3P100
拡張された時間依存多配置波動関数法によるCH3OHの電子−プロトン波動関数の実時間発展
東大院理○木村瑠杜，加藤毅，山内薫

Real-time propagation of electro-protonic wavefunctions of CH3OH byextended multiconfiguration time-dependent Hartree-Fock method○Ryuto Kimura, Tsuyoshi Kato, Kaoru Yamanouchi
Department of Chemistry, The University of Tokyo, Japan[Abstract] In order to describe theoretically the ultrafast motion of protons within ahydrocarbon molecule induced by an intense laser field, we developed a calculation code ofthe extended multiconfiguration time-dependent Hartree-Fock (MCTDHF) method, in which
both electrons and protons are treated quantum mechanically using electro-protonic wave
functions, and perform real-time propagation of the electro-protonic wave functions ofCH3OH exposed to an intense laser field using a parallelized code.
[Introduction] Recent experimental studies on ionization and dissociation processes ofhydrocarbon molecules induced by an ultrashort intense laser field revealed that the hydrogenatoms and/or protons migrate rapidly within the hydrocarbon molecules prior to the breakingof skeletal chemical bonds. For example, it was revealed that a hydrogen atom in methylgroup in methanol migrates to hydroxy group within 25 fs by the pump-probe coincidence
momentum imaging measurements of the Coulomb explosion processes of methanol dication,
CH3OH2+, using few-cycle intense near-IR laser pulses [1]. When the ultrafast hydrogen atom
migration proceeds, the interatomic potential along the C-O bond should be influenced by themotion of the hydrogen atom, competing with the C-O bond breaking.
For describing theoretically the ultrafast motion of hydrogen atoms and/or protons within ahydrocarbon molecule, we need to treat the motion of hydrogen atoms separately from theheavier atoms such as carbon and oxygen atoms. In 2009, we proposed a method called
extended multiconfiguration time-dependent Hartree-Fock (MCTDHF) [2], in which not only
the motion of electrons but also the motion of protons are described by time-dependent
orbitals, and showed that the ground-state electro-protonic wave function of methanol,
CH3OH, can be derived using the extended MCTDHF method [3].
In order to perform efficiently real-time propagation of orbitals for electrons and protons inCH3OH, we developed a program code of the extended MCTDHF method, and examined its
performance in describing the interaction between the electronic and protonic orbitals and anintense laser field.
[Theory] In the extended MCTDHF method, the molecular wave function of CH3OH, F, isexpanded by electro-protonic multiconfiguration wave functions asF({x j }, {y p },t; R) = Â Â CIA (t)FI ({x j },t; R)YA ({y p },t; R),I<latexit sha1_base64="/exNTXX7UYCeoGEbmFkPN392X88=">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</latexit>A(1)where xj = (rj, sj) denotes the spatial and spin coordinates of the j-th electron (1 ≤ j ≤ 18), yp =(rp, sp) denotes the spatial and spin coordinates of the p-th proton (1 ≤ p ≤ 4), R is theinter-nuclear distance between C and O, FI and YA are the electronic and protonic Slater
determinants, respectively, and CIA is a time-dependent configuration interaction (CI)
coefficient. The equation of motion of the CI coefficients and spin orbitals are derived basedon the Dirac-Frenkel time-dependent variational principle.
[Results and Discussion] For the real-time propagation of the spin orbitals, we apply the split
operator technique [4]. First, we write the equation of motion of electronic spin orbitals as"#∂i} f j (x,t) = h j (t) + Â Vk j (t) f j (x,t),∂tk(2)<latexit sha1_base64="7R9dHEiUUK36A9BtuDSr/sdtRew=">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</latexit>

where hj(t) is an orbital-independent operator and Vkj(t) is a non-liner operator that depends onthe spin orbitals. When propagating the orbitals from time t to t + dt starting from the initialorbital fj(t), we regard hj and Vkj to be time-independent and compute the orbital after thepropagation, fj(t + dt), asdt dti<latexit sha1_base64="n+snHo80aAD3veeEECMtV8AjIXw=">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</latexit>

idt

idt dtf j (t + dt) ⇡ e } h j (t+ 2 ) 2 e } Âk Vk j (t+ 2 )dt e } h j (t+ 2 ) 2 f j (t).

(3)

The exponential operation of the orbital-independent operators are performed by theCrank-Nicolson method and the exponential operation of the non-liner operators areperformed by Lanczos method [5]. Because electrons and protons are treated in a parallelmanner in the electro-protonic Hamiltonian, we propagate the protonic spin orbitals using thesame algorithm adopted for the electronic orbitals.
In order to parallelize the calculation code, we use the library of OpenMP. Using thecalculation times, T(n), required when the number of threads n takes the values of n = 1, 2,and 4, we obtain the parallelization ratio, p, of p = 0.92 for the imaginary-time propagation bya least-squares fit to the Amdahl’s law written as⇣p
⌘T (n) =+ 1 p t1 ,(4)n<latexit sha1_base64="tAP47xAg3PEggqtPBnDl+2ehFsQ=">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</latexit>

where t1 is the calculation time when n = 1. When n = 8, for example, the calculation time of
the imaginary-time propagation can be 5.1 times shorter than when n = 1. The calculation
code of real-time propagation is also parallelized in the same manner as in the imaginary-time
propagation, and the real-time propagation of the electro-protonic wave functions of CH3OH
exposed to an intense laser field is performed using the parallelized code.
[References][1] T. Ando, A. Shimamoto, S. Miura, K. Nakai, H. Xu, A. Iwasaki, and K. Yamanouchi, Chem. Phys. Lett. 624,
78 (2015).
[2] T. Kato and K. Yamanouchi, J. Chem. Phys. 131, 164118 (2009).
[3] T. Kato and K. Yamanouchi, Phys. Rev. A 85, 034504 (2012).
[4] E. Lötstedt, T. Kato and K. Yamanouchi, Phys. Rev. A 97, 013423 (2018).
[5] R. J. LeVeque, “Finite difference methods for ordinary and partial differential equations”, (Society for
Industrial and Applied Mathematics, Philadelphia, 2007).