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Accelerating Realtime TDDFT with BlockOrthogonalized Manby-Miller Embedding Theory Kevin J. Koh, Triet S. Nguyen-Beck, and John Parkhill J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b00494 • Publication Date (Web): 19 Jul 2017 Downloaded from http://pubs.acs.org on July 20, 2017

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Journal of Chemical Theory and Computation

Accelerating Realtime TDDFT with Block-Orthogonalized Manby-Miller Embedding Theory Kevin J. Koh, Triet S. Nguyen-Beck, and John Parkhill∗ 251 Nieuwland Science Hall, Notre Dame, IN 46556 E-mail: [email protected]

Abstract Realtime time-dependent density-functional theory (RT-TDDFT) is one of the most practical techniques available to simulate electronic dynamics of molecules and materials. Promising applications of RT-TDDFT to study non-linear spectra and energy transport, demand simulations of large solvated systems over long timescales, which are computationally quite costly. In this paper, we apply an embedding technique developed for ground-state SCF methods by Manby and Miller to accelerate realtime TDDFT. We assess the accuracy and speed of these approximations by studying the absorption spectra of solvated and covalently split chromophores. Our embedding approach is also compared with less accurate, less costly QM/MM charge embeddings. We find that by mixing levels of detail the embedded mean-field theory scheme is a simple, accurate, and effective way to accelerate RT-TDDFT simulations.

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1

Introduction

Realtime time-dependent density functional theory (RT-TDDFT) has become a popular choice for propagating electronic densities of medium-sized systems in response to one-body time-dependent perturbations. It can be used to simulate materials which transform light into electronic energy. 1–3 With multiple propagations, the technique can be used to simulate non-linear spectra with promising advantages over response-theory based approaches. 4–14 Although RT-TDDFT is a relatively affordable approach, it is still prohibitively expensive to routinely reach picosecond timescales, especially with functionals that feature a highaccuracy treatment of exchange. 15 In this work, we incorporate a ground-state DFT embedding approach to RT-TDDFT, showing that it is a simple and accurate way to obtain high quality results at nearly the cost of a low-quality propagation. Subsystem embedding has gained momentum as a computationally efficient strategy for modeling electronic structures of complex systems at mixed levels of detail, methods including QM/MM, 16,17 density-matrix embedding theory, 18 dynamical mean-field theory, 19 and other density-based embedding methods. 20–34 Manby and Miller have introduced several variations of embedding theory, 29,35 all of which produce effective Fock matrices for a subsystem-of-interest including effects of electronic environments at lower levels of detail. In this study, we apply the DFT embedding techniques developed by the groups of Manby and Miller to improve the efficiency of realtime TDDFT. We employ the block orthogonalized partitioning variant 29 that was introduced to produce more physical embedded density matrices. We show that when the method, hereafter referred to as the block-orthogonalized Manby-Miller embedding (BOMME), is used in place of the ordinary Fock build for RTTDDFT, explicit electronic solvation dynamics can be treated at little additional cost. The accuracy of absorption spectra produced using this multiresolution dynamics approximation is in good agreement with high-level calculations performed on the whole system and experimental spectra. The advantage of this approach is the possibility of obtaining highlevel quality simulations with nearly the speed of a low-level approximation. We present an 2 ACS Paragon Plus Environment

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open-source implementation where 90% of the ideal speedup is achieved for just eight water molecules solvating a benzene. Investigating the accuracy of the technique further, we also compare with QM/MM point-charge embedding to demonstrate that the BOMME spectra are significantly closer to high-level calculations and experimental results than point-charge embedding. We examine the possibility of using the technique to mix levels of detail within a single molecule as well, and also discuss how to avoid artifacts from partitioning of the system.

2

Methods

The basic formalism of RT-TDDFT 36–39 and the BOMME technique 29 are described in previous work. We will discuss a density matrix-based realtime implementation, intended for studying non-equilibrium electronic relaxation. Realtime TDDFT and Hartree-Fock lead to a Liouville equation for the one-particle density matrix: 40,41 i γ˙ = − [Fˆ (γ), γ] h ¯

(1)

where γ is the time-dependent one-electron density matrix, and F is the time-dependent Fock matrix of HF (or KS) theory expressed in an orthonormal basis. The Fock matrix can be written as: F [γ] = h0 + G[γ] in which h0 is the core Hamiltonian, and G contains the two-electron contributions:

G[γ] = J[γ] + cx K[γ] + Vxc [γ]

(2)

J and K are the density-dependent Coulomb and exchange operators, Vxc is the KS exchangecorrelation contribution to the Fock matrix, 42 and cx is the fraction of the exact exchange for hybrid KS theories. The bottleneck of integrating Eq.1 comes from the formation of J, Vxc and especially K which is done several times a femtosecond. Techniques which decrease



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the cost of the Fock build can easily save days of propagation time. 43–52 The BOMME method 29,35 describes a subsystem with a high-level Fock matrix and the remaining degrees of freedom with a less expensive Fock matrix by reducing the quality of basis and exchange. The block orthogonalized (BO) partitioning scheme uses a projected basis in place of the conventional atomic-orbital (AO) partitioning to define the high- and low-level components of the system. Ding and coworkers have shown that this reduces artifacts in properties related to the embedding approximation:

˜ Low [˜ H˜0 = U T H0 U , G γ ] = U T GLow [γ]U , and γ˜ = U T γU

(3)

In Eq.3, U is the transformation matrix from the nonorthogonal AO basis set to the BO basis set:

 AA

 I U=

−P

AB

IBB

0

  

In this blocked matrix, the identity matrices of subsystems A and B, IAA and IBB , have the dimensions of na and nb which are the number of basis functions in each subblock, and P AB is the projection matrix, PAB = (SAA )−1 SAB , in which SAB is the AO overlap between the subsystems. The effective Fock matrix in the BO scheme is then constructed as:

˜ Low [˜ ˜ High [˜ ˜ Low [˜ F˜ [˜ γ ] = h˜0 + G γ ] + (G γ AA ] − G γ AA ])

(4)

˜ High and G ˜ Low represent the where quantities in tildes are expressed in the BO basis. G two-electron contributions to the Fock matrix from the high- and low-level of theories, and γ˜ AA denotes the density matrix blocks that belong to the subsystem with high-level theory (hereafter called the AA block). The composition of the Fock matrix formed by BOMME is illustrated in Figure 1. We prefer the simplest scheme to calculate EEX [γ AA ], which only


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considers the exact exchange interaction within the AA block:

EEX0 = EEX [γ AA ] = −

1 X AA AA (µκ|υλ)γµυ γκλ 4 µυλκ∈A

(5)

Ding et al. also discussed two more complex schemes, but determined that EEX0 has the best balance between accuracy and speed, and we find it sufficient for producing accurate spectra. We have implemented RT-TDDFT propagation with BOMME method in the PySCF

F

high = H,J + Vxc +

F

= H,J + Vxc

K

High-Level Theory

Low-Level Theory

low

Approximate Fock Matrix

~ F

= H,J + Vxc +

K

BOMME

Figure 1: The Fock matrix composition in the BOMME scheme compared with its counterparts in purely high- and low-level theory. chemistry package, 53 integrating the equation of motion with a midpoint step. 34,39,54,55 Pseudocodes and a link to the GitHub repository for the code used in this work are included in the Supporting Information (SI). As demonstrated in Figure 2, the method achieves significant speedup compared with simulations of purely high-level theory, and becomes relatively affordable as more explicit solvent molecules are included in the simulation. It is important to note that the combined BOMME/RT-TDDFT scheme as described above only evolves electrons in real time, and does not propagate explicit nuclear trajectories. Consequently, the method is inappropriate for electronic dynamics that involve significant rearrangement of the nuclei. Some more costly propagations and QM/MM calculations used our implementations in the Q-Chem quantum chemistry package. 56 All propagations 5 ACS Paragon Plus Environment


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Figure 2: Speed performance (wall-hours per picosecond of simulation) with and without embedding is demonstrated on a benzene surrounded with an increasing number of water molecules. Ratios of simulation time are calculated with respect to the high-level theory scheme. The functional/basis set combinations are PBE0/6-31G* on high-level theory (red line and red shaded region), PBE/STO-3G on low-level theory (green line and green shaded region), and mixed bases on BOMME method (blue line and red/green shaded region). were performed on ground-state DFT optimized geometries. Each system experiences an impulsive electric field with a strength of 0.01 a.u. and duration of 0.07 atomic time units. In the figures provided with each spectrum, the atoms belong to high- and low-level theory regions are specified by the red and green shaded regions, respectively.

3 3.1

Results Mixed Functional Propagation

Figure 3 shows the absorption spectra obtained by performing RT-TDDFT on a system of one methanol molecule surrounded by three water molecules treated at the PBE0 level and PBE level, respectively. This simple system demonstrates the overall features of the BOMME method. Stability of dipole moments, density trace, and total entropy are shown in Figures S3, S4, and S5, respectively (SI). Note that the entropy of the high- and lowlevel subsystems are not constant, since electrons and coherence are exchanged between these regions throughout the propagation. As expected for embedding, the absorption spectrum of the whole system is a mixture of the high-quality methanol 6 ACS Paragon Plus Environment

BOMME BOMME High-level Theory High-level Theory Low-level Theory Low-level Theory Isolated Methanol Isolated Methanol

Absorption (arb. units)

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Figure 3: Absorption spectra of one methanol with 3 water molecules: (a) Low-energy region, and (b) high-energy region. Note that the spectra are plotted at different scales for clarity. High theory employs PBE0, and low-level theory employs PBE. Both methods used a mixed basis of 6-31G*/STO-3G on methanol and water, respectively. spectrum and low-level water spectra. The BOMME spectrum visibly matches peaks of the high-level spectrum in the low-energy region (up to 18 eV), and it matches peaks of the low-level spectrum in the higher-energy region. All peaks of the BOMME method under 18 eV, other than peaks at 13.44 eV and 17.32 eV, correspond to the PBE0/6-31G* methanol peaks, whereas absorption above 18 eV and peaks at 13.44 eV and 17.32 eV correspond to PBE/STO-3G water peaks. For this particular geometry the solvation shift is small. A more strongly coupled geometry was also examined (Section 3 of SI). At this geometry, a bright singlet of the isolated methanol is at 11.85 eV, whereas the high-level theory places this state at 11.66 eV, and BOMME at 11.61 eV. No sort of unphysical collapse, instability, or artifactual transitions are observed and the dipole induced by the impulse agrees well with the high-level calculation. 29 Stepping up the system size, we examined the errors of the BOMME method that increase with the size of environment or polarizability of the solvated chromophore. We obtained the absorption spectra of a benzene with one water molecule (Figure 4.a) and ten water molecules (Figure 4.b). In Figure 4.a, the lowest π → π ∗ transition energy of BOMME, high-, and low-level theory are 7.37 eV, 7.40 eV and 8.80 eV, respectively. The good agreement is maintained in Figure 4.b where the peaks are 7.24 eV, 7.35 eV and 8.74 eV, respectively. 7 ACS Paragon Plus Environment


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Figure 4: Absorption spectra of benzene with (a) one and (b) ten water molecules. Highlevel theory used PBE0/6-31G*, low-level theory used PBE/STO-3G, and isolated methanol used PBE0/6-31G*. In the presence of larger number of environment, the BOMME method still performs close to the high-level theory as shown in Figure 4.b while the computing cost remains close to low-level theory (Figure 2). We also tested an intramolecular embedding, which we expected to be significantly


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0.5

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68 7 10 7 8 6 EnergyEnergy (eV) (eV) Energy (eV)

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Figure 5: Absorption spectra of 3,4-Dimethylpyrazole. High-level theory used PBE0/6-31G*, and low-level theory used PBE/6-31g*. more challenging and possibly reveal false absorption peaks due to non-physical motion between high- and low-level regions. We chose 3,4-dimethylpyrazole as the first test system. Indeed, naively applying the same propagation described so far lead to spurious peaks in the BOMME method. Analyzing the time-dependent electron density, we found that, as expected, characteristic DFT over-delocalization of the density from high- to low-level regions 8 ACS Paragon Plus Environment

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caused these poles. The exciting pulse delocalized the electron and hole density into the two different regions. 57 We found a simple solution to this problem was to only perturb the high-quality block, by restricting all fields to the BO high-quality block. After a modified perturbation was applied on the system, the resulting spectra has no visible artifacts. The first significant excited-state peak is at 7.43 eV, 7.88 eV and 7.98 eV for low-level theory, the BOMME method, and high-level theory, respectively (Figure 5). Simply applying the perturbation to only the high-level region appears to effectively

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Figure 6: Absorption spectra of all-trans-retinal. High-level theory used PBE0 and low-level theory used PBE. All methods used mixed basis of 6-31G*/STO-3G. solve the problem of spurious embedding absorbances. Figure 6 shows the absorption spectra of all-trans-retinal. For this larger intramolecular example, BOMME also functions well when we choose a reasonable definition of high- and low-level regions that do not cleave the optically active parts of the chromophore. There is a low-level treatment of exchange energy between high- and low-level regions; therefore, when mixing exchange functionals, users should plot the electron and hole densities and verify that both densities are contained in the high-quality region. If either quasi-particle leaks into the low-quality region, the coupling between regions is physically too strong to treat accurately with BOMME, and the user should choose a different partitioning scheme. The code we have posted to GitHub is able to generate densities for this purpose.



3.2

BOMME vs. point-charge embedding

In this section, we compared the BOMME method with point-charge QM/MM embedding for a few systems, to demonstrate the value of explicit electron solvation. Figure S6 shows the low-lying absorption spectrum of benzene and ten water molecules generated by purely highlevel theory, BOMME, and QM/MM. The solvation shift of the lowest lying state in highlevel theory and BOMME were roughly -0.17 eV, whereas the QM/MM spectrum is actually blue-shifted +0.02 eV. Figure 7 shows the absorption spectra of a neutral GFP chromophore obtained using the BOMME method, QM/MM method and experiment. Point-charge embedding produces a first bright state approximately 40 nm away from the experimental peak, which is at 370 nm, whereas BOMME is in fortuitously good agreement with the experiment. In both cases, the solvation shift is captured well by BOMME, whereas point-charge embedding is totally inadequate. BOMME Experiment QM/MM Isolated GFP

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BOMME BOMME Experiment Experiment QM/MM QM/MM Isolated GFP GFP Isolated

1.0

0.0 0.5 300 0.5

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400 Wavelength (nm)

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Figure 7: Absorption spectra of a neutral GFP chromophore with ten water molecules calculated with BOMME 0.0 (6-31G/PBE0, STO-3G/PBE) and QM/MM, and of an isolated 0.0 300 300 350 350 400 400 450 450 500 500 GFP neutral chromophore (PBE0/6-31G),Wavelength compared with experimental spectrum from Ref. (nm) (nm) Wavelength 58.

4

Conclusions

In this work, we investigate the usefulness of BOMME applied to RT-TDDFT. The BO variation of the BOMME scheme is found to produce spectra that are in good agreement 10 ACS Paragon Plus Environment

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with high-level calculations and experimental data at a cost much closer to the low-level calculations. We have shown that the method works as both intermolecular and intramolecular embedding schemes, and usefully captures the shift of a solvated chromophore. However, the case of intramolecular embedding requires careful attention to the way density moves in the propagation, and to verify that the electron and hole do not significantly leak into the low-quality region. It is also advantageous to only perturb the high-quality region of the spectrum, to avoid driving electrons non-physically across the boundary between highand low-quality regions. Because of its simplicity, the method can easily be combined with a multiple timestepping scheme to obtain further speedups; 59,60 we may explore these directions in future work. It can also be combined with our methods for electronic dissipation to study electronic relaxation phenomena and non-linear spectra.

5

Associated Content

Pseudocode and schematic for BOMME, additional plots for a system of one methanol and three water molecules, and spectra for a system of one benzene and ten water molecules. This material is available free of charge via the Internet at http://pubs.acs.org.

6

Author Information

Corresponding author: John Parkhill, E-mail: [email protected]

7

Notes

The authors declare no competing financial interest.



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Acknowledgement We thank the University of Notre Dame’s College of Science and Department of Chemistry and Biochemistry for generous start-up funding. T. S. N.-B. acknowledges the support of the NSF Graduate Research Fellowship under grant DGE-1313583 and the Patrick and Jana Eilers Graduate Student Fellowship for Energy Related Research.

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8

For Table of Contents use only RT-TDDFT with Manby-Miller Embedding 1.0

0.5 0.5

0.5


1.0 1.0

Absorption (arb. units) Absorption (arb. units)

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0.0 300 0.0 0.0 300 300

BOMME Experiment BOMME BOMME QM/MM Experiment Experiment Isolated GFP QM/MM QM/MM Isolated GFP GFP Isolated

350 400 450 350 350 400 (nm) 450 450 400 Wavelength Wavelength (nm) (nm) Wavelength

~ F

H

J,Vxc

K

500 500 500

Figure 8: The embedded mean-field theory scheme and the absorption spectra of a GFP neutral chromophore solvated in water computed with BOMME/RT-TDDFT and QM/MM.


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