along the b2g coordinate (elongation of the carbon or silicon square ring), and with the excited Eg term resulting in eg (puckering) distortion. A rhombic distortion b1g of the ring is also possible due to the coupling between excited A1g and B1g terms. For C4F4, ab initio calculations of the energy profiles allowed us to evaluate the PJTE constants and to show that the two instabilities, square-to-tetragonal b2g and puckering eg coexist, thus explaining the origin of the observed geometry of this system in the ground state. The preferred cis−trans (eg type) puckering in C4F4 versus trans−trans puckering (b2u distortion) in Si4H4 follows from the differences in the energy gaps to their excited electronic Eg and A1u terms causing different PJTE in these two cases. al.7 showing that C4F4 is not aromatic. Schleyer et al.11 carried out a detailed comparison of their results with previous work and concluded that the reduced antiaromaticity and reduced FC−CF repulsions favor the nonplanar C2h configuration. These authors state that “the possibility of having a secondorder Jahn−Teller effect (SOJTE) in C4F4 is unlikely” because of the relatively large HOMO−LUMO gap (0.15 au in D2h and 0.13 au in C2h), and the SOJTE requirement is not met due to the different symmetry of the HOMO(ag) and LUMO(bg) orbitals in C2h geometry. On the other hand, it was rigorously proved in a formulation based on first principles that the only source of instability of high-symmetry configurations of molecular systems and solids in nondegenerate states is the pseudo Jahn−Teller effect (PJTE) (see in ref 14; for degenerate states the instability may also be of Jahn−Teller or Renner−Teller origin14). The PJTE has been shown to be instrumental in solving a variety of molecular and solid state problems (see earlier publications15−21 and more recent developments,14,22−26 with the first application of the PJTE to solve a structural problem in ref 18). These methods have also been applied to closely related problems like the spectroscopy of the C4H4−.27 It incorporates

1. INTRODUCTION In spite of relatively simple composition, the structure and properties of four-membered ring molecules based on carbon and silicon have not been fully rationalized so far resulting in intensive discussion.1−13 The simplest stable representative of this group, cyclobutadiene C4H4, was shown to undergo a distortion of its high-symmetry square-planar D4h configuration to the planar-rectangular D2h structure,2,4,5,9,10 whereas a series of its analogs produced by substituting carbon with silicon or hydrogen with halogen have different geometries. For instance, for Si4H4 the planar-rectangular D2h configuration is just a saddle point on the adiabatic potential energy surface (APES) with the stable equilibrium geometry having a puckered D2d structure,1,10,13 whereas the silicon four-membered ring in Si4(EMind)4 (Emind = bulky 1,1,7,7-tetraethyl-3,3,5,5-tetramethyl-s-hydrindacen-4-yl) was found to be planar-rhombic.12,13 In this respect the structure of tetrafluorocyclobutadiene, C4F4, is even more challenging. Its first experimental study by Petersson et al.3 detected the unstable puckering vibrational mode and concluded that C4F4 prefers C2h to D2h symmetry. After that, several papers tried to explain the nonplanarity of C4F4 from the perspective of aromaticity and antiaromaticity of the cyclic four-electron π system. Seal and Chakrabarti6,8 pointed out that the unusual aromaticity and second-order Jahn−Teller Effect are the driving forces behind its structure. The aromaticity claim was challenged by Herges et © 2012 American Chemical Society

Received: April 6, 2012 Revised: June 2, 2012 Published: June 25, 2012 7564

dx.doi.org/10.1021/jp3032836 | J. Phys. Chem. A 2012, 116, 7564−7570

The Journal of Physical Chemistry A

Article

all the particular interactions, often used to explain the origin of structural instabilities, in one rigorously formulated vibronic coupling effect. The qualitative chemical interpretation of this effect is that it leads to additional (better) covalent bonding by distortion.14 As mentioned above, some previous studies6,8 indicated the PJTE as a possible cause for distortion of C4F4 system, and others11 disagreed on its significance. This uncertainty is based on earlier papers17,19 that showed that the PJTE is one of possible sources of instability and distortions of molecular systems. But in later developments14 it was proved that the PJTE is a general quantum-mechanical effect of vibronic mixing of electronic states (belonging to different irreducible representations), and it is the only source of distortions of high-symmetry geometries of molecular systems. As such it incorporates all the other particular interactions forming a direct tool for rationalizing the origin of specific structures of polyatomic systems. In the present paper we show that the correctly formulated PJTE problem including the vibronic coupling between the ground and several excited states of C4F4 in combination with ab initio calculations explains the origin of instability and distortions of its high-symmetry square-planar configuration producing the observed nonplanar (puckered) ring-rectangular geometry. The general PJTE problem (1B2g + 1A1g + 1B1g + 1Eg) ⊗ (b1g + b2g + eg) including all the possible instabilities induced by several PJT active excited states is formulated and, by comparison with the ab initio calculated APES and its corresponding cross sections ((1B2g + 1Eg) ⊗ eg and (1B2g + 1 A1g) ⊗ b2g problems), the PJTE constants were evaluated and the conditions of instability verified. The PJTE explains also why the cis−trans puckering along the double-bonded ring members is realized in this molecule instead of occurring at single-bonded ring members or the trans−trans distortion expected due to lesser steric repulsion. A qualitative explanation is given also to the differences in the structures of the similar four-membered ring molecules mentioned above. These problems and their solutions are similar to those obtained earlier26 in a more general treatment of possible molecular geometries generated by the PJTE.

Figure 1. Electronic energy level diagram of C4F4 with indication of the pseudo Jahn−Teller couplings resulting in the possible geometrical distortions.

Table 1. Relative Energy Levels, Main Electronic Configurations, and Some Important Molecular Orbitals for the Low-Lying Electronic States of C4F4 Based on the 3A2g D4h Equilibrium Geometry Obtained with the MRCI+Q Method with an Active Space of CAS(4,5)

2. PJTE PROBLEM FOR C4F4: GENERAL ANALYSIS AND RESULTS As required by the general theory,14 the PJTE in the C4F4 system should be formulated as a vibronic coupling problem applied to the high-symmetry D4h configuration. For this purpose the energy levels and wave functions of the ground and several lowest excited electronic states in this nuclear configuration should be calculated first. In the D4h geometry all the four-membered ring molecules of the type C4H4, C4F4, Si4H4, etc. have the valence eg2 electronic configuration. The four lowest electronic states that emerge from this electronic configuration span the representations eg ⊗ eg = 3A2g + 1A1g + 1 B2g + 1B1g. In Figure 1 the corresponding energy scheme of such systems is given, whereas Table 1 lists the main electronic configurations for each state, as well as the most relevant frontier orbitals. Above the e2 manifold, the lowest electronic singlet state in the square-planar structure is 1Eg, formed by an excitation of the π electron to the antibonding σC−F* orbital. From this energy scheme the qualitative consequences of the PJTE follow directly: the vibronic coupling of the lowest energy state 1B2g to the excited state 1A1g may distort the square-planar system in the Q(b2g) direction producing the rectangular

geometry, the coupling of 1A1g and 1B1g state results in the rhombic geometry, and the vibronic influence of the higher 1Eg state, if strong enough, triggers the puckering modes Qθ and Qε (Figure 1). Whether or not any of these possible distortions takes place depends on the energy gap Δ to the corresponding excited state and the PJTE coupling constants F. If only two states are engaged in the vibronic coupling, the condition of instability is

Δ < 2F 2/K 0

(1)

where K0 is the primary force constant characterizing the rigidity of the system without the vibronic coupling. If more 7565

dx.doi.org/10.1021/jp3032836 | J. Phys. Chem. A 2012, 116, 7564−7570

The Journal of Physical Chemistry A

Article

excited states are contributing to the instability of the lowest energy state, their effect should be summarized in the righthand side of eq 1. The energy level scheme in Figure 1 is very similar to that obtained in our previous study of cis−trans conformational changes in molecular systems containing a double bond.26 There, as above, it was shown that the PJTE interaction of the states arising from both e2 and a1e1 electronic configurations describe all the possible conformations of the system. Distinct from the molecules with double bonds where a dominant b1 mode distorts the high-symmetry geometrical configuration (that has no double bond) to a planar structure (with the actually observed double bond), which may be altered in a second step by the e mode coupling to the E states producing cis−trans (or boat-chair) conformations, in C4F4 the b2g and eg modes have similar strengths and lead the molecule to a global minimum with both distortions. Moreover, as the puckering mode alters the positions of carbon atoms on the opposite sides of the square-ring, the effect of a small superimposed b2g distortion may be concealed, unobserved without careful analysis of the geometries. It is important to note that the combination of b2g and eg modes leads to two chemically different C2h conformers; in one of them the cis fluorine atoms are bonded to two carbon atoms with a short double bond between them, whereas in the other conformer these two carbons form a longer single bond (with the double bond between two carbons to which the attached fluorine atoms form a cis−trans pair). A simple diagram of both possibilities is shown in Figure 1. By employing the PJTE, we can also predict which of them is the actual stable equilibrium geometry. As shown in Figure 1, the movement along the b2g mode breaks the molecular symmetry and lowers it from D4h to D2h. This displacement induces the splitting of the Eg excited state into 1 B2g and 1B3g components (in D2h labels) whose energies separate linearly with the distortion due to the Eg ⊗ b2g JTE. According to our choice of axes these electronic states represent, respectively, a double bond oriented along x (1B2g) or y ( 1 B 3g ) that is degenerate for the square-planar configuration (Figure 1 and Table 1). Similarly, the puckering distortion breaks down under the b2g displacement into b3g and b2g distortions along, respectively, the C2(x) or C2(y) axis. Hence, if for Qb2g > 0 (a shortening of the C−C bonds along x and an elongation of those along y), the 1B3g state is lower in energy than 1B2g, which means that the coupling of the excited 1 B3g electronic state with the 1Ag ground state via b3g distortions will be stronger than the coupling of the same ground state with 1 B2g through b2g mode. A reversed situation will occur when Qb2g < 0, but in both cases the cis-puckering of the F atoms linked to double-bonded carbons is more stable. An important feature of this qualitative PJTE analysis is that among the feasible distortions of the systems there are no b2u type ones that describe the puckering in which the nearneighbor fluorine atoms are out-of-plane displaced in opposite directions (conventional, trans−trans-puckering). This type of puckering could be expected due to significant steric repulsion between the highly charged fluorine atoms. According to the PJTE the b2u instability of the D4h geometry is possible when there is a sufficiently strong vibronic coupling of the ground state with an excited electronic 1A1u state according to (1B2g + 1 A1u) ⊗ b2u. To check this possibility, we calculated the excited 1 A1u state of C4F4 and found that it is at Δ ∼ 8.10 eV, which is much higher than the Eg term at Δ ∼ 4.53 eV (Table 2). This

Table 2. Relative Energies (eV) of Several Electronic Terms for C4F4, C4H4, and Si4H4 Molecules Based on Their 3A2g D4h Equilibrium Geometry Calculated at the MRCI+Q Level with an Active Space of CAS(4,7) energy electronic states 1

B2g 3 A2g 1 A1g 1 B1g 1 Eg 1 A1u

C4 F 4

C4H4

Si4H4

0.00 0.20 1.77 2.27 4.53 8.10

0.00 0.20 1.57 2.01 6.61 6.85

0.00 0.0007 0.77 0.90 2.83 3.63

explains why the latter (conventional, cis−trans) puckering is observed, not the former, showing also that the PJTE is a stronger criterion of instability and distortions different from any considerations of particular chemical interactions like steric effects, aromaticity, hyperconjugation, etc. As mentioned above, the PJTE incorporates all possible interactions, rigorously formulating (in a summarized way) the conditions when they lead to instability (as pointed out above, the PJTE produces additional covalence by distortion14). Another interesting comparison of the structures of the above-mentioned similar four-membered ring systems C4F4, C4H4, and Si4H4 can be made by examining their energy level diagrams of Figure 1. The three systems stabilize in different structures: C4F4 displays two types of distortions, square-torectangular and cis−trans puckering, C4H4 has no puckering at all and shows just rectangular elongation, whereas Si4H4 has trans−trans puckering and out-of-plane ring deformation, but no Si−Si bond elongation. Table 2 provides some clues toward the elucidation of this question, although a full comparison would require ab initio calculation, estimation of the vibronic coupling constants, and checking the condition of instability, as performed below for C4F4. We see from Figure 1 and Table 2 that the Eg term responsible for the puckering is very high in C4H4 as compared with Si4H4 and C4F4. This explains their difference in puckering. Moreover, Si4H4 has a relatively very low 1A1u state, much lower than the other two complexes (Table 2), and this explains why its puckering is of trans−trans type, distinguished from the cis−trans puckering in C4F4. Seemingly, this puckering in Si4H4 is so strong, that it involves the Si4 core, distorting it accordingly.1 On the other hand, the energy gap Δ from 1B2g to the A1g term in C4F4 and C4H4 (which have the same C4 skeleton) is almost the same (somewhat lower in C4H4), and therefore the C4 ring is rectangular distorted in both cases. A similar comparison between these two distorted C4 skeletons and the Si4 skeleton in Si4H4 cannot be done solely on the basis of the A1g term position because they have significantly different rigidity (K0 values) and vibronic coupling constants with respect to this distortion. Note that the square-to-rhombus distortion of the Si4 ring in Si4(EMind)4 (Emind = bulky 1,1,7,7-tetraethyl3,3,5,5-tetramethyl-s-hydrindacen-4-yl),12,13 follows also from our PJTE analysis: it results from the b1g instability produced by the possible PJT coupling of the A1g state to the excited B1g term, the (A1g + B1g) ⊗ b1g PJTE problem. 7566

dx.doi.org/10.1021/jp3032836 | J. Phys. Chem. A 2012, 116, 7564−7570

The Journal of Physical Chemistry A

Article

space. Dunning’s correlation consistent basis set cc-pVTZ32,33 is employed. The ab initio SA-CASSCF and MRCI calculations performed on the C4F4 molecule fully support the PJTE presentation of the previous section. We found, as in previous reports,3,6,11 that the geometrical configuration at the global minimum has C2h symmetry. The analysis of this configuration and comparison with a reference one of D4h symmetry shows that compared to the latter the molecule undergoes both puckering, involving movement of carbon and fluorine atoms along the Cartesian zaxis, and rectangular distortion in the xy plane. Because this molecule has no rhombic distortion, we ignored at first the b1g coordinates. In Figure 2 the APES of the ground singlet state of

3. QUANTITATIVE FORMULATION AND COMPARISON WITH AB INITIO CALCULATIONS Quantitatively, the PJTE is formulated as a perturbation problem in which all the active electronic states are mixed by the vibronic coupling to all three possible modes of distortion termed as the (1B2g + 1A1g + 1B1g + 1Eg)⊗(b2g + b1g + eg) PJTE problem,14 with the matrix form (2)

W − εI = 0 where ε are the energy levels, I is a 5 × 5 unit matrix, and W= ⎛WB ⎜ 2 ⎜ F2Q b2 ⎜ ⎜0 ⎜ ⎜ F3Q ε ⎜ ⎜F Q ⎝ 3 θ

F2Q b

2

WA1

F3Q ε

0

FQ 1 b F4Q ε 1

FQ 1 b

WB1

F4Q ε

F5Q θ WE

F4Q θ

F5Q ε Vb1Q b + Vb2Q b

1

⎞ ⎟ ⎟ F4Q θ ⎟ ⎟ F5Q ε ⎟ Vb1Q b + Vb2Q b ⎟ 1 2⎟ ⎟ WE ⎠ F3Q θ

F5Q θ

1

2

(3)

In this interaction matrix Vb1 and Vb2 are respectively the linear vibronic coupling constants of the 1E state with the b1g and b2g displacements, and F1, F2, and F3,4,5 are respectively the PJT coupling constants to the rhombic, rectangular, and puckering eg (cis−trans) vibrations. The energy levels Δi and the primary (ib ) (ib ) force constants K0 1 , K0 2 , and K(ie) for b1g, b2g, and eg 0 vibrations in state i are contained in the Wi terms with i labeling the various states: Wi = Δi + +

1 (i b1) 2 1 (i b2) 2 K0 Q b + K0 Q b 1 2 2 2

1 (ie) K 0 (Q θ 2 + Q ε 2 ) 2

(4)

All of the parameters in eqs 3 and 4 are defined at the highsymmetry D4h configuration. In principle, they can be calculated as well-defined matrix elements,14,25 but this is a very computercostly problem. More easily, they can be evaluated by fitting the appropriate eigenvalues of the matrix (3) with those obtained by ab initio calculations of the APES of the C4F4 molecule along these relevant coordinates. The 3A2g equilibrium geometry with D4h configuration was fully optimized first with the Hartree−Fock method to provide the initial geometrical coordinates and normal coordinates for further energy profiles (APES cross sections) calculations. The energy profiles along the distortion coordinates were calculated by the multireference configuration interaction (MRCI) method28 based on the state-average complete active space self-consistent field (SA-CASSCF)29,30 wave functions as implemented in the Molpro 2010 package.31 The APES of the ground and excited states are calculated by the SA-CASSCF method. The MRCI+Q28 method including the Davidson correction used in the evaluation of the relative energies of C 4 F 4 , C 4 H 4 , and Si 4 H 4 based on their respective 3 A 2g equilibrium geometry. The active space is composed of four carbon π electrons and five active orbitals, denoted as CAS(4,5), which includes additionally the antibonding empty orbital σ*C−F of C4F4 as shown in Table 1 (σ*C−H for C4H4 and σ*Si−H for Si4H4). This active space produces qualitatively the same results as a larger active space, like CAS(4,7) with adding a degenerate eu empty orbitals, CAS(4,12) with the full valence

Figure 2. Ground state APES (a) and contour map (b) of the C4F4 molecule showing the distortions from the high-symmetry D4h configuration along the puckering (δFCCC) and rectangular-ring (ΔR) displacements.

C4F4 (1B2g at D4h and 1Ag at C2h) is shown in two dimensions, along the puckering and rectangular-ring distortions (Figure 2a), together with the corresponding contour map (Figure 2b) where the critical points of the APES can clearly be seen. For simplicity, the dihedral angle δFCCC and the CC bond distance change (ΔR) are used to present the puckering and rectangular-ring displacements, respectively. It is seen that the D4h configuration at δFCCC = 180° and ΔR = 0 has a secondorder saddle-point instability whereas a C2h minimum appears at δFCCC ≈ 152° and ΔR ≈ 0.18 Å corresponding to a 7567

dx.doi.org/10.1021/jp3032836 | J. Phys. Chem. A 2012, 116, 7564−7570

The Journal of Physical Chemistry A

Article

combined distortion along eg and b2g modes. The scanned structure at this minimum is close to the fully optimized geometry (δFCCC = 156°; ΔR = 0.20 Å at the Mk-MRCCSD/ cc-pVTZ11 level, and δFCCC = 168°; ΔR = 0.23 Å at the B3LYP/cc-pVDZ3 level). To calculate the cross section of the APES along the normal coordinates of puckering distortion (akin to solving the twolevel (1B2g + 1Eg) ⊗ eg PJTE problem), we used a higher level MRCI method. The results are shown in Figure 3. The

Figure 4. Energy profiles (APES cross sections) of low-lying electronic states of C4F4 (in D4h denotations) along the rectangular-ring distortion normal coordinate Qb2g calculated at the MRCI level.

Figure 3. Energy profiles (APES cross sections) of low-lying electronic states of C4F4 (in D4h denotations) along the normal coordinates of the puckering distortion Qθ(eg) calculated at the MRCI level.

decrease in energy of the 1B2g and 1A1g states and its increase in the 1Eg states at the high-symmetry configuration along one component of the eg coordinate confirm the predicted PJTE interaction between the 1B2g, 1A1g, and 1Eg states and the instability along the puckering distortion. For larger distortions these curves are less informative because normal displacements for D4h symmetry hardly describe well the bent configuration that has a different symmetry and is described by other normal coordinates. In Figure 4 the cross section of the APES along the b2g coordinate (the rectangular-elongation distortion in normal coordinates) is shown. It is seen that the 1A1g curve increases sharply in this direction, whereas the 1B2g state decreases, as predicted by the PJTE (1B2g + 1A1g) ⊗ b2g problem. As the eg and b2g distortions are essentially independent, they may coexist resulting in the global minimum of C2h symmetry in which the C4F4 has the puckered structure with a rectangular C4 skeleton. Finally, and for comprehensiveness, we have plotted the cross-section of the APES along the rhombus distortion with b1g symmetry in Figure 5. We can see how the distortion destabilizes the lower 1A1g state and stabilizes the higher 1B1g one as predicted by the PJTE. Although no relevant minima or transition states are associated with this mode in the ground state of C4F4, it may be effective in its B1g excited state influencing both the geometry and spectra, and it can be used as an explanation of the rhombic configuration of the Si4 skeleton in the above-mentioned system Si4(EMind)4 (Emind = bulky 1,1,7,7-tetraethyl-3,3,5,5-tetramethyl-s-hydrindacen-4yl). In this case, if the vibronic coupling is sufficiently strong, the A1g state may cross the lower terms and become the ground state by the rhombic distortion, similar to several other cases of strong PJTE in two excited states.26

Figure 5. Energy profiles (APES cross sections) of low-lying electronic states of C4F4 (in D4h denotations) along the rhombic-ring distortion normal coordinates Qb1g calculated at the MRCI level.

By fitting the vibronic interaction matrix (3) with the ab initio cross sections shown in Figure 3 (see, for example, in refs 21, 22, 24−26, and 34), we evaluated the vibronic constants in Table 3. With these constants and the energy gaps Δ between the PJT interacting terms from Table 1 we can check the instability condition (1). For the puckering, 2(F3)2/K0B2e = 10.18 and Δ = 4.68, whereas for the rectangular distortion we get 2(F2)2/K0B2b2 = 4.19 with Δ = 1.77, so they all satisfy the eq Table 3. Vibronic Coupling Parameters for C4F4 Including the Primary Force Constants K0 (eV/Å2) and Linear Coupling Constants F and V (eV/Å) along puckering eg displacement K0B2e = 7.8, K0A1e = 8.7, K0B1e = 13.2, K0Ee = 10.0; F3 = 6.3, F4 = 4.2, F5 = 3.4 along rectangular b2g displacement K0B2b2 = 122.2, K0A1b2 = 87.0, K0B1b2 = 123.3, K0Eb2 = 109.6; F2 = 16.0, Vb2 = 8.1 along rhombic b1g displacement K0B2b1 = 24.1, K0A1b1 = 23.8, K0B1b1 = 23.8, K0Eb1 = 12.7; F1 = 4.7, Vb1 = 0.9 7568

dx.doi.org/10.1021/jp3032836 | J. Phys. Chem. A 2012, 116, 7564−7570

The Journal of Physical Chemistry A

Article

1 (Δ < 2F2/K0), and the condition of instability is realized for both distortions. In addition to the qualitative results given in the previous section, this gives us a full rationalization of the origin of the structure of this molecule with two types of distortions as based on the PJTE.

4. CONCLUSIONS The following general conclusions can be made on the basis of the results of this work. 1. Using the C4F4 and some analogous systems as examples, it is shown that the PJTE in combination with ab initio calculations rationalizes the origin and differences in the structures of molecular systems without involving any qualitative chemical concepts like steric effects, aromaticity, hyperconjugation, etc.; in the PJTE they are all incorporated under a rigorous definition based on the electronic structure of the ground and several excited states in the high-symmetry configuration. In pure chemical terminology, the PJTE produces additional covalency by distortion. 2. For C4F4 and the similar C4H4 and Si4H4 molecules, the ground and several low-lying excited electronic states with D4h geometry emerge from e2 and e1a1 electronic configurations that form a general trend outlined in our previous work. There are three excited states of different symmetry that may be involved in the vibronic coupling with the lowest energy state to produce three types of PJTE instabilities and distortions termed respectively as planar-rectangular, planar-rhombic, and out-of-plane puckering. The puckering may be of two types: cis− trans and trans−trans. Whether or not some of these possible distortions are realized depends on the satisfaction of the quantitative criterion of PJTE instability, which includes the position of the energy levels of the corresponding excited state and the vibronic coupling constants. In C4F4 two of the four possible distortions are realized, ring-rectangular and cis−transpuckering, in C4H4 only the rectangular distortion of the C4 core is effective, whereas in Si4H4 only strong trans− trans-puckering is seen. The rhombic distortion of the core has been observed in Si4(EMind)4 (Emind = bulky 1,1,7,7-tetraethyl-3,3,5,5-tetramethyl-s-hydrindacen-4-yl). 3. The PJTE explains directly why the puckering in C4F4 is of (unexpected) cis−trans type. Indeed, because of steric repulsion, the out-of-plane displacements of the fluorine atoms are expected to alternate in opposite directions realizing trans−trans puckering. Contrary to this expectation, the observed puckering takes place as two by two parallel (cis) out-of-plane displacements trans to each other at the long site of the C4 core (Figure 1). According to the PJTE the alternate out-of-plane displacements of b2u type can be realized by the vibronic coupling of the lowest energy state to the excited 1A1u term which is shown to be much higher in energy than the Eg term. Hence it does not compete with the latter in producing instability in spite of weaker steric repulsions. On the contrary, in Si4H4 the puckering is of trans−trans type which, again, follows from the PJTE explanation: the excited electronic 1A1u term is very low in this system. These are some of the features that cannot be explained without the PJTE.

■

4. The PJTE provides clear arguments for which of the two possible cis−trans C2h conformers is more stable. In particular, we have seen that the splitting of the Eg excited state due to the Eg ⊗ b2g JTE modulates the coupling strength of these states with the ground one through puckering modes along the x or y directions. The PJTE prediction is that the cis-puckered fluorine atoms connected to the double-bonded carbon atoms is lower in energy, in accordance with experimental observations. 5. By comparison of the expected PJTE distortions with those obtained from ab initio calculations, the PJTE parameters are evaluated and the quantitative criterion of instability is shown to be fulfilled for the observed distortions. Altogether with the qualitative results, outlined above, this serves to fully rationalize the origin of the structures of the systems under consideration and to characterize the PJTE as a general tool for understanding the origins of molecular structures.

AUTHOR INFORMATION

Corresponding Author

*Telephone: +1 512-471-4671. Fax: +1 512-471-8696. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work has been partially supported by the Spanish Ministerio of Ciencia y Tecnologia under Project FIS200907083.

■

REFERENCES

(1) Yates, B. F.; Clabo, D. A., Jr; Schaefer, H. F., III. Cyclic isomers of singlet Si4H4 related to tetrasilacyclobutadiene. Chem. Phys. Lett. 1988, 143 (5), 421−427. (2) Koseki, S.; Toyota, A. Energy Component Analysis of the Pseudo-Jahn−Teller Effect in the Ground and Electronically Excited States of the Cyclic Conjugated Hydrocarbons: Cyclobutadiene, Benzene, and Cyclooctatetraene. J. Phys. Chem. A 1997, 101 (31), 5712−5718. (3) Petersson, E. J.; Fanuele, J. C.; Nimlos, M. R.; Lemal, D. M.; Ellison, G. B.; Radziszewski, J. G. Nonplanarity of Tetrafluorocyclobutadiene. J. Am. Chem. Soc. 1997, 119 (45), 11122−11123. (4) Wiberg, K. B.; Marquez, M. Effect of Fluorine Substitution on the Energies of Small Ring Compounds. J. Am. Chem. Soc. 1998, 120 (12), 2932−2938. (5) Shiota, Y.; Kondo, M.; Yoshizawa, K. Role of molecular distortions in the spin-orbit coupling between the singlet and triplet states of the 4π electron systems C4H4, C5H5+, and C3H3−. J. Chem. Phys. 2001, 115 (20), 9243−9254. (6) Seal, P.; Chakrabarti, S. Origin of the Nonplanarity of Tetrafluoro Cyclobutadiene, C4F4. J. Phys. Chem. A 2007, 111 (4), 715−718. (7) Koehler, F.; Herges, R.; Stanger, A. Comment on “Origin of the Nonplanarity of Tetrafluoro Cyclobutadiene, C4F4. J. Phys. Chem. A 2007, 111 (23), 5116−5118. (8) Seal, P.; Chakrabarti, S. Reply to “Comment on ‘Origin of the Nonplanarity of Tetrafluoro Cyclobutadiene, C4F4'”. J. Phys. Chem. A 2007, 111 (23), 5119−5121. (9) Pierrefixe, S. C. A. H.; Bickelhaupt, F. M. Aromaticity and Antiaromaticity in 4-, 6-, 8-, and 10-Membered Conjugated Hydrocarbon Rings. J. Phys. Chem. A 2008, 112 (50), 12816−12822. (10) Nazari, F.; Doroodi, Z. The substitution effect on heavy versions of cyclobutadiene. Int. J. Quantum Chem. 2010, 110 (8), 1514−1528. 7569

dx.doi.org/10.1021/jp3032836 | J. Phys. Chem. A 2012, 116, 7564−7570

The Journal of Physical Chemistry A

Article

dynamics: the photoelectron spectrum of CH3F. J. Chem. Phys. 2005, 123 (23), 231103.

(11) Wu, J. I.; Evangelista, F. A.; Schleyer, P. v. R. Why Are Perfluorocyclobutadiene and Some Other (CF)nq Rings Non-Planar? Org. Lett. 2010, 12 (4), 768−771. (12) Apeloig, Y. The Diamond Within a Silicon Analog of Cyclobutadiene. Science 2011, 331 (6022), 1277−1278. (13) Suzuki, K.; Matsuo, T.; Hashizume, D.; Fueno, H.; Tanaka, K.; Tamao, K. A Planar Rhombic Charge-Separated Tetrasilacyclobutadiene. Science 2011, 331 (6022), 1306−1309. (14) Bersuker, I. B., The Jahn-Teller Effect; Cambridge University Press: Cambridge, U.K., 2006. (15) Opik, U.; Pryce, M. H. L. Studies of the Jahn-Teller Effect. I. A Survey of the Static Problem. Proc. R. Soc. London., A 1957, 238 (1215), 425−447. (16) Longuet-Higgins, H. C. Chem. Soc. Spec. Publ. 1959, 12, 17. (17) Bader, R. F. W. An interpretation of potential interaction constants in terms of low-lying excited states. Mol. Phys. 1960, 3 (2), 137−151. (18) Bersuker, I. B. On the origin of ferroelectricity in perovskite-type crystals. Phys. Lett. 1966, 20 (6), 589−590. (19) Pearson, R. G. Concerning Jahn-Teller Effects. Proc. Natl. Acad. Sci. 1975, 72 (6), 2104−2106. (20) Bersuker, I. B.; Gorinchoi, N. N.; Polinger, V. Z. On the origin of dynamic instability of molecular systems. Theor. Chem. Acc. 1984, 66 (3), 161−172. (21) Köuppel, H.; Domcke, W.; Cederbaum, L. S. Multimode Molecular Dynamics Beyond the Born-Oppenheimer Approximation. Adv. Chem. Phys. 1984, 57, 59−246. (22) Garcia-Fernandez, P.; Bersuker, I. B.; Boggs, J. E. Orbital disproportionation and spin crossover as a pseudo Jahn-Teller effect. J. Chem. Phys. 2006, 125 (10), 104102. (23) Bersuker, I. B., Recent Developments in the Jahn−Teller Effect Theory: The Hidden Jahn-Teller Effect. In Köppel, H.; Yarkony, D. R.; Barentzen, H. E. The Jahn-Teller Effect: Fundamentals and Implications for Physics and Chemistry; Springer-Verlag: Berlin Heidelberg, 2009. (24) Liu, Y.; Bersuker, I. B.; Zou, W.; Boggs, J. E. Combined Jahn− Teller and Pseudo-Jahn−Teller Effect in the CO3 Molecule: A SevenState Six-Mode Problem. J. Chem. Theory Comput. 2009, 5 (10), 2679−2686. (25) Liu, Y.; Bersuker, I. B.; Zou, W.; Boggs, J. E. Pseudo Jahn-Teller versus Renner-Teller effects in the instability of linear molecules. Chem. Phys. 2010, 376 (1−3), 30−35. (26) Garcia-Fernandez, P.; Liu, Y.; Bersuker, I. B.; Boggs, J. E. Pseudo Jahn-Teller origin of cis-trans and other conformational changes. The role of double bonds. Phys. Chem. Chem. Phys. 2011, 13 (8), 3502− 3513. (27) Gunion, R. F.; Koppel, H.; Leach, G. W.; Lineberger, W. C. Photoelectron spectroscopy of C4H4−: Ab initio calculations and dynamics of the 1,2-hydrogen shift in vinylvinylidene. J. Chem. Phys. 1995, 103 (4), 1250−1262. (28) Knowles, P. J.; Werner, H.-J. Internally contracted multiconfiguration-reference configuration interaction calculations for excited states. Theor. Chem. Acc. 1992, 84 (1), 95−103. (29) Knowles, P. J.; Werner, H.-J. An efficient second-order MC SCF method for long configuration expansions. Chem. Phys. Lett. 1985, 115 (3), 259−267. (30) Werner, H.-J.; Knowles, P. J. A second order multiconfiguration SCF procedure with optimum convergence. J. Chem. Phys. 1985, 82 (11), 5053−5063. (31) Werner, H.-J.; Knowles, P. J.; R.Lindh; Manby, F. R.; Schütz, M., et al. MOLPRO, version 2010.1, a package of ab initio programs; see http://www.molpro.net. (32) Dunning, T. H. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys. 1989, 90 (2), 1007−1023. (33) Woon, D. E.; Dunning, T. H., Jr. Gaussian basis sets for use in correlated molecular calculations. III. The atoms aluminum through argon. J. Chem. Phys. 1993, 98 (2), 1358−1371. (34) Mahapatra, S.; Vallet, V.; Woywod, C.; Koppel, H.; Domcke, W. Remarkable impact of intermode couplings on multimode vibronic 7570

dx.doi.org/10.1021/jp3032836 | J. Phys. Chem. A 2012, 116, 7564−7570