# American Institute of Mathematical Sciences

March  2017, 10(1): 263-298. doi: 10.3934/krm.2017011

## On the classical limit of a time-dependent self-consistent field system: Analysis and computation

 1 Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706, USA 2 Department of Mathematics, Institute of Natural Sciences and MOE Key Lab in Scientific, and Engineering Computing, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai, 200240, China 3 Department of Mathematics, Statistics, and Computer Science, M/C 249, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607, USA 4 Department of Mathematics, Duke University, Box 90320, Durham NC 27708, USA

Dedicated to Peter Markowich on the occasion of his 60th birthday

Received  October 2015 Revised  January 2016 Published  November 2016

Fund Project: This work was partially supported by NSF grants DMS-1522184 and DMS-1107291: NSF Research Network in Mathematical Sciences KI-Net: Kinetic description of emerging challenges in multiscale problems of natural sciences. C.S. acknowledges support by the NSF through grant numbers DMS-1161580 and DMS-1348092.

We consider a coupled system of Schrödinger equations, arising in quantum mechanics via the so-called time-dependent self-consistent field method. Using Wigner transformation techniques we study the corresponding classical limit dynamics in two cases. In the first case, the classical limit is only taken in one of the two equations, leading to a mixed quantum-classical model which is closely connected to the well-known Ehrenfest method in molecular dynamics. In the second case, the classical limit of the full system is rigorously established, resulting in a system of coupled Vlasov-type equations. In the second part of our work, we provide a numerical study of the coupled semi-classically scaled Schrödinger equations and of the mixed quantum-classical model obtained via Ehrenfest's method. A second order (in time) method is introduced for each case. We show that the proposed methods allow time steps independent of the semi-classical parameter(s) while still capturing the correct behavior of physical observables. It also becomes clear that the order of accuracy of our methods can be improved in a straightforward way.

Citation: Shi Jin, Christof Sparber, Zhennan Zhou. On the classical limit of a time-dependent self-consistent field system: Analysis and computation. Kinetic and Related Models, 2017, 10 (1) : 263-298. doi: 10.3934/krm.2017011
##### References:
 [1] W. Bao, S. Jin and P. A. Markowich, Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes, SIAM J. Sci. Comput., 25 (2003), 27-64.  doi: 10.1137/S1064827501393253. [2] W. Bao, S. Jin and P. A. Markowich, On time-splitting spectral approximations for the Schrödinger equation in the semi-classical regime, J. Comput. Phys., 175 (2002), 487-524.  doi: 10.1006/jcph.2001.6956. [3] C. Bayer, H. Hoel, A. Kadir, P. Plechac, M. Sandberg and A. Szepessy, Computational error estimates for Born-Oppenheimer molecular dynamics with nearly crossing potential surfaces, Appl Math Res Express, 2015 (2015), 329–417, arXiv: 1305.3330. doi: 10.1093/amrx/abv007. [4] R. H. Bisseling, R. Kosloff, R. B. Gerber, M. A. Ratner, L. Gibson and C. Cerjan, Exact time-dependent quantum mechanical dissociation dynamics of I2 He: Comparison of exact time-dependent quantum calculation with the quantum time-dependent self-consistent field (TDSCF) approximation, J. Chem. Phys., 87 (1987), 2760-2765. [5] F. Bornemann, P. Nettesheim and C. Schütte, Quantum-classical molecular dynamics as an approximation to full quantum dynamics, J. Chem. Phys., 105 (1996), 1074-1083.  doi: 10.1063/1.471952. [6] F. Bornemann and C. Schütte, On the singular limit of the quantum-classical molecular dynamics model, SIAM J. Appl. Math., 59 (1999), 1208-1224.  doi: 10.1137/S0036139997318834. [7] R. Carles, On Fourier time-splitting methods for nonlinear Schrödinger equations in the semi-classical limit, SIAM J. Numer. Anal., 51 (2013), 3232-3258.  doi: 10.1137/120892416. [8] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics vo. 10, New York University, 2003. doi: 10.1090/cln/010. [9] K. Drukker, Basics of surface hopping in mixed quantum/classical simulations, J. Comput. Phys., 153 (1999), 225-272.  doi: 10.1006/jcph.1999.6287. [10] I. Gasser and P. A. Markowich, Quantum hydrodynamics, Wigner transforms and the classical limit, Asymptotic Analysis, 14 (1997), 97-116. [11] P. Gérard, P. Markowich, N. Mauser and F. Poupaud, Homogenization Limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379.  doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C. [12] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations vo. 31, Springer, Heidelberg, 2010. [13] J. Hinze, MC-SCF. I. The multi-configurational self-consistent-field method, J. Chem. Phys., 59 (1973), 6424-6432.  doi: 10.1063/1.1680022. [14] S. Jin, C. D. Levermore and D. W. McLaughlin, The behavior of solutions of the NLS equation in the semi-classical limit, Singular limits of dispersive waves, Springer US, 320 (1994), 235-255. [15] S. Jin, P. Markowich and C. Sparber, Mathematical and computational methods for semi-classical Schrödinger equations, Acta Num., 20 (2011), 121-209.  doi: 10.1017/S0962492911000031. [16] S. Jin and Z. Zhou, A semi-Lagrangian time splitting method for the Schrödinger equation with vector potentials, Comm. Inf. Syst., 13 (2013), 247-289.  doi: 10.4310/CIS.2013.v13.n3.a1. [17] C. Klein, Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equations, Electronic Trans. Num. Anal., 29 (2008), 116-135. [18] Z. Kotler, E. Neria and A. Nitzan, Multiconfiguration time-dependent self-consistent field approximations in the numerical-solution of quantum dynamic problems, Comput. Phys. Comm., 63 (1991), 243-258. [19] Z. Kotler, A. Nitzan and R. Kosloff, Multiconfiguration time-dependent self-consistent field approximation for curve crossing in presence of a bath. A Fast Fourier Transform study, Chem. Phys. Lett., 153 (1988), 483-489.  doi: 10.1016/0009-2614(88)85247-3. [20] P. L. Lions and T. Paul, Sur les measures de Wigner, Rev. Math. Iberoamericana, 9 (1993), 553-618.  doi: 10.4171/RMI/143. [21] N. Makri and W. H. Miller, Time-dependent self-consistent field (TDSCF) approximation for a reaction coordinate coupled to a harmonic bath: Single and multiple configuration treatments, J. Chem. Phys., 87 (1987), 5781-5787.  doi: 10.1063/1.453501. [22] P. Markowich and N. Mauser, The classical limit of a self-consistent quantum-Vlasov equation in 3-D, Math. Models Methods Appl. Sci., 3 (1993), 109-124.  doi: 10.1142/S0218202593000072. [23] P. A. Markowich, P. Pietra and C. Pohl, Numerical approximation of quadratic observables of Schrödinger type equations in the semi-classical limit, Numer. Math., 81 (1999), 595-630.  doi: 10.1007/s002110050406. [24] C. Sparber, P. A. Markowich and N. J. Mauser, Wigner functions versus WKB-methods in multivalued geometrical optics, Asymptotic Analysis, 33 (2003), 153-187. [25] H. Spohn and S. Teufel, Adiabatic decoupling and time-dependent born oppenheimer theory, Comm. Math. Phys., 224 (2001), 113-132.  doi: 10.1007/s002200100535. [26] X. Sun and W. H. Miller, Mixed semi-classical classical approaches to the dynamics of complex molecular systems, J. Chem. Phys., 106 (1997), 916-927. [27] A. Szepessy, Langevin molecular dynamics derived from Ehrenfest dynamics, Math. Models Methods Appl. Sci., 21 (2011), 2289-2334.  doi: 10.1142/S0218202511005751. [28] E. Wigner, On the quantum correction for the thermodynamic equilibrium, The Collected Works of Eugene Paul Wigner, A/4 (1997), 110-120.  doi: 10.1007/978-3-642-59033-7_9.

show all references

Dedicated to Peter Markowich on the occasion of his 60th birthday

##### References:
 [1] W. Bao, S. Jin and P. A. Markowich, Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes, SIAM J. Sci. Comput., 25 (2003), 27-64.  doi: 10.1137/S1064827501393253. [2] W. Bao, S. Jin and P. A. Markowich, On time-splitting spectral approximations for the Schrödinger equation in the semi-classical regime, J. Comput. Phys., 175 (2002), 487-524.  doi: 10.1006/jcph.2001.6956. [3] C. Bayer, H. Hoel, A. Kadir, P. Plechac, M. Sandberg and A. Szepessy, Computational error estimates for Born-Oppenheimer molecular dynamics with nearly crossing potential surfaces, Appl Math Res Express, 2015 (2015), 329–417, arXiv: 1305.3330. doi: 10.1093/amrx/abv007. [4] R. H. Bisseling, R. Kosloff, R. B. Gerber, M. A. Ratner, L. Gibson and C. Cerjan, Exact time-dependent quantum mechanical dissociation dynamics of I2 He: Comparison of exact time-dependent quantum calculation with the quantum time-dependent self-consistent field (TDSCF) approximation, J. Chem. Phys., 87 (1987), 2760-2765. [5] F. Bornemann, P. Nettesheim and C. Schütte, Quantum-classical molecular dynamics as an approximation to full quantum dynamics, J. Chem. Phys., 105 (1996), 1074-1083.  doi: 10.1063/1.471952. [6] F. Bornemann and C. Schütte, On the singular limit of the quantum-classical molecular dynamics model, SIAM J. Appl. Math., 59 (1999), 1208-1224.  doi: 10.1137/S0036139997318834. [7] R. Carles, On Fourier time-splitting methods for nonlinear Schrödinger equations in the semi-classical limit, SIAM J. Numer. Anal., 51 (2013), 3232-3258.  doi: 10.1137/120892416. [8] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics vo. 10, New York University, 2003. doi: 10.1090/cln/010. [9] K. Drukker, Basics of surface hopping in mixed quantum/classical simulations, J. Comput. Phys., 153 (1999), 225-272.  doi: 10.1006/jcph.1999.6287. [10] I. Gasser and P. A. Markowich, Quantum hydrodynamics, Wigner transforms and the classical limit, Asymptotic Analysis, 14 (1997), 97-116. [11] P. Gérard, P. Markowich, N. Mauser and F. Poupaud, Homogenization Limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379.  doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C. [12] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations vo. 31, Springer, Heidelberg, 2010. [13] J. Hinze, MC-SCF. I. The multi-configurational self-consistent-field method, J. Chem. Phys., 59 (1973), 6424-6432.  doi: 10.1063/1.1680022. [14] S. Jin, C. D. Levermore and D. W. McLaughlin, The behavior of solutions of the NLS equation in the semi-classical limit, Singular limits of dispersive waves, Springer US, 320 (1994), 235-255. [15] S. Jin, P. Markowich and C. Sparber, Mathematical and computational methods for semi-classical Schrödinger equations, Acta Num., 20 (2011), 121-209.  doi: 10.1017/S0962492911000031. [16] S. Jin and Z. Zhou, A semi-Lagrangian time splitting method for the Schrödinger equation with vector potentials, Comm. Inf. Syst., 13 (2013), 247-289.  doi: 10.4310/CIS.2013.v13.n3.a1. [17] C. Klein, Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equations, Electronic Trans. Num. Anal., 29 (2008), 116-135. [18] Z. Kotler, E. Neria and A. Nitzan, Multiconfiguration time-dependent self-consistent field approximations in the numerical-solution of quantum dynamic problems, Comput. Phys. Comm., 63 (1991), 243-258. [19] Z. Kotler, A. Nitzan and R. Kosloff, Multiconfiguration time-dependent self-consistent field approximation for curve crossing in presence of a bath. A Fast Fourier Transform study, Chem. Phys. Lett., 153 (1988), 483-489.  doi: 10.1016/0009-2614(88)85247-3. [20] P. L. Lions and T. Paul, Sur les measures de Wigner, Rev. Math. Iberoamericana, 9 (1993), 553-618.  doi: 10.4171/RMI/143. [21] N. Makri and W. H. Miller, Time-dependent self-consistent field (TDSCF) approximation for a reaction coordinate coupled to a harmonic bath: Single and multiple configuration treatments, J. Chem. Phys., 87 (1987), 5781-5787.  doi: 10.1063/1.453501. [22] P. Markowich and N. Mauser, The classical limit of a self-consistent quantum-Vlasov equation in 3-D, Math. Models Methods Appl. Sci., 3 (1993), 109-124.  doi: 10.1142/S0218202593000072. [23] P. A. Markowich, P. Pietra and C. Pohl, Numerical approximation of quadratic observables of Schrödinger type equations in the semi-classical limit, Numer. Math., 81 (1999), 595-630.  doi: 10.1007/s002110050406. [24] C. Sparber, P. A. Markowich and N. J. Mauser, Wigner functions versus WKB-methods in multivalued geometrical optics, Asymptotic Analysis, 33 (2003), 153-187. [25] H. Spohn and S. Teufel, Adiabatic decoupling and time-dependent born oppenheimer theory, Comm. Math. Phys., 224 (2001), 113-132.  doi: 10.1007/s002200100535. [26] X. Sun and W. H. Miller, Mixed semi-classical classical approaches to the dynamics of complex molecular systems, J. Chem. Phys., 106 (1997), 916-927. [27] A. Szepessy, Langevin molecular dynamics derived from Ehrenfest dynamics, Math. Models Methods Appl. Sci., 21 (2011), 2289-2334.  doi: 10.1142/S0218202511005751. [28] E. Wigner, On the quantum correction for the thermodynamic equilibrium, The Collected Works of Eugene Paul Wigner, A/4 (1997), 110-120.  doi: 10.1007/978-3-642-59033-7_9.
The diagram of semi-classical limits: the iterated limit and the classical limit
Reference solution: $\Delta x=\Delta y= \frac{2\pi}{32768}$ and $\Delta t=\frac{0.4}{4096}$. Upper picture: fix $\Delta y= \frac{2\pi}{32768}$ and $\Delta t=\frac{0.4}{4096}$, take $\Delta x=\frac{2\pi}{16384}$, $\frac{2\pi}{8192}$, $\frac{2\pi}{4096}$, $\frac{2\pi}{2048}$, $\frac{2\pi}{1024}$, $\frac{2\pi}{512}$, $\frac{2\pi}{256}$, $\frac{2\pi}{128}$, $\frac{2\pi}{64}$, $\frac{2\pi}{32}$, $\frac{2\pi}{16}$, $\frac{2\pi}{8}$. Lower Picture: fix $\Delta x= \frac{2\pi}{32768}$ and $\Delta t=\frac{0.4}{4096}$, take $\Delta y=\frac{2\pi}{16384}$, $\frac{2\pi}{8192}$, $\frac{2\pi}{4096}$, $\frac{2\pi}{2048}$, $\frac{2\pi}{1024}$, $\frac{2\pi}{512}$, $\frac{2\pi}{256}$, $\frac{2\pi}{128}$, $\frac{2\pi}{64}$, $\frac{2\pi}{32}$, $\frac{2\pi}{16}$, $\frac{2\pi}{8}$. These results show that, when $\delta=O(1)$ and $\varepsilon \ll 1$, the meshing strategy $\Delta x= O(\delta)$ and $\Delta y=O(\varepsilon )$ is sufficient for obtaining spectral accuracy.
Reference solution: $\Delta x=\frac{2\pi}{512}$, $\Delta y= \frac{2\pi}{16348}$ and $\Delta t=\frac{0.4}{4096}$. SSP2: fix $\Delta x=\frac{2\pi}{512}$, $\Delta y= \frac{2\pi}{16348}$, take $\Delta t=\frac{0.4}{1024}$, $\frac{2\pi}{512}$, $\frac{2\pi}{256}$, $\frac{2\pi}{128}$, $\frac{2\pi}{64}$, $\frac{2\pi}{32}$, $\frac{2\pi}{16}$, $\frac{2\pi}{8}$. These results show that, when $\delta=O(1)$ and $\varepsilon \ll 1$, the SSP2 method is unconditionally stable and is second order accurate in time
Fix $\Delta t=0.05$. For $\varepsilon =1/64$, $1/128$, $1/256$, $1/512$, $1/1024$, $1/2048$ and $1/{4096}$, $\Delta x=2\pi\varepsilon/16$, respectively. The reference solution is computed with the same $\Delta x$, but $\Delta t={\varepsilon }/{10}$. These results show that, $\varepsilon$-independent time steps can be taken to obtain accurate physical observables, but not accurate wave functions
$\varepsilon=\frac{1}{512}$. First row: position density and current density of $\varphi^e$;
$\varepsilon=\frac{1}{2048}$. First row: position density and flux density of $\varphi^e$; second row: position density and current density of $\psi^e$
Fix $\Delta$ t=0.005. For $\varepsilon=\frac{1}{256}$, $\frac1{512}$, $\frac1{1024}$, $\frac1{2048}$, $\frac1{4096}$, $\Delta x=\frac{\varepsilon}{8}$, respectively. The reference solution is computed with the same $\Delta x$, but $\Delta t=\frac{0.54\varepsilon}{4}$. These results show that, $\varepsilon$-independent time steps can be taken to obtain accurate physical observables, but not accurate wave functions.
Fix $\varepsilon=\frac{1}{256}$ and $\Delta t=\frac{0.4 \varepsilon}{16}$. Take $\Delta x= \frac{2\pi\varepsilon}{32}$, $\frac{2\pi\varepsilon}{16}$, $\frac{2\pi\varepsilon}{8}$, $\frac{2\pi\varepsilon}{4}$, $\frac{2\pi\varepsilon}{2}$ and $\frac{2\pi\varepsilon}{1}$ respectively. The reference solution is computed with the same $\Delta t$, but $\Delta x=\frac{2\pi\varepsilon}{64}$. These results show that, when $\delta=\varepsilon \ll 1$, the meshing strategy $\Delta x= O(\varepsilon )$ and $\Delta y=O(\varepsilon )$ is sufficient for obtaining spectral accuracy
Fix $\varepsilon=\frac{1}{1024}$ and $\Delta x=\frac{2 \pi}{16}$. Take $\Delta t= \frac{0.4}{32}$, $\frac{0.4}{64}$, $\frac{0.4}{128}$, $\frac{0.4}{256}$, $\frac{0.4}{512}$ and $\frac{0.4}{1024}$, respectively. The reference solution is computed with the same $\Delta x$, but $\Delta t=\frac{0.4}{8192}$. These results show that, when $\delta=\varepsilon \ll 1$, the SSP2 method is unconditionally stable and is second order accurate in time
Fix $\Delta t=\frac{0.4}{64}$. For $\delta=\frac1{256}$, $\frac1{512}$, $\frac1{1024}$, $\frac1{2048}$, $\frac1{4096}$, $\Delta x=2\pi\varepsilon/16$, respectively. The reference solution is computed with the same $\Delta x$, but $\Delta t=\frac{\delta}{10}$. These results show that, $\delta$-independent time steps can be taken to obtain accurate physical observables and classical coordinates, but not accurate wave functions
Fix $\delta=\frac{1}{1024}$ and $\Delta x=\frac{2 \pi}{16}$. Take $\Delta t= \frac{0.4}{32}$, $\frac{0.4}{64}$, $\frac{0.4}{128}$, $\frac{0.4}{256}$, $\frac{0.4}{512}$ and $\frac{0.4}{1024}$, respectively. The reference solution is computed with the same $\Delta x$, but $\Delta t=\frac{0.4}{8192}$. These results show that, the SVSP2 method is unconditionally stable and is second order accurate in time
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