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An effective potential theory for time-dependent multi-configuration wave function
Tsuyoshi Kato
Department of Chemistry, School of Science, The University of Tokyo, Japan
Email:[email protected]
1. Introduction After more than a decade of the formulation of multi-configuration time-dependent HartreeFock (MCTDHF) method to treat electronic dynamics in atoms and molecules induced by the interaction with
intense ultrafast laser pulses from first principles [1], the theoretical efforts exerted on the developments of
the method has been changed their aspects from the basic formulations and the proof-of-principle type
calculations to practical calculations in order to elucidate the many electron dynamics by comparisons with
experimental results [2]. Recently, efforts have been made to improve the numerical performance of the
MCTDHF method aiming to reduce the size of the configuration space by restricting the orbital excitation
schemes [3,4], although a different approximation of factorized configuration interaction coefficients [5] as
well as the multi-layer formulation of MCTDHF [6] have been introduced recently.
In the present study, we propose an alternative formulation for the time propagation of a time-dependent
multi-configuration wave function in which the spin-orbitals follow a single-particle time-dependent
Schrödinger equation (TDSE) specified by a multiplicative time-dependent local effective potential 𝑣eff (𝒓, 𝑡).
2. Theory We consider an N-electron time-dependent wave function 𝛹(1,2, ⋯ , 𝑁, 𝑡) perturbed by a timedependent external field. The wave function is assumed to be represented by
Ψ(1,2,3, ⋯ , 𝑁, 𝑡) = ∑ 𝐶𝐾 (𝑡)Φ𝐾 (1,2,3, ⋯ , 𝑁, 𝑡),
(1)
𝐾=1
where {𝐶𝐾 (𝑡)} represent time-dependent configuration interaction coefficients and {Φ𝐾 (𝑡)} time-dependent
Slater determinants. The time-dependence of each Slater determinant is due to the time dependence of the
̂ (𝑡) = 𝑇̂ + 𝑉̂ext (𝑡) + 𝑉̂ee,
constituent spin-orbitals. The total Hamiltonian of the system is represented by 𝐻
𝑁
̂
̂
̂
where 𝑇 , 𝑉ext (𝑡) = ∑𝑗=1 𝑣ext (𝒓𝑗 , 𝑡) , and 𝑉ee represent the kinetic energy operator, the sum of nuclear
attraction potential and the time-dependent external perturbation, and the electron-electron repulsion potential,
respectively. The spin-orbitals are assumed to obey a single-particle TDSE expressed by
𝜕
ℏ2 𝜕 2
(2)
+ 𝑣eff (𝒓, 𝑡))] 𝜙𝑘 (𝑥, 𝑡) = 0 (𝑘 = 1,2, ⋯ , 𝑁, ⋯ ),
[𝑖ℏ − (−
𝜕𝑡
2𝑚e 𝜕𝒓2
where 𝑥 = (𝒓, σ) denotes the spatial and spin-coordinates of an electron, and 𝑣eff (𝒓, 𝑡) is the effective
potential to be calculated. We define an effective Hamiltonian for the relevant system as
̂eff (𝑡) = 𝑇̂ + ∑
𝐻
𝑁
𝑗=1
𝑣eff (𝒓𝑗 , 𝑡) = 𝑇̂ + 𝑉̂eff (𝑡).
(3)
The effective potential is formulated by using McLachlan’s minimization principle in which the difference of
̂ (𝑡)
the time-evolution of the wave function 𝛹(1,2, ⋯ , 𝑁, 𝑡) is minimized between the TDSEs specified by 𝐻
̂eff (𝑡).
and 𝐻
3. Discussion We report the detailed theoretical analysis of the properties of the effective potential associated
with an exact wave function. Furthermore, as an elementary application of the present formalism, we propose
a direct method to calculate the so-called Brueckner orbitals [7] as a special solution of a set of spin-orbitals
calculated as eigenfunctions for a single-particle Schrödinger equation specified by a time-independent
effective potential 𝑣eff (𝒓) that is associated with an exact ground-state wave function.
References:
1. For example, T. Kato and H. Kono, Chem. Phys. Lett. 392 (2004) 533-540.
2. K.L. Ishikawa and T. Sato, IEEE J. Sel. Topics Quantum Electron. 21 (2015) 8700916-1-16.
3. H. Miyagi and L.B. Madsen, Phys. Rev. A 87 (2013) 062511-1-12.
4. T. Sato and K. L. Ishikawa, Phys. Rev. A 91 (2015) 023417-1-15.
5. E. Lötstedt, T. Kato, and Y. Yamanouchi, J. Chem. Phys. 144 (2016) 154116-1-13.
6. H. Wang and M. Thoss, J. Chem. Phys. 131 (2009) 024114-1-14.
7. R.K. Nesbet, Phys. Rev. 109 (1958) 1632-1638.
Presentation Method (Invited ):