Momentum-resolved TDDFT algorithm in atomic basis for real time tracking of electronic excitation
Chao Lian Shi-Qi Hu Meng-Xue Guan Sheng Meng

	
	
	
	
	
	Beijing National Laboratory for Condensed Matter
		Physics and Institute of Physics, Chinese Academy of Sciences,
		Beijing, 100190, P. R. China
	
	
	smeng@iphy.ac.cn
	Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences,
		Beijing, 100190, P. R. China
	Collaborative Innovation Center of Quantum
		Matter, Beijing, 100190, P. R. China
	
	

    
    	
Ultrafast electronic dynamics in solids lies at the core of modern condensed matter and materials physics. To build up a practical ab initio method for studying solids under photoexcitation,
we develop a momentum-resolved real-time time dependent density functional theory (rt-TDDFT)
algorithm using numerical atomic basis, together with the implementation of both the length and
vector gauge of the electromagnetic field. When applied to simulate elementary excitations in
two-dimensional materials such as graphene, different excitation modes, only distinguishable in
momentum space, are observed. The momentum-resolved rt-TDDFT is important and computationally efficient for the study of ultrafast dynamics in extended systems.
    

    Momentum-resolved TDDFT algorithm in atomic basis for real
		time tracking of electronic excitation
    Sheng Meng
    December 30, 2023
===================================================================================================



§ INTRODUCTION

    Real-time (rt) time dependent density functional theory
    (TDDFT) is an efficient ab initio method to study electron dynamics in complex electron-nuclear systems in both
    the ground state and excited state. Compared with other
    widely used approaches such as frequency domain TDDFT,
    quasi-particle GW, and Bethe-Salpeter equations, rt-TDDFT
    has two major advantages: (i) Time-dependent Kohn-Sham
    (TDKS) equations in rt-TDDFT include all nonlinear effects
    and are intrinsically non-perturbative, making rt-TDDFT a
    better tool to describe materials in a strong field and (ii)
    rt-TDDFT directly provides complete information on real
    time evolution of electronic wavefunctions together with ionic
    movements, presenting a unique way for real-time tracking ultrafast dynamics and complex phenomena far from equilibrium. Thus, rt-TDDFT is a natural choice for the exploration of strong field physics and ultrafast phenomena. Motivated by the rapid developments in ultrafast experimental
    techniques, e.g., attosecond based spectroscopy <cit.>, ultrastrong laser sources <cit.> and free electron X-ray lasers <cit.>, rt-TDDFT
    is drawing more and more attention as a method to simulate ultrafast phenomena in the current line of research
    frontiers.
    
Nevertheless, rt-TDDFT is not widely used as the method
of choice in the literature, being much less popular than other
density functional theory (DFT) based approaches such as
ΔSCF, DFT+U, frequency-domain TDDFT, etc. 
Thus, numerical atomic orbitals (NAO) have been
a common choice to dramatically reduce computation cost for
simulating complex materials and have been widely used in DFT codes such as SIESTA <cit.> and OpenMX <cit.> and rt-TDDFT implementations by A. Tsolakidis <cit.> and X. Li <cit.>. The biggest advantage of using NAO is the extremely small computational cost. To describe a system with N_a atoms, only about 10 × N_a NAOs are required, while 10^3 - 10^4 × N_a real space grids or plane waves have to be invoked. In addition, with a relatively small real-space cutoff for NAOs, the order-N linear
scaling with respect to system size can be achieved. Since a
major difficulty in developing rt-TDDFT is its extreme time
consumption due to the use of ultrasmall time step (on the order of ∼1 attosecond), NAO based 𝐤-resolved rt-TDDFT is very promising for simulating realistic condensed matter systems, complex materials, and interfaces with a long simulation time.
    
    Most previous rt-TDDFT investigations focus on the
    photoabsorption and related properties of finite-size zero-dimensional (0D) systems (atoms/molecules/nanoparticles)
    including optical spectra, <cit.>, excited state dynamics <cit.>, solvation effect <cit.>, relativistic effect variationally <cit.>, photochemical stability <cit.>, and recently plasmonic excitations <cit.>. In 0D systems, only single Γ point is needed in
    the reciprocal space sampling. Thus, the Γ-only algorithm
    is overwhelming, as commonly implemented and used in
    the majority of rt-TDDFT simulations. However, to study
    photoexcitation and electronic dynamics in extended systems, Γ-only k-point sampling is insufficient and momentum-resolved (𝐤-resolved) sampling in the reciprocal space is required. 

    
    An important advantage of using 𝐤-resolved rt-TDDFT is computational efficiency. With Γ-only TDDFT, to get the
    accurate charge density and ionic forces, an extraordinary large supercell has to be invoked. Many previous studies on extended systems belong to this scenario, <cit.> including our
    recent studies on ultrafast electron-hole dynamics in dye-sensitized solar cells, <cit.> charge separation in van der Waals heterojunctions, <cit.> and nonthermal melting of silicon. <cit.> Using 𝐤-resolved algorithms, and at the same accuracy level, the supercell size as well as the computational cost, can be largely reduced, as will be demonstrated later. 
    
    Besides technical advantages, 𝐤-resolved algorithm introduces the important 𝐤-space resolution and a new degree
    of freedom, which is essential to describe key quantities and important physics in condensed matter materials such as time-dependent band structures, quasiparticles, and valley dynamics. Only rt-TDDFT with 𝐤-resolved sampling can provide essential information concerning the real time evolution of
    material properties.
    
    Although 𝐤-resolved rt-TDDFT algorithms have been implemented by several groups <cit.> and applied for both semiconductors, <cit.> and metals, <cit.> these implementations employ either real space grids or planewaves as basis sets. With
    a much smaller basis set, the implementation of 𝐤-resolved rt-TDDFT algorithms with NAO basis has advantages in efficiency. To take the advantages of NAOs, a new framework and a more complicated implementation of rt-TDDFT are required.
    
    In this work, we strive to tackle the major challenges mentioned above in NAO-based rt-TDDFT. We have successfully developed the 𝐤-resolved rt-TDDFT algorithm based on local atomic basis sets using numerical atomic orbitals.
    Both the length and vector gauge of the electromagnetic
    field have been implemented. This approach enables rt-TDDFT calculations of solids and surfaces using rather
    simple unit cells, reducing computational cost by several orders of magnitudes. Moreover, momentum-resolved electron dynamics in the excited states can be tackled by this approach. For instance, 𝐤 selective photoexcitations in
    graphene are demonstrated here, where three distinct photoexcitation modes located at different 𝐤points in the reciprocal space are induced upon laser illumination. This kind of 𝐤-dependent electronic dynamics is ubiquitous in extended systems such as periodic solids and interfaces. Therefore, we expect highly efficient 𝐤-resolved rt-TDDFT algorithms
    employing local bases be an important development and will be widely used in first-principles simulations of ultrafast phenomena under a strong field and optimal control of quantum materials.
    
    
    

§ METHODOLOGY

    The main framework of 𝐤-resolved rt-TDDFT algorithm is inherited from an earlier single-Γ version of Time Dependent ab initio Package (), <cit.> which is based on the  <cit.> package. In such a rt-TDDFT algorithm, the
    flowchart of a given ionic step is shown in Fig. <ref>. Each process is described in detail in the Secs. II A-II G, marked with the same labels as in Fig. <ref>. Here atomic units ħ = m_e = e = 1 are used throughout this work.
    
     
    
    

 §.§ Hamiltonian and overlap matrix

    Adopting periodical boundary conditions, the lattice of an extended system are denoted as 𝐑_s (s = 1, 2, 3, ..., N) and the atoms i in the unit cell are located at positions 𝐛_i, where N is truncated to construct a finite supercell. A set of numerical atomic-centered orbitals (NAOs) {ξ_iα} is associated with each atom in the simulated system, where α denotes both the orbital and angular quantum number of an atomic orbital,  each expressed in multiple radial basis functions ζ <cit.>. Here, since all the operators and functions are time-dependent, we only denote the explicit dependence on t as f(t) and omit t for implicit dependence.
    
    Overlap matrix S_𝐤 and Hamiltonian H_𝐤 at the each 𝐤 point in the reciprocal space are expressed with NAOs:
    
    S_iα,jβ,𝐤 = ∑_s e^-i 𝐤·𝐑_s⟨ξ_iα(𝐫 + 𝐑_s + 𝐛_i) | ξ_jβ(𝐫 + 𝐛_j)|,⟩

    
    H_iα,jβ,𝐤 = ∑_s e^-i 𝐤·𝐑_s⟨ξ_iα(𝐫 + 𝐑_s + 𝐛_i) |Ĥ| ξ_jβ(𝐫 + 𝐛_j)|,⟩

    where 
    
    Ĥ =   T̂ + ∑ V_I^local(𝐫) + ∑ V_I^KB + V^H(𝐫, ρ(𝐫))
       + V^XC(𝐫, ρ(𝐫)) + V^ext(𝐫)

    is the Hamiltonian operator. Here T̂ = 1/2∇_𝐫^2 is the kinetic energy operator, I is the index for atoms, V_I^local and V_I^KB are the local and Kleinman-Bylander parts of the pseduopotential for the Ith atom, V^H is the Hartree potential, V^XC is the exchange-correlation (XC) potential and V^ext is the potential of external field. Details in the calculation of ⟨ξ_iα(𝐫 + 𝐑_s + 𝐛_i)  |Ĥ| ξ_jβ(𝐫 + 𝐛_j)|$⟩ are described in Ref. Ordejon1996. Within adiabatic local density approximation (LDA) and generalized gradient approximations (GGA) <cit.> for the exchange-correlation functional,V^XCdoes not depend explicitly on timet, i.e.V^XC[ρ(𝐫,t),t] = V^XC[ρ(𝐫,t)]. Thus, most XC functionals in ground-state DFT such as Perdew-Wang <cit.>, Perdew-Burke-Ernzerhof <cit.>, Becke-Lee-Yang-Parr <cit.>, and van der Waals density functional <cit.> are compatible in this implementation of rt-TDDFT.
    
    

 §.§ External field

    To simulate the laser-matter interactions, time-dependent electric field𝐄(t)is introduced to the Hamiltonian to represent the external time-dependent laser field in two different scenarios: the length gauge and vector gauge. 
    
    Within the length gauge, the effect of electric field𝐄(t)is added toV^extas a scalar potential
    
    V^ext(𝐫, t) = - 𝐄(t)·𝐫.

    Time dependent𝐄(t)can be tuned adopting any shape in time evolution. A most popular example is using the shape of a Gaussian wave packet
    
    𝐄(t)=𝐄_0cos(2π f t + ϕ)exp[-(t-t_0)^2/2σ^2],

    wherefis the laser frequency,t_0is the peak time, andϕis the phase factor. 
    
    We note that, the translational symmetry of Hamiltonian is broken by the introduction of finite external field𝐄in the length gauge, since 
    
    V^ext(𝐫 + 𝐑_s, t) = - 𝐄(t) · (𝐫 + 𝐑_s) ≠ - 𝐄(t) ·𝐫.

    Thus, a common solution is using a sawtooth field along spatial directionμ ∈{x,y,z}E_μ(𝐫,t)=
        E_μ(t)   ϵ < x_μ < L_μ - ϵ,
    
        - E_μ(t)L_μ/2ϵ     -ϵ < x_μ < + ϵ.

    whereL_μis the length of unit cell alongμandϵ→0. Thus,-E_μ(t)L_μ/2ϵ→∞, which requires that charge density vanishesρ(x_μ) = 0in the region-ϵ< x_μ < + ϵ, otherwise the energy diverges. Thus, a vacuum layer is essential alongμ. The requirement for a vacuum layer limits the application of theoretical approaches using the length gauge field to study the extended systems. Since there is no vacuum layer in the extended bulk systems, the translational symmetry of the Hamiltonian is broken, H(𝐫 + 𝐑_s) ≠ H(𝐫), using the length gauge field.Plus, length gauge field is invalid in large systems and in short wavelength perturbation <cit.>.
    
    
    Dynamical electric field in the vector gauge by introducting vector potential𝐀could preserve the transitional symmetry of Hamiltonian, thus removes the requirement of the vacuum layer. <cit.> The relation between𝐄and𝐀is
    
    𝐄 = -1/c∂𝐀/∂ t; 𝐀 = -c∫𝐄 dt.

    	The Hamiltonian with the presence of𝐀is then
    	
    H = 1/2m (ħ𝐤 - e/c𝐀)^2 = 1/2m (ħ k + e ∫𝐄 dt)^2 = ħ^2/2m ( 𝐤 + 𝐤_𝐀 )^2,
 
    	where
    	
    𝐤_𝐀 = e/ħ∫𝐄 dt = √(2)∫𝐄 dt.

    	
    	
    	
    	within Rydberg atomic unit, wheree = √(2),ħ= 1andt = ħ/Ry.
    	
    	
    	
    	
    	
    	
    	
    	
    	The unit of𝐤_𝐀isBohr^-1, the same as the unit of𝐤.
    	
    	
    	
    	
    	
    	
    	
    	

 §.§ Propagation

    	With time-dependent (TD) Hamiltonian and overlap matrix, TDKS equation is solved to obtain|u_n𝐤(𝐫,t+Δt)⟩from the state|u_n𝐤(𝐫,t)⟩at the previous time step: 
    |u_n𝐤(𝐫, t_2)⟩ = exp[-i S^-1_𝐤(t') H_𝐤(t')Δ t ] |u_n𝐤(𝐫, t_1)⟩.
 
    	whereΔt = t_2 - t_1is the length of time step,|u_n𝐤(𝐫,t)⟩is Bloch function andt' ≈(t_1 + t_2)/2. 
    	
    	It is rather difficult to evaluateH_𝐤(t')andS_𝐤(t')directly. BecauseΔtis quite small (< 0.05fs), the ion positions barely changes fromt_1tot_2. SinceS_𝐤(t)is only determined by ionic positions (Eq. (<ref>)), it is accurate enough to assumeS_𝐤(t') ≈S_𝐤(t_2). However,H_𝐤(t)may largely change due to the rapid evolution of electrons. To approximate H_𝐤(t') properly, mid-point technique has been widely used <cit.>.
    	











    	
    	Note that, |u_n𝐤(𝐫)(t_2)⟩ is not explicitly dependent on other TDKS orbitals |u_n'𝐤'(𝐫)(t_1)⟩ (n' ≠ n or 𝐤' ≠𝐤), as a result of the v-representativity of the TDKS equations <cit.>. It decouples the evolution equations of different TDKS orbitals and make TDDFT calculations practical. However, it  nevertheless can account for both interband and intraband scatterings. Because H_𝐤 is determined by the total charge density, which is a weighted summation of all the occupied orbitals, there still exists an implicit coupling between different TDKS orbitals.
    	
    	Numerically, the time propagatorexp(-i S^-1_𝐤H_𝐤Δt)in Eq. (<ref>) is expanded using first-order Crank-Nicholson scheme:
    	
    |u_n𝐤(𝐫, t_2)⟩ = 1-i S^-1_𝐤H_𝐤Δ t/2/1+i S^-1_𝐤H_𝐤Δ t/2|u_n𝐤(𝐫, t_1)⟩.

    	Technically, since computingS^-1_𝐤is the most time-consuming part in the calculation of Eq. (<ref>), we minimize the times for its computing.S^-1_𝐤is only updated when atomic positions, thus the center of NAOs,𝐛_iare changed. Consequently, when ions are fixed,S^-1_𝐤is computed only once at the first ionic step. Even with ions moving,S^-1_𝐤only need to be updated once for each ionic step.
    	
    	

 §.§ Updating charge density

    	With|u_n𝐤(𝐫,t_2)⟩solved in Eq. (<ref>), 
    	
    	
    	
    	
    	
    	
    	
    	the density matrix (DM)ρ_iα,jβ(t_2)is computed accordingly as:
    	
    ρ_iα,jβ(t_2)    = ∑_n∑_𝐤 q_n,𝐤|u_n𝐤(𝐫, t_2)⟩⟨u_n𝐤(𝐫, t_2)|
        = ∑_n∑_𝐤 q_n,𝐤 c^*_n,iα,𝐤(t_2) c_n,jβ,𝐤(t_2),

    	whereq_n,𝐤is the electronic population of the bandnat𝐤, andc_n,jβ,𝐤(t_2)is the coefficient of|u_n𝐤(𝐫, t_2)⟩in NAO basis:
    	
    |u_n𝐤(𝐫, t_2)⟩ = ∑_jβ c_n,jβ,𝐤(t_2) ξ_jβ(𝐫).


 §.§ Self-consistent evolution

    	We use the self-consistent process described in Ref. <cit.> during the time evolution of charge density. This process substantially increases the numerical stability <cit.>. All the criteria for convergence test developed in SIESTA are compatible with the current approach, such as using the maximum element of the DM difference, the energy difference, or the Harris energy difference, etc. as a criterion for achieving self-consistency. <cit.>.
    	
    	Here, we use DM difference as an example. Convergence in charge density during time evolution is reached when
    	
    max{|ρ^new_iα,jβ - ρ_iα,jβ|} < η,

    	whereηis about 10^-4. 
    	
    	

 §.§ Mixing

    	If not converged, the linear mixing of DM is needed to generate the input DM for computing charge densityρ_nextat the next cycle, instead of usingρ_newdirectly,
    	
    ρ = (1-w)ρ + wρ_new,

    	where theρon the right side of Eq. (<ref>) is the input DM andρ_newis the output DM, andwis the mixing weight, usuallyw = 0.1 - 0.5.
    	
    	

 §.§ Postprocessing

    	If self-consistent time evolution of charge density is converged, the postprocessing steps are evoked, including the calculation of total energy, Hellmann-Feynman forces, ionic movements, etc. These functions are implemented in SIESTA <cit.> and compatibly used in TDAP <cit.>. We note that, rt-TDDFT in atomic orbital basis gives rise to additional Pulay terms that contribute to the force evaluations<cit.>. The total force is the combination of Hellmann-Feynman force and Pulay term.With the calculated forces, the coupled electron-ion motion can be simulated based on classical ionic trajectories, in the framework of Ehrenfest dynamics. In Ehrenfest dynamics, the forces on the ions are averaged over the adiabatic electronic states along all possible ionic paths. If one path is dominating or many similar potential energy surfaces are involved, Ehrenfest dynamics works very well <cit.>; otherwise, classical trajectory approximations in Ehrenfest dynamics become less accurate <cit.>. Furthermore, detailed balance for quantum electronic states is absent in the Ehrenfest dynamics. Thus, the applications of the present method are limited to the  cases where the averaged potential energy surfaces yields a reasonable description of coupled electron-ion dynamics. Since we focus on the dynamics of excited electrons in this work, the ions are fixed in the simulations.Here we introduce some analysis in detail for typical rt-TDDFT simulations. First, we could evaluate the state-to-state transition probabilities between TDKS orbitals during time evolution <cit.>:
    	
    P_nn'𝐤 = |C_nn'𝐤|^2 = |⟨v_n𝐤|S_k|u_n'𝐤|⟩|^2,

    	where|v_n𝐤⟩is the adiabatic basis satisfying
    	
    H_𝐤|v_n𝐤(𝐫)⟩ = E_n𝐤 S_𝐤|v_n𝐤(𝐫)⟩.

    	The population𝓆_n𝐤of the adiabatic staten𝐤is thus projected from the TDKS orbitals at a given time as:
    	
    𝓆_n𝐤 =  ∑_n'∈ n_𝐤,occ q_n'𝐤 P_nn'𝐤,

    	wheren_𝐤,occis the occupied state at𝐤point.
    	
    	
    	
    	
    	For finite systems and surface slabs, we can calculate time-dependent dipole moment along the direction.
    	
    	
    	
    	
    	
    	For periodic systems, the dipole moment is ill-defined. Instead, we calculate time dependent current,   
    	
    𝐣 = -ieħ/m∑_n ( ⟨u_n𝐤|∇|u_n𝐤|-⟩⟨u_n𝐤|∇|u_n𝐤|^⟩*  ),

    	as the response function.
    	
    	
    	
    	
    	
    	
    	
    	
    	
    	
    	
    	
    	
    	
    	
    	
    	
    	
    	
    	
    	

§ RESULTS AND DISCUSSION


 §.§ Momentum-resolved versus supercell approaches

    	To demonstrate the𝐤-resolved algorithm, we choose graphene as the model system (see Fig. <ref>(a)). An exotic property of graphene (also of other Dirac materials) is the linear dispersion near K point, namely,E(𝐤) = v_F𝐤, whereEthe band energy andv_Fis the Fermi velocity which could reach10^6m/s. To describe all the Bloch electrons, especially those near the Fermi energy, two kinds of strategies are used: unit cell calculations with𝐤-resolved reciprocal space samplings or a supercell approach withΓ-only𝐤-sampling. To demonstrate the advantages of the𝐤-resolved algorithm, we compare three cases: 
    	
 (i) unit cell with the Monkhorst-Pack <cit.>N_k ×N_k ×1𝐤point mesh, to cover all important special𝐤-pointsM,ΓandK, facilitating a line-mode analysis alongM →Γ→K →M[Fig. <ref>(b)]; 
    	
 (ii)N_c×N_c ×1supercell with singleΓpoint; and 

    	(iii)N_c×N_c ×1supercell with singleKpoint.
    	
    	
    	
    	To compare the computation accuracy of these three cases, we define an error functionΔas,
    	
    Δ = 1/T∫_0^T |E_ex(t) - E^ref_ex(t)|dt,

    	whereTis the total simulation time,E^ref_exis the excitation energy of the reference case andE_ex(t)is the excitation energy
    	
    E_ex(t) = E_KS(t) - E_KS(t=0),

    	whereE_KSis the total energy of the system. 
    	
    	
    	Here, we evaluateΔunder such settings: the Gaussian-shaped laser pulse [Eq. (<ref>)] withϕ= 0,t_0 = 7.0fs,σ= 2.0fs,f = 21.93eV is applied; the total simulation time isT = 20fs; and the reference energyE^ref_exis calculated with60×60×1𝐤-point mesh. A diagram to illustrate the definition ofΔis shown in the inset of Fig. <ref>. The time step is chosen to Δ t = 0.02 fs and the total time is 20 fs. Troullier-Martin pseudopotentials <cit.>, adiabatic local density approximation (ALDA) exchange-correlation functional <cit.> and an auxiliary real-space grid equivalent to a plane-wave cutoff of 75 Ry are used. In description of C atoms, we use a basis set of 8 double-ζ orbitals {2s(2ζ), 2p_x(2ζ), 2p_y(2ζ), 2p_z(2ζ)} and 5 polarization orbitals {P_d_xy, P_d_yz, P_d_z^2, P_d_xz, P_d_x^2-y^2}. We calculate the test cases with one 8-core Intel(R) Xeon(R) CPU E5-2650@2.00GHz. 
    	
    	We plotΔof these three cases in Fig. <ref>. The errorΔdecreases asN = N_k(orN_c) increases. The absolute value ofΔon the the same scale is achieved withN_c = N_k. That is to say, the unit cell approach withN ×N ×1𝐤-point mesh is as accurate as the approach using aN×N ×1supercell. To achieve an accuracy with theΔ≤2meV/atom,N_k  = 24is needed. Thus, it can be predicted thatN_C = 24is needed for the supercell approach. 
    	
    	
    	
    	However, we emphasize that the computational cost for calculatingN_c×N_c ×1supercell is extremely heavy. As shown in Fig. <ref>, solving Eq. (<ref>) dominates (∼80%) the computer time consumption at largeN_k(N_c), which scales linearly with the total number of𝐤points,N_k^2, and quadratically with the total number of atoms,N_c^2. The CPU clock timet_capproximately scales as O(N^2_k×N_c^4). Thus,t_c = N_c^4for supercell calculation, whilet_c = N_k^2for𝐤-resolved calculations at the same level of accuracy. 
    	
    	AsNincreases, this difference become more significant. For supercell calculations, we are able to only compute supercells up toN_C = 9, which already costs over2×10^3min. At the same accuracy level,N_k = 9calculation costs only20min, which is only 1/100 of that forN_C = 9case, consistent with the time complexity analysisN_k^2/N_c^4 = 1/81. As mentioned above,N_k  = 24orN_C = 24is needed for relatively accurate calculations. To fulfill this requirement, calculation withN_k = 24costs only about 1 hour, showing that it is readily accessible and efficient. In contrast, calculating aN_c = 24supercell would require a computer time over 576 hours (24 days) and thus heavy in real applications. Regarding the computational accuracy and efficiency, we choose𝐤-point mesh24×24×1to achieve an extremely dense sampling of the Brillouin zone. 
    	
    	With the small unit cell of graphene, the number of real space grids is ∼1000, which is 30 times of the number of NAOs used. Considering the evolution algorithm has the computational complexity of O(n^2), where n is the number of basis functions, the computer time for wavefunction evolution using NAO basis is largely reduced to 1/90 of that using real space grid basis. In practical calculations using the same number of message-passing-interface (MPI) processes, the reduction in the total computer time is tested to be about 1/5 to 1/10 depending on the systems under consideration <cit.>.

 §.§ Out-of-plane excitation

    	We then adopt a laser field perpendicularly polarized to the graphene plane to excite electrons in graphene, i.e. in a set-up of small angle scattering. Since there is a vaccum layer along the out-of-plane direction, the laser field in the length gauge can be used. 
    	
    	We first calculate the dielectric function of graphene to locate the photon energy for resonant excitation,α_μ,ν, whereμ,νdenote the spatial directionμ,ν∈{x,y,z}. Theα_μ,νdescribes the response of dipole momentP_μ(ω)to the electric fieldE_ν(ω)in the frequency domain,
    	
    P_μ(ω) = α_μ,ν(ω) E_ν(ω).

    	
    	
    	In rt-TDDFT calculations, we apply the electric fieldE_ν(t)and obtain the dipole momentP_μ(t)in time domain. Then we carry out the Fourier transform to get Eq. [<ref>],
    	
    ∫ P_μ(t)exp(iω t) dt = α_μ,ν(ω) ∫  E_ν(t)exp(iω t) dt.

    	We then obtain
    	
    α_μ,ν(ω) = ∫  P_μ(t)exp(iω t) dt/∫  E_ν(t)exp(iω t) dt.

    	In principleE_ν(t)can be in an arbitrary shape with time. However, in practice, it is better to choose Dirac functionE^δ_ν(t) = E_ν0 δ(t), or the Heaviside step functionE^θ_ν(t) = E_ν0 [1 - θ(t)]to include componentsE_ν(ω)at allω, since we have
    	
    E^θ_ν(ω) = ∫ E_ν0[1 - θ(t)]exp(iω t) dt = E^0_ν/iω.

    	Here we choose the latter form:
    	
    E^θ_ν(t)= E^0_ν [1 - θ(t)] =  E^0_ν    t ≤ 0  
     0    t > 0 ,

    	which leads to 
    	
    α_μ,ν(ω) = iω/E^0_ν∫ P_μ(t)exp(iω t) dt.

    	Importantly,Im{α_μ,μ(ω)}characterizes the optical absorbance atωalong theμdirection. 
    	
    	
    	We calculate the imaginary part of the dielectric function along the out-of-planezdirection of graphene,Im{α_z,z(ω)}. As shown in Fig. <ref>,Im{α_z,z(ω)}are almost the same with the increase ofE^0from0.05to0.5V/Å, indicating the linear response theory is appropriate in this range of light illumination. The absorption peaks are located at relative high energies (> 20eV). The first absorption peak is located at21.93eV. We choose this photon energy to simulate the resonant excitation of graphene in the perpendicular direction. 
    	
    	
    	
    	We demonstrate the excitation dynamics of graphene at a resonant light frequency ofω_r = 21.93eV, and compare to the case at the non-resonant light frequency ofω_nr = 2.0eV. 
    	We characterize the overall excitation through tracking the number of excited electrons, as well as the total energy change during the excitation process as a function of time. The number of excited electronsn(t)is calculated as,
    	
    n(t) = ∑_unocc𝓆_n𝐤(t),

    	where𝓆_n𝐤(t)is obtained from Eq. (<ref>), andunoccdenotes the unoccupied TDKS states. 
    	
    	As shown in Fig. <ref>, different behaviors are observed for the two excitation conditions. The excited electronsn(t)and excitation energyE_ex(t)increases atω_r, while no response is observed atω_nr. The same results are obtained at other non-resonant light frequencies of1.0,2.0,4.0eV. It verifies that the calculated Im{α_z,z(ω)} characterizes well selectivity in optical absorption: only the light with the right photon energy ω, at which  Im{α_z,z(ω)} peaks, has a strong absorption.
    	
    	We discuss the resonant case here. In general,n(t)is similar to the shape of the laser pulse, while two special features are observed. Firstly, the time variation inn(t)has a1.4fs delay from the laser field. This delay represents the intrinsic response time of graphene to laser field, namely, the time needed for light absorption and electronic transitions. Secondly, n(t) decreases but does not vanish after the end of light pulse. Thus, we propose that two kinds of excitation process exist: one is the transient excited electrons, which quickly vanishes after the laser pulse is off; another is the residual excited electrons, which live relatively longer. 
    	
    	Residualn(t)would decrease with the occurrence of electron-electron and further electron-phonon scatterings at the time scale of100fs, thus is not observed in our short-time simulation (<20fs).
    	We note that, the dependence on history is absent in the calculations with adiabatic exchange-correlation functionals, which causes less accurate prediction of the lifetime of excited states and ionic forces on a long time scale <cit.>.
    	
    	To verify our assumption about the existence of two kinds of excitation processes, we further distinguish the excitation with𝐤-point resolution. We choose six snapshots of𝓆_n𝐤(t)defined in Eq. (<ref>), as shown in Fig. <ref>. Att = 2.0fs with the absence of laser pulse, no excitation is observed at all𝐤points. Att = 4.0fs, the excitation is still ignorable, while the laser field is just turned on, due to the delay in electronic response we discussed above. At the peak time of the laser pulset = 7.0fs,𝓆_n𝐤(t)shows a significant distribution over many𝐤-points. We mark the dominant excitation mode as L, which involving bondingπand antibondingπbands. Withtincreases from7fs to12fs, the L mode excitation rapidly decreases. In contrast, two new modes (labeled by their locations in the reciprocal space, K_1and K_2) increases and become dominant. K_1and K_2modes maintain within20fs while L mode gradually vanishes. Thus, with the assistance of newly developed𝐤-resolved algorithm, we are able to successfully distinguish these two kinds of excitation processes: L mode produce the transient excited electrons while K_1and K_2modes produce the residual excited electrons. 
    	
    	
    	
    	Although K_1and K_2are both long-lived excitations, their time-dependence is quite different. We plot𝓆_n𝐤(t)as a function oftat three𝐤pointsΓ, K_1and K_2, as shown in Fig. <ref>. For L mode (represented by photoexcitation atΓpoint), the clear transient character is demonstrated. The excitation only exists when the laser field is present, consistent with the observations in Fig. <ref>. However, for K_1and K_2modes, new differences are observed. Excited electrons in K_1mode increases monotonically, while𝓆_n𝐤(t)at K_2shows an oscillation with a periodicity ofT_K_2 ∼5fs. These different behaviors are due to different excitation energies of three modes, originated from different band structures at the different𝐤-point. For instance, the oscillation ofK_2mode is analogous to the beating,
    	
    𝓆_nK_1(t) = A_0 cos(ω_K_2 - ω_r /2 t) cos(ω_K_2 + ω_r /2 t).

    	For K_2mode,ω_K_2 = 22.75eV is the energy difference between the two electronic bands involved in the optical transition at K_2(initial and final states), andω_2 = ω_r = 21.93eV is the driving photon energy. Beat frequencyT_b = 4π/(ω_1 - ω_2 ) = 5.07fs, which is close to the observed oscillation periodicityT_K_2. Thus, the oscillation of K_2mode is the beat formed by the intrinsic band energy difference and the driving laser frequency. In contrast,K_1mode excitation has very close energies:ω_K_1 = 21.59eV and ω_r = 21.93eV, thus only a half period of the beat (T_K_1 = 12.4fs) is observed in our simulation. For L mode, the excitation energy is19.32eV, far below theω_r. A non-resonant interference shows up instead of beating. The rich photoexcitation phenomena discussed above and the associated complex dynamic behaviors hint for the needs for developing efficient rt-TDDFT algorithms with momentum resolution. By introducing a new degree of freedom in the reciprocal space, the𝐤-resolved dynamics labels the distinct excitation processes as well as final distribution of excited states in the Brillouin zone after the incidence of laser pulses.  
    	
    	

 §.§ In-plane excitation

    	
    	
    	For a laser pulse with its field polarization lying parallel to the atomic plane of graphene (i.e., normal incidence), adding a vacuum layer along the laser polarization direction is not possible for the periodical extended system such as graphene. Therefore we adopt the vector potential approach to simulate the in-plane laser-graphene interaction. The graphene sheet is illuminated with a linearly polarized laser pulse, as shown in Fig. <ref>. We note that in-plane excitation is well described by the Fermi's golden rule. Only the bands with an energy gapΔE_g(𝐤)equal to the photon energyωwill be excited. As a result, in-plane polarized laser excites electrons near the Dirac point for photon energies≤5 eV, see Fig. <ref>. The momentum-resolved simulation will distinguish the photoexcitation induced by a laser pulse with different photon energiesω.
    	
    	Here, we use four different wavelengthsλ=1200 nm, 600 nm, 400 nm, and 300 nm for the laser pulse, corresponding to photon energiesω=1.03, 2.06, 3.10, 4.13 eV, respectively, to excite graphene in the in-plane direction. For simplicity the laser field is polarized perpendicular to the C-C bond of graphene lattice (referred to asydirection). The momentum resolved excitation patterns in the reciprocal space are shown in Fig. <ref> (a), with the corresponding band energy differenceΔE_g(𝐤)shown in Fig. <ref> (b). It is clear that only the𝐤points withΔE_g(𝐤) = ωare excited. This agreement justifies the validation of the vector gauge used in the current TDDFT implementation. 
    	
    	
    	
    	Furthermore, we note that the presence of strong laser field breaks the six-fold rotational symmetry of the graphene lattice. For instance, withω= 4.13eV, photoexcitation at twoM'points are absent, while excitations at other symmetricMpoints are observed. This symmetry breaking is caused by presence of linearly polarized laser field along theydirection. 
    	
    	
    	It can be explained with a two band model of graphene (see appendix). The excellent agreement on the excitation outcome between the model Hamiltonian and first-principles quantum dynamics simulations justify the validity of our rt-TDDFT algorithm with a vector gauge field. We therefore expect that it is readily applicable to investigate quantum dynamics of a variety of electronic phases such as charge/spin density waves, Mott insulators, valley electronics, and electronic melting in two-dimensional materials and conventional semiconductors. 
    	

    	
    	

    	To demonstrate the general applicability of the present approach, we tackle photoexcitation induced electron dynamics in a complex material. The layered transition-metal dichalcogenides such as 1T-TaS_2have been widely studied in literature to understand charge density wave (CDW) physics in real materials, whose structure is shown in Fig. <ref>(a). The 1T-TaS_2is a typical quasi two-dimensional CDW material with a pristine lattice constant of 3.36 Å in the undistorted 1T phase. In ground state, the lattice undergoes a structural reconstruction forming a√(13)×√(13)superstructure with star-of-David patterns. Laser induced phase dynamics in 1T-TaS_2has been investigated in recent experiments, where its responses to ultrashort laser pulses play a critical role.      	
    	Here we study the carrier distribution in 1T-TaS_2upon ultrafast laser excitation. As shown in Fig. <ref>(b), the excitation energy strongly oscillates with the field of laser pulse. The excitation energy deposited by the laser pulse is∼12 eV/cell after laser illumination with a photon energy ofħω= 1.55eV and pulse width of 8 fs. The carrier distribution at 20 fs after the passing of the laser pulse is shown in Fig. <ref>(c). The majority of excited electrons and holes are located at energies ranging from-2 to 2 eV near the Fermi level. It indicates that the photoexcitation mainly consists of single-photon processes as well as a minor fraction of two-photon processes (with excited electrons located at∼3 eV and holes at-3eV).
    	
    	
    	

§ CONCLUSIONS

    	In conclusion, we have developed𝐤-resolved rt-TDDFT algorithms using efficient numerical atomic basis. It enables large-scale rt-TDDFT simulations of extended systems including solids, interfaces, and two-dimensional materials with a rather small unit cell, significantly reducing the heavy computational cost of typically rt-TDDFT simulations. Consequently,𝐤-resolved excitation dynamics in periodical crystal materials are observed. The key advantages of this unique approach includes:
 i) The 𝐤-resolved real-time evolution algorithm introduces the important 𝐤-space resolution and a new degree of freedom, which is essential to describe key quantities and important physics in photoexcited condensed matter materials. The use of many 𝐤-points with a rather small unit cell also significantly improves the computational efficiency of rt-TDDFT calculations for photoexcitation in solids.
 ii)  Different from approaches using real space grids and all-electron full-potential linearized augmented-planewaves, the adoption of numerical atomic basis in the present implementation reduces the number of required basis functions to one-hundredth of its original value, making rt-TDDFT computation of realistic large systems (comprising∼500 atoms and lasting for∼1000 fs) plausible. In addition, with a relatively small real-space cutoff for NAOs, the order-Nlinear scaling with respect to the system size can be achieved. 
 iii) Both electronic and ionic degree of freedoms are evolved, therefore a complete information on  electronic wavefunctions and ionic movements during real time evolution can be provided for simulations of complex materials and rich phenomena far from equilibrium.
    
    	
    	
    	When applied to study photoexcitation dynamics of a prototypical model material–graphene, the𝐤-resolved algorithm enables the observation of𝐤selective excitation modes. Three distinct modes are excited, located at different𝐤. In-plance excitation of the Dirac electrons in graphene can be understood by assuming an effective vector field of laser field, via taking into account the angular dependence of optical transition matrix elements. This kind of𝐤dependent electronic dynamics are ubiquitous in solids. Thus,𝐤-resolved rt-TDDFT algorithm is an important development for investigating ultrafast photoexcitation dynamics and electron-electron scattering, and is expected to be widely used in the future. 
    	
    	

§ ACKNOWLEDGEMENT

    	We acknowledge partial financial support from MOST (Grant Nos. 2016YFA0300902 and 2015CB921001), NSFC (Grant Nos. 11774396, 11474328, and 91850120), and CAS (Grant No. XDB07030100).

		

§ APPENDIX: THE TWO-BAND MODEL OF GRAPHENE

		
		
		
		The ground state Hamiltonian of two-band model of graphene reads,
		
    H_0(k_x, k_y) = v_F(k_x σ_x + k_y σ_y),

		wherek_x,k_yis the𝐤coordinate,σis the Pauli matrices,v_Fis the Fermi velocity.v_F = 1eV·Bohr for simplification. The units ofk_xandk_yare chosen as Bohr^-1. The energy unit is thus eV. The eigenvalues and eigenvectors are solved as,
		
    E_0 = -√(k_x^2 + k_y^2), 
    		ϕ_0 = √(2)/2([                           -1; k_x + i k_y/√(k_x^2 + k_y^2) ]),

    E_1 = √(k_x^2 + k_y^2), 
    		ϕ_1 = √(2)/2([                            1; k_x + i k_y/√(k_x^2 + k_y^2) ]).

		Thus, the initial state wavefunction is the ground stateψ(t=0) = ϕ_0. 
		
		A vector field polarized alongycan be introduced as,
		
    H'(t) = A(t) σ_y,

		whereA(t)is the vector gauge field. The time-dependent Hamiltonian is thus, 
		
    H(t)  =  H_0 + H'(t).

		The wavefunction at timetcan be obtained from time-dependent Schrödinger equation, 
		
    |ψ(t)⟩ = exp[-iH(t)t]|ϕ_0⟩,

		which can be expanded with|ϕ_0⟩and|ϕ_1⟩basis,
		
    |ψ(t)⟩ = c_0(t) |ϕ_0⟩  +  c_1(t) |ϕ_1⟩,

		where
		
    c_i(t) = ⟨ϕ_i | ψ(t)|⟩

		is the time-dependent coefficients. All equations are solved numerically with the  package <cit.>.
		
		
		
		
		We can reproduce the symmetry breaking in the distribution of excited electrons in𝐤space induced by linearly polarized laser. We analyze the coefficients of|c_2(t)|^2withk_x = cosθ, k_y = sinθ, under the vector field𝐀 = 0.2Bohr^-1, whereθis the angle between𝐤and𝐀, as shown in Fig. <ref>. Thus, the energy differenceΔE_g(𝐤) = 2.0eV. These𝐤points are only excited withω= 2.0eV, consistent with the results from TDDFT and Fermi's golden rule. 
		
		To explain the origin of the symmetry breaking, 
		
		the excited electrons at different𝐤points at the end of laser pulse|c_2(t=50 fs)|^2are shown in Fig. <ref>(a). 
		It suggests that, the effect of linearly polarized laser on point𝐤is not solely characterized by the𝐀field, but also related to the angleθbetween𝐤and𝐀. Withθ= 0andπ, i.e. the𝐤is parallel/anti-parallel to the𝐀field, the excitation is fully suppressed, while the excitation is the maximum withθ= π/2and3π/2. An effective fieldA_eff = Asinθ, always perpendicular to the vector𝐤, is thus introduced to induce electronic transitions at𝐤=(k_x = cosθ, k_y = sinθ), as shown in Fig. <ref>(b). It explains the origin of the symmetry breaking in TDDFT simultions (Fig. <ref>). The excitations at𝐤points are the results of the combined effects of energy match and the angleθbetween the𝐤 - 𝐊and𝐀field, where𝐊is the coordinates of the adjacent Dirac point. Since𝐀field is alongy,sinθ=0for all the𝐤points with𝐤 - 𝐊parallel to the polarization direction. Thus, this is no effective field to introduce photoexcitations at the twoM'points. 
		
    	
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
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