^{1,a)}, Adrian E. Roitberg

^{2}, Tammie Nelson

^{3}and Sergei Tretiak

^{3}

### Abstract

Radiationless transitions between electronic excited states in polyatomic molecules take place through unavoided crossings of the potential energysurfaces with substantial non-adiabatic coupling between the respective adiabatic states. While the extent in time of these couplings are large enough, these transitions can be reasonably well simulated through quantum transitions using trajectory surface hopping-like methods. In addition, complex molecular systems may have multiple “trivial” unavoided crossings between noninteracting states. In these cases, the non-adiabatic couplings are described as sharp peaks strongly localized in time. Therefore, their modeling is commonly subjected to the identification of regions close to the particular instantaneous nuclear configurations for which the energysurfaces actually cross each other. Here, we present a novel procedure to identify and treat these regions of unavoided crossings between non-interacting states using the so-called Min-Cost algorithm. The method differentiates between unavoided crossings between interacting states (simulated by quantum hops), and trivial unavoided crossings between non-interacting states (detected by tracking the states in time with Min-Cost procedure). We discuss its implementation within our recently developed non-adiabaticexcited statemolecular dynamics framework. Fragments of two- and four-ring linear polyphenylene ethynylene chromophore units at various separations have been used as a representative molecular system to test the algorithm. Our results enable us to distinguish and analyze the main features of these different types of radiationless transitions the molecular system undertakes during internal conversion.

This work was partially supported by CONICET, UNQ, ANPCyT (PICT-2010-2375) and the National Science Foundation Grant No. CHE-0239120). S.T. and T.N. acknowledge the support of the U.S. Department of Energy through the Los Alamos National Laboratory (LANL) LDRD Program. LANL is operated by Los Alamos National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy (DOE) under contract DE-AC52-06NA25396. We acknowledge support of Center for qIntegrated Nanotechnology (CINT) and Center for Nonlinear Studies (CNLS) at LANL.

I. INTRODUCTION

II. METHODS

A. The NA-ESMD background

B. Analysis of the spatial localization of electronic transition densities

C. Identification of unavoided crossings

D. Molecular dynamics simulations

III. RESULTS AND DISCUSSION

IV. CONCLUSIONS

### Key Topics

- Excited states
- 34.0
- Non adiabatic couplings
- 22.0
- Molecular dynamics
- 19.0
- Energy transfer
- 15.0
- Surface states
- 13.0

## Figures

(a) Chemical structure of the dendritic molecule consisting of two two-ring, one three-ring, and one four-ring linear poly(phenylene ehynylene) (PPE) units linked by meta-substitution. (b) Fragments of the dendrite (2,2-PPE and 4-PPE molecules) used to test the numerical procedure to identify unavoided crossings.

(a) Chemical structure of the dendritic molecule consisting of two two-ring, one three-ring, and one four-ring linear poly(phenylene ehynylene) (PPE) units linked by meta-substitution. (b) Fragments of the dendrite (2,2-PPE and 4-PPE molecules) used to test the numerical procedure to identify unavoided crossings.

Variation of the adiabatic energies obtained during a short-time NA-ESMD dynamics of the 2,2-PPE and 4-PPE molecules separated by 100 Å. Blue and red lines correspond to the excited states fully localized on 2,2-PPE and 4-PPE fragments, respectively, as determined by the transition density matrix analysis.

Variation of the adiabatic energies obtained during a short-time NA-ESMD dynamics of the 2,2-PPE and 4-PPE molecules separated by 100 Å. Blue and red lines correspond to the excited states fully localized on 2,2-PPE and 4-PPE fragments, respectively, as determined by the transition density matrix analysis.

Typical trivial unavoided crossings between non-interacting adiabatic states obtained from a NA-ESMD simulation of the 2,2-PPE and 4-PPE fragments separated by 500 Å. (a) Time evolution of the adiabatic state energies. The molecular fragments where the transition densities are localized for each state, are also shown. The arrows indicate the dynamics of the system that remains on the same diabatic state. (b) Variation of the respective time-dependent nonadiabatic coupling (NACT) between S_{2} and S_{3} states.

Typical trivial unavoided crossings between non-interacting adiabatic states obtained from a NA-ESMD simulation of the 2,2-PPE and 4-PPE fragments separated by 500 Å. (a) Time evolution of the adiabatic state energies. The molecular fragments where the transition densities are localized for each state, are also shown. The arrows indicate the dynamics of the system that remains on the same diabatic state. (b) Variation of the respective time-dependent nonadiabatic coupling (NACT) between S_{2} and S_{3} states.

Variation of the time-dependent fraction of the transition densities localized at the 2,2-PPE (blue lines) and 4-PPE (red lines) fragments when they are separated by different distances. (a), (b), and (c) are obtained from the NA-ESMD simulations without considering any specific treatment of unavoiding crossings. (d), (e), and (f) are obtained from the NA-ESMD simulations including the Min-Cost algorithm dealing with trivial unavoiding crossings.

Variation of the time-dependent fraction of the transition densities localized at the 2,2-PPE (blue lines) and 4-PPE (red lines) fragments when they are separated by different distances. (a), (b), and (c) are obtained from the NA-ESMD simulations without considering any specific treatment of unavoiding crossings. (d), (e), and (f) are obtained from the NA-ESMD simulations including the Min-Cost algorithm dealing with trivial unavoiding crossings.

Comparison of the variation of the time-dependent fraction of the transition densities localized at the 2,2-PPE (blue lines) and 4-PPE (red lines) fragments with *r* _{ f } ∼ 100 Å obtained using different procedures: the method proposed in this article (solid lines), a simple projection scheme to detect the trivial unavoided crossings (dashed lines), and an alternative algorithm to deal with cases of s_{βα}(*t*; *t* + Δ*t*) < S_{lim} (dotted lines).

Comparison of the variation of the time-dependent fraction of the transition densities localized at the 2,2-PPE (blue lines) and 4-PPE (red lines) fragments with *r* _{ f } ∼ 100 Å obtained using different procedures: the method proposed in this article (solid lines), a simple projection scheme to detect the trivial unavoided crossings (dashed lines), and an alternative algorithm to deal with cases of s_{βα}(*t*; *t* + Δ*t*) < S_{lim} (dotted lines).

Time evolution of (a) the energy gap , and (b) the overlap with (*n* = 0,…, *N* _{c}(*N* _{ q } − 1)) (see Eq. (7)) throughout the time-interval defined by a classical time step Δ*t* at which a potential unavoided crossing between non-interacting states takes place. The value of the overlap s_{αβ}(*t*; *t* + Δ*t*) ≡ ** ϕ **

_{α}(

**;**

*r***(**

*R**t*)) ·

*ϕ*_{β}(

**;**

*r***(**

*R**t*− Δ

*t*)) is shown as a dashed line.

Time evolution of (a) the energy gap , and (b) the overlap with (*n* = 0,…, *N* _{c}(*N* _{ q } − 1)) (see Eq. (7)) throughout the time-interval defined by a classical time step Δ*t* at which a potential unavoided crossing between non-interacting states takes place. The value of the overlap s_{αβ}(*t*; *t* + Δ*t*) ≡ ** ϕ **

_{α}(

**;**

*r***(**

*R**t*)) ·

*ϕ*_{β}(

**;**

*r***(**

*R**t*− Δ

*t*)) is shown as a dashed line.

Distribution of (a) the energy gaps ΔE_{αβ} and (b)NACT_{αβ} at the moment of S_{β} → S_{α} hops, and distribution of (c) energy gaps ΔE_{αβ} and (d)NACT_{αβ}at the moment of S_{β} → S_{α} unavoided crossings.

Distribution of (a) the energy gaps ΔE_{αβ} and (b)NACT_{αβ} at the moment of S_{β} → S_{α} hops, and distribution of (c) energy gaps ΔE_{αβ} and (d)NACT_{αβ}at the moment of S_{β} → S_{α} unavoided crossings.

Histogram of the overlaps s_{αβ}(*t*; *t* + Δ*t*) taken at S_{β} → S_{α} hops, and at trivial S_{β} → S_{α} unavoided crossings.

Histogram of the overlaps s_{αβ}(*t*; *t* + Δ*t*) taken at S_{β} → S_{α} hops, and at trivial S_{β} → S_{α} unavoided crossings.

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