_{1}

^{*}

The goal of this brief partly review paper is to summarize the results of the works published over the last few years regarding the origin of the out-of-plane distortions (puckering) of heterocyclic compounds. In all the papers devoted to this problem, it is shown that the instability of planar configurations of heterocyclic molecules leading to symmetry breaking and distortions is induced by the pseudo Jahn-Teller effect (PJTE). Special attention in this work is paid to the mechanism of suppression and enhancement of the PJTE distortions of heterocycles by oxidation, reduction, and chemical substitutions. It is demonstrated that oxidation of 1,4-dithiine containing compounds leads to suppression of the PJTE and to restoration of their planar nuclear configurations. An example of a dibenzo[1,2]dithiine molecule is used to demonstrate the mechanism of enhancement of the PJTE by reduction. It is shown that the reduction of the neutral C
_{12}H
_{8}S
_{2 }molecule up to the dianion (C
_{12}H
_{8}S
_{2})
^{2-} enhances the PJTE, followed by the S-S bond cleavage and significant structural distortions of the system. The change of the PJTE by chemical substitutions, accompanied either by puckering or by planarization of heterocyclic compounds, is discussed using as examples 1,4-ditinine and its S-oxygenated derivatives.

From general space-symmetry considerations, one would expect that in an isotropic environment any molecular system should have the highest possible symmetry, but in fact we see that in many cases the symmetry of the system is much lower (see [

In this paper, we first summarize the results of studies published over the last few years regarding the PJTE origin of the puckered structures of heterocyclic compounds. Then, the possibility of enhancement and suppression of the pseudo Jahn-Teller effect by oxidation, reduction and chemical substitutions and therefore the possibility of changing the structure of the systems is discussed in more detail using a series of heterocyclic compounds as examples.

Chart 1 lists the heterocyclic systems which were studied recently with regard to the PJTE in the origin of their puckering. The figures in the first column of Chart 1 show schematically the initial planar and equilibrium configurations of the compounds under study, with the indication of the symmetry of the configurations and the type of distortions. The second column indicates the PJTE problem, which was solved in the corresponding work.

The first group of compounds includes 1,4-dithiine molecules C_{4}S_{2}L_{4} [_{6}S_{8} with two S-atoms in the 1,4-position of the central six-membered ring [_{4}N_{2}H_{4}L_{2} [_{2h} symmetry, in the equilibrium configuration of C_{2v} symmetry they are bent at the S-S axis. The exception is the tricyclic C_{6}S_{8} molecule, which has C_{2h} symmetry in the planar configuration and C_{2} symmetry in the equilibrium.

The next group of molecules includes 1,2-dichalcogenins, 1,2-C_{4}X_{2}L_{4}, with X = O, S, Se, Te and L = H, F [_{6}S_{8} and C 6 S 8 2 − containing two S-atoms in the 1,2-positions of the central six-membered ring and one

Chart 1. List of heterocyclic systems studied with respect to the PJTE origin of their puckering.

thione (C=S) bond in the five-membered rings on its either side [_{12}H_{8}S_{2} and (C_{12}H_{8}S_{2})^{2−} [this work], 3,6-pyridazinedione derivatives N_{2}C_{4}H_{2}Y_{2}Z_{2} (Y = O, S, Se; Z = H, F, Cl, Br) [_{2}N_{2}L_{4} with L = H, F, Cl, Br [_{2} symmetry that leads to a puckered equilibrium structure of C_{2} symmetry.

Finally, one can distinguish the third group of five-membered heterocyclic molecules of the type C_{2}X_{3}Y_{2} (X = O, S, Se, Te; Y = H, F) [_{4}XY_{5} (X = N, P, As; Y = H, F, Cl) [_{6}Y_{6}O and C_{6}Y_{7}N with Y = H, F, Cl, Br [_{2v} symmetry is distorted along the out-of-plane b_{1} coordinate of instability that leads to equilibrium puckered structure of Cs symmetry.

In the next Section 2, we give the basic formulas of the PJTE theory, which will be needed in discussing the results. In Section 3, the possibility of flattening the 1,4-dithiine containing compounds by suppressing the PJTE via oxidation is discussed. The enhancement of the PJTE induced distortion by reduction is analyzed in Section 4 using the dibenzo[1,2]dithiine molecule as an example. In the last Section 5, the change of the PJTE by chemical substitution, accompanied by a change in the structure of heterocyclic compounds, is discussed using, as examples, 1,4-ditinine and its S-oxygenated derivatives. Geometry optimization and vibration frequency analysis for all the molecules considered in Sections 4 and 5 were performed at the B3LYP level of the DFT method [_{12}H_{8}S_{2} and (C_{12}H_{8}S_{2})^{2−} species was composed of eight electrons and nine active orbitals, while that for the 1,4-ditinine S-oxygenated derivatives included ten electrons and ten active states. All the calculations were carried out using the GAUSSIAN 09 program package [

The theory of the PJTE is well developed (see, for example, [_{0} for which the first derivatives are zero. At the minimum of the APES K = ( ∂ 2 E ( Q ) / ∂ Q 2 ) 0 > 0 , while K < 0 if the high-symmetry configuration is unstable.

The exact expression for K of any molecular system in nondegenerate ground state Ψ_{0} can be obtained in the second order perturbation theory with respect to small nuclear displacements Q:

K = 〈 Ψ 0 | ( ∂ 2 H / ∂ Q 2 ) 0 | Ψ 0 〉 − 2 ∑ i | 〈 Ψ 0 | ( ∂ H / ∂ Q ) 0 | Ψ i 〉 | 2 ( E i − E 0 ) . (1)

The first term in Equation (1), K_{0}, is the so-called primary force constant. It determines the restoring force arising when the nuclei are displaced with respect to the “frozen” electron distribution. The second term, K_{v},

K v = − 2 ∑ i F 0 i 2 E i − E 0 (2)

is the vibronic contribution to the curvature coming from the vibronic interaction of the ground Ψ_{0} and excited Ψ_{i} states under the nuclear displacements, E_{0} and E_{i} are their energies. For the ground state this term is always negative. The matrix elements F_{0i} in Equations (1) and (2),

F 0 i = 〈 Ψ 0 | ( ∂ H / ∂ Q ) 0 | Ψ i 〉 , (3)

are the off-diagonal vibronic coupling constants. They are non-zero only if the product of irreducible representations of the wave functions of the ground and excited states Γ Ψ 0 × Γ Ψ i contains the irreducible representation Γ Q of the displacement Q.

It was proved analytically and confirmed by a series of numerical calculations (see in [_{0} > 0. This means that the negative value of the curvature and, therefore, structural instabilities and distortions of high-symmetry nuclear configurations are only due to the vibronic contribution K_{v}. The instability takes place if the inequality

| K v | > K 0 or Δ < 2 F 2 / K 0 (4)

holds.

In the simplest two-level problem, when only one nondegenerate excited state contributes significantly to the instability of the ground state, we get the following secular equation for the two states (the energy is read of the ground state level):

| ( 1 / 2 ) K 0 Q 2 − ε F Q F Q ( 1 / 2 ) K 1 Q 2 − ε + Δ | = 0 (5)

where K_{0} and K_{1} are the primary force constants in the ground and excited states, respectively.

The solution of this 2 × 2 equation is straightforward:

ε 1 , 2 = 1 4 ( K 0 + K 1 ) Q 2 + Δ 2 ± 1 2 [ 1 2 ( K 0 − K 1 ) Q 2 − Δ ] 2 + 4 F 2 Q 2 (6)

Direct calculation of the vibronic constants F and K_{i} involved in this PJT model is rather difficult mathematically. Usually, the values of the parameters are estimated by fitting the solutions of the secular equations to the ab initio calculated energy profiles along the coordinate of instability. In the more complex case of the three-level problem, when two excited states are active in the PJT mixing, the matrix equation is

| 1 2 K 0 Q 2 − ε F 01 Q F 02 Q F 01 Q 1 2 K 1 Q 2 + Δ 01 − ε 0 F 02 Q 0 1 2 K 2 Q 2 + Δ 02 − ε | = | a − ε f g f b − ε 0 g 0 c − ε | = 0 (7)

Here K_{0}, K_{1}, and K_{2} are the primary force constants for the ground and two excited states, respectively, Δ_{01}, and Δ_{02} are the energy gaps between the ground and the two excited states, and F_{01} and F_{02} are the corresponding vibronic coupling constants,

F 01 = 〈 0 | ( ∂ H / ∂ Q ) 0 | 1 〉 and F 02 = 〈 0 1 | ( ∂ H / ∂ Q ) 0 | 2 〉 . (8)

The constants a, b, c, f, and g are simplifying denotations. Solving for the 3×3 secular determinant, one can get the following equation for the energies of the three states (1 ground state and 2 excited states) along the distortion coordinate:

ε 3 − ( a + b + c ) ε 2 + ( a b + a c + b c − f 2 − g 2 ) ε − a b c + c f 2 + b g 2 = 0 (9)

A similar equation can also be obtained for the four-level PJT problem. The procedure for estimating the vibronic constants in these cases can be found in the works [

Note that Equations (5)-(7) are correct for small Q values only; they characterize the changes in the K values induced by the term interactions under the Q displacements.

For a qualitative interpretation of phenomena related to the PJTE, it is worthwhile to use a more visual orbital scheme that provides information on the role of individual orbitals in the origin of instability. First of all we note that in Equation (1) ∂H/∂Q = ∂V/∂Q where V is the part of the Hamiltonian that depends on the nuclear coordinates

V = − ∑ α = 1 N ∑ i = 1 n Z α | r i − R α | + 1 2 ∑ α > β N Z α Z β | R α − R β | (10)

Here N is the number of atoms, Z_{α} are the nuclear charges, and R_{α} and r_{i} are the nuclear and electron coordinates respectively. Further, assume that the wavefunction of the excited state Ψ_{i} is the Slater determinant which differs from that of the ground state just by one-electron excitation α → m from the double occupied MO |φ_{α}ñ to the unoccupied MO |φ_{m}ñ. Then, the vibronic coupling constant F_{0i} in Equation (3) can be simply expressed by means of the off-diagonal orbital vibronic coupling constant (OVCC) f Q α m :

F 0 i = 〈 Ψ 0 | ( ∂ H / ∂ Q ) 0 | Ψ i 〉 = 2 f α m , f α m = 〈 φ α | ( ∂ V / ∂ Q ) 0 | φ m 〉 , (11)

and f Q α m are nonzero only if the product of symmetries of the two MOs |φ_{α}ñ and |φ_{m}ñ contains the symmetry of distortion Q. If |φ_{α}ñ is the single occupied MO then

F 0 i = 〈 Ψ 0 | ( ∂ H / ∂ Q ) 0 | Ψ i 〉 = f α m (12)

Thus, a complicated PJT problem involving many excited states can be reduced to the study of the orbital pairs which are mixed by distortion thereby destabilizing the ground state.

The reference nuclear configuration of all the considered systems from this group of compounds (see Introduction) is a planar structure of D_{2h} symmetry. In this configuration they have one imaginary frequency corresponding to the out-of-plane distortion of b_{1u} symmetry that leads to the C_{2v} butterfly-like equilibrium geometry [_{1u} displacements (note that this distortion is of b_{1u}-type if the symmetry axis of the second order is perpendicular to the plane of the molecule [_{3u}-type [^{1}A_{g} one. According to the PJTE (Equation (3)) the excited states that cause the instability of the ground state should have the ^{1}B_{1u} symmetry (A_{g} × B_{1u} = b_{1u}). In all the systems there is only one low-lying ^{1}B_{1u} excited state. Therefore, in the studied compounds we have a two-level (A_{g} + B_{1u}) Ä b_{1u} PJTE problem. An exception is the C_{6}S_{8} molecule in which three excited states of A_{u} symmetry contribute to the instability of the ground state trough the four-level (A_{g} + 1A_{u} + 2A_{u} + 3A_{u}) Ä a_{u} PJTE problem [

Then the energy profiles (cross-sections of the APES) along the coordinate of instability b_{1u} were calculated for the ground and the B_{1u} excited states. The vibronic coupling constants were estimated by means of fitting the solutions of the 2 × 2 secular equations (Equation (6)) to the corresponding energy profiles. Thus obtained numerical values of the PJTE parameters allow one to verify the condition of PJTE instability (4). Indeed, for small values of Q the expression for the energy can be approximately written as

ε g r = 1 2 K Q 2 = 1 2 ( K 0 − 2 F 2 Δ ) Q 2 (13)

Substituting in this formula the numerical data from the works [_{2h} nuclear configuration of the 1,4-dithiins with respect to the b_{1u} puckering modes and their distortions are due to the PJT mixing of the ground ^{1}A_{g} and the first excited ^{1}B_{1u} states.

In all the considered 1,4-dithiins these ^{1}B_{1u} excited states are mainly formed by one-electron excitations from the HOMO b_{1u} to the unoccupied a_{g} MO. Then the removal of an electron from this orbital reduces by half the PJTE negative contribution K_{v} to the curvature of the APES compared with a neutral molecule. Indeed, as follows from Equations (11) (12), the vibronic contribution to the curvature for the neutral molecules is approximately equal to K v ≈ − 4 f 2 / Δ , whereas for the radical cations K v ≈ − 2 f 2 / Δ where f = 〈 b 1 u | ( ∂ H / ∂ Q b 1 u ) 0 | a g 〉 .

The decrease in the vibronic contribution leads to a positive value for the curvature K of the APES, indicating that the cations of 1,4-dithiine containing compounds should have a planar nuclear configurations. To verify this statement, we have performed calculations of the oxidized molecules, which have shown the absence of any imaginary frequency in the ground state of their planar configuration, that is, the planar geometry of the cations is an equilibrium one.

Thus, one can conclude that bending of all the considered 1,4-dithiine containing molecules is due to the PJT coupling between the ground A_{g} and the excited B_{1u} states and restoration of their planar nuclear configurations upon oxidation is directly related to the decrease in the orbital vibronic coupling between these states, and hence, suppression of the PJTE.

In [_{6}S_{8} and its dianion C_{6}S_{8}^{2−} containing two S-atoms in the 1,2-positions of the central six-membered ring and one thione (C=S) bond in the five-membered rings on its either side. We have shown [_{6}S_{8} and C 6 S 8 2 − systems the out-of-plane distortions are due to the PJT coupling between the ground ^{1}A_{1} and two excited electronic states of ^{1}A_{2} symmetry and that reduction of C_{6}S_{8} up to C 6 S 8 2 − leads to enhancement of the PJTE and as a result, to a much stronger distortion of the dianion compared with the neutral molecule. In the present paper we present the results of the electronic structure and PJTE investigation of dibenzo[1,2]dithiine molecule, C_{12}H_{8}S_{2}.

For this purpose the electronic structure calculations and vibrational frequency analysis for C_{12}H_{8}S_{2} and (C_{12}H_{8}S_{2})^{2−} molecules were carried out in their reference planar nuclear configuration of C_{2v} symmetry. In both cases, the calculations have shown the presence of one imaginary frequency of a_{2} symmetry with the values of 146.3 cm^{−1} and 236.7 cm^{−1} for C_{12}H_{8}S_{2} and (C_{12}H_{8}S_{2})^{2−}, respectively. This means that the planar configuration is unstable with respect to the symmetrized out-of-plane a_{2} displacement of the atoms. The calculations give also the directions and relative values of displacements (

Both systems have a nondegenerate ground electronic state ^{1}A_{1}. Therefore, only the excited states of ^{1}A_{2} symmetry can produce the instability of the ground state (Equation (3)). In ^{1}A_{1} and the low-lying excited ^{1}A_{2} states of C_{12}H_{8}S_{2} and (C_{12}H_{8}S_{2})^{2−} molecules along the instability coordinate a_{2} are presented. In the neutral C_{12}H_{8}S_{2} compound the first 1^{1}A_{2} state (∆ = 6.36 eV) is formed by one-electron excitation a 2 ( HOMO ) → a 1 ( LUMO + 3 ) , while the 2^{1}A_{2} state corresponds to the excitation b 1 ( HOMO − 2 ) → b 2 ( LUMO + 2 ) . This 2^{1}A_{2} state gives only a slight negative contribution to the curvature of the APES firstly due to the large energy gap (8.05 eV) and, secondly, because of small value of the vibronic constant (F_{02} = 0.51 eV/Å). In the dianion (C_{12}H_{8}S_{2})^{2−} the first excited 1^{1}A_{2} state b 1 ( HOMO ) → b 2 ( LUMO ) does not contribute to the instability of the ground state. Although the energy gap is only 2.5 eV, the vibronic constant is very small (F_{01} = 0.23 eV/Å) due to the fact that the mixed orbitals are localized in different parts of the molecule (

By fitting the solutions of the secular Equations (5) to the ab initio calculated energy profiles along the coordinate of instability we can estimate the values of the PJTE parameters. As a result we obtain: for the neutral molecule K_{0} = 0.69 eV/Å^{2}, F_{01} = 1.56 eV/Å, ∆ = 6.36 eV, and according to Equation (13) the resulting value of the curvature K = −0.075 eV/Å^{2}; for the dianion K_{0} = 0.63 eV/Å^{2},

F_{02} = 1.51 eV/Å, ∆ = 5.52 eV, and K = −0.196 eV/Å^{2}. We see that the absolute value of the curvature for the reduced form estimated via the PJTE is almost two and a half times larger than this value for the neutral molecule. Thus, the reduction of the C_{12}H_{8}S_{2} molecule enhances significantly the PJTE and, as a consequence, enhances the distortions induced by it.

It should be noted that reduction does not always lead to enhancement of the PJTE and to an increase of the PJTE-induced distortions. In those cases when the excited state which is active in the PJTE is formed by excitation to the lowest unoccupied molecular orbital, the population of the latter in the process of reduction leads to suppression of the PJT effect and to the planarization of the distorted molecule. One such example is a cyclic molecule P_{6} which is distorted in a free state but it becomes planar being coordinated in the triple-decker sandwich complexes CpMoP_{6}MoCp due to the transfer of the electron density from the metal to the lowest unoccupied MO of P_{6} molecule [

In [_{2h} symmetry. In this configuration it does not have any low-lying excited states of B_{1u} symmetry which produce the PJTE instability with respect to the folding of the molecule. In comparison, the substituted 9,10-dihydroanthracene (9, 10-H_{2}An) has a folded geometry of C_{2v} symmetry. The addition of two hydrogen atoms in this system leads to the appearance of a new excited B_{1u} state, the vibronic mixing of which with the ground state results in the enhancement of the PJTE and the distortion [_{12}H_{8}S_{2}N_{4} molecule [

A similar suppression of the PJTE was shown to take place in the S-oxygenated derivative of 1,4-dithiine, C_{4}H_{4}(SO_{2})_{2}, which are planar [_{2} groups are due to enhancement and suppression of the PJTE.

For this purpose we studied a series of six molecules: 1,4-dithiine and five its oxygenated derivatives. Starting with the planar configuration, ab initio calculations of the electronic structure and vibrational frequencies of the systems in their high-symmetry nuclear configurations were carried out. One can see from _{2} groups have a planar conformation (no imaginary frequency), while the other four compounds have an imaginary frequency of b_{1u}/b_{1} symmetry. This means that the planar nuclear configurations of these molecules are unstable and undergo symmetry breaking along the normal coordinates Q(b_{1u}/b_{1}) transforming the planar D_{2h}/C_{2v} structures to the bending C_{2v}/C_{2} ones. The calculations give also the directions and relative values of these displacements (

All the considered molecules have a nondegenerate ground electronic states, ^{1}A_{g} in systems with D_{2h} reference configuration and ^{1}A_{1} for molecules with C_{2v} planar nuclear configuration (

states which produce the instability of the ground state should have the B_{1u}/B_{1} symmetry. It can be seen from

In 1,4-dithiine molecule, the main vibronic contribution to the instability comes from only one low-lying excited B_{1u} state which is formed by one-electron excitation from the HOMO b_{1u} to the (LUMO + 4) a_{g} (_{0} = 0.37 eV/Å^{2}, ∆ = 6.77 eV, and F = 1.32 eV/Å. They agree rather well with those calculated in [_{1u} puckering coordinate is negative, K = −0.14 eV/Å^{2}. Hence the condition of puckering instability (Equation (4)) is fully satisfied.

Passing to the C_{4}H_{4}(SO_{2})S and C_{4}H_{4}(SO_{2})_{2} molecules, one can see that the energy gaps between the ground and the excited B_{1}/B_{1u} states significantly increase (_{4}H_{4}(SO_{2})S molecule and F = 2 f b 1 u a g for the C_{4}H_{4}(SO_{2})_{2} one. Since the highest occupied b_{1} MO in the first molecule and b_{1u} MO in the second one have a zero value at one and, respectively, at both sulfur atoms (

orbital vibronic constants f b 1 a 1 / f b 1 u a 1 g . Both these factors (an increase of the energy gaps and a decrease of the vibronic coupling constants) lead to suppression of the PJTE, the condition of Equation (4) is not satisfied, so these molecules have planar conformations.

In the derivatives with one and two SO groups, oxygen atoms bring their 2p_{π} electrons to the π-electron system of the heterocycle, thereby providing the appearance of additional occupied b_{1} MOs (HOMO-1 in _{1} states (_{1} + 1B_{1} + 2B_{1}) Ä b_{1}. Following the procedure outlined in Section 1, and using calculated APES cross-sections for these compounds, we can estimate the values of the PJT parameters and the resulting values of the curvature of the APES in these cases. Such, for the C_{4}H_{4}SOS molecule we obtain K_{0} = 0.44 eV/Å^{2}, F_{01} = 1.26 eV/Å, Δ_{01} = 5.14 eV, F_{02} = 1.39 eV/Å, Δ_{02} = 8.57 eV. At small values of Q_{b}_{1} the resulting value of the curvature of the APES is equal to K ≈ K 0 − 2 F 01 2 / Δ 01 − 2 F 02 2 / Δ 02 = − 0.63 eV/Å^{2}. For the C_{4}H_{4}SOSO molecule these values are: K_{0} = 0.56 eV/Å^{2}, F_{01} = 1.20 eV/Å, Δ_{01} = 4.16 eV, F_{02} = 1.48 eV/Å, Δ_{02} = 6.42 eV, and K ≈ K 0 − 2 F 01 2 / Δ 01 − 2 F 02 2 / Δ 02 = − 0.81 eV/Å^{2}. We see that the absolute values of the curvature of the APES for these molecules estimated via the PJTE are much larger than that for the 1,4-dithiine molecule. At last, in the C_{4}H_{4}SO_{2}SO molecule, in comparison with the C_{4}H_{4}(SO_{2})S one, the energy gap between the ground and the excited B_{1} states decreases, while the value of the vibronic coupling constant F = 〈 A 1 | ( ∂ H / ∂ Q ) 0 | 1 B 1 〉 becomes quite large (F = 1.38 eV/Å) due to the fact that the mixed orbitals (HOMO b_{1} and (LUMO + 1) a_{1}) are localized on the OSC_{2} fragment of the molecule which undergoes the most significant distortion (see _{v} and to negative value of the curvature of the APES, so the heterocycle is distorted.

Thus, the oxygenation of the 1,4-dithiine molecule can lead to both the enhancement of the PJTE accompanied by puckering of the heterocycles (if oxygen atoms donate their 2p_{π} electrons to the π-system of the heterocycle) and to the PJTE suppression and subsequent flattening of derivatives with one and two SO_{2} groups.

Analysis of the results of published works and the results obtained in this paper allows us to draw the following conclusions:

1) The instability of planar nuclear configurations of all the considered heterocyclic molecules and their out-of-plane distortions are due to the pseudo Jahn-Teller effect.

2) If the instability of the ground state is provided mainly by the excited state formed by one-electron excitation from the HOMO to the appropriate by symmetry unoccupied MO, the oxidation of the systems by removing electrons from this MO leads to the suppression of the PJTE. In the case of 1,4-dithiine containing molecules, this results in the restoration of planar configuration.

3) In 1,2-dithiin containing molecules, reduction leads to enhancement of the PJTE followed by S-S bond cleavage and significant structural rearrangements of the systems.

4) Changes of the PJTE in the series of 1,4-ditinin and its S-oxygenated derivatives are accompanied either by out-of-plane distortions of their heterocycles (if oxygen atoms donate their 2p_{π} electrons to the π-system of a heterocycle) or by their flattening in derivatives with one and two SO_{2} groups.

Gorinchoy, N. (2018) Pseudo Jahn-Teller Effect in Puckering and Planarization of Heterocyclic Compounds. International Journal of Organic Chemistry, 8, 142-159. https://doi.org/10.4236/ijoc.2018.81010