Abstract
We report the formulation of a new, cost-effective approximation method in the time-dependent optimized coupled-cluster (TD-OCC) framework [T. Sato et al., J. Chem. Phys. 148, 051101 (2018)] for first-principles simulations of multielectron dynamics in an intense laser field. The method, designated as TD-OCCD(T), is a time-dependent, orbital-optimized extension of the “gold-standard” CCSD(T) method in the ground-state electronic structure theory. The equations of motion for the orbital functions and the coupled-cluster amplitudes are derived based on the real-valued time-dependent variational principle using the fourth-order Lagrangian. The TD-OCCD(T) is size extensive and gauge invariant, and scales as O(N7) with respect to the number of active orbitals N. The pilot application of the TD-OCCD(T) method to the strong-field ionization and high-order harmonic generation from a Kr atom is reported in comparison with the results of the previously developed methods, such as the time-dependent complete-active-space self-consistent field (TD-CASSCF), TD-OCC with double and triple excitations (TD-OCCDT), TD-OCC with double excitations (TD-OCCD), and the time-dependent Hartree-Fock (TDHF) methods.
1 Introduction
Recent years witnessed unprecedented progress in laser technologies, which made it possible to observe the motions of electrons at the attosecond time scale (; ; ; ). On the other hand, various theoretical and numerical methods have been developed for interpreting, understanding, and predicting the experiments.
The multi-configuration time-dependent Hartree-Fock (MCTDHF) method (; ; ; ; ), and the time-dependent complete-active-space self-consistent-field (TD-CASSCF) method (; ; Sato et al. (2018a)) are the most rigorous approaches to solve time-dependent Schrödinger equation (TDSE) of many-electron systems, where the wavefunction is given by the full configuration interaction (FCI) expansion,with both CI coefficients {CI(t)} and orbital functions {ψp(t)} constituting Slater determinants {ΦI(t)} are propagated in time according to the time-dependent variational principle (TDVP). The TD-CASSCF method broadens the applicability of the MCTDHF method by flexibly classifying the orbital subspace into frozen-core, dynamical-core, and active. Unfortunately, the factorial computational scaling impedes large-scale applications. There are reports of various affordable size-inextensive methods (, ; ; Sato and Ishikawa (2015)) developed by limiting the CI expansion of the wavefunction. Alternatively, the size-extensive coupled-cluster method, which relies on an exponential wavefunction, is a superior choice to address these problems with a polynomial cost-scaling (; Shavitt and Bartlett (2009)). We have developed an explicitly time-dependent coupled-cluster method considering optimized orthonormal orbitals within the flexibly chosen active space, called the time-dependent optimized coupled-cluster (TD-OCC) method, (Sato et al. (2018b)) including double (TD-OCCD) and double and triple excitation amplitudes (TD-OCCDT). Our method is a time-dependent formulation of the stationary optimized coupled-cluster method (Scuseria and Schaefer (1987); Sherrill et al. (1998); ). Kvaal () also developed an orbital adaptive time-dependent coupled-cluster (OATDCC) method using biorthogonal orbitals. We take note of a few reports on the time-dependent coupled-cluster methods (; ; ), using time-independent orbitals, and their interpretation (; ), including the very initial attempts (Schonhammer (1978); , ).
The TD-OCCDT scales as O(N8) (N= the number of active orbitals), not ideally suited for applications to larger chemical systems. Therefore, we have developed a few lower cost methods in the TD-OCC framework (,,, ). We find triple excitations are necessary, including perfect optimization of the orbitals. Therefore, we are interested in developing affordable TD-OCC methods retaining a part of the triples. The most popular coupled-cluster method that treats the triple excitation amplitudes approximately is called CCSD(T) (; Watts et al. (1993)). Bozkaya et al, () included various symmetric and asymmetric triple excitation corrections to their optimized double (OD) method.
In this communication, we report the formulation and implementation of the CCSD(T) method in the time-dependent optimized coupled-cluster framework, TD-OCCD(T). Following our previous works (Sato et al. (2018b); ,, ), we exclude single excitation amplitudes but optimize the orbitals according to time-dependent variational principle (TDVP). As the first application of this method, we study electron dynamics in Kr using intense near-infrared laser fields.
2 Methods
The second quantization representation of the Hamiltonian, including the laser field, is as follows,where represents a creation (annihilation) operator in a complete, orthonormal set of 2nbas time-dependent spin-orbitals {ψμ(t)}. nbas is the number of basis functions used for expanding the spatial part of ψμ, which, in the present real-space implementation, corresponds to the number of grid points, andwhere xi = (ri, σi) represents a composite spatial-spin coordinate. h0 is the field free one-electronic Hamiltonian and Vext = A(t)pz in the velocity gauge, A(t) = −∫tE(t′)dt′ is the vector potential, with E(t) being the laser electric field linearly polarized along the z axis.
The complete set of 2nbas spin-orbitals (labeled with μ, ν, γ, λ) is divided into noccoccupied (o, p, q, r, s) and 2nbas − noccvirtual spin-orbitals. The coupled-cluster (or CI) wavefunction is constructed only with occupied spin-orbitals, which are time-dependent in general, and virtual spin-orbitals form the orthogonal complement of the occupied spin-orbital space. The occupied spin-orbitals are classified into ncorecore spin-orbitals, which are occupied in the reference Φ and kept uncorrelated, and N = nocc − ncoreactive spin-orbitals (t, u, v, w) among which the active electrons are correlated. The active spin-orbitals are further split into those in the hole space (i, j, k, l) and the particle space (a, b, c, d), which are defined as those occupied and unoccupied, respectively, in the reference Φ. The core spin-orbitals can further be split into frozen-core space (i′′, j′′), fixed in time and the dynamical-core space (i′, j′), propagated in time () (See. Figure 1 in Sato et al. (2018b) for a pictorial illustration).
FIGURE 1
The real action formulation of the TDVP with orthonormal orbitals is our guiding principle, (Sato et al. (2018b))where are (de-)excitation amplitudes, and . The stationary conditions, δS = 0, with respect to the variation of the parameters of the wavefunction (, , and δψμ) gives us the corresponding equations of motions (EOMs), δψμ is orthonormality-conserving orbital variation.
For deriving the TD-OCCD(T) method, we first construct a fourth-order Lagrangian defined in . We make a further approximation to the Lagrangian and write separating it into two parts,where , , , and , , and is anti-Hermitian. The double amplitudes are obtained by making of Eq. 9a stationary with respect to , , the triples by making Eq. 9b stationary with respect to , and ,where p(μν) and p(μ|νγ) are the permutation operators; p(μν)Aμν = Aμν − Aνμ, and p(μ/νγ) = 1 − p(μν) − p(μγ).
The EOM for the orbitals can be written down in the following form ,where is the identity operator within the space spanned by the given basis, is the projector onto the occupied spin-orbital space, andwhere D and P are Hermitialized one- (1RDM) and two- (2RDM) particle reduced density matrices defined in Sato et al. (2018b), and is the mean-field operator (). The matrix element includes orbital rotations among various subspaces. Non-redundant orbital rotations are determined by . Redundant orbital rotations , , and can be arbitrary antiHermitian matrix elements. The general expressions for the RDMs are the same as in the TD-OCCDT(4) method ().
3 Numerical results and discussion
Our numerical implementation has an interface with the Gaussian09 program () for checking ground state energy with the standard Gaussian basis results. We study BH molecule with double-ζ plus polarization (DZP). We have reported ground state energy computed by propagating in the imaginary time for OCCD and OCCD(T) methods in Table 1 and compared those with the optimized double and asymmetric triple excitation corrections for the orbital-optimized doubles method of Bozkaya et al., . We also compare our OCCD ground state energy result with Krylov et al., within the chosen active space of six electrons correlated among the six optimized active orbitals. We obtained a perfect agreement for all available values.
TABLE 1
| Method | This work | References | |
|---|---|---|---|
| OCCDb | − 25.225 591 67 | − 25.225 592 | |
| OCCD(T)b | − 25.226 913 29 | − 25.226 913 | |
| OCCDc | − 25.178 285 70 | − 25.178 286 | |
| OCCD(T)c | − 25.178 301 00 |
Comparison of the ground state energy of BH (re=2.4 bohr) molecule in DZP basisa.
Gaussian09 program () is used to generate the required one-electron, two-electron, and overlap integrals, required for the imaginary time propagation of EOMs in the orthonormalized Gaussian basis. A convergence cut-off of 10–15 Hartree of energy difference is chosen in subsequent time steps.
Six electrons correlated within the full basis set.
Six electrons correlated within the six optimized active orbitals.
We have used a spherical-finite-element-discrete-variable representation (FEDVR) basis for representing orbital functions, ; where Ylm and fk(r) are spherical harmonics and the normalized radial-FEDVR basis function, respectively. The expansion of the spherical harmonics continued up to the maximum angular momentum Lmax, and the radial FEDVR basis supports the range of radial coordinate 0 ≤ r ≤ Rmax, with cos1/4 mask function used as an absorbing boundary for avoiding unphysical reflection from the wall of the simulation box. We have used lmax = 72, and the FEDVR basis supporting the radial coordinate 0 < r < 300 using 78 finite elements each containing 25 DVR functions. The absorbing boundary is switched on at r = 180 in all our simulations. The Fourth-order exponential Runge-Kutta method () is used to propagate the EOMs with 20000 time steps for each optical cycle. We run the simulations for a further 6,000 time steps after the end of the pulse. In all correlation calculations, eight electrons of 4s4p orbitals are considered as active and correlated among thirteen active orbitals. We report simulation results computed using a three-cycle laser pulse with a central wavelength of 800 nm having intensity 2 × 1014 W/cm2 and a period of T = 2π/ω0 ∼ 2.67 fs.
We report the time evolution of dipole moment of Kr in Figure 1 and in Figure 2 single electron ionization probability. Time-dependent dipole moment is evaluated as a trace using 1RDMs. For the single electron ionization probability, we computed the probability of finding an electron outside a sphere of a radius of 20 a.u. using RDMs defined in Refs. 19; 20; 37. We compare the results of TD-CASSCF, TD-OCCD(T), TD-OCCD, and TDHF methods.
FIGURE 2
We observe a substantial underestimation (both in Figure 1, and Figure 2) by the TDHF method due to the lack of correlation treatment. All correlation methods perform according to their ability to treat electron correlation. We also computed results using the TD-OCCDT method but not reported here since those results are not identifiable from the TD-CASSCF results within the graphical resolution.
Next, we report high-harmonic generation in Figure 3. It is calculated by squaring the modulus I(ω) = |a(ω)|2 of the Fourier transform of the expectation value of the dipole acceleration with a modified Ehrenfest expression (). In panel (c) of Figure 3, we plot the absolute relative deviation (δ(ω), of the spectral amplitude a(ω) from the TD-CASSCF value for each method. All methods qualitatively predict similar HHG spectra with TDHF underestimates the spectral intensity. The relative deviation of results from TD-CASSCF ones follows the general trend TDHFTD-OCCDTD-OCCD(T)TD-OCCDT, the same as what we observe for the time-dependent dipole moment and single ionization probability. We also simulated results with lower and higher intensity. However, the trend remains the same.
FIGURE 3
Finally, we make a tally of computational costs for all the methods considered in this article. All simulations performed using an Intel(R) Xeon(R) Gold 6,230 central processing unit (CPU) with 40 processors with a clock speed of 2.10 GHz, and report total simulations time in Table 2. Further, we report a reduction in the computational cost for various TD-OCC methods relative to the TD-CASSCF. We see a massive 63% cost reduction for the TD-OCCD(T) method, which is larger than for the TD-OCCDT method (58%), and a minimal increase from the TD-OCCD method.
TABLE 2
| Method | Time (min) | Cost reduction (%) |
|---|---|---|
| TD-CASSCF | 47303 | … |
| TD-OCCDT | 19697 | 58 |
| TD-OCCD(T) | 17504 | 63 |
| TD-OCCD | 17494 | 63 |
Comparison of the total simulation timea (in min) spent for TD-CASSCF, TD-OCCDT, TDCCD(T), and TD-OCCD methods.
Time spent for the simulation of Kr atom for 66000 time steps (0 ≤ t ≤ 3.3T) of a real-time simulation (I0 = 2 × 1014 W/cm2 and λ = 800 nm), using an Intel(R) Xeon(R) Gold 6230 CPU with 40 processors having a clock speed of 2.10 GHz.
4 Concluding remarks
We have reported the formulation and implementation of the TD-OCCD(T) method. As the first application, we employed this method to study laser-driven dynamics in Kr exposed to an intense near-infrared laser pulse. We observe a 63% cost reduction in comparison to the TD-CASSCF method without losing much accuracy. Therefore, we conclude that TD-OCCD(T) method will certainly be beneficial in exploring highly accurate ab initio simulations of electron dynamics in larger chemical systems.
Statements
Data availability statement
The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.
Author contributions
HP and TS formulated the method. HP numerically implemented the method and performed simulations. All the authors analyzed the results and contributed to the submitted version of the manuscript.
Funding
This research was supported in part by a Grant-in-Aid for Scientific Research (Grants No. JP18H03891 and No. JP19H00869) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. This research was also partially supported by JST COI (Grant No. JPMJCE1313), JST CREST (Grant No. JPMJCR15N1), and by MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) Grant Number JPMXS0118067246.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
References
1
BakerS.RobinsonJ. S.HaworthC.TengH.SmithR.ChirilăC.et al (2006). Probing proton dynamics in molecules on an attosecond time scale. Science312, 424–427. 10.1126/science.1123904
2
BozkayaU.SchaeferH. F.III (2012). Symmetric and asymmetric triple excitation corrections for the orbital-optimized coupled-cluster doubles method: Improving upon ccsd (t) and ccsd (t) λ: Preliminary application. J. Chem. Phys.136, 204114. 10.1063/1.4720382
3
CaillatJ.ZanghelliniJ.KitzlerM.KochO.KreuzerW.ScrinziA. (2005). Correlated multielectron systems in strong laser fields: A multiconfiguration time-dependent Hartree-Fock approach. Phys. Rev. A . Coll. Park.71, 012712. 10.1103/physreva.71.012712
4
CorkumP. á.KrauszF. (2007). Attosecond science. Nat. Phys.3, 381–387. 10.1038/nphys620
5
[Dataset]FrischM.TrucksG.SchlegelH.ScuseriaG.RobbM.CheesemanJ.et al (2009). Gaussian 09, revision d. 01.
6
HaxtonD. J.LawlerK. V.McCurdyC. W. (2011). Multiconfiguration time-dependent Hartree-Fock treatment of electronic and nuclear dynamics in diatomic molecules. Phys. Rev. A . Coll. Park.83, 063416. 10.1103/physreva.83.063416
7
HaxtonD. J.McCurdyC. W. (2015). Two methods for restricted configuration spaces within the multiconfiguration time-dependent Hartree-Fock method. Phys. Rev. A . Coll. Park.91, 012509. 10.1103/physreva.91.012509
8
HochbruckM.OstermannA. (2010). Exponential integrators. Acta Numer.19, 209–286. 10.1017/s0962492910000048
9
HochstuhlD.BonitzM. (2011). Two-photon ionization of helium studied with the multiconfigurational time-dependent Hartree–Fock method. J. Chem. Phys.134, 084106. 10.1063/1.3553176
10
HoodbhoyP.NegeleJ. W. (1978). Time-dependent coupled-cluster approximation to nuclear dynamics. i. application to a solvable model. Phys. Rev. C18, 2380–2394. 10.1103/physrevc.18.2380
11
HoodbhoyP.NegeleJ. W. (1979). Time-dependent coupled-cluster approximation to nuclear dynamics. ii. general formulation. Phys. Rev. C19, 1971–1982. 10.1103/physrevc.19.1971
12
HuberC.KlamrothT. (2011). Explicitly time-dependent coupled cluster singles doubles calculations of laser-driven many-electron dynamics. J. Chem. Phys.134, 054113. 10.1063/1.3530807
13
ItataniJ.LevesqueJ.ZeidlerD.NiikuraH.PépinH.KiefferJ. C.et al (2004). Tomographic imaging of molecular orbitals. Nature432, 867–871. 10.1038/nature03183
14
KatoT.KonoH. (2004). Time-dependent multiconfiguration theory for electronic dynamics of molecules in an intense laser field. Chem. Phys. Lett.392, 533–540. 10.1016/j.cplett.2004.05.106
15
KrauszF.IvanovM. (2009). Attosecond physics. Rev. Mod. Phys.81, 163–234. 10.1103/revmodphys.81.163
16
KrylovA. I.SherrillC. D.ByrdE. F.Head-GordonM. (1998). Size-consistent wave functions for nondynamical correlation energy: The valence active space optimized orbital coupled–cluster doubles model. J. Chem. Phys.109, 10669–10678. 10.1063/1.477764
17
KümmelH. G. (2003). A biography of the coupled cluster method. Int. J. Mod. Phys. B17, 5311–5325. 10.1142/s0217979203020442
18
KvaalS. (2012). Ab initio quantum dynamics using coupled-cluster. J. Chem. Phys.136, 194109. 10.1063/1.4718427
19
MiyagiH.MadsenL. B. (2013). Time-dependent restricted-active-space self-consistent-field theory for laser-driven many-electron dynamics. Phys. Rev. A . Coll. Park.87, 062511. 10.1103/physreva.87.062511
20
MiyagiH.MadsenL. B. (2014). Time-dependent restricted-active-space self-consistent-field theory for laser-driven many-electron dynamics. ii. extended formulation and numerical analysis. Phys. Rev. A . Coll. Park.89, 063416. 10.1103/physreva.89.063416
21
NascimentoD. R.DePrinceA. E.III (2016). Linear absorption spectra from explicitly time-dependent equation-of-motion coupled-cluster theory. J. Chem. Theory Comput.12, 5834–5840. 10.1021/acs.jctc.6b00796
22
NestM.KlamrothT.SaalfrankP. (2005). The multiconfiguration time-dependent Hartree–Fock method for quantum chemical calculations. J. Chem. Phys.122, 124102. 10.1063/1.1862243
23
OrimoY.SatoT.ScrinziA.IshikawaK. L. (2018). Implementation of the infinite-range exterior complex scaling to the time-dependent complete-active-space self-consistent-field method. Phys. Rev. A. Coll. Park.97, 023423. 10.1103/physreva.97.023423
24
PathakH.SatoT.IshikawaK. L. (2020a). Study of laser-driven multielectron dynamics of ne atom using time-dependent optimised second-order many-body perturbation theory. Mol. Phys.118, e1813910. 10.1080/00268976.2020.1813910
25
PathakH.SatoT.IshikawaK. L. (2021). Time-dependent optimized coupled-cluster method for multielectron dynamics iv: Approximate consideration of the triple excitation amplitudes. J. Chem. Phys.154, 234104. 10.1063/5.0054743
26
PathakH.SatoT.IshikawaK. L. (2020b). Time-dependent optimized coupled-cluster method for multielectron dynamics. ii. a coupled electron-pair approximation. J. Chem. Phys.152, 124115. 10.1063/1.5143747
27
PathakH.SatoT.IshikawaK. L. (2020c). Time-dependent optimized coupled-cluster method for multielectron dynamics. iii. a second-order many-body perturbation approximation. J. Chem. Phys.153, 034110. 10.1063/5.0008789
28
PedersenT. B.KristiansenH. E.BodensteinT.KvaalS.SchøyenØ. S. (2021). Interpretation of coupled-cluster many-electron dynamics in terms of stationary states. J. Chem. Theory Comput.17, 388–404. 10.1021/acs.jctc.0c00977
29
PedersenT. B.KvaalS. (2019). Symplectic integration and physical interpretation of time-dependent coupled-cluster theory. J. Chem. Phys.150, 144106. 10.1063/1.5085390
30
PiggD. A.HagenG.NamH.PapenbrockT. (2012). Time-dependent coupled-cluster method for atomic nuclei. Phys. Rev. C86, 014308. 10.1103/physrevc.86.014308
31
RaghavachariK.TrucksG. W.PopleJ. A.Head-GordonM. (1989). A fifth-order perturbation comparison of electron correlation theories. Chem. Phys. Lett.157, 479–483. 10.1016/s0009-2614(89)87395-6
32
SatoT.IshikawaK. L.BřezinováI.LacknerF.NageleS.BurgdörferJ. (2016). Time-dependent complete-active-space self-consistent-field method for atoms: Application to high-order harmonic generation. Phys. Rev. A . Coll. Park.94, 023405. 10.1103/physreva.94.023405
33
SatoT.IshikawaK. L. (2013). Time-dependent complete-active-space self-consistent-field method for multielectron dynamics in intense laser fields. Phys. Rev. A . Coll. Park.88, 023402. 10.1103/physreva.88.023402
34
SatoT.IshikawaK. L. (2015). Time-dependent multiconfiguration self-consistent-field method based on the occupation-restricted multiple-active-space model for multielectron dynamics in intense laser fields. Phys. Rev. A . Coll. Park.91, 023417. 10.1103/physreva.91.023417
35
SatoT.OrimoY.TeramuraT.TugsO.IshikawaK. L. (2018a). “Time-dependent complete-active-space self-consistent-field method for ultrafast intense laser science,” in Progress in ultrafast intense laser science XIV (Berlin, Germany: Springer), 143–171.
36
SatoT.PathakH.OrimoY.IshikawaK. L. (2018b). Communication: Time-dependent optimized coupled-cluster method for multielectron dynamics. J. Chem. Phys.148, 051101. 10.1063/1.5020633
37
SchonhammerK.GunnarssonO. (1978). Time-dependent approach to the calculation of spectral functions. Phys. Rev. B18, 6606. 10.1103/physrevb.18.6606
38
ScuseriaG. E.SchaeferH. F.III (1987). The optimization of molecular orbitals for coupled cluster wavefunctions. Chem. Phys. Lett.142, 354–358. 10.1016/0009-2614(87)85122-9
39
ShavittI.BartlettR. J. (2009). Many-body methods in chemistry and physics: MBPT and coupled-cluster theory. Cambridge: Cambridge University Press.
40
SherrillC. D.KrylovA. I.ByrdE. F.Head-GordonM. (1998). Energies and analytic gradients for a coupled-cluster doubles model using variational brueckner orbitals: Application to symmetry breaking in o 4+. J. Chem. Phys.109, 4171–4181. 10.1063/1.477023
41
WattsJ. D.GaussJ.BartlettR. J. (1993). Coupled-cluster methods with noniterative triple excitations for restricted open-shell Hartree–Fock and other general single determinant reference functions. energies and analytical gradients. J. Chem. Phys.98, 8718–8733. 10.1063/1.464480
Summary
Keywords
multielectron dynamics, time-dependent optimized coupled-cluster, high harmonic generation, strong laser field, strong field ionization
Citation
Pathak H, Sato T and Ishikawa KL (2022) Time-dependent optimized coupled-cluster method with doubles and perturbative triples for first principles simulation of multielectron dynamics. Front. Chem. 10:982120. doi: 10.3389/fchem.2022.982120
Received
30 June 2022
Accepted
09 August 2022
Published
13 September 2022
Volume
10 - 2022
Edited by
Yuichi Fujimura, Tohoku University, Japan
Reviewed by
Xiaowei Sheng, Anhui Normal University, China
Shu Ohmura, Nagoya Institute of Technology, Japan
Updates
Copyright
© 2022 Pathak, Sato and Ishikawa.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Takeshi Sato, sato@atto.t.u-tokyo.ac.jp
This article was submitted to Physical Chemistry and Chemical Physics, a section of the journal Frontiers in Chemistry
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.