Marek Pederzoli J. Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, v.v.i.,

Transcription

1 Supplementary Material: A New Approach to Molecular Dynamics with Non-adiabatic and Spin-orbit Effects with Applications to QM/MM Simulations of Thiophene and Selenophene Marek Pederzoli J. Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, v.v.i., Dolejškova 3, Prague 8, Czech Republic and Charles University in Prague, Faculty of Science, Department of Physical and Macromolecular Chemistry, Hlavova 8, Prague 2, Czech Republic Jiří Pittner J. Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, v.v.i., Dolejškova 3, Prague 8, Czech Republic Electronic address: marek.pederzoli@jh-inst.cas.cz Electronic address: jiri.pittner@jh-inst.cas.cz,corresponding~author 1

2 Contents I. Treatment of the crossings of states of different multiplicity in the surface hopping dynamics 3 Dynamics in the MCH representation 3 Dynamics in the diagonal representation without the 3-step propagator 4 Dynamics in the diagonal representation using the 3-step propagator 5 II. Computational details of the CASSCF calculation 6 III. Supplementary figures 7 References 11 2

3 I. TREATMENT OF THE CROSSINGS OF STATES OF DIFFERENT MULTI- PLICITY IN THE SURFACE HOPPING DYNAMICS Proper treatment of the state crossings of states of different multiplicity is crucial for surface hopping dynamics, yet, to our knowledge, this topic has not been addressed in sufficient detail in the literature. The states in the surface hopping simulations are usually ordered by energy, hence the central question is, whether the population of the crossing states is supposed to be exchanged, and how such exchange can be carried out in various implementations of the surface hopping dynamics. When dealing only with states of the same multiplicity the crossings are either avoided, or, in general triatomic and larger molecules, where the non-crossing rule does not apply, there is at most (n 2)-dimensional intersection. In this case, the populations never need to be exchanged in the surface hopping scheme and the changes in the character of the adiabatic states are automatically treated by the quantum-chemical method via the nonadiabatic coupling terms. On the other hand, states of different multiplicities may cross freely in (n 1)-dimensional seams. In this case, the populations need to be treated within the surface hopping scheme in order to avoid unphysical, non-continuous, changes in the populations. In the following sections, we will discuss how these crossings might be detected, and how they should be treated in various implementations of surface hopping dynamics with states of different multiplicity. We will discuss only crossing of singlet and triplet states; the generalization to higher multiplets is straightforward. Dynamics in the MCH representation For dynamics in the MCH representation, the detection of crossings is trivial. If the states are ordered by S and M s (canonical ordering), no reordering is necessary. If all the states are sorted by energy, the detection can be easily achieved by comparing the energies of the singlet and triplet states in consecutive steps. When a crossing is detected, all the variables corresponding to the swapped states need to be reordered to assure continuity. In standard surface hopping, this would mean coefficients, and their derivatives, eigenvectors in the U matrix, and matrices containing non-adiabatic and spin-orbit couplings. The most convenient approach is to reorder all the quantities of the previous step (or steps, if multiple 3

4 last steps are used in the integration of the electronic Schrödinger equation); this way, the reordering needs to be done only once per crossing. If one of the states participating in the switch is the currently active state, the active state should be changed as well. This change of the active state, however, should be not implemented as a forced hopping, because doing so would effectively prohibit hopping in the current step. Dynamics in the diagonal representation without the 3-step propagator For the dynamics using mixed spin states, the treatment of the crossings is more complicated and we have implemented several methods of their detection. If spin-orbit coupling is low, and the components of the multiplets are very close in energy, the multiplets can be identified simply by counting the number energies within a given threshold of each other. This method may fail, if the multiplets are very close or overlapping, in that case, we can resolve the situation more accurately by comparing the relative position of the energetic center of mass in the consecutive steps. Another method, one that is applicable in general case, when the splitting is large, is to use the overlap of the U matrix with the one from the previous step. If no crossing occurred, S = U(t) U(t + t) should be close to diagonal matrix, or, for the case with degenerate eigenvalues, to a block diagonal matrix with block sizes corresponding to the eigenvalue degeneracies. If a crossing occurred, the order of the eigenvectors is changed and S will have a block-permutation matrix form. A disadvantage of both these approaches is that it requires numerical thresholds for block structures and further problem may arise in the case of accidental degeneracies, causing the blocks not only to permute, but change their sizes. To solve this problem, we transform the coefficients and coupling terms of the time step t t to the new basis given by U(t), using equation (19) or Ũ using equation (29). These transformations automatically handle the swaps of states without any numerical thresholds and hence are the default choice in our implementation. 4

5 Dynamics in the diagonal representation using the 3-step propagator The 3-step propagator computes the time evolution of the coefficients in the MCH representation, which does not require state swapping, as long as the canonical ordering of states is used. Furthermore, the transformation matrices U (t+ t) and U(t) in Eq(13) of the main manuscript do not need to have identical order of states; the coefficients are transformed to the MCH representation where the coefficients are propagated and then transformed back to the diagonal representation with the new order of the states. On the first glance, the 3-step propagator thus does not need any phase alignment of the U matrix whatsoever. This was also claimed in Ref [1], however, in our opinion it is still preferable to perform the phase adjustment for the case of (almost) degenerate eigenvalues, which occurs when SO coupling is small. The eigenvectors corresponding to the degenerate eigenvalue may arbitrarily mix among themselves, which leads to random fluctuations in the populations of the components of multiplets; only the total population of the whole multiplet would remain continuous. A rigorously correct treatment of the degenerate components of multiplets would require to include spin-spin coupling, which we did not attempt, however, these non-continuous population transfers may result into unnecessary hops, which goes against the spirit of the fewest switches approach and would cause numerical problems if the derivatives of the coefficients need to be calculated. The SHARC code actually does implement the phase alignment by default, but we believe it needs to be generalized especially if states of higher multiplicity than triplets are involved. In our opinion, the correct alignment procedure should take into account that in the presence of near-degeneracies the overlap S = U(t) U(t + t) is close to a block-diagonal matrix when the order of eigenvectors is the same and has a block-permutation form, when the order is different, rather than having diagonal or permutation matrix form. As discussed earlier, in the 3-step propagator the state reordering is resolved automatically. When going back to diagonal representation after propagation, the states are reordered and if the active state was reordered as well, it will most likely result in an indirectly induced hop. Lack of such induced hop actually means normal hop would have happened if the surfaces didn t cross, which should be logged and considered in the analysis. It is also possible to implement an approach where the states are not sorted by energy, but remain in their original order. This approach was also implemented to Newton-X; it does not exhibit 5

6 noncontinuous changes of coefficients nor crossing-induced hops, both of which complicate analysis of the trajectories. On the other hand the order of states may not be well defined, when the states mix, thus it is safer to remove the discontinuities in the post-processing of the trajectories. II. COMPUTATIONAL DETAILS OF THE CASSCF CALCULATION In selection of the active space of the CASSCF calculation, especially for molecular dynamics where hundreds of single point calculations are necessary in each trajectory, one often needs to consider a trade-off between accuracy and computational cost. In order to properly describe the electronic structure of thiophene and selenophene, the active space certainly needs to contain the two π orbitals together with two π orbitals, n orbital of sulphur/selenium and also a σ and σ orbital to account for the possibility of ring opening. Hence, we have selected the active space of 8 electrons in 7 orbitals, also employed for thiophene in Ref [2]. Three singlet and three triplet states have been included into the state-averaging procedure with equal weights. The resulting molecular orbitals are visualized in Figure S1. All CASSCF calculations have been performed using Molpro [3] and the implementation details can be found in Refs [4, 5] 6

7 III. SUPPLEMENTARY FIGURES Figure S1: Molecular orbitals of thiophene: three π-type orbitals, followed by two π orbitals, a σ and σ orbital. 7

8 Figure S2: Absorption spectrum of thiophene and its breakdown into the individual excited states (a) (b) Figure S3: Potential energies averaged over the trajectories starting in S 1 (a) and S 2 (b) 8

9 (a) (b) Figure S4: Time evolution of spin-orbit coupling for an example trajectory of thiophene (a) and selenophene (b) Figure S5: Time evolution of average adiabatic population in additional trajectories of thiophene with SOC matrix elements multiplied by a factor of 5. 9

10 (a) (b) Figure S6: Example trajectory comparison of adiabatic populations obtained using two distinct implementations: SVD approach in Newton-X (a) and 3-step approach in SHARC (b) The same initial conditions and same random numbers have been used during the surface hopping procedure 10

11 [1] S. Mai, P. Marquetand, and L. Gonzalez, Int. J. Quant. Chem. 115, 1215 (2015). [2] G. Cui and W. Fang, J. Phys. Chem. A 115, (2011). [3] H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, M. Schütz, P. Celani, W. Györffy, D. Kats, T. Korona, R. Lindh, A. Mitrushenkov, G. Rauhut, K. R. Shamasundar, T. B. Adler, R. D. Amos, A. Bernhardsson, A. Berning, D. L. Cooper, M. J. O. Deegan, A. J. Dobbyn, F. Eckert, E. Goll, C. Hampel, A. Hesselmann, G. Hetzer, T. Hrenar, G. Jansen, C. Köppl, Y. Liu, A. W. Lloyd, R. A. Mata, A. J. May, S. J. McNicholas, W. Meyer, M. E. Mura, A. Nicklass, D. P. O Neill, P. Palmieri, D. Peng, K. Pflüger, R. Pitzer, M. Reiher, T. Shiozaki, H. Stoll, A. J. Stone, R. Tarroni, T. Thorsteinsson, and M. Wang, MOLPRO, version , a package of ab initio programs, 2015, see [4] H.-J. Werner and P. J. Knowles, J. Chem. Phys. 82, 5053 (1985). [5] P. J. Knowles and H.-J. Werner, Chem. Phys. Letters 115, 259 (1985). 11