Conical intersection

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Ideal Conical Intersection PeakedCI.jpg
Ideal Conical Intersection

In quantum chemistry, a conical intersection of two or more potential energy surfaces is the set of molecular geometry points where the potential energy surfaces are degenerate (intersect) and the non-adiabatic couplings between these states are non-vanishing. In the vicinity of conical intersections, the Born–Oppenheimer approximation breaks down and the coupling between electronic and nuclear motion becomes important, allowing non-adiabatic processes to take place. The location and characterization of conical intersections are therefore essential to the understanding of a wide range of important phenomena governed by non-adiabatic events, such as photoisomerization, photosynthesis, vision and the photostability of DNA. The conical intersection involving the ground electronic state potential energy surface of the C6H3F3+ molecular ion is discussed in connection with the Jahn–Teller effect in Section 13.4.2 on pages 380-388 of the textbook by Bunker and Jensen. [1]

Contents

Conical intersections are also called molecular funnels or diabolic points as they have become an established paradigm for understanding reaction mechanisms in photochemistry as important as transitions states in thermal chemistry. This comes from the very important role they play in non-radiative de-excitation transitions from excited electronic states to the ground electronic state of molecules. [2] For example, the stability of DNA with respect to the UV irradiation is due to such conical intersection. [3] The molecular wave packet excited to some electronic excited state by the UV photon follows the slope of the potential energy surface and reaches the conical intersection from above. At this point the very large vibronic coupling induces a non-radiative transition (surface-hopping) which leads the molecule back to its electronic ground state. The singularity of vibronic coupling at conical intersections is responsible for the existence of Geometric phase, which was discovered by Longuet-Higgins [4] in this context.

Degenerate points between potential energy surfaces lie in what is called the intersection or seam space with a dimensionality of 3N-8 (where N is the number of atoms). Any critical points in this space of degeneracy are characterised as minima, transition states or higher-order saddle points and can be connected to each other through the analogue of an intrinsic reaction coordinate in the seam. In benzene, for example, there is a recurrent connectivity pattern where permutationally isomeric seam segments are connected by intersections of a higher symmetry point group. [5] The remaining two dimensions that lift the energetic degeneracy of the system are known as the branching space.

Experimental observation

In order to be able to observe it, the process would need to be slowed down from femtoseconds to milliseconds. A novel 2023 quantum experiment, involving trapped-ion quantum computer, slowed down interference pattern of a single atom (caused by a conical intersection) by a factor of 100 billion, making a direct observation possible. [6]

Local characterization

Conical intersections are ubiquitous in both trivial and non-trivial chemical systems. In an ideal system of two dimensionalities, this can occur at one molecular geometry. If the potential energy surfaces are plotted as functions of the two coordinates, they form a cone centered at the degeneracy point. This is shown in the adjacent picture, where the upper and lower potential energy surfaces are plotted in different colors. The name conical intersection comes from this observation.

In diatomic molecules, the number of vibrational degrees of freedom is 1. Without the necessary two dimensions required to form the cone shape, conical intersections cannot exist in these molecules. Instead, the potential energy curves experience avoided crossings if they have the same point group symmetry, otherwise they can cross.

In molecules with three or more atoms, the number of degrees of freedom for molecular vibrations is at least 3. In these systems, when spin–orbit interaction is ignored, the degeneracy of conical intersection is lifted through first order by displacements in a two dimensional subspace of the nuclear coordinate space.

The two-dimensional degeneracy lifting subspace is referred to as the branching space or branching plane. This space is spanned by two vectors, the difference of energy gradient vectors of the two intersecting electronic states (the g vector), and the non-adiabatic coupling vector between these two states (the h vector). Because the electronic states are degenerate, the wave functions of the two electronic states are subject to an arbitrary rotation. Therefore, the g and h vectors are also subject to a related arbitrary rotation, despite the fact that the space spanned by the two vectors is invariant. To enable a consistent representation of the branching space, the set of wave functions that makes the g and h vectors orthogonal is usually chosen. This choice is unique up to the signs and switchings of the two vectors, and allows these two vectors to have proper symmetry when the molecular geometry is symmetric.

The degeneracy is preserved through first order by differential displacements that are perpendicular to the branching space. The space of non-degeneracy-lifting displacements, which is the orthogonal complement of the branching space, is termed the seam space. Movement within the seam space will take the molecule from one point of conical intersection to an adjacent point of conical intersection. The degeneracy space connecting different conical intersections can be explored and characterised using band and molecular dynamics methods. [7]

For an open shell molecule, when spin-orbit interaction is added to the,[ clarification needed ] the dimensionality of seam space is reduced. [8]

The presence of conical intersections can be detected experimentally. [9] It has been proposed that two-dimensional spectroscopy can be used to detect their presence through the modulation of the frequency of the vibrational coupling mode. [10] A more direct spectroscopy of conical intersections, based on ultrafast X-ray transient absorption spectroscopy, was proposed, offering new approaches to their study. [11]

Categorization by symmetry of intersecting electronic states

Conical intersections can occur between electronic states with the same or different point group symmetry, with the same or different spin symmetry. When restricted to a non-relativistic Coulomb Hamiltonian, conical intersections can be classified as symmetry-required, accidental symmetry-allowed, or accidental same-symmetry, according to the symmetry of the intersecting states.

A symmetry-required conical intersection is an intersection between two electronic states carrying the same multidimensional irreducible representation. For example, intersections between a pair of E states at a geometry that has a non-abelian group symmetry (e.g. C3h, C3v or D3h). It is named symmetry-required because these electronic states will always be degenerate as long as the symmetry is present. Symmetry-required intersections are often associated with Jahn–Teller effect.

An accidental symmetry-allowed conical intersection is an intersection between two electronic states that carry different point group symmetry. It is called accidental because the states may or may not be degenerate when the symmetry is present. Movement along one of the dimensions along which the degeneracy is lifted, the direction of the difference of the energy gradients of the two electronic states, will preserve the symmetry while displacements along the other degeneracy lifting dimension, the direction of the non-adiabatic couplings, will break the symmetry of the molecule. Thus, by enforcing the symmetry of the molecule, the degeneracy lifting effect caused by inter-state couplings is prevented. Therefore, the search for a symmetry-allowed intersection becomes a one-dimensional problem and does not require knowledge of the non-adiabatic couplings, significantly simplifying the effort. As a result, all the conical intersections found through quantum mechanical calculations during the early years of quantum chemistry were symmetry-allowed intersections.

An accidental same-symmetry conical intersection is an intersection between two electronic states that carry the same point group symmetry. While this type of intersection was traditionally more difficult to locate, a number of efficient searching algorithms and methods to compute non-adiabatic couplings have emerged in the past decade. It is now understood that same-symmetry intersections play as important a role in non-adiabatic processes as symmetry-allowed intersections.

See also

Related Research Articles

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In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the best-known mathematical approximation in molecular dynamics. Specifically, it is the assumption that the wave functions of atomic nuclei and electrons in a molecule can be treated separately, based on the fact that the nuclei are much heavier than the electrons. Due to the larger relative mass of a nucleus compared to an electron, the coordinates of the nuclei in a system are approximated as fixed, while the coordinates of the electrons are dynamic. The approach is named after Max Born and his 23-year-old graduate student J. Robert Oppenheimer, the latter of whom proposed it in 1927 during a period of intense ferment in the development of quantum mechanics.

In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Hamiltonian. The phenomenon was independently discovered by S. Pancharatnam (1956), in classical optics and by H. C. Longuet-Higgins (1958) in molecular physics; it was generalized by Michael Berry in (1984). It is also known as the Pancharatnam–Berry phase, Pancharatnam phase, or Berry phase. It can be seen in the conical intersection of potential energy surfaces and in the Aharonov–Bohm effect. Geometric phase around the conical intersection involving the ground electronic state of the C6H3F3+ molecular ion is discussed on pages 385–386 of the textbook by Bunker and Jensen. In the case of the Aharonov–Bohm effect, the adiabatic parameter is the magnetic field enclosed by two interference paths, and it is cyclic in the sense that these two paths form a loop. In the case of the conical intersection, the adiabatic parameters are the molecular coordinates. Apart from quantum mechanics, it arises in a variety of other wave systems, such as classical optics. As a rule of thumb, it can occur whenever there are at least two parameters characterizing a wave in the vicinity of some sort of singularity or hole in the topology; two parameters are required because either the set of nonsingular states will not be simply connected, or there will be nonzero holonomy.

<span class="mw-page-title-main">Rotational spectroscopy</span> Spectroscopy of quantized rotational states of gases

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In molecular physics, the molecular term symbol is a shorthand expression of the group representation and angular momenta that characterize the state of a molecule, i.e. its electronic quantum state which is an eigenstate of the electronic molecular Hamiltonian. It is the equivalent of the term symbol for the atomic case. However, the following presentation is restricted to the case of homonuclear diatomic molecules, or other symmetric molecules with an inversion centre. For heteronuclear diatomic molecules, the u/g symbol does not correspond to any exact symmetry of the electronic molecular Hamiltonian. In the case of less symmetric molecules the molecular term symbol contains the symbol of the group representation to which the molecular electronic state belongs.

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<span class="mw-page-title-main">Avoided crossing</span> Quantum physics and chemistry phenomenon

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<span class="mw-page-title-main">Degenerate energy levels</span> Energy level of a quantum system that corresponds to two or more different measurable states

In quantum mechanics, an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give the same value of energy upon measurement. The number of different states corresponding to a particular energy level is known as the degree of degeneracy of the level. It is represented mathematically by the Hamiltonian for the system having more than one linearly independent eigenstate with the same energy eigenvalue. When this is the case, energy alone is not enough to characterize what state the system is in, and other quantum numbers are needed to characterize the exact state when distinction is desired. In classical mechanics, this can be understood in terms of different possible trajectories corresponding to the same energy.

<span class="mw-page-title-main">MOLCAS</span> Computational chemistry software

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The Renner–Teller effect is observed in the spectra of molecules having electronic states that allow vibration through a linear configuration. For such molecules electronic states that are doubly degenerate at linearity will split into two close-lying nondegenerate states for non-linear configurations. As part of the Renner–Teller effect, the rovibronic levels of such a pair of states will be strongly Coriolis coupled by the rotational kinetic energy operator causing a breakdown of the Born-Oppenheimer approximation. This is to be contrasted with the Jahn–Teller effect which occurs for polyatomic molecules in electronic states that allow vibration through a symmetric nonlinear configuration, where the electronic state is degenerate, and which further involves a breakdown of the Born-Oppenheimer approximation but here caused by the vibrational kinetic energy operator.

<span class="mw-page-title-main">Molecular symmetry</span> Symmetry of molecules of chemical compounds

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The pseudo Jahn–Teller effect (PJTE), occasionally also known as second-order JTE, is a direct extension of the Jahn–Teller effect (JTE) where spontaneous symmetry breaking in polyatomic systems occurs even when the relevant electronic states are not degenerate. The PJTE can occur under the influence of sufficiently low-lying electronic excited states of appropriate symmetry. "The pseudo Jahn–Teller effect is the only source of instability and distortions of high-symmetry configurations of polyatomic systems in nondegenerate states, and it contributes significantly to the instability in degenerate states".

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<span class="mw-page-title-main">Isaac B. Bersuker</span> Biography of a prominent scientist known for his works in chemical physics and solid-state theory

Isaac B. Bersuker is a Soviet-Moldоvan-American theoretical physicist and quantum chemist whose principal research is in chemical physics, solid-state physics, and theoretical chemistry. Known for his "life-long years of experience in theoretical chemistry" working on the electronic structure and properties of coordination compounds, Isaac B. Bersuker is “one of the most widely recognized authorities” in the theory of the Jahn–Teller effect (JTE) and the pseudo-Jahn–Teller effect (PJTE). His accomplishments include explaining the polarization of the atomic core in Rydberg atoms, the effect of tunneling splitting in molecules and solids with a strong JTE, and the discovery of the PJTE origin of ferroelectricity in cubic perovskites. Known as the leading expert in JTE and PJTE, Bersuker is the permanent chairman of the international steering committee of the Jahn–Teller symposia. His present affiliation is with the Oden Institute for Computational Engineering and Science of the Department of Chemistry of the University of Texas at Austin.

References

  1. ↑ Molecular Symmetry and Spectroscopy, 2nd ed. Philip R. Bunker and Per Jensen, NRC Research Press, Ottawa (1998) ISBN   9780660196282
  2. ↑ Todd J. Martinez (September 2010). "Physical chemistry: Seaming is believing". Nature. 467 (7314): 412–413. Bibcode:2010Natur.467..412M. doi: 10.1038/467412a . PMID   20864993. S2CID   205058988.
  3. ↑ Kang, Hyuk; Kang Taek Lee; Boyong Jung; Yeon Jae Ko; Seong Keun Kim (October 2002). "Intrinsic Lifetimes of the Excited State of DNA and RNA Bases". J. Am. Chem. Soc. 124 (44): 12958–12959. doi:10.1021/ja027627x. PMID   12405817.
  4. ↑ H. C. Longuet Higgins; U. Öpik; M. H. L. Pryce; R. A. Sack (1958). "Studies of the Jahn-Teller effect .II. The dynamical problem". Proc. R. Soc. A. 244 (1236): 1–16. Bibcode:1958RSPSA.244....1L. doi:10.1098/rspa.1958.0022. S2CID   97141844.See page 12
  5. ↑ Lluís Blancafort (November 2010). "A global picture of the S1/S0 conical intersection seam of benzene" (PDF). Chemical Physics. 377 (1): 60–65. Bibcode:2010CP....377...60L. doi:10.1016/j.chemphys.2010.08.016. hdl: 10044/1/10099 .
  6. ↑ C. H. Valahu, V. C. Olaya-Agudelo, R. J. MacDonell, T. Navickas, A. D. Rao, M. J. Millican, J. B. Pérez-Sánchez, J. Yuen-Zhou, M. J. Biercuk, C. Hempel, T. R. Tan, I. Kassal. "Direct observation of geometric-phase interference in dynamics around a conical intersection." Nature Chemistry, 2023; DOI: 10.1038/s41557-023-01300-3
  7. ↑ Laino, T.; D. Passerone (2004). "Pseudo-dynamics and band optimizations: shedding light into conical intersection seams". Chemical Physics Letters. 389 (1–3): 1–6. doi:10.1016/j.cplett.2004.02.110.
  8. ↑ Matsika, Spiridoula; David R Yarkony (1 August 2001). "On the effects of spin-orbit coupling on conical intersection seams in molecules with an odd number of electrons. I. Locating the seam". Journal of Chemical Physics. 115 (5): 2038. Bibcode:2001JChPh.115.2038M. doi:10.1063/1.1378324.
  9. ↑ Stolow, Albert; Michael S. Schuurman (2018). "Dynamics at Conical Intersections". Annual Review of Physical Chemistry. 69: 427–459. doi:10.1146/annurev-physchem-052516-050721. PMID   29490199.
  10. ↑ Farag, M. H.; T. L. C. Jansen; J. Knoester (2016). "Probing the Inter-State Coupling near a Conical Intersection by Optical Spectroscopy". Journal of Physical Chemistry Letters. 7 (17): 3328–3334. doi:10.1021/acs.jpclett.6b01463. PMID   27509384.
  11. ↑ Neville, Simon P.; Majed Chergui; Albert Stolow; Michael S. Schuurman (2018). "Ultrafast X-Ray Spectroscopy of Conical Intersections". Physical Review Letters. 120 (24): 243001. arXiv: 1805.05733 . doi:10.1103/PhysRevLett.120.243001. PMID   29956989. S2CID   206312264.
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