Simulating Electronic Stopping with RT-TDDFT

According the Runge-Gross theorem [70], it is possible for electron dynamics to be treated efficiently using time-dependent density functional theory (TDDFT) [7,23]. This

presents opportunities for studying the electronic stopping process without reliance on analytical models. For metals, the electronic stopping power is a linear function of ion velocity in the low velocity limit, and the slope is called the friction coefficient. Nazarov and co-workers demonstrated the calculation of the friction coefficients for a series of metals with various projectile ions using TDDFT and its current density functional theory variation [71,72]. Campillo and co-workers reported the stopping power calculation over a wide range of ion velocity using the LR-TDDFT [73].

In the last several years, scientists have started to explore the use of RT-TDDFT simulations to obtain electronic stopping power by directly simulating the electronic response to the projectile ion. This approach was found to be quite successful for obtaining the linear part of the electronic stopping power [74,75], and it also seemed promising for obtaining the electronic stopping power curve even for higher ion velocities [76,77] beyond the Bragg peak (peak in stopping power curve).

In order to simulate the non-equilibrium quantum dynamics of electrons in response to an energetic ion, we need to balance computational accuracy and efficiency. Although several methods and codes already exist for performing RT-TDDFT calculations (see Refs. [24,33,34,43,78-84]) this work requires a numerical implementation that scales efficiently on thousands of processors in massively parallel computers because large system sizes with thousands of electrons needs to be simulated for accurate modeling (as discussed in subsection 2.4.1). The computational approach we employ in this work is the direct calculation of the stopping power via its definition as the deposited energy as a function of the ion displacement x at a constant velocity v:

where E is the electronic energy in the system and the bracket indicates a classical ensemble average of ion trajectories at constant ion velocity [69,85]. For calculating this quantity, one performs a series of non-equilibrium electron dynamics simulation with different ion

velocities. The projectile ion is introduced in the simulation cell after the equilibrium electronic structure is obtained for the system using a standard DFT calculation. The RT- TDDFT simulation is then performed with the projectile ion moving at a constant velocity. By constraining the ion velocity, the total electronic energy increases, allowing for

calculation of the electronic stopping power for the given velocity.

Fig. 2.2. Electron density change in response to an energetic proton traveling with a velocity of 2.0 a.u. in a homogeneous electron gas. A steady state needs to be reached in the simulation before the energy derivative (stopping power) is obtained to calculate the electronic stopping power.

The non-equilibrium electronic response to the ion movement is obtained in the RT- TDDFT simulation only after the steady state is reached in the simulation, requiring a large simulation supercell. A case for the homogeneous electron gas is shown as an example in

Figure 2.2. The electronic energy derivative with respect to the ion displacement (Eq. 2.39) is then numerically calculated and the ensemble average is taken. In the electronic stopping simulation, only the projectile ion is moved and no other atoms are allowed to move, and thus no contributions from nuclear stopping (elastic collision) enter in the simulation although it would be negligible in any case at the velocities that we are interested in.

In a realistic system (unlike the homogeneous electron gas widely studied in the literature), atomic nuclei of the system complicate the numerical calculation of the electronic energy change (Eq. 2.39), especially because non-local pseudo-potentials are typically used for treating core electrons. The pseudo-potentials are given as the sums of local potentials and separable non-local operators

(2.40) where 𝜓_˜™šV _{is an atomic pseudo-wavefunctions and 𝛿𝑉}

˜ is an angular momentum-specific

potentials as defined in Kleinman-Bylander representation [40]. The construction of the pseudo-potentials needs to be performed more carefully than for standard DFT calculations, minimizing the cut-off radius as detailed in Ref. [69]. The pseudo-potentials are generated by inverting the atomic Kohn-Sham equation for a specific XC potential [86], with a rather small cut-off radius of ( typically ~1 a.u.). Most of the error resulting from the use of non- local pseudo-potentials can be removed by performing separate Born-Oppenheimer MD calculations on the same path [87], but a separate procedure of baseline fitting is generally required for calculating the energy derivative. Another approach that we found to remediate this problem is to calculate the electronic stopping power from the average nonadiabatic “drag” force (NA force) on the projectile rather than the total change in energy of the system.

r' ˆV_ext r =V_local(r)δ(r'−r)+ r'ψlm PS δV_l δV_lψlm PS r ψlm PS δV_l ψlm PS l,m

∑

This approach is discussed in more detail in Chapter 4 and in Appendix B. Of course, there are many more physical and numerical complications in the simulation of electronic stopping and calculation of stopping power with RT-TDDFT, such as time step, plane-wave cutoff energy, etc. These issues are the primary subject of the work detailed in Chapter 4.

Although this work focuses on first-principles RT-TDDFT simulations as a means to go beyond analytical models, the computational cost of our approach is very high and it is desirable to advance analytical models using the first-principles simulations at the same time. First-principles simulations provide us with great details of the non-equilibrium electronic structure in electronic stopping processes, which allows us to systematically examine the validity of the plane-wave Born approximation in the analytical models (see, in particular, Chapters 3, 5, and 6). This key approximation results in a mathematically closed expression with a projectile charge that is independent of its velocity. Here, we have made progress in tackling this challenge by employing the first-principles quantum mechanical simulation to quantify the velocity-dependent effective charge. Such efforts will also help connect the wealth of existing understanding and recent increasingly popular efforts in simulating electronic stopping from first-principles quantum mechanical theory.

REFERENCES

[1] P. Hohenberg and W. Kohn, Physical review 136, B864 (1964). [2] M. Born and R. Oppenheimer, Annalen der Physik 389, 457 (1927). [3] C. D. Sherrill, School of Chemistry and Biochemistry Georgia Institute of

Technology (2000).

[4] B. L. Hammond, W. A. Lester, and P. J. Reynolds, Monte Carlo methods in ab initio quantum chemistry (World Scientific, 1994), Vol. 1.

[5] A. Williams, P. J. Feibelman, and N. Lang, Physical Review B 26, 5433 (1982). [6] P. J. Knowles and H.-J. Werner, Chemical physics letters 145, 514 (1988). [7] C. A. Ullrich, Time-Dependent Density-Functional Theory, Concepts and

Applications (Oxford Graduate Texts, 2011).

[8] W. Kohn and L. J. Sham, Physical review 140, A1133 (1965).

[9] W. Koch, M. C. Holthausen, and M. C. Holthausen, A chemist's guide to density functional theory (Wiley Online Library, 2001), Vol. 2.

[10] J. P. Perdew, K. Burke, and M. Ernzerhof, Physical review letters 77, 3865 (1996). [11] J. P. Perdew, A. Ruzsinszky, G. I. Csonka, L. A. Constantin, and J. Sun, Physical

Review Letters 103, 026403 (2009).

[12] S. Kurth, J. P. Perdew, and P. Blaha, International journal of quantum chemistry 75, 889 (1999).

[13] A. D. Becke, The Journal of chemical physics 98, 5648 (1993).

[14] J. P. Perdew, M. Ernzerhof, and K. Burke, The Journal of chemical physics 105, 9982 (1996).

[15] P. Stephens, F. Devlin, C. Chabalowski, and M. J. Frisch, The Journal of Physical Chemistry 98, 11623 (1994).

[16] E. Runge and E. K. Gross, Physical Review Letters 52, 997 (1984).

[17] N. T. Maitra, I. Souza, and K. Burke, Physical Review B 68, 045109 (2003).

[18] S. Botti, A. Schindlmayr, R. Del Sole, and L. Reining, Reports on Progress in Physics

70, 357 (2007).

[20] J. I. Fuks, L. Lacombe, S. E. Nielsen, and N. T. Maitra, Physical Chemistry Chemical Physics 20, 26145 (2018).

[21] M. E. Casida, C. Jamorski, K. C. Casida, and D. R. Salahub, The Journal of chemical physics 108, 4439 (1998).

[22] S. Baroni and R. Gebauer, in Fundamentals of Time-Dependent Density Functional Theory (Springer, 2012), pp. 375.

[23] M. A. L. Marques and E. K. U. Gross, Annual Review of Physical Chemistry 55, 427 (2004).

[24] K. Yabana and G. F. Bertsch, Physical Review B 54, 4484 (1996). [25] K. Yabana and G. Bertsch, Physical Review A 60, 1271 (1999).

[26] K. Yabana and G. Bertsch, International journal of quantum chemistry 75, 55 (1999). [27] M. R. Provorse and C. M. Isborn, International Journal of Quantum Chemistry 116,

739 (2016).

[28] F. Gygi, IBM Journal of Research and Development 52, 137 (2008). [29] F. Gygi, Qbox Open Source Code Project, 2014.

[30] A. Schleife, E. W. Draeger, Y. Kanai, and A. A. Correa, The Journal of chemical physics 137, 22A546 (2012).

[31] A. Schleife, E. W. Draeger, V. M. Anisimov, A. A. Correa, and Y. Kanai, Computing in Science & Engineering 16, 54 (2014).

[32] E. W. Draeger, X. Andrade, J. A. Gunnels, A. Bhatele, A. Schleife, and A. A. Correa, Journal of Parallel and Distributed Computing 106, 205 (2017).

[33] W. Liang, C. T. Chapman, and X. Li, The Journal of Chemical Physics 134, 184102 (2011).

[34] K. Lopata and N. Govind, Journal of Chemical Theory and Computation 7, 1344 (2011).

[35] X. Andrade et al., Physical Chemistry Chemical Physics 17, 31371 (2015). [36] M. Noda et al., Computer Physics Communications 235, 356 (2019).

[37] J. Hutter, M. Iannuzzi, F. Schiffmann, and J. VandeVondele, Wiley Interdisciplinary Reviews: Computational Molecular Science 4, 15 (2014).

[39] D. Hamann, M. Schlüter, and C. Chiang, Physical Review Letters 43, 1494 (1979). [40] L. Kleinman and D. M. Bylander, Physical Review Letters 48, 1425 (1982). [41] D. Hamann, Physical Review B 88, 085117 (2013).

[42] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes 3rd edition: The art of scientific computing (Cambridge university press, 2007).

[43] A. Castro, M. A. L. Marques, and A. Rubio, The Journal of Chemical Physics 121, 3425 (2004).

[44] S. Filippone, in International Workshop on Applied Parallel Computing (Springer, 1996), pp. 247.

[45] M. Frigo and S. G. Johnson, in Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP'98 (Cat. No. 98CH36181) (IEEE, 1998), pp. 1381.

[46] L. S. Blackford et al., ScaLAPACK users' guide (Siam, 1997), Vol. 4. [47] H. Bethe, Annalen der Physik 397, 325 (1930).

[48] G. F. V. G. Giuliani, Quantum Theory of the Electron Liquid (Cambridge, 2005). [49] L. E. Porter, International Journal of Quantum Chemistry 95, 504 (2003).

[50] L. E. Porter, International Journal of Quantum Chemistry 100, 973 (2004). [51] J. J. Thomson, Philosophical Magazine Series 6 23, 449 (1912).

[52] C. G. Darwin, Philosophical Magazine Series 6 23, 901 (1912). [53] M. Inokuti, Reviews of Modern Physics 43, 297 (1971).

[54] E. Fermi and E. Teller, Physical Review 72, 399 (1947).

[55] C. P. Race, D. R. Mason, M. W. Finnis, W. M. C. Foulkes, A. P. Horsfield, and A. P. Sutton, Reports on Progress in Physics 73, 116501 (2010).

[56] L. Porter, International journal of quantum chemistry 90, 684 (2002). [57] L. C. Northcliffe, Physical Review 120, 1744 (1960).

[58] W. Brandt and M. Kitagawa, Physical Review B 25, 5631 (1982). [59] L. C. Northcliffe, Physical Review 120, 1744 (1960).

[60] J. Lindhard and M. Scharff, Physical Review 124, 128 (1961).

[61] P. M. Echenique, R. M. Nieminen, J. C. Ashley, and R. H. Ritchie, Physical Review A 33, 897 (1986).

[62] P. M. Echenique, R. M. Nieminen, and R. H. Ritchie, Solid State Communications

37, 779 (1981).

[63] J. R. Sabin, J. Oddershede, and S. P. A. Sauer, in Advances in Quantum Chemistry, edited by B. Dževad (Academic Press, 2013), pp. 63.

[64] S. P. A. Sauer, J. Oddershede, and J. R. Sabin, in Advances in Quantum Chemistry, edited by R. S. John, and B. Erkki (Academic Press, 2011), pp. 215.

[65] S. P. A. Sauer, J. Oddershede, and J. R. Sabin, The Journal of Physical Chemistry C

114, 20335 (2010).

[66] J. R. Sabin, J. Oddershede, R. Cabrera-Trujillo, S. P. A. Sauer, E. Deumens, and Y. Öhrn, Molecular Physics 108, 2891 (2010).

[67] J. R. Sabin, J. Oddershede, and S. P. A. Sauer, AIP Conference Proceedings 1080, 138 (2008).

[68] R. Cabrera-Trujillo, J. R. Sabin, and J. Oddershede, in Advances in Quantum Chemistry (Academic Press, 2004), pp. 121.

[69] A. Schleife, Y. Kanai, and A. A. Correa, Physical Review B 91, 014306 (2015). [70] E. Runge and E. K. U. Gross, Physical Review Letters 52, 997 (1984).

[71] V. U. Nazarov, J. M. Pitarke, C. S. Kim, and Y. Takada, Physical Review B 71, 121106 (2005).

[72] V. U. Nazarov, J. M. Pitarke, Y. Takada, G. Vignale, and Y. C. Chang, Physical Review B 76, 205103 (2007).

[73] I. Campillo, J. M. Pitarke, and A. G. Eguiluz, Physical Review B 58, 10307 (1998). [74] J. M. Pruneda, D. Sánchez-Portal, A. Arnau, J. I. Juaristi, and E. Artacho, Physical

Review Letters 99, 235501 (2007).

[75] M. A. Zeb, J. Kohanoff, D. Sánchez-Portal, A. Arnau, J. I. Juaristi, and E. Artacho, Physical Review Letters 108, 225504 (2012).

[76] A. V. Krasheninnikov, Y. Miyamoto, and D. Tománek, Physical Review Letters 99, 016104 (2007).

[78] O. Sugino and Y. Miyamoto, Physical Review B 59, 2579 (1999).

[79] X. Li, J. C. Tully, H. B. Schlegel, and M. J. Frisch, The Journal of Chemical Physics

123, 084106 (2005).

[80] X. Qian, J. Li, X. Lin, and S. Yip, Physical Review B 73, 035408 (2006).

[81] C.-L. Cheng, J. S. Evans, and T. Van Voorhis, Physical Review B 74, 155112 (2006). [82] C. M. Isborn, X. Li, and J. C. Tully, The Journal of Chemical Physics 126, 134307

(2007).

[83] S. K. Min, Y. Cho, and K. S. Kim, The Journal of Chemical Physics 135 (2011). [84] P. López-Tarifa, M.-P. Gaigeot, R. Vuilleumier, I. Tavernelli, M. Alcamí, F. Martín,

M.-A. Hervé du Penhoat, and M.-F. Politis, Angewandte Chemie 125, 3242 (2013). [85] A. Schleife, E. W. Draeger, V. Anisimov, A. A. Correa, and Y. Kanai, Computing in

Science and Engineering 16, 54 (2014).

[86] M. Fuchs and M. Scheffler, Computer Physics Communications 119, 67 (1999). [87] R. Car and M. Parrinello, Physical Review Letters 55, 2471 (1985).