Ecient all-electron time-dependent density functional theory calculations using an enriched nite element basis

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Eﬃcient all-electron time-dependent density

functional theory calculations using an enriched

ﬁnite element basis

Bikash Kanungo,†Nelson D. Rufus,†and Vikram Gavini∗,†,‡

†Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan

48109, USA

‡Department of Materials Science and Engineering, University of Michigan, Ann Arbor,

Michigan 48109, USA

E-mail: vikramg@umich.edu

Abstract

We present an eﬃcient and systematically convergent approach to all-electron real-

time time-dependent density functional theory (TDDFT) calculations using a mixed

basis, termed as enriched ﬁnite element (EFE) basis. The EFE basis augments the

classical ﬁnite element basis (CFE) with compactly supported numerical atom cen-

tered basis, obtained from atomic groundstate DFT calculations. Particularly, we

orthogonalize the enrichment functions with respect to the classical ﬁnite element ba-

sis to ensure good conditioning of the resultant basis. We employ the second-order

Magnus propagator in conjunction with an adaptive Krylov subspace method for eﬃ-

cient time evolution of the Kohn-Sham orbitals. We rely on a priori error estimates to

guide our choice of an adaptive ﬁnite element mesh as well as the time-step to be used

in the TDDFT calculations. We observe close to optimal rates of convergence of the

dipole moment with respect to spatial and temporal discretization. Notably, we attain

arXiv:2210.14421v1 [physics.chem-ph] 26 Oct 2022

a 50 −100×speedup for the EFE basis over the CFE basis. We also demonstrate

the eﬃcacy of the EFE basis for both linear and nonlinear response by studying the

absorption spectrum in sodium clusters, the linear to nonlinear response transition in

green ﬂuorescence protein chromophore, and the higher harmonic generation in magne-

sium dimer. Lastly, we attain good parallel scalability of our numerical implementation

of the EFE basis for up to ∼1000 processors, using a benchmark system of 50-atom

sodium nanocluster.

1 Introduction

An accurate description of electron excitations and dynamics under the inﬂuence of external

stimuli is fundamental to our understanding of a range of physical and chemical processes.

To that end, time-dependent density functional theory (TDDFT) oﬀers a powerful tool by

extending the key ideas of groundstate density functional theory (DFT) to electron excita-

tions and dynamics. Analogous to the Hohenberg-Kohn theorem1in groundstate density

functional theory (DFT), TDDFT relies on the Runge-Gross2and van Leeuwen3theorems

to establish, for an initial state, a one-to-one correspondence between the time-dependent ex-

ternal potential and the time-dependent electron density. This, in turn, provides a formally

exact reduction of the complicated many-electron time-dependent Schr¨odinger equation to

a set of eﬀective single electron equations, called the time-dependent Kohn-Sham (TDKS)

equations. At the heart of this simpliﬁcation lies the exchange-correlation (XC) functional,

which captures all the quantum many-electron interactions as a mean-ﬁeld of the time-

dependent electron density. Similar to DFT, in practice, TDDFT has remained far from

exact due to the unavailability of the exact XC functional, thereby, necessitating the use of

approximations. However, the available XC approximations has lent TDDFT a great balance

of accuracy and eﬃciency, allowing the study of a wide array of time-dependent processes—

optical4and higher-order responses,5,6multi-photon ionization,7–11 electronic stopping,12–16

core electron excitations,17–21 surface plasmons,22,23 higher-harmonic generation,24,25 elec-

tron transport,26,27 charge-transfer excitations,28,29 dynamics of chemical bonds,30 to name

a few.

While the XC approximation remains an unavoidable approximation in TDDFT, a typ-

ical TDDFT calculation also employs the pseudopotential approximation to attain greater

computational eﬃciency by modeling the eﬀect of the singular nuclear potential and the core

electrons into a smooth eﬀective potential, known as the pseudopotential. Despite tremen-

dous success in predicting a wide range of materials properties, pseudopotentials remain

sensitive to the choice of core-valence split and also tend to oversimplify the treatment of

core electrons as chemically inert for various systems and conditions. Within the context of

groundstate DFT, pseudopotentials are known to have inaccurate predictions for the phase

transition properties of transition metal oxides and semiconductors31–34 ; ionization poten-

tials,35 magnetizability,36 and spectroscopic properties37–39 of heavy atoms; excited state

properties40,41 etc. More importantly, given that the construction of the pseudopotentials

have happened in the context groundstate DFT, their deﬁciencies are expected to be more

pronounced in TDDFT. Several time-dependent processes involving the use of a strong exter-

nal ﬁeld rely on core electron excitations, wherein the use of pseudopotential approximation

is impractical. Thus, all-electron TDDFT calculations are necessary for an accurate descrip-

tion of the time-dependent phenomena in such systems and conditions. Additionally, an

eﬃcient and robust all-electron TDDFT method can also aid in studying the transferability

of various pseudopotentials for TDDFT calculations.

The initial use of TDDFT relied on the linear response (LR) formulation, known as

LR-TDDFT,42,43 applicable to the perturbative regime (i.e., weak interaction between the

external ﬁeld and the electrons), wherein the ﬁrst-order response functions (e.g., absorp-

tion spectrum) can be directly evaluated from the groundstate. Subsequently, the real-time

formulation of TDDFT, known as RT-TDDFT, provided a generic framework to electronic

dynamics in real-time, thereby, allowing to handle both perturbative and non-perturbative

regimes (e.g., harmonic generation, electron transport) in a uniﬁed manner. This work per-

tains to the more general RT-TDDFT, and hence, we simply refer to RT-TDDFT as TDDFT.

There exists a growing body of work on eﬃcient numerical schemes for TDDFT calculations

as extensions to widely used groundstate DFT packages, leveraging on the underlying spatial

discretization. These include planewave basis in QBox;44,45 atomic orbital basis in Siesta,46,47

GPAW,48 NWChem,49,50 and FHI-aims;51,52 linearized augmented planewave (LAPW) basis in

exciting53,54 and Elk;55 and ﬁnite-diﬀerence (FD) based approaches in Octopus56 and

GPAW.44,57 Among these available methods, planewave and FD based approaches are lim-

ited to pseudopoential calculations, owing to their lack of spatial adaptivity that is war-

ranted to capture the sharp electronic ﬁelds in an all-electron calculation. The augmented

planewave58,59 family of methods, namely the augmented planewave (APW),60,61 linearized

augmented planewave (LAPW),62–64 APW+lo (localized orbitals),53,65,66 and LAPW+lo,67,68

remedy the lack of adaptivity in planewaves by describing the electronic ﬁelds as products

of radial functions and spherical harmonics inside muﬃn-tins (MTs) surrounding each atom,

and in terms of planewaves in the interstitial regions between atoms. Although eﬃcient for

all-electron calculations, the quality of the augmented planewave basis remains sensitive to

various parameters, such as the choice of the MT radius, the core-valence split, the matching

constraints at MT boundary, the energy parameter used in constructing the radial functions,

etc. Moreover, they inherit certain notable disadvantages of planewaves, such as their restric-

tions to periodic boundary conditions and the limited parallel scalability owing to the the

extended nature of planewaves. Above all, from a theoretical standpoint, the use of periodic

boundary conditions limits the use of planewaves to only periodic external potentials, so as

to satisfy the assumptions of Runge-Gross theorem. Alternatively, the periodic case needs to

be handled in a more generic and formal way through time-dependent current density func-

tional theory (TDCDFT),69 where one uses the current density as the fundamental quantity

instead of the density. The atomic orbital basis provide an eﬃcient description of the sharp

electronic ﬁelds in all-electron DFT/TDDFT through atom speciﬁc analytical or numerical

orbitals. However, their lack of completeness limits systematic convergence, leading to sig-

niﬁcant basis set errors in groundstate properties, especially for metallic systems.70–72 More

importantly, given that the atomic orbitals are constructed for groundstate DFT, the eﬀects

of incompleteness become more pronounced when they are employed in TDDFT calculations,

leading to large basis set errors of 0.1−0.6 eV in the excitation energies.52

The ﬁnite element (FE) basis,73,74 comprising of local piecewise continuous polynomi-

als, presents an alternative with several desirable features—completeness which guarantees

systematic convergence, locality that aﬀords good parallel scalability, ease of adaptive spa-

tial resolution, and the ability to handle arbitrary boundary conditions. In the context

of groundstate DFT, several past eﬀorts75–88 have established the promise of the FE basis.

Particularly, recent eﬀorts86–88 at eﬃcient and scalable FE based DFT calculations have out-

performed planewaves by 5−10×, for pseudopotential based DFT calculations, and have been

employed in various studies involving large-scale DFT calculations.89–91 These developments

have also enabled systematically converged inverse DFT calculations to obtain exchange-

correlation potentials from electron densities.92,93 In the context of TDDFT calculations, a

few recent eﬀorts94–96 have established the competence of the FE basis. Notably, as shown

in,96 for pseudopotential based TDDFT calculation, the FE basis signiﬁcantly outperforms

the widely used ﬁnite diﬀerence approach in Octopus. However, the success of the FE basis

for pseudopotential calculations does not trivially extend to the all-electron case. As shown

in,84,97 for all-electron DFT calculations, the FE basis remains an order of magnitude or

more ineﬃcient than the gaussian basis in computational time, owing to the requirement of

large number of basis functions to capture the sharp variations in electronic ﬁelds near the

nuclei in an all-electron calculation. As will be shown in the work, this shortcoming of FE

basis for the all-electron case, as expected, also extends to TDDFT calculations.

Given the various shortcomings of existing basis sets for all-electron TDDFT calculations,

an eﬃcient, systematically convergent, and highly parallelizable basis is desirable. The

current work seeks to address this gap by extending the ideas of enriched ﬁnite element (EFE)

basis, recently developed in the context of DFT calculations,97–99 to TDDFT. The EFE basis

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Ecientall-electrontime-dependentdensityfunctionaltheorycalculationsusinganenrichedniteelementbasisBikashKanungo,yNelsonD.Rufus,yandVikramGavini,y,zyDepartmentofMechanicalEngineering,UniversityofMichigan,AnnArbor,Michigan48109,USAzDepartmentofMaterialsScienceandEngineering,UniversityofMichigan,Ann...

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Ecient all-electron time-dependent density functional theory calculations using an enriched nite element basis

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