# Superadiabatic Forces via the Acceleration Gradient in Quantum Many-Body Dynamics

^{*}

## Abstract

**:**

**2015**, 143, 174108) to a one-dimensional Hooke’s helium model atom. The physical dynamics are described on the one-body level beyond the density-based adiabatic approximation. We show that gradients of both the microscopic velocity and acceleration field are required to correctly describe the effects due to interparticle interactions. We validate the proposed analytical forms of the superadiabatic force and transport contributions by comparison to one-body data from exact numerical solution of the Schrödinger equation. Superadiabatic contributions beyond the adiabatic approximation are important in the dynamics and they include effective dissipation.

## 1. Introduction

## 2. Theoretical Background

## 3. Hooke’s Atom

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Hohenberg, P.; Kohn, W. Inhomogeneous electron gas. Phys. Rev.
**1964**, 136, B864. [Google Scholar] [CrossRef] - Burke, K. Perspective on density functional theory. J. Chem. Phys.
**2012**, 136, 150901. [Google Scholar] [CrossRef] [PubMed] - Vignale, G.; Rasolt, M. Density-functional theory in strong magnetic fields. Phys. Rev. Lett.
**1987**, 59, 2360. [Google Scholar] [CrossRef] [PubMed] - Vignale, G.; Rasolt, M. Current- and spin-density-functional theory for inhomogeneous electronic systems in strong magnetic fields. Phys. Rev. B
**1988**, 37, 10685. [Google Scholar] [CrossRef] [PubMed] - Runge, E.; Gross, E.K.U. Density-functional theory for time-dependent systems. Phys. Rev. Lett.
**1984**, 52, 997. [Google Scholar] [CrossRef] - Marques, M.A.L.; Maitra, N.T.; Nogueira, F.M.S.; Gross, E.K.U.; Rubio, A. (Eds.) Fundamentals of Time-Dependent Density Functional Theory; Springer: Berlin, Germany, 2012. [Google Scholar]
- Ullrich, C.A.; Yang, Z.-H. A Brief Compendium of Time-Dependent Density Functional Theory. Braz. J. Phys.
**2014**, 44, 154. [Google Scholar] [CrossRef] - Maitra, N.T. Perspective: Fundamental aspects of time-dependent density functional theory. J. Chem. Phys.
**2016**, 144, 220901. [Google Scholar] [CrossRef] [PubMed] - Vignale, G.; Kohn, W. Current-dependent exchange–correlation potential for dynamical linear response theory. Phys. Rev. Lett.
**1996**, 77, 2037. [Google Scholar] [CrossRef] [PubMed] - Vignale, G.; Ullrich, C.A.; Conti, S. Time-dependent density functional theory beyond the adiabatic local density approximation. Phys. Rev. Lett.
**1997**, 79, 4878. [Google Scholar] [CrossRef] - Gosh, S.K.; Dhara, A.K. Density-functional theory of many-electron systems subjected to time-dependent electric and magnetic fields. Phys. Rev. A
**1988**, 38, 1149. [Google Scholar] [CrossRef] - Vignale, G. Mapping from current densities to vector potentials in time-dependent current density functional theory. Phys. Rev. B
**2004**, 70, 201102(R). [Google Scholar] [CrossRef] - Van Leeuwen, R. Key concepts in time-dependent density-functional theory. Int. J. Mod. Phys. B
**2001**, 15, 1969. [Google Scholar] [CrossRef] - Thiele, M.; Gross, E.K.U.; Kümmel, S. Adiabatic approximation in nonperturbative time-dependent density-functional theory. Phys. Rev. Lett.
**2008**, 100, 153004. [Google Scholar] [CrossRef] [PubMed] - Thiele, M.; Kümmel, S. Hydrodynamic perspective on memory in time-dependent density-functional theory. Phys. Rev. E
**2009**, 79, 052503. [Google Scholar] [CrossRef] - Tokatly, I.V. Quantum many-body dynamics in a Lagrangian frame: I. Equations of motion and conservation laws. Phys. Rev. B
**2005**, 71, 165104. [Google Scholar] [CrossRef][Green Version] - Tokatly, I.V. Quantum many-body dynamics in a Lagrangian frame: II. Geometric formulation of time-dependent density functional theory. Phys. Rev. B
**2005**, 71, 165105. [Google Scholar] [CrossRef][Green Version] - Tokatly, I.V. Time-dependent current density functional theory via time-dependent deformation functional theory: A constrained search formulation in the time domain. Phys. Chem. Chem. Phys.
**2009**, 11, 4621. [Google Scholar] [CrossRef] [PubMed] - Diaw, A.; Murillo, M.S. A viscous quantum hydrodynamics model based on dynamic density functional theory. Sci. Rep.
**2017**, 7, 15352. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bousquet, D.; Hughes, K.H.; Micha, D.A.; Burghardt, I. Extended hydrodynamic approach to quantum-classical nonequilibrium evolution. I. Theory. J. Chem. Phys.
**2011**, 134, 064116. [Google Scholar] [CrossRef] - Wijewardane, H.O.; Ullrich, C.A. Time-Dependent Kohn-Sham Theory with Memory. Phys. Rev. Lett.
**2005**, 95, 086401. [Google Scholar] [CrossRef][Green Version] - Ullrich, C.A. Time-dependent density-functional theory beyond the adiabatic approximation: Insights from a two-electron model system. J. Chem. Phys.
**2006**, 125, 234108. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hessler, P.; Maitra, N.T.; Burke, K. Correlation in time-dependent density-functional theory. J. Chem. Phys.
**2002**, 117, 72. [Google Scholar] [CrossRef] - Ullrich, C.A.; Tokatly, I.V. Nonadiabatic electron dynamics in time-dependent density-functional theory. Phys. Rev. B
**2006**, 73, 235102. [Google Scholar] [CrossRef][Green Version] - Evans, R. The nature of the liquid-vapour interface and other topics in the statistical mechanics of non-uniform, classical fluids. Adv. Phys.
**1979**, 28, 143. [Google Scholar] [CrossRef] - Archer, A.J.; Evans, R. Dynamical density functional theory and its application to spinodal decomposition. J. Chem. Phys.
**2004**, 121, 4246. [Google Scholar] [CrossRef] [PubMed] - Marconi, U.M.B.; Tarazona, P. Dynamic density functional theory of fluids. J. Chem. Phys.
**1999**, 110, 8032. [Google Scholar] [CrossRef] - Schmidt, M.; Brader, J.M. Power functional theory for Brownian dynamics. J. Chem. Phys.
**2013**, 138, 214101. [Google Scholar] [CrossRef][Green Version] - Fortini, A.; de las Heras, D.; Brader, J.M.; Schmidt, M. Superadiabatic forces in Brownian many-body dynamics. Phys. Rev. Lett.
**2014**, 113, 167801. [Google Scholar] [CrossRef] - Bernreuther, E.; Schmidt, M. Superadiabatic forces in the dynamics of the one-dimensional Gaussian core model. Phys. Rev. E
**2016**, 94, 022105. [Google Scholar] [CrossRef] - De las Heras, D.; Schmidt, M. Velocity Gradient Power Functional for Brownian Dynamics. Phys. Rev. Lett.
**2018**, 120, 028001. [Google Scholar] [CrossRef][Green Version] - Stuhlmüller, N.C.X.; Eckert, T.; de las Heras, D.; Schmidt, M. Structural nonequilibrium forces in driven colloidal systems. Phys. Rev. Lett.
**2018**, 121, 098002. [Google Scholar] [CrossRef] [PubMed] - Schmidt, M. Power functional theory for Newtonian many-body dynamics. J. Chem. Phys.
**2018**, 148, 044502. [Google Scholar] [CrossRef] [PubMed][Green Version] - Schmidt, M. Quantum power functional theory for many-body dynamics. J. Chem. Phys.
**2015**, 143, 174108. [Google Scholar] [CrossRef] [PubMed] - Elliott, P.; Fuks, J.I.; Rubio, A.; Maitra, N.T. Universal dynamical steps in the exact time-dependent exchange–correlation potential. Phys. Rev. Lett.
**2012**, 109, 266404. [Google Scholar] [CrossRef] - Maitra, N.T.; Burke, K. Demonstration of initial-state dependence in time-dependent density-functional theory. Phys. Rev. A
**2001**, 63, 042501. [Google Scholar] [CrossRef][Green Version] - Wijewardane, H.O.; Ullrich, C.A. Real-time electron dynamics with exact-exchange time-dependent density-functional theory. Phys. Rev. Lett.
**2008**, 100, 056404. [Google Scholar] [CrossRef] - Ullrich, C.A.; Gossmann, U.J.; Gross, E.K.U. Time-dependent optimized effective potential. Phys. Rev. Lett.
**1995**, 74, 872. [Google Scholar] [CrossRef] [PubMed] - Van Leeuwen, R. The Sham-Schlüter equation in time-dependent density-functional theory. Phys. Rev. Lett.
**1996**, 76, 3610. [Google Scholar] [CrossRef] - Van Leeuwen, R. Causality and symmetry in time-dependent density-functional theory. Phys. Rev. Lett.
**1998**, 80, 1280. [Google Scholar] [CrossRef] - Fuks, J.I.; Luo, K.; Sandoval, E.D.; Maitra, N.T. Time-Resolved Spectroscopy in Time-Dependent Density Functional Theory: An Exact Condition. Phys. Rev. Lett.
**2015**, 114, 183002. [Google Scholar] [CrossRef] - Luo, K.; Fuks, J.I.; Sandoval, E.D.; Elliott, P.; Maitra, N.T. Kinetic and interaction components of the exact time-dependent correlation potential. J. Chem. Phys.
**2014**, 140, 18A515. [Google Scholar] [CrossRef] [PubMed][Green Version] - Fuks, J.I.; Elliott, P.; Rubio, A.; Maitra, N.T. Dynamics of charge-transfer processes with time-dependent density functional theory. J. Phys. Chem. Lett.
**2013**, 4, 735. [Google Scholar] [CrossRef] [PubMed] - Hodgson, M.J.P.; Ramsden, J.D.; Chapman, J.B.J.; Lillystone, P.; Godby, R.W. Exact time-dependent density-functional potentials for strongly correlated tunneling electrons. Phys. Rev. B
**2013**, 88, 241102. [Google Scholar] [CrossRef][Green Version] - Ruggenthaler, M.; Bauer, D. Rabi oscillations and few-level approximations in time-dependent density functional theory. Phys. Rev. Lett.
**2009**, 102, 233001. [Google Scholar] [CrossRef] - Madelung, E. Eine anschauliche Deutung der Gleichung von Schrödinger. Naturwissenschaften
**1926**, 14, 1004. [Google Scholar] [CrossRef] - Madelung, E. Quantentheorie in hydrodynamischer Form. Z. Phys. A Hadron. Nucl.
**1927**, 40, 322. [Google Scholar] [CrossRef] - Askar, A.; Cakmak, A.S. Explicit integration method for the time-dependent Schrödinger equation for collision problems. J. Chem. Phys.
**1978**, 68, 2794. [Google Scholar] [CrossRef] - Visscher, P.B. A fast explicit algorithm for the time-dependent Schrödinger equation. Comp. Phys.
**1991**, 5, 596. [Google Scholar] [CrossRef] - De las Heras, D.; Renner, J.; Schmidt, M. Custom flow in overdamped Brownian dynamics. Phys. Rev. E
**2019**, 99, 023306. [Google Scholar] [CrossRef][Green Version] - Hansen, J.-P.; McDonald, I.R. Theory of Simple Liquids, 4th ed.; Academic Press: Oxford, UK, 2013. [Google Scholar]

Sample Availability: Not available. |

**Figure 1.**Time evolution of (

**a**) the reduced density ${n}^{*}=n\sqrt{\hslash /m\omega}$ and (

**b**) the reduced current density ${J}^{*}=J/\omega $ as a function of the reduced box coordinate ${x}^{*}=x\sqrt{m\omega /\hslash}$ for $\u03f5/\hslash \omega =0.5$ and ${\alpha}^{*}=1$. For comparison, at reduced time ${t}^{*}\equiv t\omega =3$ the curves for ${\alpha}^{*}=4$ are also shown (dashed line). Illustration of the total force density and the different contributions according to Equation (3) for ${t}^{*}=0.05$ (

**c**) and ${t}^{*}=2$ (

**d**) given in reduced units as ${F}^{*}=F/m{\omega}^{2}$. The encircled arrows indicate the force direction at selected space points. The total size of the box is $30\sqrt{\hslash /m\omega}$.

**Figure 2.**Reduced kinetic stress tensor ${\tau}^{*}=\tau /\sqrt{m{\omega}^{3}\hslash}$ (blue solid line) and its splitting into ideal (red dashed line) and excess (solid yellow line) parts, Equation (6), and the theoretical prediction (yellow circles), Equation (8), for interaction lengths ${\alpha}^{*}=0.2$ (

**a**), ${\alpha}^{*}=1$ (

**b**) and ${\alpha}^{*}=4$ (

**c**). Internal (green solid line), adiabatic (blue dashed line) and superadiabatic (yellow solid line) force densities, cf. Equation (7), and theoretical prediction (yellow circles), Equation (9), in reduced units, i.e., ${F}^{*}=F/m{\omega}^{2}$ for interaction lengths ${\alpha}^{*}=0.2$ (

**d**), ${\alpha}^{*}=1$ (

**e**) and ${\alpha}^{*}=4$ (

**f**). Data taken at ${t}^{*}=3$. The model parameters and reduced units are identical to those in Figure 1. The insets in panels (

**a**,

**c**,

**d**) are enlarged views of the superadiabatic terms. The inset in panel (

**e**) shows the splitting of the superadiabatic force density (circles) into ${F}_{v}$ (pink) and ${F}_{a}$ (blue).

**Figure 3.**Scaled amplitudes ${C}_{0}^{*}={C}_{0}\sqrt{m\omega /{\hslash}^{3}}$ (yellow), ${C}_{1}^{*}={C}_{1}\sqrt{m\omega /{\hslash}^{3}}$ (violet) and ${C}_{2}^{*}={C}_{2}\sqrt{m{\omega}^{3}/{\hslash}^{3}}$ (blue) of the functional approximations for ${\tau}_{\mathrm{exc}}$, ${F}_{v}$ and ${F}_{a}$, cf. Equations (8), (10) and (11), as a function of the scaled time ${t}^{*}=t\omega $ for three different values of the interaction length, as indicated. Note the different scales (left and right vertical axes). The data set ${C}_{1}^{*}$ (${\alpha}^{*}=4$) has been scaled with a factor $0.1$, as indicated, for a better visualization.

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**MDPI and ACS Style**

Brütting, M.; Trepl, T.; de las Heras, D.; Schmidt, M. Superadiabatic Forces via the Acceleration Gradient in Quantum Many-Body Dynamics. *Molecules* **2019**, *24*, 3660.
https://doi.org/10.3390/molecules24203660

**AMA Style**

Brütting M, Trepl T, de las Heras D, Schmidt M. Superadiabatic Forces via the Acceleration Gradient in Quantum Many-Body Dynamics. *Molecules*. 2019; 24(20):3660.
https://doi.org/10.3390/molecules24203660

**Chicago/Turabian Style**

Brütting, Moritz, Thomas Trepl, Daniel de las Heras, and Matthias Schmidt. 2019. "Superadiabatic Forces via the Acceleration Gradient in Quantum Many-Body Dynamics" *Molecules* 24, no. 20: 3660.
https://doi.org/10.3390/molecules24203660