Bifurcation analysis in terms of second-order Jahn—Teller effect

ELSEVIER

THEO CHEM Journal of Molecular Structure (Theochem) 130 (1994) 169-176

Bifurcation analysis in terms of second-order Jahn-Teller effect Tetsuya Takctsugu", Tsuneo Hirano b,* a Department

of Industrial Chemistry, Faculty of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan of Chemistry, Faculty of Science, Ochanomizu University. 2-1-1 Otsuka, Bunkyo-ku, Tokyo 112, Japan

b Department

(Received 19 July 1993; accepted 7 October 1993)

Abstract It is known that the reaction path for the dissociation reaction H 2CS - t H 2 + CS has Cs symmetry while the reaction path for the isomerization reaction H 2CS - t HCSH has C. symmetry at the Hartree-Fock (HF) level. In addition, the former reaction path is known to have a bifurcating region. In order to clarify the bifurcation mechanism in terms of the second-order Jahn-Teller (SOJT) effect, the intrinsic reaction coordinates (lRCs) and energies of the first-excited state (A") along the IRC were calculated for both reactions under the constraint of Cs symmetry with 6-31 G** basis sets at the HF level and at the complete active space SCF (CASSCF) level. At the HF level, bifurcation was proven to occur in both dissociation and isomerization reactions due to a vibronic interaction through the SOJT effect. At the CASSCF level, however, the energy separations between the ground and the A" excited states in both reactions are large enough to avoid state intermixing due to the SOJT effect, and the bifurcation disappears from each IRe. For the purpose of comparison, the IRCs of H 2CO for the dissociation to H 2 + CO and for the isomerization to HCOH were also calculated with 631G** basis sets at the HF level. As expected from the SOJT theory for the calculated large energy separation, bifurcation does not occur in both dissociation and isomerization reactions of H 2CO .

1. Introduction

When a chemical reaction is studied from a quantum chemical approach, the concept of reaction path is very useful. The reaction path should be located in a representative region around which a reaction system proceeds. Fukui [1] defined the intrinsic reaction coordinate (IRq along the steepest descent path from a saddle point in the mass-weighted coordinate system. Since the IRC should be of total symmetry [1], the points on the IRC path are located on the bottom of valleys, ridges or shelves perpendicular to the non-totally

• Corresponding author.

symmetric directions. Thus, the IRC path never loses its nuclear symmetry except in stationary points. Metiu et a1. [2] illustrated bifurcation as a transition from a valley into a ridge along the reaction path. Taking molecular vibrations perpendicular to the reaction path into consideration, the symmetry of the nuclear configuration is not necessarily conserved for the bifurcating region . Thus, the IRC path is not the real reaction path for the bifurcating region since the reaction system on the IRC path should proceed even on a ridge in the case of bifurcation. Bifurcation reflects the instability of the IRC. When the IRC contains the bifurcating region, molecular dynamic behavior becomes of interest.

0166-1280/94/507.00 © 1994 Elsevier Science B.V. All rights reserved SSDl 0166-1280(94)03529-G

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As to the bifurcation, its dynamic effect and geometrical character have been the focus of several studies [3-6], but there is no study of the mechanism of bifurcation. Using the second-order lahn-Teller (SOJT) effect [7-14], this paper proposes that bifurcation occurs because of vibronic interaction. Tachibana et al. [15] obtained the IRC of Cs symmetry with bifurcation for the dissociation reaction H 2CS ---+ H 2 + CS, and the IRC of C\ symmetry without bifurcation for the isomerization reaction H 2CS ---+ HCSH(tralls) [16,17]. For both reactions, we calculated the IRCs and the energies of the first-excited state along the IRC to demonstrate our SOlT theory for bifurcation. For the purpose of comparison, the same calculations were also performed for the dissociation reaction H 2CO ---+ H 2 + CO, and for the isomerization reaction H 2CO ---+ HCOH(tralls).

2. Theory According to the SOJT theory [7-14] based on the perturbational approach, the ground state energy of a molecule, distorted from a certain geometrical configuration along the non-totally symmetric ith normal mode, can be expressed as

(1) where Eo is the energy of the undistorted molecule, Q; is the ith normal coordinate and thef; terms, i.e. the force constants along the ith normal mode, are given by (2)

(3) Here '1/10 and 1/Jk are the electronic ground-state and kth excited-state wavefunctions, respectively, E k is the electronic kth excited-state energy and H is the electronic hamiltonian. The first term,loo,;, always has a positive sign. The second term, !ok,;, means a relaxation term with a negative sign, owing to the mixing of the kth excited state into the ground state through

the ith normal vibration. Thus, this second term corresponds to a vibronic interaction. When the second term overwhelms the first term, the feature of the potential surface changes from the valley to a ridge perpendicular to this direction. The most important state in the summation of !ok,; in Eq. (1) is in most cases the first-excited state due to the smallest value of the denominator, Eo - Ek [8]. Since the numerator of 10k,; has a non-zero value only when the direct product of the representations of 1/Jo and 1/Jk contains the representation of Q;, we should examine the stability of the molecular system in the direction of normal modes belonging to the symmetry representation of the non-totally symmetric first-excited state. When the separation of the energy levels of the ground and first-excited states is large, the bottom of the valley for the electronic ground state stays as a valley because the vibronic interaction with the first-excited state is expected to be weak. However, when the energy separation between the ground and the first-excited states is small, the bottom of the valley for the electronic ground state changes itself as a ridge owing to the mixing with the firstexcited state because the vibronic interaction is expected to be strong. The vibronic interaction changes as the reaction develops. When the vibronic interaction becomes strong enough, because of the small energy separation between the ground and the first-excited states, a bifurcation occurs and the valley along the reaction path turns out to be a ridge. Therefore, we can conclude that bifurcation occurs because of changes of vibronic interaction along thc reaction path and that the SOlT theory should be a powerful tool to predict and analyze the bifurcation.

3. Computations The molecular orbital calculations were carried out for the dissociation and isomerization reactions of H 2CS with 6-31G** basis sets at the HartreeFock (HF) and complete "active space SCF (CASSCF) levels using GAMESS [I8] and GAUSSIAN 90 programs [19]. Geometric structures at the minima and saddle points were located with the analytical gradient method. The IRCs for both

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T. Taketsugu, T. HiraI/oIl. Mol. Struct. (Theochem) 310 (1994) 169-176

reactions were calculated at the HF and the CASSCF levels. In order to clarify the bifurcation mechanism, a saddle point and an IRC for the isomerization reaction were calculated under the constraint of C, symmetry. The energy of the first-excited state was also calculated for geometries along the IRC at both computation levels. Similar calculations were also performed for the dissociation and isomerization reactions of H 2CO with the same basis sets at the HF level.

Here, we consider the bifurcation mechansim in terms of the SOlT effect. From this viewpoint, as described in the theory section, the bifurcation should occur due to the strong vibronic interaction, i.e. the 10k,; term in Eq. (3). In order to estimate the magnitude of the vibronic interaction, we should take both the numerator and denominator of 10k,; into consideration. The numerator indicates, according to a group theoretical consideration, whether or not 10k,; has a nonzero value. The denominator is the difference of energy levels of the ground and the kth excited states, and a smaller difference means a stronger vibronic interaction. Thus, we can roughly estimate the degree of vibronic interaction between the states in question judging from the symmetries of the states and the difference in their energy levels. In this case, the numerator of 10k,; has a non-zero value for the vibronic interaction of the electronic ground state (AI) with the excited singlet state (All) through the out-of-plane vibration (All), since AI x All X All = AI. Thus, this vibronic interaction should become greater around the bifurcating region. As shown in Fig. 1, the energy levels of the AI and All states get closer to each other around the bifurcating region in conformity with our prediction. It follows that our

4. Results and discussion

4.1 Dissociation reaction

01 H 2CS

Figure I shows changes of the force constant>. of the out-of-plane mode (All), and energy profiles of the ground (AI) and the first-excited (All) singlet states along the IRC (C s symmetry) calculated for the dissociation reaction of H 2CS at the HF level. In the region of the reaction coordinate s from -1.1 to -O.7bohru l / 2 , the force constant turns out to be of negative value and the IRC path changes from valley to ridge running perpendicular to the out-ofplane mode direction, which indicates the instability of the IRC [15] and the occurrence of bifurcation.

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T. Taket sugu, T. HiranojJ, Mol. Struct, [Th eochem} 3/0 (/99-1) /69-/76

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SOJT theory works well for the HF potential surface. The effect of electron correlation on the bifurcation can be discussed at the post-HF calculations such as CASSCF and CI. For the closed shell system, electron correlation lowers the energy of the ground state more than that of the first-excited state. Thus, the separation of the energy levels becomes larger even for the bifurcating region of the HF potential surface, resulting in the decrease of vibronic interaction necessary to invoke bifurcation. Thus, the ridge path with bifurcation demonstrated by the HF calculations may disappear on the potential surface obtained by electron correlation calculations. Figure 2 shows changes of the force constant Xof the out-of-plane mode (A"), and energy profiles of the ground (A') and the first-excited (A") singlet states along the IRC at the CASSCF level. The energy separation between the A' and A" states becomes larger than the energy separation at the bifurcating region in the HF energy profiles . The force con stant keeps a plus sign throughout the whole IRC, i.e. the bifurcation disappears. These results are in conformity with our SOJT theory.

4.2 Isom erization reaction of H 2CS

The transition state (TS) geometry for the isomerization reaction has C 1 symmetry at the HF level [15]. In order to discuss a bifurcation mechanism we searched a saddle point of the isomerization under the Cs symmetry constraint. The thus obtained HF saddle point turned out to be a second-order saddle point (SOSP), which has two negative eigenvalues of the hessian matrix. One eigenvalue corresponds to a totally symmetric normal coordinate, while the other to a nontotally symmetric normal coordinate, i.e. an out-of-plane normal coordinate. When the out-ofplane movement of atoms is not allowed under the symmetry constraint, the saddle point characterized as the SOSP behaves as the TS with only one negative eigenvalue, and the IRC can be determined straightforwardly. Figure 3 shows changes of the force constant Xof the out-of-plane mode (A"), and energy profiles of the ground (A') and the first-excited (A") singlet states along the IRC of Cssymmetry for the isomerization reaction of H 2CS at the HF level. The force constant ,\ shows a minus sign in the range of reaction coordinate s from -0.1 to 0.3 bohru 1j 2 across the SOSP. It follows that the

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although the energy separation between the A' and A" states remains small throughout the whole IRe. The bifurcating region requires a small energy separation between two interacting electronic states, but a small energy separation does not always result in bifurcation. We should also take the numerator of fOk,i into consideration. That is,

bifurcating region is localized around the saddle point. Our SOJT theory for bifurcation suggests that, in this bifurcating region, the energy of the A' and A" states are so close that the vibronic interaction becomes strong, resulting in the bifurcation. As shown in Fig. 3, the energy separation is not especially small in the bifurcating region

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T. Taketsugu, T. HiranojJ, Mol. Struct, (Theochem) 3/0 (1994) 169-176

174

disappearance of the bifurcation at the CASSCF level can be understood in terms of our SOJT theory.

the bifurcating region requires not only a small energy separation, but also a large numerator of fOk,i' For the reaction path considered here, the numerator should be large around the SOSP, since we observe bifurcation across the SOSP. At the CASSCF level, the TS geometry for the isomerization reaction is proved to have C, symmetry, not C I symmetry, and is located on the Cs symmetry IRC path. This suggests that bifurcation, which occurs around the SOSP of Cs symmetry at the HF level, disappears and the SOSP at the HF level becomes the TS with only one negative eigenvalue of the hessian matrix at the CASSCF level by the electron correlation effect. Figure 4 shows changes of the force constant >. of the out-ofplane mode (A"), and energy profiles of the ground (A') and the first-excited singlet (A") states along the IRC at the CASSCF level. Since the force constant keeps a plus sign even for the bifurcating regions at the HF level, bifurcation is proved to disappear at the CASSCF level as in the case of the dissociation reaction. As shown in Fig. 4, the separation of energy levels of the ground and the first-excited states by the CASSCF method is large in comparison with that at the HF level, resulting in the weak vibronic interaction. Thus,

4.3 Comparison with dissociation and isomerization reactions of H 2CO We also applied our SOJT theory to the dissociation and isomerization reactions of H 2CO, which are counterparts to be compared with those of H 2CS. The potential surfaces and the transition state geometries for these reactions of H 2CO have already been reported [20-26]. Figures 5 and 6 show changes of the force constant>. of the outof-plane mode (A") and energy profiles of the ground (A') and the first-excited singlet (A") states along the IRC for the dissociation and isomerization reactions of H 2CO, respectively. In both cases, the force constant>. keeps a plus sign throughout the whole IRC, which means that the IRC path has no bifurcating region. From our SOlT theory for bifurcation, the vibronic interaction should be weak throughout the reaction. As shown in Figs. 5 and 6, the separation of energy between the ground state (A') and the first-excited

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T. Taketsugu, Tc HiranojJ, Mol. Struct. {Theochem} 3/0 (1994) 169-/76

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state (A") are large even at the HF level of calculations. Therefore, our SOlT theory for bifurcation also works well for the two reactions of H 2CO in question.

5. Conclusion

This paper clarified the mechanism of bifurcation along the reaction path in terms of the SOlT theory. The bifurcation was proved to occur due to a vibronic interaction, i.e. the mixing of the nontotally symmetric first-excited state into the ground state through the molecular vibration of the same non-total symmetry. For the dissociation and isomerization reactions of H 2CS, we showed that a bifurcation occurs at the HF level where the A" excited state gets close to the A' ground state. We also discussed the effect of electron correlation on bifurcation. The electron correlation effect lowers the energy of the ground state more than that of the first-excited state; the separation of energy levels between the ground and the firstexcited states becomes large, and the vibronic interaction becomes weak. For the dissociation reaction, the bifurcation occurs at the HF level while it disappears at the CASSCF level. For

the isomerization reaction, the HF level calculations predicted the TS and IRC of C I symmetry as the results of the bifurcation, while the TS and IRC were proven to have C, symmetry at the CASSCF level as the result of the disappearance of bifurcation. Our SOlT theory also works well for the IRCs of the dissociation and isomerization reactions of H 2CO, which have no bifurcating region. Once we know an energy separation between the interacting states, we can predict the stability of the reaction path, i.e. the possible occurrence of bifurcation. Alternatively, once the bifurcating region is picked up, the information about the separation of energy levels along the reaction path can be derived. References [I] K. Fukui, J. Phys. Chern., 74 (1970) 4161. [2] H. Metiu, J. Ross, R. Silbey and T.F. George, J. Chern. Phys., 61 (1974) 3200. [3] P. Valtazanos and K. Ruedenberg, Theor. Chirn. Acta, 69 (1986) 281. [4] \V.A. Kraus and A.E. DePristo, Theor, Chirn. Acta, 69 (1986) 309. [5] M.V. Basilevsky, TheoL Chirn. Acta, 72 (1987) 63.

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