Evaluation of the Time-Derivative Coupling for Accurate Electronic

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Evaluation of the Time-Derivative Coupling for Accurate Electronic State Transition Probabilities from Numerical Simulations Garrett A. Meek, and Benjamin G. Levine J. Phys. Chem. Lett., Just Accepted Manuscript • Publication Date (Web): 16 Jun 2014 Downloaded from http://pubs.acs.org on June 16, 2014

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The Journal of Physical Chemistry Letters

Evaluation of the Time-Derivative Coupling for Accurate Electronic State Transition Probabilities from Numerical Simulations Garrett A. Meek, Benjamin G. Levine* Department of Chemistry, Michigan State University, East Lansing, MI 48824 * to whom correspondence should be addressed: [email protected] Abstract Spikes in the time-derivative coupling (TDC) near surface crossings make the accurate integration of the time-dependent Schrodinger equation in nonadiabatic molecular dynamics simulations a challenge. To address this issue, we present an approximation to the TDC based on a norm-preserving interpolation (NPI) of the adiabatic electronic wavefunctions within each time step. We apply NPI and two other schemes for computing the TDC in numerical simulations of the Landau-Zener model, comparing the simulated transfer probabilities to the exact solution. Though NPI does not require the analytical calculation of nonadiabatic coupling matrix elements, it consistently yields unsigned population transfer probability errors of ~0.001, while analytical calculation of the TDC yields errors of 0.0-1.0 depending on the time step, the offset of the maximum in the TDC from the beginning of the time step, and the coupling strength. The approximation of Hammes-Schiffer and Tully yields errors intermediate between NPI and the analytical scheme. TOC Graphic

Keywords: Ab initio multiple spawning, surface hopping, trivial unavoided crossing, local diabatization, unitary transformation 1 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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The accurate theoretical modeling of molecular dynamics on multiple electronic states has allowed the elucidation of the ultrafast motions following the electronic excitation of molecules and materials.

Classical-trajectory-1-16 and wavepacket-based17-22 approaches allow

the simulation of such phenomena on accurate potential energy surfaces (PESs) determined by ab initio calculations performed on-the-fly. Both of these classes of methods generally rely on the numerical integration of the time-dependent Schrodinger equation (TDSE), which in turn requires the discretization of time into steps of finite size. Numerical schemes for the integration of differential equations like the TDSE are well known,23 but such approaches only provide accurate results when the terms in the equation vary on timescales shorter than the time step. Unfortunately, the simulation of nonadiabatic dynamics in the adiabatic representation requires the treatment of population transfer through conical intersections, points of degeneracy between electronic states.24 Such intersections introduce spikes in the time-derivative coupling (TDC), defined

σ kj = φk ( r; t ) ) ∂φj ( r; t ) / ∂t ,

(1)

where φ j is the electronic wavefunction of adiabatic state j. These spikes may be infinitesimally narrow in time and infinitely large in magnitude, and their accurate integration is essential to a correct prediction of the probability of population transfer.25-27

Working in the diabatic

representation can reduce these issues, but no unique diabatic representation exists, and the accuracy of simulations performed in the diabatic representation can be lower than those based on the adiabatic representation.7,28 Approaches to numerically integrating the TDSE in the adiabatic basis can be split into two broad categories: schemes based on analytic calculations of the nonadiabatic coupling matrix

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element (NACME) vector, and those based on the numerical differentiation of the wavefunction in time. The NACME— d kj (R ) = φk (r;R ) | ∇ Rφj ( r; R ) ,

(2)

where r and R represent the electronic and nuclear coordinates, respectively—can now be computed for many approximate electronic structure methods using analytical gradient techniques.29-37

Treating nuclear motion classically, the TDC matrix element can then be

computed according to

σ kj = R& ⋅ d kj (R ) ,

(3)

& represents the time derivative of R. Though efficient and widely used, this approach where R

suffers when the TDSE is numerically integrated with a finite time step, as illustrated in Figure 1. When σ kj spikes on a timescale shorter than the integration time step—e.g. at a conical intersection or weakly avoided crossing—the simulation may step on the spike, resulting in a coupling which is far larger than the average coupling over the time step, or step over the spike, thus missing the coupling altogether. In these cases, numerical integration of the TDSE results in an erroneous population transfer probability not because the computed analytical TDC is in error, but instead because the TDC is known at only a few discrete points in time. These errors may be especially large in cases where systematically weak coupling between diabats leads to (N-1)-dimensional trivial unavoided crossings, such as in long-range energy transfer,25-27 longrange charge transfer,38 and intersystem crossing resulting from weak spin-orbit interactions. Therefore, though analytical derivative techniques provide the exact TDC at discrete times, discretization of time itself is a coarse approximation which can lead to very large errors. Adaptive time step integration (as implemented in ab initio multiple spawning; AIMS) alleviates this problem,38,39 but at a significant computational cost. 3 ACS Paragon Plus Environment


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Figure 1. Cartoon representation of the cases where the simulation steps over (left) or steps on (right) the maximum of the TDC. Dotted lines indicate calculations of the TDC.

An appealing alternative is to use information about the electronic wavefunction at the beginning ( t 0 ) and end ( t0 + ∆t ) of the time step to approximate the TDC. In such approaches, the true change in the electronic wavefunctions over the entire time step (rather than the derivative at a single time) is resolved by computing overlap integrals between the adiabatic wavefunctions at times t 0 and t0 + ∆t . Hammes-Schiffer and Tully (HST) proposed such an overlap-based approach,40 approximating the TDC according to

φk ( t ) ∂φ ( t ) / ∂t j

t0 +∆t /2

≈

1  φk ( t0 ) φj ( t0 + ∆t ) − φk ( t0 + ∆t ) φj ( t0 )  .  2 ∆t 

(4)

Though generally thought of as an approximation to the analytical scheme, the HST method has been demonstrated to predict a similar probability for population transfer upon averaging over large ensembles of surface hopping trajectories.41,42

The HST and other overlap-based


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approaches have the additional advantage that they can be applied in conjunction with electronic structure methods for which analytic NACMEs are not implemented,43 and in other cases can be less expensive to compute than such analytic calculations.41,42 In a similar spirit, several overlapbased schemes have been developed for directly determining the hopping probability for surface hopping simulations, bypassing the TDC altogether.10,25-27 These approaches have been shown to address the trivial unavoided crossing problem, but because they do not provide an explicit definition for the TDC they cannot be applied in wavepacket-based dynamical schemes. Given the advantages of overlap-based approaches, we propose a new such approach, the norm-preserving interpolation (NPI) method. The NPI method is based on an approximate definition for the adiabatic electronic wavefunction which varies continuously with time, with the TDC computed analytically from this wavefunction. Therefore, the NPI TDC includes contributions from the change in the wavefunction at all times, thus eliminating the discretization error inherent to the analytical approach, but accepting some numerical error relative to the analytical scheme in return. Because NPI provides a recipe for the explicit calculation of the TDC it can be applied in conjunction with both wavepacket-based and surface hopping methods. We will analyze the accuracy of simulations utilizing NPI compared to those based on analytical derivative and HST approaches by applying these methods to integrate the TDSE for the adiabatized Landau-Zener (LZ) model,44,45 for which the exact solution is known. In the NPI approach, we approximate the electronic wavefunction of adiabatic state j at time, τ , on the interval t0 ≤ τ ≤ t 0 + ∆t as the product of a time-dependent transformation matrix, U (τ ) , with the wavefunction at the beginning of the time step, t0 ,

φj (τ ) = U (τ ) φj ( t0 ) .

(5)

The diagonal and off-diagonal elements of U (τ ) are respectively defined 5 ACS Paragon Plus Environment


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 cos −1 (U jj ( t0 + ∆t ) )  U jj (τ ) = cos  (τ − t0 )    ∆t  

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(6)

and

 sin −1 (U jk ( t0 + ∆t ) )  U jk (τ ) = sin  (τ − t0 )  .   ∆t  

(7)

with U jk ( t0 + ∆t ) = φj ( t0 ) φk ( t0 + ∆t ) .

(8)

Note that in the complete Hilbert space U (τ ) is a unitary transformation for all τ, and thus Eq. (5) represents an interpolation of the adiabatic electronic wavefunction between its known values at the beginning and end of the time step, φj ( t0 ) and φj ( t0 + ∆t ) , that maintains the normalization of the wavefunction at all times, τ. More specifically, this definition of U (τ ) represents an interpolation of the vector, φj (τ ) , in which the angle of that vector relative to its initial value, φj ( t0 ) , varies linearly with time. This formulation thus represents that most direct and gradual transformation of the wavefunction from φj ( t0 ) to φj ( t0 + ∆t ) which maintains normalization. We can thus approximate the TDC between states k and j within each time step as

φk (τ ) ∂φj (τ ) / ∂τ = φk ( t0 ) U † (τ )

∂ U (τ ) φj ( t0 ) . ∂τ

(9)

Averaging (9) over the interval [t0 , t0 + ∆t ] yields the NPI approximation to the TDC at the center of the time step,


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φk ( t )

∂φ j ( t ) ∂t

≈ t0 +∆t /2

∫

t0 +∆t

t0

dτ φk ( t0 ) U† (τ ) ∆t

∂ U (τ ) φ j ( t0 ) ∂τ .

(10)

In a sense, this is not an approximation to the TDC at a specific point in time so much as a value of the TDC that represents the full change in the wavefunction over the time step. Given our simple definition of U (τ ) , the time integral in equation (10) can be computed analytically, leading to an algebraic expression for the NPI TDC. This expression and comments about its proper numerical implementation are presented in Supporting Information.

Though the

expression is somewhat more complicated than Eq. (4), it requires essentially the same information from the electronic structure calculation: the overlaps of the adiabatic wavefunctions at the beginning of each time step with those at the end. The Landau-Zener44,45 two-state model has been chosen as a test case for the performance of the NPI method because the exact solution for the population transfer probability is known, thus allowing for direct quantification of errors. In the Landau-Zener two-state model the timedependent energies of diabatic electronic states µ and ν are expressed as Eµ ( t ) = −α t and Eν ( t ) = α t , respectively, where α is the slope of the potential energy surfaces. These states are coupled by a time-independent coupling, ε µν . A set of time-dependent adiabatic states are defined by diagonalizing the resulting 2x2 Hamiltonian. The exact probability of population 2 / hα ) . All numerical simulations of transfer between adiabats is known to be PLZ = exp(−πε µν

population transfer performed here have been compared with this exact solution to determine the population transfer probability error, Perror = PSimulation − PLZ . The TDSE is numerically integrated to determine the population transfer probability. Three approximations to the TDC have been tested. The analytical scheme corresponds to the 7 ACS Paragon Plus Environment


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exact value of the TDC defined by Eq. (1) at each time step. The NPI and HST schemes are computed according to Eqs. (10) and (4), respectively. Integration of the TDSE has been performed using the fourth-order Runge-Kutta method, with a multiple time step algorithm similar to that employed in the software implementation of ab initio multiple spawning (AIMS).22 Additional details of the numerical integration scheme are presented in Supporting Information.

Figure 2. The population transfer probability error, Perror , for each method of evaluating the timederivative coupling in the steps-over case is presented as a function of the length of the time step.

In Figure 2 the error in the computed population transfer probability is plotted as a function of the length of the time step for the passage through a weakly avoided crossing (

ε µν = 10 meV and α = 0.1 eV/fs ). These parameters were drawn from our group’s recent AIMS study of the non-radiative decay of silicon epoxide defects, where similar slopes and several 8 ACS Paragon Plus Environment

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nonadiabatic events involving a gap of 10 meV or smaller were observed.46 Note that a similar size coupling was employed in a recent study of the behavior of surface hopping at trivial unavoided crossings as well.27 The exact LZ population transfer in this case is 0.995. For time steps in the range typically chosen for realistic dynamical simulations (0.25-0.5 fs) the NPI method results in a smaller error than the HST and analytical schemes. At a time step of 0.5 fs, the error in the computed population transfer probability from the NPI-based simulations has a negligible value of -0.001. Application of the HST scheme yields a larger magnitude error of 0.093, while the analytical method results in an error of very large magnitude: -0.457. When a very short time step is employed (∆t < 0.15 fs) the error in all methods is small ( Perror < 0.010 ), but it is interesting to note that the NPI scheme still yields the smallest error in these cases (Figure S1). We have also tested the NPI approach by application to various LZ-like models involving three interacting states, with similarly strong results (presented in Figures S2-4). It is important to note that in all of the simulations reported in Figure 2 the initial time was set such that the maximum of the TDC falls at the center of a time step, whereas the calculation of the potential, wavefunction, and (for the analytical method) the analytically computed TDC are performed at the beginning and end of the time step. In more graphical language, the simulations in Figure 2 correspond to the case where the discretized trajectories step over the intersection. In this case (weakly avoided crossing, stepping over the intersection), HST and NPI have the advantage of fully resolving the change in the electronic wavefunction, and thus exhibit smaller errors. In contrast, the analytical simulations do not resolve the spike in the TDC, resulting in large errors. Of course whether the discretized trajectory steps on or steps over the intersection in a dynamical simulation is determined by chance, and thus we consider how the population transfer



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error depends on the offset of the maximum in the TDC relative to the beginning of the time step. In Figure 3, we show the error in the simulated population transfer probability for the same LZ model as a function of this offset. All simulations were conducted with a 0.5 fs time step, so a 0.25 fs offset corresponds to the case where the trajectory symmetrically steps over the spike in the coupling (Figure 1 left), 0.0 and 0.5 fs offsets correspond to the steps-on case (Figure 1 right), and the other offsets correspond to the continuum of possibilities between these two extremes.

Figure 3. The population transfer error is plotted as a function of the offset of the time step from the maximum of the TDC. The symmetrical steps-over case falls at 0.25 fs, while the steps-on case falls at 0.0 and 0.5 fs.


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The NPI scheme yields population transfer probability errors between -0.001 and 0.000 for all offsets. In contrast, the error in the population transfer varies wildly when analytical TDCs are employed, changing from -0.865 in the steps-on case, to nearly zero at a 0.10 fs offset, to -0.457 in the previously discussed steps-on case. This is because the analytically calculated TDC is far too large at the steps-on point to result in a correct prediction of the population transfer probability, and far too small at the steps-over point. One might expect the error to average out over many simulations, but this is not the case for a weakly avoided crossing. At a weakly avoided crossing, the probability of population transfer is near unity. Any error, whether it be that the TDC is too large or too small, results in a reduction of the predicted population transfer probability. Thus, at weakly avoided crossings, the analytic scheme will systematically underpredict the probability of nonadiabatic population transfer. In the context of realistic simulations, this could lead to, for example, the prediction of unphysical long-range charge and energy transfer events where diabatic motion through an intersection would be physically correct. Again, the HST scheme results in smaller population transfer errors than the analytical approach, but the errors rise as large as -0.093 at the steps-over point. The errors in the HST approach can be understood by demonstrating how the HST definition of the TDC (Eq. (4)) can be derived from the same unitary formalism used to derive NPI. The HST expression arises from the approximation

φk ( t )

∂φ j ( t ) ∂t

≈ t0 +∆t /2

∫

t0 +∆t

t0

t0 +∆t ∂ ∂ U (τ ) φ j ( t0 ) ∫ dτ φk ( t0 + ∆t ) U (τ ) φ j ( t0 ) t ∂τ ∂τ . + 0 2∆t 2∆t

dτ φk ( t0 )

(11)

This expression differs from the NPI definition of the TDC (Eq. (10)) in that only the ket wavefunction is interpolated in time, while the bra wavefunction is frozen at t 0 and t 0 + ∆ t in 11 ACS Paragon Plus Environment


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the first and second terms, respectively.

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These definitions are essentially equal when

φk ( t0 ) φj ( t0 + ∆t ) is very small, but in the case that φk ( t0 ) φj ( t0 + ∆t ) approaches 1, the definitions of the TDC in Eqs. (4) and (10) approach 1/ ∆t and π / 2∆t , respectively, and only integration of the latter value results in the prediction of the correct LZ transition probability for very weakly avoided crossings.

Figure 4. The population transfer error is plotted as a function of the coupling strength between the diabatic states, ε µν .

We continue by investigating the utility of NPI at more strongly avoided crossings. Figure 4 presents the error in the population transfer as a function of the coupling between diabats, ε µν . In all cases a time step of 0.5 fs is used, α is 0.1 eV/fs, and the offset is chosen such that the trajectory steps over the intersection symmetrically. For all values of ε µν the use of NPI results in the smallest population transfer errors, with all errors between -0.007 and 0.000. 12 ACS Paragon Plus Environment

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For larger values of ε µν (strongly avoided crossings) HST and the analytical method both provide suitable accuracy, but as the coupling strength decreases these approaches yield significant errors. Similar results are obtained when we consider the steps-on case, though HST yields considerably smaller errors in this case (see Figure S5 for details). Thus, by using a time-dependent unitary transformation to interpolate the adiabatic electronic wavefunction across each time step and computing the TDC between these interpolated wavefunctions, NPI is able to provide TDCs which result in the accurate integration of the TDSE even at very weakly avoided crossings. When applied in numerical simulations based on the adiabatized LZ model, the NPI approach results in population transfer errors of ~0.001 relative to the exact LZ solutions over a wide range of time steps and LZ model parameters, whereas application of the HST and analytical approaches results in much larger errors which depend strongly on both the choice of time step and the model parameters. In addition, the probability of population transfer at weakly avoided crossings predicted by NPIbased simulations does not depend strongly on the offset of the spike in the TDC relative to the beginning of the time step, whereas the HST and analytical schemes yield errors which are strongly dependent on this uncontrollable factor. For the majority of offsets and realistic time steps, simulations based on HST yield smaller population transfer errors at weakly avoided crossings than do those based on analytical calculation of the TDC. Lastly, NPI is trivially implemented into surface hopping and wavepacket-based simulation schemes, and does not require the analytical calculation of NACMEs. Thus NPI can be employed with a wide range of electronic structure methods for which such analytic calculations have not yet been implemented.



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Acknowledgement We gratefully acknowledge startup funds provided by Michigan State University. Many thanks to Todd Martínez and Aaron Virshup for fruitful discussion. Supporting Information Available: Integrated expressions for the NPI approximation to the TDC, details of the numerical integration scheme, and supporting figures are presented. This material is available free of charge via the Internet at http://pubs.acs.org.

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Evaluation of the Time-Derivative Coupling for Accurate Electronic

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