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Conical intersections induced by the Renner effect: Selected systems
Chemical Physics Letters 504 (2011) 20–23
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Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Conical intersections induced by the Renner effect: Selected systems A. Papp a, G.J. Halász b, M.C. Bacchus-Montabonel c, Á. Vibók a,⇑ a
Department of Theoretical Physics, University of Debrecen, H-4010 Debrecen, P.O. Box 5, Hungary Department of Information Technology, University of Debrecen, H-4010 Debrecen, P.O. Box 12, Hungary c Laboratoire de Spectrométrie Ionique et Moléculaire, Université Lyon 1, CNRS, UMR5579, 43 Bd. du 11 Novembre 1918, 69622 Villeurbane Cedex, France b
a r t i c l e
i n f o
Article history: Received 1 December 2010 In final form 19 January 2011 Available online 22 January 2011
a b s t r a c t Conical intersections (CIs) play an important role in nonadiabatic molecular processes. We pointed out, that the Renner effect in polyatomic molecules is accompanied by symmetry-allowed CIs. Ab initio calculations and a perturbational approach have confirmed that two aligned CIs are created in the slightly bent C2 Hþ 2 molecule. In contrast, different results were obtained for the H2 CN molecule. To understand this new phenomenon, additional more detailed investigations have been performed for selected Renner– Teller systems. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction The Born–Oppenheimer approximation relies on the fact that the electrons, because of their smaller mass, move much faster than the heavier nuclei, so they follow the motion of the nuclei adiabatically [1–3]. In this picture, the nuclei move on a single potential energy surface created by the faster moving electrons. Although this approximation is sufficient to describe several chemical and physical processes, still in many important cases this picture breaks down. These are the so-called nonadiabatic processes where the nuclear and electronic motion can couple and so-called conical intersections (CIs) arise. In this situation the energy exchange between the electrons and nuclei can become significant [4–7]. Nonadiabatic processes are ubiquitous in photophysics and photochemistry. It was demonstrated for a variety of examples that conical intersections can provide the mechanism for extremely fast chemical processes, e.g. photodissociation, photoisomerization and internal conversion to the electronic ground state [8–13]. Therefore they may be considered as photochemically relevant decay channels. CIs can be formed between different electronic states starting from triatomic systems to truly large polyatomic molecules. Points of conical intersection are not isolated but are continuously connected, forming seams. Several important books, review articles and publications have demonstrated the existence and relevance of such intersections in recent years [6,7,14,15]. As is known the nonadiabatic coupling terms (NACT) couple the different electronic states in a molecule and play a crucial role in the close vicinity of a CI. Being singular at the CI, these couplings may become the source of numerous (statical and) dynamical phe⇑ Corresponding author. E-mail address: [email protected] (Á. Vibók). 0009-2614/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2011.01.050
nomena [4,3]. Couplings between different Born–Oppenheimer states become increasingly more important at higher energies, when the density of electronic states increases. Another type of electronic intersection is the Renner or Renner– Teller (RT) intersection [16]. As is known, in linear molecules, a doubly degenerate state arises when the orbital angular momentum around the molecular axis, K, is greater than zero. The degeneracy is spit on bending the molecule. In this geometry, the shape of the PES is quadratic in the vicinity of the degeneracy, as the coupling between the states is caused by second-order terms, while for the CIs the crossed PES are expected to behave linearly around of these points. Recently an unexpected connection between CIs and RT electronic intersections has been found [17–21]. We obtained that CIs may appear in polyatomic molecules exhibiting the Renner– Teller effect when these molecules are distorted from a linear configuration loosing both their axis and their plane of symmetry. Several ab initio calculations [22,23] and a perturbational approach [24] have confirmed that not only one but two aligned CIs will be created in a planar arrangement of the slightly bent Renner–Teller molecule. In these works the originally collinear C2 Hþ 2 molecule [25,26] was chosen as a sample system. Another tetra-atomic RT-type molecule, namely, the H2 CN was investigated by Baer and his coworkers [27]. Although initially they planned to study this molecule in order to obtain similar topological properties as for C2 Hþ 2 previously, surprisingly for the bent configuration of this compound they did not find any CI. In contrast, they have obtained that the H2 CN system keeps its Renner–Teller character even for the case of bent configurations. Since this result is (completely opposite) to those obtained from previous similar studies, our aim is to carry out more detailed investigations. Therefore, several different Renner–Teller type systems including the H2 CN molecule, are studied here. We simply calculate the energy differences in these systems between the ground
21
A. Papp et al. / Chemical Physics Letters 504 (2011) 20–23
and first excited states for several different distorted geometrical arrangements. The results obtained clearly confirm the ab initio and perturbational findings earlier presented namely, shifting one of the atoms from the collinear arrangement of the original Renner–Teller system, the bent molecule loses its Renner–Teller character, but at the same time, two CIs arise in the system. The Letter is organized as follows. Section 2 gives a brief description of the methodology and computational details of the ab initio calculations. In Section 3 the results and discussion are presented. Conclusions are given in the final Section. 2. Methodology and computational details This section will briefly summarize the geometrical arrangements of the systems studied, namely H2 CN, HC2 O, H2 Bþ 2 , and HC2 S. The quantum chemical and computational details of the numerical calculations will also be described.
(ii) For the molecule HC2 O, the C–C distance: RCC ¼ 1:26 Å; the C–H distance: RCH ¼ 1:06 Å, and the C–O distance: RCO ¼ 1:18 Å were chosen. The active space was used including all fifteen valence electrons distributed on thirteen orbitals. (iii) For the case of H2 Bþ 2 , the following collinear geometry was applied: the B–B distance: RBB ¼ 1:614 Å, and the B–H distance: RBH ¼ 1:175 Å. The active space was used including all seven valence electrons distributed on ten orbitals. (iv) For the molecule HC2 S, the C–C distance: RCC ¼ 1:225 Å; the C–H distance: RCH ¼ 1:06Å, and the C–S distance: RCS ¼ 1:625 Å were chosen. The active space was used including all 15 valence electrons distributed on thirteen orbitals. The computation were performed at the Jülich Supercomputing Centre on an Intel Xeon X5570 Cluster JuRoPA.
3. Results and discussion
In Figure 1, the geometrical configurations of the molecular systems studied are displayed. The arrangement of the atoms in the particular molecule is the following: if the molecule has two hydrogens they are the outer atoms while if it has two carbons they are the two inner atoms. Panel a shows the initial collinear configuration of the four-atomic system which possesses Renner–Teller character. In the actual numerical computations the two inner atoms define at the molecular axis, while the two outer atoms are shifted from the axis (see panel b). In these off-axis displacements ðq1 ; q2 Þ one of the outer atoms is kept fixed ðq1 Þ, while for the other atom the distance from the molecular axis ðq2 Þ is continuously changed. 2.2. Computational details In this Letter, we focus on locating the positions of the CIs formed by the two lowest energy electronic states of the distorted molecular systems. Four different systems were selected and studied by us. These are H2 CN; HC2 O; H2 Bþ 2 , and HC2 S in which the two lowest electronic energy states are degenerate and possess X 2 Pu symmetry in the linear configuration. Practically, the energy difference between the two states for several different bent configurations were calculated in Cs symmetry. The calculations were carried out by employing the MOLPRO program package [28] at the MCSCF and MRCI ðH2 Bþ 2 Þ levels of theory, using the 6-311G basis set. The particular computational details for the different systems are specified as follows: (i) For the case of H2 CN, the following collinear geometry was applied: the C–N distance: RCN ¼ 1:22 Å; the C–H distance: RCH ¼ 1:11 Å, and the N–H distance: RNH ¼ 1:03 Å. The active space was used including all eleven valence electrons distributed o ten orbitals.
a b
1
2
3
4
1 q1
4
q2
2
3
Figure 1. The two studied configurations of the studied four atomic molecules: (a) the collinear arrangement; (b) the nonsymmetric form created by two displaced atoms (1 and 4), where atom (1) is clamped. The numbers from left to right denote the atoms H, C, N, and H for the H2 CN; O, C, C, and H for the HC2 O; H, B, B, and H for the H2 Bþ 2 ; S, C, C, and H for the HC2 S molecules.
Figure 2 presents the energy difference curves between the 12 A0 and 12 A00 electronic states of the H2 CN molecule as a function of q2 . Here, q2 is the distance of the H atom (next to the C atom) from the molecular axis. The two electronic states 12 A0 and 12 A00 evolve from the collinear X 2 Pu state. The three different curves belong to three different q1 distances. In case of q1 ¼ 0; and q2 ¼ 0, one obtains back the Renner–Teller system and for this situation the energy difference between the 12 A0 and 12 A00 states is zero. Moving away from point q2 ¼ 0, the energy difference curve shows a quadratic dependence. It can also be seen that each of the two other curves intersects at two different points the q2 axis at the zero value of the corresponding energy differences. As the curves behave linearly in the close vicinity of the degeneracy points, a clear justification is thus obtained for the appearance of two real CIs. It is also observed that one of the intersection point occurs in the negative side of the q2 axis, while the other one occurs in the positive direction. Likewise, as in the previous case of the C2 Hþ 2 molecule [22,23], one of the CIs is placed much closer to the molecular axis than the other. Increasing the value of q1 the two degeneracy points are moving away from each other. Based on our earlier results, we assume that the CI located further away from the axis will finally dis-
0.20
2 2 E(1 A’)−E(1 A") [eV]
2.1. Geometrical arrangement
q1=0.0Å q1=0.2Å q1=0.4Å
0.00
−0.20
−0.40 −1.0
−0.5
0.0
q2 [Å]
0.5
1.0
Figure 2. Energy difference curves for the H2 CN molecule as a function of q2 related to two electronic states: the 12 A00 state and 12 A0 state (both evolving from the two components of the degenerate X 2 Pu state). These two states are the lower ones for the collinear configuration and at regions close to it. The geometrical arrangement is specified as: the two inner atoms (C and N) are clamped at the molecular axis, while one of the H atoms (next to the C atom) is shifted from the axis but keeping fixed ðq1 Þ, and the distance ðq2 Þ of the other H from the molecular axis is continuously changed. The three curves are calculated for the three different values of q1 .
A. Papp et al. / Chemical Physics Letters 504 (2011) 20–23
appear for a particular value of q1 . However, these kinds of studies are out of our present scope. Since the current system is not symmetric, as C2 Hþ 2 was, such a kind of ratio between the q1 and q2 displacements was not obtained as before [22,23]. Namely, the q2ð1Þ =q1 and q2ð2Þ =q1 values are not reciprocal to each other any more. (Here the q2ð1Þ and q2ð2Þ are the positions of the first and second CI points, respectively. Both of them belong to a certain q1 value for the off-axis displacement of the clamped H atom.) Similar results were obtained for the three remaining HC2 O, H2 Bþ 2 , and HC2 S systems. These curves are displayed in Figures 3–5. The qualitative results concerning the most important message of this Letter are the same, namely that two aligned CIs are created in a planar arrangement of the slightly bent Renner–Teller molecule. However, some minor differences can be found in each of
0.15
q1=0.0Å q1=0.2Å q1=0.4Å
0.10
E(12A’)−E(12A") [eV]
22
0.05
0.00
−0.05 −1.0
−0.5
0.0
0.5
1.0
q2 [Å] 0.15
Figure 5. Energy difference curves for the HC2 S molecule as a function of q2 related to two electronic states: the 12 A00 state and 12 A0 state (both evolving from the two components of the degenerate X 2 Pu state). These two states are the lower ones for the collinear configuration and at regions close to it. The geometrical arrangement is specified as: the two inner C atoms are clamped at the molecular axis, while the S atom is shifted from the axis but keeping fixed ðq1 Þ, and the distance ðq2 Þ of the H from the molecular axis is continuously changed. The three curves are calculated for the three different values of q1 .
q1=0.00Å q1=0.20Å q1=0.25Å
2 2 E(1 A’)−E(1 A") [eV]
0.10
0.05
0.00
−0.05 −0.5
0.0
q2 [Å]
0.5
1.0
Figure 3. Energy difference curves for the HC2 O molecule as a function of q2 related to two electronic states: the 12 A00 state and 12 A0 state (both evolving from the two components of the degenerate X 2 Pu state). These two states are the lower ones for the collinear configuration and at regions close to it. The geometrical arrangement is specified as: the two inner C atoms are clamped at the molecular axis, while the O atom is shifted from the axis but keeping fixed ðq1 Þ, and the distance ðq2 Þ of the H from the molecular axis is continuously changed. The three curves are calculated for the three different values of q1 .
the individual figures. First of all, the resulting two CIs (concerning their relative locations to the axis of the molecule) may appear on the same or on opposite sides of the molecule. From our examination of these systems it seems that for the case of symmetrical molþ ecules like H2 Bþ 2 and the earlier studied H2 C2 [22,23] the two CIs occur on the same side of the molecule, while for the antisymmetrical systems as the H2 CN, HC2 O, and HC2 S compounds, the two CIs appear on different sides of the molecule. For the symmetrical molecule H2 Bþ 2 , similarly to the case of C2 Hþ 2 , one can also find a reciprocal relation between the q2ð1Þ =q1 and q2ð2Þ =q1 quantities. However, these values are now significantly different from those obtained previously. In this case the relations are q2ð1Þ q1 =2:2 and q2ð2Þ q1 2:2, while for C2 Hþ 2 we obtained the relations q2ð1Þ q1 =16 and q2ð2Þ q1 16. 4. Conclusions
0.05
2 2 E(1 A’)−E(1 A") [eV]
0.00
q1=0.0Å q1=0.2Å q1=0.4Å
−0.05
−0.10
−0.15
−0.20 −0.5
0.0
0.5
1.0
q2 [Å] Figure 4. Energy difference curves for the H2 Bþ 2 molecule as a function of q2 related to two electronic states: the 12 A00 state and 12 A0 state (both evolving from the two 2 components of the degenerate X Pu state). These two states are the lower ones for the collinear configuration and at regions close to it. The geometrical arrangement is specified as: the two inner B atoms are clamped at the molecular axis, while one of the H atom is shifted from the axis but keeping fixed ðq1 Þ, and the distance ðq2 Þ of the other H from the molecular axis is continuously changed. The three curves are calculated for the three different values of q1 .
In some recent publications it was shown that certain CIs must exist in a polyatomic molecule exhibiting the Renner–Teller effect when these molecules are distorted from a linear configuration. Several ab initio calculations and also a perturbational approach have confirmed that not only one but two aligned CIs are formed in a planar arrangement of the slightly bent Renner–Teller active C2 Hþ 2 cation. Contrary to these results obtained earlier, a new finding has been reported by Baer and his collaborators investigating another Renner–Teller type system, namely the H2 CN molecule. Initially they planned to study this molecule so as to obtain similar topological properties as for the case of C2 Hþ 2 previously, but surprisingly they have obtained quite different results. Namely, they have found that the H2 CN system keeps its Renner–Teller character even for the case of bent configuration and the so-called symmetry-allowed CIs do not (at all) form. Since this result is completely contradictory to those which were obtained in the previous above mentioned studies, it seemed to be important to carry out some more detailed investigations. Therefore, selected Renner–Teller type sample systems were chosen including the H2 CN molecule, too, and the energy differences in these systems between the ground and first excited states were calculated for a few different distorted geometrical arrangements.
A. Papp et al. / Chemical Physics Letters 504 (2011) 20–23
Our new results confirmed again undoubtedly the already published interesting phenomena already published. Namely, in the slightly bent molecules two conical intersection seams are induced as a consequence of the disappearance of the Renner effect. Acknowledgment Á.V. acknowledges the OTKA Grant No. 80095 and the computational resources provided by the John-von-Neumann Institute, Research Centre Juelich (Project ID ehu01). The financial support by the COST Action CM0702 CUSPFEL is greatly acknowledged. References [1] M. Born, Ann. Phys. 84 (1927) 457. [2] M. Born, K. Huang, The Dynamical Theory of Crystal Lattices, Oxford University Press, Oxford, UK, 1954. [3] M. Baer, Beyond Born Oppenheimer: Electronic Non-Adiabatic Coupling Terms and Conical Intersections, Wiley, Hoboken, NJ, 2006. [4] H. Köppel, W. Domcke, L.S. Cederbaum, Adv. Chem. Phys. 57 (1984) 59. [5] M. Baer, G.D. Billing, The Role of Degenerate States in Chemistry Advances in Chemical Physics, vol. 124, Wiley-Interscience, New York, 2002. [6] G.A. Worth, L.S. Cederbaum, Annu. Rev. Phys. Chem. 55 (2004) 127.
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[7] W. Domcke, D.R. Yarkony, H. Köppel (Eds.), Conical Intersections: Electronic Structure, Dynamics and Spectroscopy, World Scientific, Singapore, 2004. [8] M. Klessinger, J. Michl, Excited States and Photochemistry of Organic Molecules, VCH Publishers Inc., New York, 1995. [9] P. Kukura et al., Science 310 (2005) 1006. [10] S. Hahn, G.J. Stock, Phys. Chem. B 104 (2000) 1146. [11] T. Schultz et al., Science 306 (2004) 1765. [12] Z. Lan, L.M. Frutos, A. Sobolewski, W. Domcke, Proc. Natl. Acad. Sci. 105 (2008) 12707. [13] T. Andruniow, N. Ferre, M. Olivucci, Proc. Natl. Acad. Sci. 101 (2004) 17908. [14] S. Matsika, Rev. Comput. Chem. 23 (2007) 83. [15] M. Baer, Phys. Rep. 358 (2002) 75. [16] E. Renner, Z. Phys. 92 (1934) 172. [17] G.J. Halász, Á. Vibók, R. Baer, M. Baer, J. Chem. Phys. 124 (2006) 081106. [18] G.J. Halász, Á. Vibók, R. Baer, M. Baer, J. Chem. Phys. 125 (2006) 094102. [19] G.J. Halász, Á. Vibók, R. Baer, M. Baer, J. Phys. A: Math. Theor. 40 (2007) 267. [20] G.J. Halász, Á. Vibók, D.K. Hoffman, D.J. Kouri, M. Baer, J. Chem. Phys. 126 (2007) 154309. [21] G.J. Halász, Á. Vibók, M. Baer, J. Chem. Phys. 127 (2007) 144108. [22] G.J. Halász, Á. Vibók, Int. J. Quant. Chem. (2010), doi:10.1002/qua.22598. [23] G.J. Halász, Á. Vibók, Chem. Phys. Lett. 494 (2010) 150. [24] T. Vértesi, R.J. Englman, Phys. B: Atom. Mol. Opt. Phys. 41 (2008) 025102. [25] M. Peric´, S. Peyerimhoff, Adv. Chem. Phys. 124 (2002) 583. [26] M. Peric´, B. Ostojic, J. Radic-Peric´, Phys. Rep. 290 (1997) 283. [27] A. Das, D. Mukhopadhyay, S. Adhikari, M. Baer, J. Chem. Phys. 133 (2010) 084107. [28] H.-J. Werner et al., MOLPRO, Version 2008.1, A Package of ab initio Programs. Available from:.
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Conical intersections induced by the Renner effect: Selected systems A. Papp a, G.J. Halász b, M.C. Bacchus-Montabonel c, Á. Vibók a,⇑ a
Department of Theoretical Physics, University of Debrecen, H-4010 Debrecen, P.O. Box 5, Hungary Department of Information Technology, University of Debrecen, H-4010 Debrecen, P.O. Box 12, Hungary c Laboratoire de Spectrométrie Ionique et Moléculaire, Université Lyon 1, CNRS, UMR5579, 43 Bd. du 11 Novembre 1918, 69622 Villeurbane Cedex, France b
a r t i c l e
i n f o
Article history: Received 1 December 2010 In final form 19 January 2011 Available online 22 January 2011
a b s t r a c t Conical intersections (CIs) play an important role in nonadiabatic molecular processes. We pointed out, that the Renner effect in polyatomic molecules is accompanied by symmetry-allowed CIs. Ab initio calculations and a perturbational approach have confirmed that two aligned CIs are created in the slightly bent C2 Hþ 2 molecule. In contrast, different results were obtained for the H2 CN molecule. To understand this new phenomenon, additional more detailed investigations have been performed for selected Renner– Teller systems. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction The Born–Oppenheimer approximation relies on the fact that the electrons, because of their smaller mass, move much faster than the heavier nuclei, so they follow the motion of the nuclei adiabatically [1–3]. In this picture, the nuclei move on a single potential energy surface created by the faster moving electrons. Although this approximation is sufficient to describe several chemical and physical processes, still in many important cases this picture breaks down. These are the so-called nonadiabatic processes where the nuclear and electronic motion can couple and so-called conical intersections (CIs) arise. In this situation the energy exchange between the electrons and nuclei can become significant [4–7]. Nonadiabatic processes are ubiquitous in photophysics and photochemistry. It was demonstrated for a variety of examples that conical intersections can provide the mechanism for extremely fast chemical processes, e.g. photodissociation, photoisomerization and internal conversion to the electronic ground state [8–13]. Therefore they may be considered as photochemically relevant decay channels. CIs can be formed between different electronic states starting from triatomic systems to truly large polyatomic molecules. Points of conical intersection are not isolated but are continuously connected, forming seams. Several important books, review articles and publications have demonstrated the existence and relevance of such intersections in recent years [6,7,14,15]. As is known the nonadiabatic coupling terms (NACT) couple the different electronic states in a molecule and play a crucial role in the close vicinity of a CI. Being singular at the CI, these couplings may become the source of numerous (statical and) dynamical phe⇑ Corresponding author. E-mail address: [email protected] (Á. Vibók). 0009-2614/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2011.01.050
nomena [4,3]. Couplings between different Born–Oppenheimer states become increasingly more important at higher energies, when the density of electronic states increases. Another type of electronic intersection is the Renner or Renner– Teller (RT) intersection [16]. As is known, in linear molecules, a doubly degenerate state arises when the orbital angular momentum around the molecular axis, K, is greater than zero. The degeneracy is spit on bending the molecule. In this geometry, the shape of the PES is quadratic in the vicinity of the degeneracy, as the coupling between the states is caused by second-order terms, while for the CIs the crossed PES are expected to behave linearly around of these points. Recently an unexpected connection between CIs and RT electronic intersections has been found [17–21]. We obtained that CIs may appear in polyatomic molecules exhibiting the Renner– Teller effect when these molecules are distorted from a linear configuration loosing both their axis and their plane of symmetry. Several ab initio calculations [22,23] and a perturbational approach [24] have confirmed that not only one but two aligned CIs will be created in a planar arrangement of the slightly bent Renner–Teller molecule. In these works the originally collinear C2 Hþ 2 molecule [25,26] was chosen as a sample system. Another tetra-atomic RT-type molecule, namely, the H2 CN was investigated by Baer and his coworkers [27]. Although initially they planned to study this molecule in order to obtain similar topological properties as for C2 Hþ 2 previously, surprisingly for the bent configuration of this compound they did not find any CI. In contrast, they have obtained that the H2 CN system keeps its Renner–Teller character even for the case of bent configurations. Since this result is (completely opposite) to those obtained from previous similar studies, our aim is to carry out more detailed investigations. Therefore, several different Renner–Teller type systems including the H2 CN molecule, are studied here. We simply calculate the energy differences in these systems between the ground
21
A. Papp et al. / Chemical Physics Letters 504 (2011) 20–23
and first excited states for several different distorted geometrical arrangements. The results obtained clearly confirm the ab initio and perturbational findings earlier presented namely, shifting one of the atoms from the collinear arrangement of the original Renner–Teller system, the bent molecule loses its Renner–Teller character, but at the same time, two CIs arise in the system. The Letter is organized as follows. Section 2 gives a brief description of the methodology and computational details of the ab initio calculations. In Section 3 the results and discussion are presented. Conclusions are given in the final Section. 2. Methodology and computational details This section will briefly summarize the geometrical arrangements of the systems studied, namely H2 CN, HC2 O, H2 Bþ 2 , and HC2 S. The quantum chemical and computational details of the numerical calculations will also be described.
(ii) For the molecule HC2 O, the C–C distance: RCC ¼ 1:26 Å; the C–H distance: RCH ¼ 1:06 Å, and the C–O distance: RCO ¼ 1:18 Å were chosen. The active space was used including all fifteen valence electrons distributed on thirteen orbitals. (iii) For the case of H2 Bþ 2 , the following collinear geometry was applied: the B–B distance: RBB ¼ 1:614 Å, and the B–H distance: RBH ¼ 1:175 Å. The active space was used including all seven valence electrons distributed on ten orbitals. (iv) For the molecule HC2 S, the C–C distance: RCC ¼ 1:225 Å; the C–H distance: RCH ¼ 1:06Å, and the C–S distance: RCS ¼ 1:625 Å were chosen. The active space was used including all 15 valence electrons distributed on thirteen orbitals. The computation were performed at the Jülich Supercomputing Centre on an Intel Xeon X5570 Cluster JuRoPA.
3. Results and discussion
In Figure 1, the geometrical configurations of the molecular systems studied are displayed. The arrangement of the atoms in the particular molecule is the following: if the molecule has two hydrogens they are the outer atoms while if it has two carbons they are the two inner atoms. Panel a shows the initial collinear configuration of the four-atomic system which possesses Renner–Teller character. In the actual numerical computations the two inner atoms define at the molecular axis, while the two outer atoms are shifted from the axis (see panel b). In these off-axis displacements ðq1 ; q2 Þ one of the outer atoms is kept fixed ðq1 Þ, while for the other atom the distance from the molecular axis ðq2 Þ is continuously changed. 2.2. Computational details In this Letter, we focus on locating the positions of the CIs formed by the two lowest energy electronic states of the distorted molecular systems. Four different systems were selected and studied by us. These are H2 CN; HC2 O; H2 Bþ 2 , and HC2 S in which the two lowest electronic energy states are degenerate and possess X 2 Pu symmetry in the linear configuration. Practically, the energy difference between the two states for several different bent configurations were calculated in Cs symmetry. The calculations were carried out by employing the MOLPRO program package [28] at the MCSCF and MRCI ðH2 Bþ 2 Þ levels of theory, using the 6-311G basis set. The particular computational details for the different systems are specified as follows: (i) For the case of H2 CN, the following collinear geometry was applied: the C–N distance: RCN ¼ 1:22 Å; the C–H distance: RCH ¼ 1:11 Å, and the N–H distance: RNH ¼ 1:03 Å. The active space was used including all eleven valence electrons distributed o ten orbitals.
a b
1
2
3
4
1 q1
4
q2
2
3
Figure 1. The two studied configurations of the studied four atomic molecules: (a) the collinear arrangement; (b) the nonsymmetric form created by two displaced atoms (1 and 4), where atom (1) is clamped. The numbers from left to right denote the atoms H, C, N, and H for the H2 CN; O, C, C, and H for the HC2 O; H, B, B, and H for the H2 Bþ 2 ; S, C, C, and H for the HC2 S molecules.
Figure 2 presents the energy difference curves between the 12 A0 and 12 A00 electronic states of the H2 CN molecule as a function of q2 . Here, q2 is the distance of the H atom (next to the C atom) from the molecular axis. The two electronic states 12 A0 and 12 A00 evolve from the collinear X 2 Pu state. The three different curves belong to three different q1 distances. In case of q1 ¼ 0; and q2 ¼ 0, one obtains back the Renner–Teller system and for this situation the energy difference between the 12 A0 and 12 A00 states is zero. Moving away from point q2 ¼ 0, the energy difference curve shows a quadratic dependence. It can also be seen that each of the two other curves intersects at two different points the q2 axis at the zero value of the corresponding energy differences. As the curves behave linearly in the close vicinity of the degeneracy points, a clear justification is thus obtained for the appearance of two real CIs. It is also observed that one of the intersection point occurs in the negative side of the q2 axis, while the other one occurs in the positive direction. Likewise, as in the previous case of the C2 Hþ 2 molecule [22,23], one of the CIs is placed much closer to the molecular axis than the other. Increasing the value of q1 the two degeneracy points are moving away from each other. Based on our earlier results, we assume that the CI located further away from the axis will finally dis-
0.20
2 2 E(1 A’)−E(1 A") [eV]
2.1. Geometrical arrangement
q1=0.0Å q1=0.2Å q1=0.4Å
0.00
−0.20
−0.40 −1.0
−0.5
0.0
q2 [Å]
0.5
1.0
Figure 2. Energy difference curves for the H2 CN molecule as a function of q2 related to two electronic states: the 12 A00 state and 12 A0 state (both evolving from the two components of the degenerate X 2 Pu state). These two states are the lower ones for the collinear configuration and at regions close to it. The geometrical arrangement is specified as: the two inner atoms (C and N) are clamped at the molecular axis, while one of the H atoms (next to the C atom) is shifted from the axis but keeping fixed ðq1 Þ, and the distance ðq2 Þ of the other H from the molecular axis is continuously changed. The three curves are calculated for the three different values of q1 .
A. Papp et al. / Chemical Physics Letters 504 (2011) 20–23
appear for a particular value of q1 . However, these kinds of studies are out of our present scope. Since the current system is not symmetric, as C2 Hþ 2 was, such a kind of ratio between the q1 and q2 displacements was not obtained as before [22,23]. Namely, the q2ð1Þ =q1 and q2ð2Þ =q1 values are not reciprocal to each other any more. (Here the q2ð1Þ and q2ð2Þ are the positions of the first and second CI points, respectively. Both of them belong to a certain q1 value for the off-axis displacement of the clamped H atom.) Similar results were obtained for the three remaining HC2 O, H2 Bþ 2 , and HC2 S systems. These curves are displayed in Figures 3–5. The qualitative results concerning the most important message of this Letter are the same, namely that two aligned CIs are created in a planar arrangement of the slightly bent Renner–Teller molecule. However, some minor differences can be found in each of
0.15
q1=0.0Å q1=0.2Å q1=0.4Å
0.10
E(12A’)−E(12A") [eV]
22
0.05
0.00
−0.05 −1.0
−0.5
0.0
0.5
1.0
q2 [Å] 0.15
Figure 5. Energy difference curves for the HC2 S molecule as a function of q2 related to two electronic states: the 12 A00 state and 12 A0 state (both evolving from the two components of the degenerate X 2 Pu state). These two states are the lower ones for the collinear configuration and at regions close to it. The geometrical arrangement is specified as: the two inner C atoms are clamped at the molecular axis, while the S atom is shifted from the axis but keeping fixed ðq1 Þ, and the distance ðq2 Þ of the H from the molecular axis is continuously changed. The three curves are calculated for the three different values of q1 .
q1=0.00Å q1=0.20Å q1=0.25Å
2 2 E(1 A’)−E(1 A") [eV]
0.10
0.05
0.00
−0.05 −0.5
0.0
q2 [Å]
0.5
1.0
Figure 3. Energy difference curves for the HC2 O molecule as a function of q2 related to two electronic states: the 12 A00 state and 12 A0 state (both evolving from the two components of the degenerate X 2 Pu state). These two states are the lower ones for the collinear configuration and at regions close to it. The geometrical arrangement is specified as: the two inner C atoms are clamped at the molecular axis, while the O atom is shifted from the axis but keeping fixed ðq1 Þ, and the distance ðq2 Þ of the H from the molecular axis is continuously changed. The three curves are calculated for the three different values of q1 .
the individual figures. First of all, the resulting two CIs (concerning their relative locations to the axis of the molecule) may appear on the same or on opposite sides of the molecule. From our examination of these systems it seems that for the case of symmetrical molþ ecules like H2 Bþ 2 and the earlier studied H2 C2 [22,23] the two CIs occur on the same side of the molecule, while for the antisymmetrical systems as the H2 CN, HC2 O, and HC2 S compounds, the two CIs appear on different sides of the molecule. For the symmetrical molecule H2 Bþ 2 , similarly to the case of C2 Hþ 2 , one can also find a reciprocal relation between the q2ð1Þ =q1 and q2ð2Þ =q1 quantities. However, these values are now significantly different from those obtained previously. In this case the relations are q2ð1Þ q1 =2:2 and q2ð2Þ q1 2:2, while for C2 Hþ 2 we obtained the relations q2ð1Þ q1 =16 and q2ð2Þ q1 16. 4. Conclusions
0.05
2 2 E(1 A’)−E(1 A") [eV]
0.00
q1=0.0Å q1=0.2Å q1=0.4Å
−0.05
−0.10
−0.15
−0.20 −0.5
0.0
0.5
1.0
q2 [Å] Figure 4. Energy difference curves for the H2 Bþ 2 molecule as a function of q2 related to two electronic states: the 12 A00 state and 12 A0 state (both evolving from the two 2 components of the degenerate X Pu state). These two states are the lower ones for the collinear configuration and at regions close to it. The geometrical arrangement is specified as: the two inner B atoms are clamped at the molecular axis, while one of the H atom is shifted from the axis but keeping fixed ðq1 Þ, and the distance ðq2 Þ of the other H from the molecular axis is continuously changed. The three curves are calculated for the three different values of q1 .
In some recent publications it was shown that certain CIs must exist in a polyatomic molecule exhibiting the Renner–Teller effect when these molecules are distorted from a linear configuration. Several ab initio calculations and also a perturbational approach have confirmed that not only one but two aligned CIs are formed in a planar arrangement of the slightly bent Renner–Teller active C2 Hþ 2 cation. Contrary to these results obtained earlier, a new finding has been reported by Baer and his collaborators investigating another Renner–Teller type system, namely the H2 CN molecule. Initially they planned to study this molecule so as to obtain similar topological properties as for the case of C2 Hþ 2 previously, but surprisingly they have obtained quite different results. Namely, they have found that the H2 CN system keeps its Renner–Teller character even for the case of bent configuration and the so-called symmetry-allowed CIs do not (at all) form. Since this result is completely contradictory to those which were obtained in the previous above mentioned studies, it seemed to be important to carry out some more detailed investigations. Therefore, selected Renner–Teller type sample systems were chosen including the H2 CN molecule, too, and the energy differences in these systems between the ground and first excited states were calculated for a few different distorted geometrical arrangements.
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Our new results confirmed again undoubtedly the already published interesting phenomena already published. Namely, in the slightly bent molecules two conical intersection seams are induced as a consequence of the disappearance of the Renner effect. Acknowledgment Á.V. acknowledges the OTKA Grant No. 80095 and the computational resources provided by the John-von-Neumann Institute, Research Centre Juelich (Project ID ehu01). The financial support by the COST Action CM0702 CUSPFEL is greatly acknowledged. References [1] M. Born, Ann. Phys. 84 (1927) 457. [2] M. Born, K. Huang, The Dynamical Theory of Crystal Lattices, Oxford University Press, Oxford, UK, 1954. [3] M. Baer, Beyond Born Oppenheimer: Electronic Non-Adiabatic Coupling Terms and Conical Intersections, Wiley, Hoboken, NJ, 2006. [4] H. Köppel, W. Domcke, L.S. Cederbaum, Adv. Chem. Phys. 57 (1984) 59. [5] M. Baer, G.D. Billing, The Role of Degenerate States in Chemistry Advances in Chemical Physics, vol. 124, Wiley-Interscience, New York, 2002. [6] G.A. Worth, L.S. Cederbaum, Annu. Rev. Phys. Chem. 55 (2004) 127.
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