196,251 research outputs found
A General Purpose Implementations of Semiclassical Molecular Dynamics for CPU and GPU hardware
No full textThe calculation of the semiclassical propagator is a Monte Carlo integration over classical trajectories. This can be accelerated either by importance sampling or by parallelization of the phase space integration. In the first case, a multiple coherent states time-averaging [1] semiclassical initial value representation (MC-TA-SC-IVR) method for spectra calculations is presented.[2, 3] The method is implemented for ab initio semiclassical simulations, i.e. a direct dynamics approach, and it is shown to faithfully reproduce all kind of quantum effects, including ZPEs, anharmonicities, tunneling splittings,[4] resonances [5] and vibrational eigenfunctions.[6] This on-the-fly approach is useful in particular for complex systems,[7, 8] where the elaboration of a pre-computed potential energy surface can turn into a formidable task. In the second case, SC-IVR is implemented for GPUs hardware. [9] An almost constant scaling for GPU calculations versus a linear scaling for CPU ones is found respect to the number of trajectories. Issues and limitations related to the GPU implementation will be discussed.
[1] A. L. Kaledin and W. H. Miller, J. Chem. Phys. 118, 7174 (2003)
[2] M. Ceotto, S. Atahan, S. Shim, G. F. Tantardini, and A. Aspuru-Guzik, Phys. Chem. Chem. Phys. 11, 3861 (2009)
[3] M. Ceotto, S. Atahan, G. F. Tantardini, and A. Aspuru-Guzik, J. Chem. Phys. 130, 234113 (2009)
[4] R. Conte, A. Aspuru-Guzik, and M. Ceotto, J. Phys. Chem. Lett. 4, 3407-3412 (2013)
[5] M. Ceotto, D. Dell'Angelo, and G. F. Tantardini, J. Chem. Phys. 133, 054701 (2010)
[6] M. Ceotto, S. Valleau, G. F. Tantardini, and A. Aspuru-Guzik, J. Chem Phys. 134, 234103 (2011)
[7] M. Ceotto, G. F. Tantardini, and A. Aspuru-Guzik, J. Chem. Phys. 135, 214108 (2011)
[8] M. Ceotto, Y. Zhuang, and W.L. Hase, J. Chem. Phys. 138, 054116 (2013)
[9] D. Tamascelli, F. D'Ambrosio, R. Conte, and M. Ceotto, in progres
Ab initio direct semiclassical molecular dynamics
No full textA multiple coherent states time-averaging [1] semiclassical initial value representation (MC-TA-SC-IVR) method for spectra calculations is presented.[2, 3] The method is implemented for ab initio semiclassical simulations, i.e. a direct dynamics approach, and it is shown to faithfully reproduce all kind of quantum effects, including ZPEs, anharmonicities, tunneling splittings, resonances [4] and vibrational eigenfunctions.[5] This on-the-fly approach is useful in particular for complex systems,[5, 6] where the elaboration of a pre-computed potential energy surface can turn into a formidable task. I will show that the method can deal with molecules with multiple wells and as complex as glycine.
[1] A. L. Kaledin and W. H. Miller, J. Chem. Phys. 118, 7174 (2003)
[2] M. Ceotto, S. Atahan, S. Shim, G. F. Tantardini, and A. Aspuru-Guzik, Phys. Chem. Chem. Phys. 11, 3861 (2009)
[3] M. Ceotto, S. Atahan, G. F. Tantardini, and A. Aspuru-Guzik, J. Chem. Phys. 130, 234113 (2009)
[4] M. Ceotto, D. Dell'Angelo, and G. F. Tantardini, J. Chem. Phys. 133, 054701 (2010)
[5] M. Ceotto, S. Valleau, G. F. Tantardini, and A. Aspuru-Guzik, J. Chem Phys. 134, 234103 (2011)
[6] M. Ceotto, G. F. Tantardini, and A. Aspuru-Guzik, J. Chem. Phys. 135, 214108 (2011)
[7] M. Ceotto, Y. Zhuang, and W.L. Hase, J. Chem. Phys. 138, 054116 (2013
Accurate and efficient pre-exponential factor approximations for the semiclassical initial value representation propagator
No full textThe semiclassical (SC) theory[1,2] is a very powerful tool to describe molecular reactivity, electronic transitions and molecular vibrations.[3-9] In semiclassical methods quantum informations are obtained by evolving classical trajectories. This is computationally less intense than grid methods. Unfortunately, when the system is complex, the calculation of the SC Herman-Kluk (HK) prefactor[10] is prohibitive, due to the trajectories' instability.[11] For this reason, it is not possible to employ the basic SC-HK approach for large systems. In the past years, several approximations to the HK prefactor have been proposed[12,13] to solve this issue, but they have never been thoroughly assessed.
In this work, first we test some of the most common prefactor approximations on small systems. Then, we put forward a new one starting from the Log-Derivative[13] formulation, and that is potentially suitable for bigger systems. This approximation depends only on the Hessian matrix and does not require the calculation of the monodromy matrix elements, which are often unstable. As a consequence, even chaotic trajectories can be employed for vibrational spectra simulations. The results show that our approximation is very reliable for molecules like H2, H2O, CO2, CH2O, CH4 and CH2D2. Future applications of our new prefactor will concern the evaluation of power spectra of large systems, for which quantum calculations are currently out of reach.
References
[1] W.H. Miller J. Phys. Chem. A 105, 2942 (2001).
[2] W.H. Miller Proc. Natl. Acad. Sci. USA 102, 6660 (2005).
[3] M. Ceotto, S. Atahan, S. Shim, G.F. Tantardini, and A. Aspuru-Guzik, Phys. Chem. Chem. Phys. 11, 3861 (2009).
[4] M. Ceotto, S. Atahan, G.F Tantardini, A. Aspuru-Guzik., J. Chem. Phys. 130, 234113 (2009).
[5] M. Ceotto, D. Dell'Angelo, G.F Tantardini, J. Chem. Phys. 133 (5), 054701 (2010).
[6] M. Ceotto, G.F. Tantardini, and A. Aspuru-Guzik, J. Chem. Phys. 135, 214108 (2011).
[7] R. Conte, A. Aspuru-Guzik, M. Ceotto J. Phys. Chem. Lett., 4, 3407 (2013).
[8] D. Tamascelli, F.S. Dambrosio, R. Conte, M. Ceotto J. Chem Phys., 140, 174109 (2014).
[9] M.L. Brewer, J.S. Hulme, and D.E. Manolopoulos J. Chem. Phys. 106, 4832 (1997).
[10] M.F. Herman, E. Kluk Chem. Phys. 91, 27 (1984).
[11] K.G. Kay J. Chem. Phys. 101, 2250 (1994).
[12] V. Guallar, V.S. Batista, and W.H. Miller J. Chem. Phys. 110, 9922 (1999).
[13] R. Gelabert, X. Gimenez, M. Thoss, H. Wang, and W.H. Miller J. Phys. Chem. A, 104, 10321 (2000)
Quantum nuclear densities from semiclassical on-the-fly molecular dynamics
No full textSemiclassical molecular dynamics is a rigorous approximation to quantum dynamics obtained from the exact quantum propagator expressed as Feynman’s path integral.[1] Recently, our group has introduced the Multiple Coherent Semiclassical Initial Value Representation (MC SCIVR) technique to reduce the number of classical trajectories required to converge vibrational spectra calculations from thousands to just a handful.[2-4] MC SCIVR has been applied successfully to several medium- and large-size molecular systems,[4-10] including fluxional and condensed phase ones.[11-13] In addition to the accurate anharmonic vibrational eigenvalue calculations, MC SCIVR yields vibrational eigenfunctions for both the ground and excited vibrational states.[14] In this talk, I will survey how we obtain the quantum anharmonic vibrational eigenfunctions from ab-initio on-the-fly trajectory simulations and how we extract the quantum nuclear densities and the geometry parameters probability distributions.[15,16] This information allows us to assign each peak in vibrational spectra, going beyond the usual harmonic normal-mode analysis. Our technique quantitatively determines how normal modes involving different functional groups cooperate to originate the spectroscopic signal. Furthermore, it allows for the visualization of the nuclear vibrations in a purely quantum picture, letting us both directly observe and quantify the effects of the full potential energy surface anharmonicity on the molecular structure. In particular, I will illustrate applications to the protonated glycine to reveal quantum mechanical and anharmonic vibrational features. The method will allow for a better rationalization of experimental spectroscopy.
[1] W.H. Miller, J. Phys. Chem. A 2001, 105, 2942.
[2] M. Ceotto, S. Atahan, S. Shim, G.F. Tantardini, A. Aspuru-Guzik, Phys. Chem. Chem. Phys. 2009, 11, 3861.
[3] M. Ceotto, S. Atahan, G.F. Tantardini, A. Aspuru-Guzik J. Chem. Phys. 2009, 130, 234113.
[4] R. Conte, M. Ceotto, In Quantum Chemistry and Dynamics of Excited States: Methods and Applications (eds L. González and R. Lindh) 2020.
[5] M. Ceotto, G. Di Liberto, R. Conte, Phys. Rev. Lett. 2017, 119, 010401.
[6] F. Gabas, R. Conte, M. Ceotto, J. Chem. Theory Comput. 2017, 13, 2378.
[7] G. Di Liberto, R. Conte, M. Ceotto, J. Chem. Phys. 2018, 148, 014307.
[8] F. Gabas, G. Di Liberto, R. Conte, M. Ceotto, Chem. Sci. 2018, 9, 7894.
[9] F. Gabas, G. Di Liberto, M. Ceotto, J. Chem. Phys. 2019, 150, 224107.
[10] F. Gabas, R. Conte, M. Ceotto, J. Chem. Theory Comput. 2020, 16, 3476.
[11] G. Bertaina, G. Di Liberto, M. Ceotto, J. Chem. Phys. 2019, 151, 114307.
[12] A. Rognoni, R. Conte, M. Ceotto, Chem. Sci., 2021, 12, 2060.
[13] M. Cazzaniga, M. Micciarelli, F. Moriggi, A. Mahmoud, F. Gabas, and M. Ceotto, J. Chem. Phys. 2020, 152, 104104.
[14] M. Micciarelli, R. Conte, J. Suarez, M. Ceotto, J. Chem. Phys. 2018 149, 064115.
[15] C. Aieta, M. Micciarelli, G. Bertaina, M. Ceotto, Nat. Commun 2020, 11, 1.
[16] C. Aieta, M. Micciarelli, G. Bertaina, M. Ceotto, J. Chem. Phys., 2020, 153, 214117
Quantum nuclear densities from semiclassical on-the-fly molecular dynamics
No full textSemiclassical molecular dynamics is a rigorous approximation to quantum dynamics obtained from the exact quantum propagator expressed as Feynman’s path integral.[1] Recently, our group has introduced the Multiple Coherent Semiclassical Initial Value Representation (MC SCIVR) technique to reduce the number of classical trajectories required to converge vibrational spectra calculations from thousands to just a handful.[2-4] MC SCIVR has been applied successfully to several medium and large-size molecular systems,[4-10] including fluxional and condensed phase ones.[11-13] In addition to the accurate anharmonic vibrational eigenvalue calculations, MC SCIVR yields vibrational eigenfunctions for both the ground and excited vibrational states.[14] In this talk, I will survey how we obtain the quantum anharmonic vibrational eigenfunctions from ab-initio on-the-fly trajectory simulations and how we extract the quantum nuclear densities and the geometry parameters probability distributions.[15,16] This information allows us to assign each peak in vibrational spectra, going beyond the usual harmonic normal-mode analysis. Our technique quantitatively determines how normal modes involving different functional groups cooperate to originate the spectroscopic
signal. Furthermore, it allows for the visualization of the nuclear vibrations in a purely quantum picture, letting us both directly observe and quantify the effects of the full potential energy surface anharmonicity on the molecular structure. In particular, I will illustrate applications to the protonated glycine to reveal quantum mechanical and anharmonic vibrational features. The method will allow for a better rationalization of experimental spectroscopy.
[1] W.H. Miller, J. Phys. Chem. A 2001, 105, 2942.
[2] M. Ceotto, S. Atahan, S. Shim, G.F. Tantardini, A. Aspuru-Guzik, Phys. Chem. Chem. Phys. 2009, 11, 3861.
[3] M. Ceotto, S. Atahan, G.F. Tantardini, A. Aspuru-Guzik J. Chem. Phys. 2009, 130, 234113.
[4] R. Conte, M. Ceotto, In Quantum Chemistry and Dynamics of Excited States: Methods and Applications (eds L. González and R. Lindh) 2020.
[5] M. Ceotto, G. Di Liberto, R. Conte, Phys. Rev. Lett. 2017, 119, 010401.
[6] F. Gabas, R. Conte, M. Ceotto, J. Chem. Theory Comput. 2017, 13, 2378.
[7] G. Di Liberto, R. Conte, M. Ceotto, J. Chem. Phys. 2018, 148, 014307.
[8] F. Gabas, G. Di Liberto, R. Conte, M. Ceotto, Chem. Sci. 2018, 9, 7894.
[9] F. Gabas, G. Di Liberto, M. Ceotto, J. Chem. Phys. 2019, 150, 224107.
[10] F. Gabas, R. Conte, M. Ceotto, J. Chem. Theory Comput. 2020, 16, 3476.
[11] G. Bertaina, G. Di Liberto, M. Ceotto, J. Chem. Phys. 2019, 151, 114307.
[12] A. Rognoni, R. Conte, M. Ceotto, Chem. Sci., 2021, 12, 2060.
[13] M. Cazzaniga, M. Micciarelli, F. Moriggi, A. Mahmoud, F. Gabas, and M. Ceotto, J. Chem. Phys. 2020, 152, 104104.
[14] M. Micciarelli, R. Conte, J. Suarez, M. Ceotto, J. Chem. Phys. 2018 149, 064115.
[15] C. Aieta, M. Micciarelli, G. Bertaina, M. Ceotto, Nat. Commun 2020, 11, 1.
[16] C. Aieta, M. Micciarelli, G. Bertaina, M. Ceotto, J. Chem. Phys., 2020, 153, 214117
How many water molecules are needed to solvate one?
No full textThe comprehension at the molecular scale of the processes involved during solvation still remains a challenge in chemistry. Remarkably, the question concerning how many solvent molecules are necessary to solvate a solute one is still open. By exploring several water clusters of increasing size, we employ semiclassical spectroscopy [1-5] to determine on quantum dynamical grounds the minimal number of surrounding water molecules to make the central one display the same vibrational features of liquid water. We find out that the minimal structure eventually responsible of proper solvation is made of 21 water molecules, and that particular care must be reserved to the quantum description of the combination of the central monomer bending mode with network low-frequency librations.[6] The results obtained with the accurate ab initio potential are then compared with the popular Caldeira-Leggett one to rationalize whether a simplified model can qualitatively and quantitatively describe the solvated system behavior.[7] An ongoing study on how genetic algorithms[8] and adiabatically switched trajectories[9] can help to deconstruct the complex spectrum of the formic acid dimer will be also presented.
[1] E. J. Heller, Acc. Chem. Res. 14, 368-375 (1981).
[2] M. F. Herman and E. Kluk, Chem. Phys. 91, 27-34 (1984).
[3] A. L. Kaledin and W. H. Miller, J. Chem. Phys. 119, 3078-3084 (2003).
[4] M. Ceotto, S. Atahan, G. F. Tantardini and A. Aspuru-Guzik, J. Chem. Phys. 130, 234113 (2009).
[5] M. Ceotto, G. Di Liberto and R. Conte, Phys. Rev. Lett. 119, 010401 (2017).
[6] A. Rognoni, R. Conte and M. Ceotto, Chem. Sci. 12, 2060 (2021).
[7] A. Rognoni, R. Conte and M. Ceotto, J. Chem. Phys. 154, 094106 (2021). [8] M. Gandolfi, A. Rognoni, C. Aieta, R. Conte and M Ceotto, J. Chem. Phys. 153, 204104 (2020). [9] R. Conte, L. Parma, C. Aieta, A. Rognoni and M. Ceotto, J. Chem. Phys. 151, 214107 (2019)
Semiclassical vibrational spectroscopy : the importance of quantum anharmonicity in supra-molecular systems
No full textSemiclassical (SC) vibrational spectroscopy has been applied successfully to several molecular systems thanks to the possibility to regain quantum effects accurately starting from short-time classical trajectories.[1-5] Larger molecular and supra-molecular systems represent instead an open challenge in the field of semiclassical spectroscopy mainly due to the necessity to work in very high dimensionality.
To start off the talk I will present some recent theoretical advances able to extend the range of applicability of SC vibrational spectroscopy to very high-dimensional systems.[6-7] Then, I will move to applications of semiclassical spectroscopy concerning the vibrational features of water clusters and two supra-molecular systems involving glycine.[8-9] These applications will point out the importance of a multi-reference, dynamical approach able to reproduce quantum anharmonicities without employing any ad-hoc scaling factor.
[1] M. F. Herman, E. Kluk, Chem. Phys. 1984, 91, 27.
[2] A. L. Kaledin, W. H. Miller, J. Chem. Phys. 2003, 118, 7174.
[3] M. Ceotto, S. Atahan, G. F. Tantardini, A. Aspuru-Guzik, J. Chem. Phys. 2009, 130, 234113.
[4] R. Conte, A. Aspuru-Guzik, M. Ceotto, J. Phys. Chem. Lett. 2013, 4, 3407.
[5] F. Gabas, R. Conte, M. Ceotto, J. Chem. Theory Comput. 2017, 13, 2378.
[6] M. Ceotto, G. Di Liberto, R. Conte, Phys. Rev. Lett. 2017, 119, 010401.
[7] G. Di Liberto, R. Conte, M. Ceotto, J. Chem. Phys. 2018, 148, 014307.
[8] G. Di Liberto, R. Conte, M. Ceotto, J. Chem. Phys. 2018, 148, 104302.
[9] F. Gabas, G. Di Liberto, R. Conte, M. Ceotto, to be submitted
Quantum nuclear densities from semiclassical on-the-fly molecular dynamics
No full textSemiclassical molecular dynamics is a rigorous approximation to quantum dynamics obtained from the exact quantum propagator expressed as Feynman’s path integral.[1] Recently, our group has introduced the Multiple Coherent Semiclassical Initial Value Representation (MC SCIVR) technique to reduce the number of classical trajectories required to converge vibrational spectra calculations from thousands to just a handful.[2-4] MC SCIVR has been applied successfully to several medium-and large-size molecular systems,[4-10] including fluxional and condensed phase ones.[11-13] In addition to the accurate anharmonic vibrational eigenvalue calculations, MC SCIVR yields vibrational eigenfunctions for both the ground and excited vibrational states.[14] In this work, we obtain the quantum anharmonic vibrational eigenfunctions from ab-initio on-the-fly trajectory simulations, and we extract the quantum nuclear densities and the geometry parameters probability distributions.[15,16] This information allows us to assign each peak in vibrational spectra, going beyond the usual harmonic normal-mode analysis. Our technique quantitatively determines how normal modes involving different functional groups cooperate to originate the spectroscopic signal. Furthermore, it allows for the visualization of the nuclear vibrations in a purely quantum picture, letting us directly observe and quantify the effects of the full potential energy surface anharmonicity on the molecular structure. In particular, for the protonated glycine molecule, our calculations reveal quantum mechanical and anharmonic vibrational features. The method will allow for a better rationalization of experimental spectroscopy.
References
[1]W. Miller, J. Phys. Chem. A, 105, 2942-2955 (2001)
[2]M. Ceotto, S. Atahan, S. Shim, G. Tantardini, A. Aspuru-Guzik, Phys. Chem. Chem. Phys., 11, 3861 (2009)
[3]M. Ceotto, S. Atahan, G. Tantardini, A. Aspuru-Guzik, The Journal of Chemical Physics, 130, 234113 (2009)
[4]R. Conte, M. Ceotto, Semiclassical Molecular Dynamics for Spectroscopic Calculations, 2020
[5]M. Ceotto, G. Di Liberto, R. Conte, Phys. Rev. Lett., 119, 010401 (2017)
[6]F. Gabas, R. Conte, M. Ceotto, J. Chem. Theory Comput., 13, 2378-2388 (2017)
[7]G. Di Liberto, R. Conte, M. Ceotto, The Journal of Chemical Physics, 148, 014307 (2018)
[8]F. Gabas, G. Di Liberto, R. Conte, M. Ceotto, Chem. Sci., 9, 7894-7901 (2018)
[9]F. Gabas, G. Di Liberto, M. Ceotto, J. Chem. Phys., 150, 224107 (2019)
[10]F. Gabas, R. Conte, M. Ceotto, J. Chem. Theory Comput., 16, 3476-3485 (2020)
[11]G. Bertaina, G. Di Liberto, M. Ceotto, J. Chem. Phys., 151, 114307 (2019)
[12]A. Rognoni, R. Conte, M. Ceotto, Chem. Sci., 12, 2060-2064 (2021)
[13]M. Cazzaniga, M. Micciarelli, F. Moriggi, A. Mahmoud, F. Gabas, M. Ceotto, J. Chem. Phys., 152, 104104 (2020)
[14]M. Micciarelli, R. Conte, J. Suarez, M. Ceotto, The Journal of Chemical Physics, 149, 064115 (2018)
[15]C. Aieta, M. Micciarelli, G. Bertaina, M. Ceotto, Nat. Commun., 11, 4348 (2020)
[16]C. Aieta, G. Bertaina, M. Micciarelli, M. Ceotto, J. Chem. Phys., 153, 214117 (2020
A quantum approximate method for the calculation of thermal reaction rate constants
No full textThe calculation of thermal reaction rate constants is a central problem in theoretical chemistry, and exact classical and quantum expressions have been formulated [1]. However, approximate approaches are necessary when dealing with complex reactions, and several techniques have been developed in recent years. They include the inclusion of quantum corrections to the classical transition state theory (TST) [2], semiclassical theories [3], and ring polymer molecular dynamics (RPMD) TST [4].
In this work, we have developed a new quantum mechanical method to compute reaction rate constants, which is related to Miller's quantum instanton [5]. Starting from the exact definition of the thermal rate constant as the time integral of the quantum flux-flux correlation function, upon introduction of a stationary phase approximation, we have derived an expression which has the same structure of the original quantum instanton but includes a contribution from real-time dynamics. This new method has been tested on the one-dimensional Eckart barrier problem, and on the two-dimensional H+H2 collinear reaction. Results over a wide range of temperatures have been found to be in agreement within 10% with exact quantum mechanical estimates.
[1] W.H. Miller, S.D. Schwartz, J.W. Tromp, J. Chem. Phys. 79, 4889 (1983); W.H. Miller, J. Phys. Chem. A 102 (5), 793, (1998)
[2] H. Eyring J. Chem. Phys. 3 (1935), p. 107; E. Wigner J. Chem. Phys, 5 (1937), p. 720
[3] W.H. Miller, J. Chem. Phys, 62, 1899 (1975); R. Hernandez, W.H. Miller Chem. Phys. Lett., 214 (1993), p. 129; T. L. Nguyen, J. F. Stanton, and J. R. Barker, Chem. Phys. Lett. 499, 9 (2010).
[4] J. O. Richardson and S. C. Althorpe, J. Chem. Phys. 131, 214106 (2009); T. J. H. Hele and S. C. Althorpe J. Chem. Phys. 138, 084108 (2013)
[5] W.H. Miller, Y. Zhao, M. Ceotto, S. Yang J. Chem. Phys. 119, 1329 (2003); M. Ceotto, S. Yang, and W.H. Miller J. Chem. Phys. 122, 044109 (2005
Semiclassical Molecular Dynamics for Spectroscopic Calculations of Complex Systems
No full textI will present some novel semiclassical methods for spectroscopic calculations. These approaches can be employed for spectroscopic calculations of gas-phase molecular and supramolecular systems
up to hundreds of degrees of freedom, as well as to condensed phase systems. Some methods are based on a “divide-and-conquer” approach, where the full dimensional spectra are obtained as a
composition of several lower dimensional ones. Others exploit hierarchically the different levels of accuracy of different semiclassical propagators. For instance, in a system-bath problem lower
semiclassical accuracy is dedicated to the bath, while the system is treated with higher accuracy and the system spectrum is eventually singled out.
All methods are amenable for ab initio molecular dynamics simulations.
References
1. F. Gabas, G. Di Liberto, R. Conte, and M. Ceotto, Chemical Science 9 (41), 7885-8026 (2018);
2. X. Ma, G. Di Liberto, R. Conte, W. L. Hase, and M. Ceotto, JCP 149, 164113 (2018)
3. M. Micciarelli, R. Conte, J. Suarez, and M. Ceotto, JCP 149, 064115 (2018);
4. M. Buchholz, F. Grossmann, and M. Ceotto, JCP 148, 114107 (2018);
5. G. Di Liberto, R. Conte, and M. Ceotto, JCP 148, 104302 (2018);
6. G. Di Liberto, R. Conte, and M. Ceotto, JCP 148, 014307 (2018);
7. M. Buchholz, F. Grossmann, and M. Ceotto, JCP 147, 164110 (2017);
8. M. Ceotto, G. Di Liberto, and R. Conte, PRL 119, 010401 (2017);
9. F. Gabas, R. Conte, and M. Ceotto, JCTC 13, 2378-2388 (2017);
10. G. Di Liberto, M. Ceotto, JCP 145, 144107 (2016);
11. M. Buchholz, F. Grossmann, M. Ceotto, JCP 144, 094102 (2016)
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