Letter Cite This: J. Phys. Chem. Lett. 2018, 9, 146−149
pubs.acs.org/JPCL
Topological Origins of Bound States in the Continuum for Systems with Conical Intersections Sarah Henshaw†,‡ and Artur F. Izmaylov*,†,‡ †
Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario M1C 1A4, Canada Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada
‡
ABSTRACT: Bound states in the continuum (BSCs) were reported in a linear vibronic coupling model with a conical intersection (CI) [Cederbaum et al. Phys. Rev. Lett. 2003, 90, 013001]. It was also found that these states are destroyed within the Born− Oppenheimer approximation (BOA). We investigate whether a nontrivial topological or geometric phase (GP) associated with the CI is responsible for BSCs. To address this question we explore modifications of the original 2D two-state linear vibronic coupling model supporting BSCs. These modifications either add GP effects after the BOA or remove the GP within a two-state problem. Using the stabilization graph technique we have shown that the GP is crucial for emergence of BSCs.
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states have different electronic symmetries.8 The other parameters were set to wx = 0.015, wy = 0.009, λ = 0.01, ε = 0.04, β = 0.5, and δ = 0.5. Even though all vibrational states of the Vb are coupled to continuum states of Vc, it was discovered that the lowest state of the Vb gives rise to a BSC. A simple way to understand this result is to inspect what continuum states can be coupled to the ground state of Vb. Because of the linear dependence of Vbc and separable x and y components of Vc, all such states can be denoted as (k, 1), where k is the quantum number along the x direction and 1 is the vibrational quantum along the y direction. Comparing the energy of the Vb ground state, (wx + wy)/2 = 0.012, with the lowest energy of the (k, 1) manifold, 3wy/2 = 0.0135, a gap that makes the bound ground state off-resonance from states of the coupled (k, 1) continuum becomes evident. This gap lifts the edge of the coupled continuum above the energy of the bound state and thus effectively breaks the state emergence in the continuum. Transforming the problem to the adiabatic representation gives rise to a conical intersection (CI) (Figure 1). Considering that CI’s energy (ECI ≈ 0.015) is higher than the energy of the ground state of the Vb potential (0.012), one can conclude that the Born−Oppenheimer approximation (BOA) may be quite adequate at least for this state. It turns out that switching to the BOA destroys the bound state and gives rise to a resonance state. This shows that purely energetic consideration is not accurate in this case and some symmetry is broken when the BOA is introduced. In general, CIs not only promote transitions between different electronic states but also introduce Berry or geometric
INTRODUCTION In most quantum-mechanical problems, the bound states emerged in the continuum of unbound states become resonances with a finite lifetime when their interaction with the continuum is accounted. However, from the dawn of quantum theory, von Neumann and Wigner1 discovered potentials that support spatially bounded, discrete states with energies within the continuum. Later, such bound states in the continuum (BSCs) were found not only in the timeindependent Schrödinger equation but also in the wave equation,2−6 where they gave rise to nanophotonic applications in lasing, sensing, and filtering.7 In 2003, BSCs were also found in a nonadiabatic model relevant to molecular dissociation.8 Cederbaum et al. considered a 2D two-state linear vibronic coupling Hamiltonian8 ij Vb Vbc yz zz Hdia = TN 12 + jjjj zz V V bc c k {
(1)
1
where TN = − 2 (wx ∂ 2x + wy∂ 2y) is the nuclear kinetic energy operator (atomic units are used throughout) and Vb and Vc are the bound and unbound potentials wy w Vb = x x 2 + y 2 (2) 2 2 Vc = εe−β(x + δ) +
wy 2
y2
(3)
coupled by a linear potential Vbc = λy. The coordinates x and y can be thought of as mass- and frequency-weighted normal modes. Such a choice of nuclear coordinates is possible for model Hamiltonians where intersecting bound and continuum © 2017 American Chemical Society
Received: October 20, 2017 Accepted: December 20, 2017 Published: December 20, 2017 146
DOI: 10.1021/acs.jpclett.7b02791 J. Phys. Chem. Lett. 2018, 9, 146−149
Letter
The Journal of Physical Chemistry Letters
e−iθ is applied to HBO to avoid working with double-valued nuclear wave functions. This results in our third Hamiltonian GP HBO = eiθHBOe−iθ
= TN + τGP + W −
(8)
where τGP =
(∇θ )2 i + i∇θ ∇ + ∇2 θ 2 2
and ∇ = ( wx ∂x,
phase (GP).9−12 The manifestation of GP is in changing a sign of the electronic wave function upon a continuous evolution along a closed path encircling a CI. To ensure that the total wave function is single-valued, the nuclear wave function must also change the sign. This sign change can be introduced via a phase factor that is position-dependent and constitutes exponential function of the GP. GP is a feature of the adiabatic representation and does not appear in the diabatic representation. The effect of the GP on motion on the single lower adiabatic potential energy surface was first described by Mead and Truhlar10 and is referred to as the Molecular Aharonov-Bohm effect.13 In several instances GP played a crucial role in obtaining qualitatively correct results when problems were considered in the adiabatic representation.14−20 Therefore, it is quite natural to inquire whether the GP is the reason for the appearance of BSCs in this system.
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METHODS To investigate the role of the GP we will consider timeindependent Schrödinger equation for four Hamiltonians. The original diabatic Hamiltonian Hdia (eq 1) is used as a reference that includes all contributions: nonadiabatic transitions and GP effects. In contrast, the Born−Oppenheimer (BO) Hamiltonian HBO does not have any of these effects. Formally, to obtain HBO one needs to diagonalize the potential matrix in Hdia using the unitary matrix
2Vbc 1 θ(x , y) = arctan 2 Vc − Vb
Table 1. Energies of the Lowest Localized State of Different Hamiltonians (in 10−3 au)
(4)
(5)
Hdia
HBO
HGP BO
HnoGP dia
10.29
9.23 − i0.10
10.23
10.02 − i0.04
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Removing the high-energy electronic state and nonadiabatic couplings gives HBO = TN + W −
wy ∂y). HGP BO includes only the GP effects
but excludes all nonadiabatic transitions. Finally, to obtain a picture where nonadiabatic transitions are preserved but GP effects are removed, we use the diabatic Hamiltonian identical to Hdia but with modified Vbc = c|y|. Introducing the absolute value function in the coupling was shown to remove the GP when the diabatic-to-adiabatic transformation is done.21 We will refer to this Hamiltonian as noGP can be a Hamiltonian obtained HnoGP dia . An alternative to Hdia by transforming Hdia to the adiabatic representation and ignoring the double-valued boundary conditions introduced by this transformation. The main advantage of HnoGP is numerical dia robustness of the diabatic representation. To determine if an eigenstate is a bound or resonant state, its lifetime is calculated using the stabilization method.22 Stabilization graphs are used to obtain complex energies in the form E = ER − iΓ/2, where ER is the real part and Γ is inversely proportional to the lifetime of the state.23 A finite lifetime of the state is an indication of its resonance character. All Hamiltonians are transformed to matrices using static basis functions. The same product basis of functions in x and y directions was used for the two electronic states of the diabatic Hamiltonians as well as for the single electronic state adiabatic Hamiltonians. Particle in a box (PB) eigenfunctions were used in the x direction. These functions introduce the length of the box as a natural stabilization parameter for the stabilization method.22 The box interval [−4,L] chosen to cover both bound and continuum parts of the potential (eqs 2 and 3). Harmonic oscillator eigenfunctions were used in the y direction. Energy of low-lying states for all Hamiltonians converged within 10−6 au with 150 PB and 20 harmonic oscillator eigenfunctions.
Figure 1. Model potentials in the adiabatic representation.
i cos θ sin θ yz zz U = jjjj z k− sin θ cos θ { where angle θ is
(9)
RESULTS Table 1 and Figures 2 and 3 summarize results of stabilization calculations for the lowest localized states of the four Hamiltonians. The Hamiltonians accounting for GP effects have only real components of energy up to a numerical error, while removing GP effects leads to a resonance character of the lowest localized state. Thus the BSC can be clearly related to the presence of the GP. An intuitive picture of this relation is that in the absence of the GP the nuclear wave packet can
(6)
where 1 1 (Vb + Vc) − (Vb − Vc)2 + 4Vbc2 (7) 2 2 To include GP in the BO representation we use the Mead and Truhlar approach,10 where the double-valued projector W− =
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DOI: 10.1021/acs.jpclett.7b02791 J. Phys. Chem. Lett. 2018, 9, 146−149
Letter
The Journal of Physical Chemistry Letters
noGP Figure 2. Stabilization graphs for the lowest localized state of various Hamiltonians: (a) Hdia, (b) HBO, (c) HGP BO, and (d) Hdia .
Figure 3. Contour of the square root of the ground electronic state nuclear probability density of the lowest localized state for different noGP Hamiltonians with box length 20.4 au: (a) Hdia, (b) HBO, (c) HGP BO, and (d) Hdia .
tunnel through a barrier separating the bound part of the W− potential from its unbound counterpart, whereas the addition of the GP creates destructive interference between the parts of the wave packet that tunnel through the barrier on different sides of the CI. The absolute overlap of the Hdia and HGP BO wave GP functions, |⟨Ψdia|ΨBO ⟩| is 0.93, which is relatively large considering that the total probability to find the system described by |Ψdia⟩ in the ground electronic state is 0.97 (see also Figure 3a,c). A common correction to the HBO Hamiltonian is the diagonal Born−Oppenheimer correction (DBOC),24 which for our model can be written as (∇θ)2/2. The DBOC diverges at the CI and can potentially reduce the lifetime of the resonance. This is indeed the case; stabilization calculations with HBO + (∇θ)2/2 provided E = (11.03 − i0.13) × 10−3 for the lowest resonance state. In conclusion, we have unambiguously shown that the nontrivial GP is the reason for bound states in the continuum in the considered model problem. This result is yet another illustration that inclusion of the GP can qualitatively change results in nonadiabatic problems and cannot be ignored a
priori. The current result is a direct analog of GP-induced localization obtained in the double-well potential problem.14,25 As in the double-well case, one can expect that there is a certain range of parameters of the linear vibronic model that supports the BSC. This is in accord with previous works on GP effects in tunneling that did not find BSCs for a very similar model with different values of corresponding parameters.19,20 Yet one can hope that it is possible to engineer a molecular system where GP will not only slow down the tunneling process but will completely freeze it by giving rise to BSCs.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Artur F. Izmaylov: 0000-0001-8035-6020 Notes
The authors declare no competing financial interest. 148
DOI: 10.1021/acs.jpclett.7b02791 J. Phys. Chem. Lett. 2018, 9, 146−149
Letter
The Journal of Physical Chemistry Letters
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(21) Izmaylov, A. F.; Li, J.; Joubert-Doriol, L. Diabatic Definition of Geometric Phase Effects. J. Chem. Theory Comput. 2016, 12, 5278− 5283. (22) Mandelshtam, V. A.; Ravuri, T. R.; Taylor, H. S. Calculation of The Density of Resonance States Using the Stabilization Method. Phys. Rev. Lett. 1993, 70, 1932−1935. (23) Moiseyev, N. Non-Hermitian Quantum Mechanics; Cambridge University Press: Cambridge, U.K., 2011. (24) Ryabinkin, I. G.; Joubert-Doriol, L.; Izmaylov, A. F. When Do We Need to Account for the Geometric Phase in Excited State Dynamics? J. Chem. Phys. 2014, 140, 214116. (25) Joubert-Doriol, L.; Izmaylov, A. F. Molecular “Topological Insulators”: A Case Study of Electron Transfer in the Bis(methylene) Adamantyl Carbocation. Chem. Commun. 2017, 53, 7365−7368.
ACKNOWLEDGMENTS We thank Ilya Ryabinkin and Loı̈c Joubert-Doriol for helpful discussions. This work was supported by a Sloan Research Fellowship, Natural Sciences and Engineering Research Council of Canada (NSERC), and an Ontario Graduate Scholarship (OGS).
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REFERENCES
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DOI: 10.1021/acs.jpclett.7b02791 J. Phys. Chem. Lett. 2018, 9, 146−149
