Open Access Article

This Open Access Article is licensed under a

Creative Commons Attribution 3.0 Unported Licence

Ryoji
Anzaki
*^{a},
Takeshi
Sato
^{ab} and
Kenichi L.
Ishikawa
^{ab}
^{a}Department of Nuclear Engineering and Management, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan. E-mail: anzaki@atto.t.u-tokyo.ac.jp
^{b}Photon Science Center, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan

Received
31st March 2017
, Accepted 25th July 2017

First published on 25th July 2017

We present a fully general time-dependent multiconfiguration self-consistent-field method to describe the dynamics of a system consisting of arbitrarily different kinds and numbers of interacting fermions and bosons. The total wave function is expressed as a superposition of different configurations constructed from time-dependent spin-orbitals prepared for each particle kind. We derive equations of motion followed by configuration–interaction (CI) coefficients and spin-orbitals for general, not restricted to full-CI, configuration spaces. The present method provides a flexible framework for the first-principles theoretical study of, e.g., correlated multielectron and multinucleus quantum dynamics in general molecules induced by intense laser fields and attosecond light pulses.

Ab initio simulations of the electronic and nuclear dynamics in atoms and molecules remain a challenge. A multiconfiguration time-dependent Hartree–Fock (MCTDHF) method^{4,5} has been developed for the investigation of multielectron dynamics in strong and/or ultrashort laser fields.^{6} In this approach, the time-dependent total electronic wave function Ψ(t) is expressed as a superposition of different Slater determinants Φ_{I}(t),

(1) |

Among successful approaches for nuclear dynamics is the multiconfiguration time-dependent Hartree (MCTDH) method.^{11} Developed for systems consisting of distinguishable particles, this method expresses the time-dependent total nuclear wave function as a superposition similar to eqn (1) but that of Hartree products. The other way around, the MCTDHF method can be viewed as an extension of the MCTDH method to fermions. By hybridizing the MCTDHF method for electrons and the MCTDH method for nuclei, one can construct a multiconfiguration electron–nuclear dynamics (MCEND) method^{12,13} to describe the non-Born–Oppenheimer coupled dynamics. Nuclei forming molecules are, however, indistinguishable particles, either fermions or bosons. Alon et al. have explored methods for systems consisting of identical particles,^{14–17}e.g., MCTDH methods for mixtures consisting of two^{14} and three^{17} kinds of identical particles, and inclusion of particle conversion.^{16} Kato and Yamanouchi have extended the MCTDHF theory to molecules composed of electrons, (fermionic) protons, and two heavy (either classical or quantum) nuclei.^{18}

In this paper, further stepping forward in this direction, we present a fully general TD-MCSCF method for a system comprising arbitrarily different kinds and numbers of interacting fermions and bosons (without particle conversion^{16}). Treating all the constituent particles on an equal footing, we expand the total wave function in terms of configurations of the whole system [see eqn (5) below]. Thus, based on the time-dependent variational principle, we derive the equations of motion (EOM) of CI coefficients and spin-orbitals for general configuration spaces, not restricted to full-CI. As a simple example, we apply the TD-MCSCF method to high-harmonic generation (HHG) from a one-dimensional (1D) model hydrogen molecular ion H_{2}^{+} induced by an intense near-infrared (NIR) laser pulse, for which numerically exact solution is available.

This paper is organized as follows. Section 2 introduces our TD-MCSCF ansatz for many-particle systems composed of different kinds of fermions and bosons, and also defines the target Hamiltonian considered in this work. In Section 3, we derive the general equations of motion, based on the time-dependent variational principle. Explicit working equations for a molecule interacting with an external laser field are shown in Section 4. We present numerical examples in Section 5. Concluding remarks are given in Section 6.

Let us define, for each kind of particles, the complete orthonormal set of spin-orbitals , which spans the one-particle Hilbert space Ω_{α}, is time-dependent in general. Then the N-particle Hilbert space is spanned by

(2) |

For rigorous and compact presentation of theory, we resort to the second quantization formulation by introducing creation and annihilation operators associated with . These operators obey the (anti-)commutation relations of bosons (fermions),

(3) |

(4) |

Within the TD-MCSCF ansatz, the complete set of spin-orbitals is split into n_{α} (≥N_{α} for fermions) occupied spin-orbitals and remaining virtual spin-orbitals . We call the subspace of Ω_{α} spanned by occupied spin-orbitals the occupied spin-orbital space Ω^{occ}_{α}, and that spanned by virtual spin-orbitals the virtual spin-orbital space Ω^{vir}_{α}, where Ω_{α} = Ω^{occ}_{α} ⊕ Ω^{vir}_{α}. The total state Ψ(t) is expressed as a superposition of configurations Φ_{I}(t) of eqn (2), but constructed from occupied spin-orbitals only. Thus we write

(5) |

|I(t)〉 = |I_{1}(t)〉 ⊗ |I_{2}(t)〉 ⊗⋯⊗ |I_{K}(t)〉, | (6) |

(7) |

It should be noted that we do not restrict the expansion eqn (5) to the full-CI one. It should also be noticed that occupied configurations are specified in terms of the whole system rather than in terms of each particle kind as,^{17}

(8) |

H = H_{1} + H_{2} +⋯+ H_{M}, M ≤ N. | (9) |

(10) |

(11) |

The Hamiltonian is equivalently expressed in the second quantization formalism as

(12) |

(13) |

(14) |

(15) |

(16) |

(ρ_{m})^{μ}_{ν} = 〈Ψ|Ê^{ν}_{μ}|Ψ〉. | (17) |

(18) |

(19) |

(20) |

(21) |

(22) |

(23) |

(24) |

(25) |

(26) |

First, the EOM for CI coefficients are obtained from δS/δC_{I}* = 0,

(27) |

(28) |

(29) |

(30) |

(31) |

(32) |

Now the RHS of eqn (31) is given by the sum over m,

(33) |

(34) |

(35) |

Finally, gathering the occupied and virtual components of the time derivative completes the derivation of the EOM for spin-orbitals

(36) |

The spin-independent molecular Hamiltonian in the coordinate representation is given by

(37) |

(38) |

V^{ext}_{α,LG}(r,r′,t) = −δ(r − r′)Z_{α}E(t)r, | (39) |

(40) |

The general formulation of Section 3 is readily applicable to the molecular Hamiltonian of eqn (37). The CI EOM reads

(41) |

(42) |

(43) |

(44) |

(45) |

(46) |

Finally, eqn (29) is formulated as the linear system of equations,

(47) |

(48) |

(49) |

Equations of motions (41) and (44), with the matrix eqn (47), define the general TD-MCSCF method, not restricted to full CI, for molecules interacting with an external field. Our formulation is very flexible; it includes as special cases both the electron dynamics at the classical-nuclei approximation (N_{n} = 0, N_{cl} = N_{atom}) and the full quantum molecular dynamics (N_{n} = N_{atom}, N_{cl} = 0). Furthermore, it allows various approaches to the same physical problem; e.g., the same nuclear species in the molecule can be treated either as identical particles or distinguishable ones to investigate the physical outcomes of the particle statistics during the course of laser-molecule interactions.

(50) |

(51) |

Let us compare the following treatments: (1) numerically exact solution of the time-dependent Schrödinger equation (TDSE) for the spatial part Ψ_{exact}(z,R,t) of the 1D × 1D wavefunction by discretizing electron coordinate z and nuclear distance R with constant grid spacing δz = 0.4 and δR = 0.2, respectively, within the simulation volume |z| < 640 and |R| < 80, (2) the electron only dynamics by solving for electronic wavefunction Ψ_{fixed}(z,t) given the first line of eqn (51) with a fixed (near to equilibrium) internuclear distance R_{0} = 2.6 [fixed (classical point) nuclei approximation], and (3) TD-MCSCF method with the total wavefunction given by

(52) |

Ψ_{frozen}(z,R,t) = χ^{e}_{1}(z,t)χ^{n}_{1}(R,0). | (53) |

Fig. 1 shows the time evolution of the dipole acceleration a(t) = d^{2}〈Ψ(t)|z|Ψ(t)〉/dt^{2} computed using the Ehrenfest theorem. As shown in Fig. 1(a), both fixed-nuclei (light blue curve) and frozen-nuclei (red) approximations strongly underestimate the nonlinear response seen in the TDSE result (black), which suggests the importance of the dynamical quantum treatment of nuclei. The TD-MCSCF method with n = 1, or TDH [Fig. 1(b)] provides improved description of the nonlinear response, but still fails to quantitatively reproduce the TDSE result. The TD-MCSCF method with n = 8 [Fig. 1(c)], on the other hand, produces the dipole acceleration that agrees with the TDSE result nearly perfectly, on the scale of the figure. The HHG spectra calculated as the modulus squared of the Fourier transform of the dipole acceleration are shown in Fig. 2. The fixed and frozen nuclei approximations underestimate the intensity of the first few harmonics, while overestimating the higher plateau (the cutoff position is estimated to be the 39th harmonic from the Lewenstein model^{26}) [Fig. 2(a)], both suggesting an important contribution from (more polarizable and loosely bound) the larger |R| region in the TDSE result. The TD-MCSCF spectrum with increasing n [shown for n = 1 (Fig. 2(b)) and n = 8 (Fig. 2(c))] shows increasingly better agreement with the TDSE one, and the TD-MCSCF method with n = 8 nicely reproduces the overall spectrum obtained using the TDSE simulations, especially for low-order harmonics, albeit with a still remaining slight overestimation of the plateau intensity which implies a strong electron–proton correlation.

Fig. 2 The HHG spectra of one-dimensional H_{2}^{+} for the same laser pulse and simulation methods as those in Fig. 1. Comparison of the results of fixed nuclei and frozen nuclei approximations (a), TD-MCSCF method with n = 1 (b), and TD-MCSCF method with n = 8 (c) with numerically exact TDSE results. Also shown in (b) is the result of the TD-MCSCF method directly applied for the Hamiltonian of eqn (50) with one electronic orbital and two protonic orbitals. |

Finally, also plotted in Fig. 2(b) with a red line is the result of TD-MCSCF method directly applied to the Hamiltonian of the original coordinate, eqn (50), with one electronic orbital and two protonic orbitals to expand the three-particle wavefunction Ψ(x,X_{1},X_{2}), using the method described in ref. 19 to eliminate the translational degree of freedom. We find a remarkable agreement between the results from the original Hamiltonian eqn (50) and the reduced Hamiltonian eqn (51). The former approach can be applied to more complex systems in general.

The present framework is highly flexible. For example, we can treat identical nuclei in spatially separated subdomains of a molecule as different particle kinds. We can also treat heavy nuclei and incident projectiles as classical particles^{18} instead of quantum ones. The latter may be handled as an external field as well. This flexibility will be useful for unraveling the physical mechanisms underlying the phenomena under investigation.^{27} As a future prospect, an introduction of a CI space that adaptively changes following the dynamics as well as an inclusion of particle conversion^{16} would enable even more efficient and flexible simulations.

Whereas our original motivation lies in ab initio simulations of the electron–nuclear dynamics in molecules driven by a laser pulse, our method will be applicable to a wide variety of problems far beyond. Especially, the Hamiltonian can contain non-local terms and involve many-body (more than two-body) interactions. Thus, it may also find applications in cold-atom/cold-molecule physics and nuclear physics.

- P. Agostini and L. F. DiMauro, Rep. Prog. Phys., 2004, 67, 1563 CrossRef.
- F. Krausz and M. Ivanov, Rev. Mod. Phys., 2009, 81, 163–234 CrossRef.
- L. Gallmann, C. Cirelli and U. Keller, Annu. Rev. Phys. Chem., 2013, 63, 447 CrossRef PubMed.
- T. Kato and H. Kono, Chem. Phys. Lett., 2004, 392, 533–540 CrossRef CAS.
- J. Caillat, J. Zanghellini, M. Kitzler, O. Koch, W. Kreuzer and A. Scrinzi, Phys. Rev. A: At., Mol., Opt. Phys., 2005, 71, 012712 CrossRef.
- K. L. Ishikawa and T. Sato, IEEE J. Sel. Top. Quantum Electron., 2015, 21, 1–16 CrossRef.
- T. Sato and K. L. Ishikawa, Phys. Rev. A: At., Mol., Opt. Phys., 2013, 88, 023402 CrossRef.
- T. Sato, K. L. Ishikawa, I. Březinová, F. Lackner, S. Nagele and J. Burgdörfer, Phys. Rev. A, 2016, 94, 023405 CrossRef.
- H. Miyagi and L. B. Madsen, Phys. Rev. A: At., Mol., Opt. Phys., 2014, 89, 063416 CrossRef.
- T. Sato and K. L. Ishikawa, Phys. Rev. A: At., Mol., Opt. Phys., 2015, 91, 023417 CrossRef.
- H.-D. Meyer, U. Manthe and L. Cederbaum, Chem. Phys. Lett., 1990, 165, 73–78 CrossRef CAS.
- I. S. Ulusoy and M. Nest, J. Chem. Phys., 2012, 136, 054112 CrossRef PubMed.
- M. Nest, Chem. Phys. Lett., 2009, 472, 171–174 CrossRef CAS.
- O. E. Alon, A. I. Streltsov and L. S. Cederbaum, Phys. Rev. A, 2007, 76, 062501 CrossRef.
- O. E. Alon, A. I. Streltsov and L. S. Cederbaum, J. Chem. Phys., 2007, 127, 154103 CrossRef PubMed.
- O. E. Alon, A. I. Streltsov and L. S. Cederbaum, Phys. Rev. A: At., Mol., Opt. Phys., 2009, 79, 022503 CrossRef.
- O. E. Alon, A. I. Streltsov, K. Sakmann, A. U. Lode, J. Grond and L. S. Cederbaum, Chem. Phys., 2012, 401, 2–14 CrossRef CAS.
- T. Kato and K. Yamanouchi, J. Chem. Phys., 2009, 131, 164118 CrossRef PubMed.
- H. Nakai, M. Hoshino, K. Miyamoto and S. Hyodo, J. Chem. Phys., 2005, 122, 164101 CrossRef PubMed.
- H. Nakai, M. Hoshino, K. Miyamoto and S. aki Hyodo, J. Chem. Phys., 2005, 123, 237102 CrossRef.
- A. Dalgarno and G. Victor, Proc. R. Soc. London, Ser. A, 1966, 291–295 CrossRef CAS.
- P.-O. Löwdin and P. Mukherjee, Chem. Phys. Lett., 1972, 14, 1–7 CrossRef.
- R. Moccia, Int. J. Quantum Chem., 1973, 7, 779–783 CrossRef.
- R. P. Miranda, A. J. Fisher, L. Stella and A. P. Horsfield, J. Chem. Phys., 2011, 134, 244101 CrossRef CAS PubMed.
- E. Khosravi, A. Abedi and N. T. Maitra, Phys. Rev. Lett., 2015, 115, 263002 CrossRef PubMed.
- M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L'Huillier and P. B. Corkum, Phys. Rev. A: At., Mol., Opt. Phys., 1994, 49, 2117 CrossRef.
- I. Tikhomirov, T. Sato and K. L. Ishikawa, Phys. Rev. Lett., 2017, 118, 203202 CrossRef PubMed.

This journal is © the Owner Societies 2017 |