Density functional theory plus dynamical mean field theory within the framework of linear combination of numerical atomic orbitals: Formulation and benchmarks

The combination of density functional theory with dynamical mean-field theory (DFT+DMFT) has become a powerful first-principles approach to tackle strongly correlated materials in condensed matter physics. The wide use of this approach relies on robust and easy-to-use implementations, and its implementation in various numerical frameworks will increase its applicability on the one hand and help crosscheck the validity of the obtained results on the other. In the work, we develop a formalism within the linear combination of numerical atomic orbital (NAO) basis set framework, which allows for merging NAO-based DFT codes with DMFT quantum impurity solvers. The formalism is implemented by interfacing two NAO-based DFT codes with three DMFT impurity solvers, and its validity is testified by benchmark calculations for a wide range of strongly correlated materials, including 3\textit{d} transition metal compounds, lanthanides, and actinides. Our work not only enables DFT+DMFT calculations using popular and rapidly developing NAO-based DFT code packages, but also facilitates the combination of more advanced beyond-DFT methodologies available in this codes with the DMFT machinery.

I. INTRODUCTION
The understanding of strong correlations among electrons in realistic materials is of great importance for both fundamental science and technological applications. The strong electron correlations trigger a variety of exotic phenomena, which provide a fruitful avenue for designing novel materials [1][2][3]. Kohn-Sham (KS) density functional theory (DFT) [4,5] under localdensity approximation (LDA) and generalized gradient approximation (GGA) has achieved remarkable success in describing a very wide range of materials. However, these approximations are found to be inadequate for correctly describing strongly correlated materials, e.g., 3d transitional metal compounds, lanthanides, actinides, quantitatively or even qualitatively [6][7][8][9]. To overcome this limitation, the so-called DFT+DMFT method that merges DFT and many-body technique based dynamical mean-field theory (DMFT) has been developed [7][8][9][10][11][12][13][14][15]. Various studies have shown the prominent strength of DFT+DMFT in describing electronic structures of strongly correlated materials. DFT+DMFT has thus becoming a promising method of choice in studying realistic strongly correlated materials with certain predictive power. [8,9,[16][17][18][19] Comparing to DFT+U [6,20,21], a method that shares similar spirit as DFT+DMFT and is available in nearly all popular DFT code packages, DFT+DMFT is not only computationally more expensive, but also relatively more difficult to access for normal users. Implementing DFT+DMFT with well-defined local orbital basis and in a way that doesn't requires much expertise from users is, thus, of practical importance for promoting the methodology in studying realistic correlated materials. One of the key issues in the implementation of DFT+DMFT scheme is the definition of local correlated orbitals, in terms of which the DMFT impurity problem is defined and solved. Many-body corrections to the KS Hamiltonian exist only in this subspace. As a result, the choice of local correlated orbitals has a noticeable influence on the obtained DFT+DMFT results at the quantitative level.
In early days, DFT+DMFT was implemented within the linear muffin-tin orbital (LMTO) basis set framework [14-16, 22, 23], where the local LMTOs were chosen as the basis orbitals hosting strongly correlated d or f electrons. Later, different Wannier-type orbitals, such as projective Wannier functions [24] and maximally localized Wannier functions, were used to define the local orbitals [25,26]. As for plane-wave based DFT, Amadon et al. [27,28] used the all-electron atomic partial waves within the projector augmented wave (PAW) framework or pseudoatomic wave functions as the local correlated orbitals. Within the linearized augmented plane-wave (LAPW) method, Aichhorn et al. [29], and Haule et al. and Kim [30] independently developed projector schemes that convert the KS orbital space to local correlated subspace. We note that, in recent years, the underlying embedding idea of DMFT has been used in broader sense arXiv:2205.03620v1 [cond-mat.str-el] 7 May 2022 whereby the embedded cluster is solved in a ab initio way using beyond-DFT approaches such as GW or coupledcluster method, whereas the environment is treated using less advanced approaches [31][32][33].
In recent years, numerical atomic orbitals (NAOs) have been employed as the basis set choice in several firstprinciples software packages [34][35][36][37][38]. Unlike most other basis sets, the linear combination of NAOs represents a versatile framework that can be used in both allelectron and pseudopotential-based DFT calculations. Past experience indicates that NAOs are an efficient basis set choice not only for conventional ground-state calculations, but also for functionalities that go beyond conventional DFT calculations whereby explicit twoelectron Coulomb integrals and/or unoccupied KS states are needed [39][40][41][42]. In this regard, it is highly desirable to develop computational schemes within the NAO basis set framework that enables a convenient merging of firstprinciples and model-Hamiltonian based approaches. In case of DFT+U , several NAO-based implementations have been reported previously [43][44][45]. However, in case of DFT+DMFT, only one recent work within the pseudoatomic orbital basis sets (i.e., NAOs tailored for pseudopotential-based calculations) was reported by Sim and Han [46]. In their work, these authors proposed to use the so-called natural atomic orbitalsthe eigen-orbitals for the local density matrix -to define the local correlated orbitals. In the present work, we develop a projection scheme that allows to conveniently interface the NAO-based DFT codes and the DMFT impurity solvers and thus enables NAO-based DFT+DMFT calculations. We test the efficacy of such a scheme for the all-electron FHI-aims code [37] and the norm-conserving pseudopotential based ABACUS code [38], interfaced with three different DMFT impurity solver packages. Consistent results are obtained for transition metal compounds, including both correlated metals (SrVO 3 ) and Mott insulators (NiO and MnO), rare-earth systems (Ce metal and Ce 2 O 3 ), and actinides (Pu 2 O 3 and PuO 2 ). Our work thus extends the reach of the NAO-based numerical framework to treat strongly correlated materials. This paper is organized as follows. Sec. II focuses on the DFT+DMFT formalism. After introducing the general DFT+DMFT formulation, we present our definition of local correlated orbitals within the NAO framework. This is followed by a discussion of the self-consistency scheme of our DFT+DMFT implementation. In Sec. III, we benchmark the validity of our DFT+DMFT formalism and implementation over a wide range of strongly correlated materials, i.e., 3d -systems SrVO 3 , MnO and NiO, 4f -systems Ce metal and Ce 2 O 3 , and 5fsystems Pu 2 O 3 and PuO 2 . Sec. IV concludes this paper.

A. General DFT+DMFT formalism
Combining DFT with DMFT is not straightforward conceptually as the two theories are formulated in different forms and in different Hilbert spaces. In KS-DFT, one needs to self-consistently solve an effective singleparticle problem, whereby (for periodic systems) a KS Hamiltonian is constructed and solved separately at each individual k point in the Brillouin zone (BZ). On the other hand, DMFT is a real-space approach developed to solve the lattice Hamiltonian models in which the onsite two-electron interactions are explicitly included. Furthermore, as a first-principles approach, DFT accounts for all the chemical details in the systems and include all (at least valence) electrons in the calculations. In contrast, only the strongly correlated electrons sitting energetically in the vicinity of the Fermi level, relevant for the low-energy physics, are included in the model Hamiltonian and treated explicitly in DMFT. As such, key quantities like charge density, density matrix, and Green functions all have different representations in DFT and DMFT, impeding a straightforward combination of the two approaches. One solution to address this challenge, as detailed below, is to define a suitable projector to convert the key quantities back and forth between the two representations. As a side note, in the following we only focus on the general formalism of DFT+DMFT, without going into a detailed account of DMFT itself in this paper. Interested readers are referred to Refs. [7-10, 12, 47] for a detailed discussion of this theory.
Briefly, in the DFT+DMFT formalism, the Green functions defined in the correlated subspace are the central objects gluing the two theories. There are two kinds of interacting Green functions defined in the corrected subspace: One is the impurity Green function G imp determined by solving the impurity problem in Anderson impurity model, describing a single site with on-site Coulomb interactions and coupled to a meanfield electron bath. The another one, denoted as G loc , is the on-site term of the lattice Green function, which is essentially the the Green function of the correlated site described by the Hubbard model and is obtained by projecting the Green function in the full Hilbert space into the correlated subspace. The fundamental requirement in DMFT is that the "on-site" Green function of the lattice problem and that of the Anderson impurity model are equal, which is achieved via the DMFT self-consistency cycle.
In the DFT+DMFT scheme, the single-particle effective Hamiltonian in the full Hilbert space can be expressed asĤ whereĤ KS is the non-interacting KS Hamiltonian,Σ the self-energy which encodes all the many-body complexities arising from strong electron correlations and are nonzero only for correlated orbitals. Finally,Ĥ dc is the doublecounting term, which is introduced to discount the interactions among correlated electrons that have been included in a static mean-field level inĤ KS . Starting from the Hamiltonian Eq. (2), the key issue in the combination of DFT and DMFT is the downfolding (projecting) of the physical quantities from the full KS space to the local correlated space, and the upfolding (embedding) of these quantities from the local space to the full space. Mathematically, the local interacting Green function G loc can be obtained through a projection procedure. In the literature, two different projection procedures have been used. The first one can be seen as a "Hamiltonian projection", whereby a tight-binding Hamiltonian is first obtained from the KS Hamiltonian by projecting the latter into the correlated subspace, Here φ mk is the Bloch-summed local orbitals with m denoting an orbital index in the correlated subspace. In the second step, the local interacting Green function can be directly calculated from this tight-binding Hamiltonian as where N k is the is the number of k points in the first Brillouin zone, equivalent to the number of unit cells in the Born-von Kármán supercell. In Eq. (4), µ is the chemical potential, I an identity matrix, and [· · · ] −1 denotes the matrix inversion which is taken within the correlated subspace. Here and below, we employ the Matsubara Green function formalism where ω n = (2n + 1)π/β are the discrete frequency points along the imaginary frequency axis, with β = 1/k B T being the inverse temperature. The spectroscopy results on the real frequency axis are obtained via analytical continuations. The above procedure to compute the local interacting Green function is the so-called tight-binding Hamiltonian method.
In the second scheme, one first constructs the interacting Green function in the full Hilbert space as where the Hamiltonian matrix H ij (k) = χ i,k |Ĥ | χ j,k is the representation of the interacting Hamiltonian operator introduced in Eq. (2) within an arbitrary orthonormal basis set | χ j,k expanding the full Hilbert space. While the matrix elements of the KS Hamiltonian H KS within such a basis set are given straightforwardly, those ofΣ andĤ dc will be discussed later in Sec. II C.
Once the full-space lattice Green function is obtained, the local interacting Green function can be obtained through a projection procedure where the projector above is given by The construction of the local Green function via Eqs. (5)-(7) is known as the projector method. The two methods discussed above to construct the local Green function differ in that the tight-binding Hamiltonian method projects the Hamiltonian matrix while the projector method projects the Green function. In the projector method, the matrix inversion is carried out in the full Hilbert space, as indicated in Eq. (5), and thus the interactions between the correlated electrons and the rest electrons are retained to some extent. On the other hand, within the tight-binding method, if one wants to describe the interaction between correlated electrons and the rest ones, e.g., studying the charge transfer process between the correlated orbitals and the bath, one has to enlarge the Hilbert space of the tight-binding Hamiltonian to encompass the extra itinerant electrons, and it will inevitably increases the computational complexity. In this work, we choose the projector method in our DFT+DMFT implementation.
The other key quantity in DFT+DMFT calculations is the interacting impurity Green function G imp in Eq. (1), corresponding to the local propagator of the effective single-impurity Anderson model, which describes a single site coupled to a bath that mimics the lattice environment at a mean-field level. Formally G imp satisfies the following relationship where E imp is the impurity energy level, ∆ (iω n ) the socalled hybridization function characterizing the influence of the environment on the embedded impurity, and Σ imp (iω n ) the impurity self-energy. It should be noted that E imp and ∆ (iω n ) are matrices when there are multiple orbitals in the correlated subspace, which is typical in DFT+DMFT calculations. In this context, it is also customary to define a so-called Weiss Green function which acts as the dynamical (energy-dependent) mean field that the impurity electrons experience, and encodes essentially the same information as the hybridization function ∆ (iω n ). When the self-consistency in the DMFT loop is reached, the local Green function G loc mm (iω n ) and the local self-energy Σ mm (iω n ) in Eq. (4) and (6) are equal to the impurity Green function G imp mm (iω n ) and the impurity self-energy Σ imp mm (iω n ), respectively.
The Weiss Green function together with the local Coulomb interactions defines the Anderson impurity model, which can be expressed in terms of an action [7] where c † l (τ ), c n (τ ), etc., should be understood as the Grassmann variables, and U lmno is the on-site Coulomb interaction expressed within a set of local orbitals (labelled by l, m, n, o) spanning the correlated subspace. The action S is essentially the integration of the Lagrangian over the imaginary time. For given S, the impurity Green function can be calculated as where i runs over all m indices, and Z is the partition function The interacting impurity Green's function defined via Eqs. (10)(11)(12) can then be obtained through a variety of numerical approaches -usually termed as impurity solvers. Up to date, several types of impurity solvers have been developed, including the quantum Monte Carlo (QMC) [13,[48][49][50], non-crossing approximation (NCA) [51][52][53], one-crossing approximation (OCA) [54,55], exact diagonalization (ED) [56,57], numerical renormalization group (NRG) [58,59], etc. Among all these impurity solvers, continuous-time quantum Monte Carlo (CTQMC) [60][61][62][63] provides access to both high and low energy scales and is effective for a wide class of realistic material calculations. Nowadays, the CTQMC, especially the hybridization expansion based CTQMC (CT-HYB), is the most popular impurity solver employed in DFT+DMFT calculations. Once the interacting impurity Green function is determined, the impurity self-energy Σ imp (iω n ) can be ob- which is usually done within the impurity solvers. This impurity self-energy will be taken as the updated local self-energy and fed into Eq. (3) and (4) or (2) and (5), from which a new local Green function and consequently a new Weiss Green's function can be obtained. This is where a new iteration starts. This self-consistency loop keeps going until the self-energy reaches convergence or the local and impurity interacting Green function satisfies the self-consistency condition, i.e., Eq. (1).

B. Construction of the projector
Within the NAO basis set framework, it's natural to take the d or f -type NAOs that contribute most to the electronic states around the Fermi level as the the local correlated orbitals. These NAOs by construction are localized and atom-centered, and thus satisfy the usual requirement of correlated orbitals. In early DFT+DMFT implementations, analogous atomic-like orbitals like LM-TOs were used. LMTOs are minimal basis sets in the sense that for each angular momentum channel there is only one radial basis function. In contrast with LMTOs, the NAO basis sets are of multi-ζ character meaning that there are more than one radial functions per angular momentum, thus offering a more accurate description of the electronic structure. In the past, the DFT+U method has been successfully implemented within NAObased DFT codes [43][44][45], whereby it turns out to be a good practice to choose the most localized d or f basis functions as the correlated orbitals to apply the Hubbard U correction. Thus the most localized d or f orbitals seems to span the correlated subspace rather well. In practice, since NAOs centering on neighboring atoms are non-orthogonal to each other, certain orthorgonalization procedure is needed to generate an orthonormal local basis set, which is convenient for DMFT calculations.
Below we shall discuss our procedure to construct the projector and local correlated orbitals to facilitate DFT+DMFT calculations within the NAO basis set framework. In analogy to the transformation between Bloch orbitals and Wannier orbitals, we define the following Bloch-summed atomic orbitals as where φ I,m (r − τ I − R) is a NAO located at atomic site I in cell R. Here the magnetic quantum number m labels the different orbitals in the correlated angular moment channel, such as the five d orbitals or seven f orbitals. Since NAOs on neighboring atomic sites are nonorthogonal to each other, it is obvious that the Φ k I,m (r)'s defined in Eq. (13) with different m are also nonorthogonal. The key next step is to employ the Löwdin orthonormalization procedure to Φ k I,m , i.e., where is the overlap matrix. The newly obtainedΦ k I,m orbitals are then orthonormal by construction. Afterwards, a Fourier transform is applied to get the correlated orbital in real space, i.e., The orthonormality of W R I,m is also guaranteed by construction.
We choose the KS states |Ψ ik as the basis sets |χ ik (cf. Eq. (7)) to span the full Hilbert space, and then Eq. (6) becomes where with ik being the KS eigenvalues. The projector, Eq. (7), then becomes where the superscript 0 denotes the central unit cell. Since in DMFT calculations, only the "on-site" Green function, where m and m orbitals are located in the same unit cell is needed, the projector is designed to project the full Green function into the central unit cell, without losing generality. Formally, the projector for a given correlated atom I and a wave vector k is a fourth-order tensor, but since it is separable and symmetric, only a second-order tensor, i.e., the overlap matrix Ψ ik | W 0 I,m needs to be computed and stored in practical implementations.
The whole DFT+DMFT scheme requires the orthonormality of local orbitals representing the correlated subspace. In the language of the projector, it requires the projector to satisfy the following orthonormal condition In principle, this condition is automatically satisfied if the summation over i goes over all the KS bands. In practical DFT+DMFT implementations, however, one truncates the full KS states into a small subset around the Fermi level, which means i just runs over bands that are located in a chosen energy window around the Fermi level (in the following, these subsets of bands are denoted as C). This truncation destroys the completeness of |Ψ ik and thus the orthonormality of the projector. To deal with this issue, one can introduce an extra transformatioñ to orthonormalize the projector. The transformation matrix in the above equation is given bỹ which is nothing but the overlap between the projections of the orthonormalized local orbitals |W 0 I,m 's within the subspace C. Mathematically, the local correlated orbitals we use above to construct the projector can be explicitly expressed as In this form, our scheme is similar in spirit to the projective Wannier-orbital scheme proposed by Anisimov et al. [24] in the context of LDA+DMFT calculations. In our case, the most localized d or f NAO plays the role of the LMTO in the work of Ref. [24] There are a few advantages of using NAOs to define the local correlated space. Firstly, this choice is physically intuitive and technologically straightforward within the NAO basis set framework. We do not need to spend extra efforts to generate a set of local orbitals and make sure they are physically reasonable atomiclike orbitals. Secondly, from both the theoretical and technical perspectives, the choice of NAOs to define the local correlated space and the resulting projection scheme are suitable for all NAO-based packages. Especially, the key quantities that are required in this formalism, e.g., the KS wave functions and the overlap matrix of the basis functions, exist naturally in NAO-based DFT code packages, and hence no additional efforts are required to calculate these quantities. Thirdly, due to its high flexibility, our DFT+DMFT infrastructure can be interfaced with a new NAO-based DFT code without much effort. We hope it can serve as a platform to enable NAO-based DFT codes to do DFT+DMFT researches on strongly correlated materials. In this work, we implement the DFT+DMFT interface and test it with two NAO-based DFT codes using different techniques, i.e., the pseudopotential-based ABACUS code [38] and full-potential all-electron FHI-aims code [37].

C. DFT+DMFT self-consistency scheme
In this section, we will explain step by step our DFT+DMFT calculation procedure, according to the workflow outlined in the flow diagram depicted in Fig. 1.
Step 1. The DFT+DMFT calculation starts from wellconverged DFT band structures. To get high-quality KS orbitals |Ψ ik , a dense k-point mesh is usually needed.
Step 2. With |Ψ ik , the projector defined in Eq. (20) can be straightforwardly constructed. To this end, the most localized d or f orbital of the correlated atoms in the NAO basis set is used to construct |W 0 I,m .
Step 3. In the DFT+DMFT iteration loop, the frequency-dependent self-energy Σ(iω n ) is determined by the impurity solver at each iteration step. To start with, the initial self-energy is set to be equal to the doublecounting term, i.e.,Σ ij (k, iω n ) = 0. Here the following choice of the double-counting term is used. Here n I is the total number of correlated electrons associated with the correlated atom I and is fixed during the DMFT cycles. In the spirit of reducing the necessity of introducing additional empirical parameters, n I is given by projecting the KS orbitals in the subset C to the local subspace as where f ik is the occupation number of KS orbital Ψ ik . This double-counting scheme is similar to the so-called fixed double-counting [22,64] scheme, which is considered to be able to improve the stability of DFT+DMFT self-consistency loop [64] by fixing the value of n I . The difference is that the nominal number of strongly correlated electrons is specified manually in the fixed double-counting scheme, whereas in our case this number is determined using Eq. (24).
Step 4. Using the projector constructed in Step 2, we embed (upfold) the self-energy back to the selected KS space subset C, which is expressed as Step 5. During the DFT+DMFT self-consistency iteration, the electronic chemical potential needs to be adjusted according to the newly obtained self-energy at each iteration, to keep the number of electrons hosted by KS bands in C conserved. Within the DMFT cycle, this condition means that where β is again the inverse temperature 1/k B T . The summation of imaginary frequency ω n should run from negative infinity to positive infinity. However, in realistic calculations, to save computational cost, the explicit summation over Matsubara frequency points is only carried within a frequency window [−ω N , ω N ], where the contribution from the frequency points outside the window is treated approximately. This is enabled by making use of the asymptotic behavior of the self-energy, i.e., lim n→∞Σii (k, iω n ) =Σ ii (k, ∞), whereΣ ii (k, ∞) is a real value. Then the Eq. (27) is approximated by When the chosen cutoff Matsubara frequency ω N is high enough so that the asymptotic behavior of the self-energy is correct, this approximation is of good accuracy.
Step 6. As all information is secured, the local interacting Green function is constructed from Eq. (17) For some CT-HYB impurity solver, the imaginary time hybridization function is needed.
Step 7. Solve the impurity problem with the determined µ, E imp I , ∆ I (iω n ) and the given on-site Coulomb interaction through the impurity solver to obtain the new self-energy and impurity Green function. In this paper, we use the Kanamori [65,66] form Coulomb interaction in which only the density-density term is included, so there are no sign problem in our CTQMC calculations.
Step 8. Check whether the self-consistency is reached. If the self-consistency is reached, stop the DFT+DMFT calcution, else return to Step 4.

A. d -electron systems
We first benchmark our DFT+DMFT implementation on three prototypical strongly correlated d -electron systems -SrVO 3 , NiO, and MnO. For the DFT part, we carried out GGA calculations using two NAO-based code packages -FHI-aims [37] and ABACUS [38]. In FHI-aims, the default tight tier 1 basis set is used for V, Ni, Mn, O, and Sr atoms, and the corresponding cutoff radii of the basis functions are 6.0 A, 6.0 A, 6.0 A, 6.0 A, and 8.0 A, respectively. In ABACUS we use the SG15 optimized norm-conserving Vanderbilt (ONCV) multi-projector pseudo-potentials [67][68][69] and the corresponding optimized double-ζ plus polarization (DZP) atomic basis sets, which comprise 4s2p2d1f basis functions with a cutoff radius of 9.0 Bohr for transition metal atoms and 2s2p1d basis functions with cutoff radii of 7.0 Bohr for O atoms and 10.0 Bohr for Sr atoms. In all DFT calculations, the Perdew-Burke-Ernzerhof (PBE) GGA exchange-correlation functional was used [70]. As for the DMFT part, we employed three different CT-HYB impurity solvers to solve the single-site impurity problem. These are PACS [71] developed by one of the present authors Gang Li, iQIST developed by Huang et al. [72][73][74], and the one used in eDMFT [30] developed by Haule at Rutgers University [62,75], which is referred to as "Rutgers" below in this paper.

SrVO3
SrVO 3 has a simple cubic perovskite structure without magnetism. DFT fails to reproduce the upper and lower Hubbard bands observed in experiments, and this calls for advanced theoretical and computational techniques [8]. In the past, this material has been extensively studied both theoretically and experimentally [25,[76][77][78][79][80][81][82], which provides abundant reference results. Therefore, SrVO 3 is an ideal test example for DFT+DMFT implementations [24,26,27,46,66]. The V 4+ cation with only one 3d electron is located in the center of the octahedron formed by its six surrouding axial O 2− ligand anions. In the presence of an octahedral crystal field, the five degenerate d orbitals split into two subsets: Three-fold degenerate t 2g orbitals (i.e., d xy , d yz and d xz ), and two-fold degenerate e g orbitals (i.e., d z 2 and d x 2 −y 2 ). The only one 3d electron of V 4+ occupies the lower-energy t 2g orbitals with the higherenergy e g orbitals being empty. As a common practice, we only consider three degenerate t 2g orbitals in our DFT+DMFT calculation.
In DFT calculations, we use 11×11×11 k-point mesh generated by the Monkhorst-Pack method [83]. The Hubbard U and Hund J parameters are set to be 4.0 eV and 0.65 eV, respectively, following the choice in the literature [26,27,46]. The DMFT calculation is carried out at a temperature of 300 K. DFT calculations give a group of bands around the Fermi level with substantial 3d characterizes, which are well separated from other bands [26,27]. We enclose the six (twelve if spin degree of freedom is taken into account) bands crossing the Fermi level in the subset of KS bands C. One of the powerful strengths of Green-function based DFT+DMFT approach is that it can deliver physically meaningful single particle excitation spectral functions, as given by the the imaginary part of Green function. Such spectral functions can be directly measured by photoemission and/or inverse photoemission experiments. In Fig. 2, the spectral functions of 3d electrons of SrVO 3 calculated by combining two NAO-based DFT packages -FHI-aims and ABACUS, and three impurity solvers -PACS, iQIST and Rutgers are presented. Despite the considerable differences underlying the implementations of the DFT codes, as well as the DMFT impurity solvers, the calculated spectral functions are remarkably similar. Most importantly, they all successfully reproduce the typical three-peak structure: A quasi-particle band located at the Fermi level with the lower and upper Hubbard bands on the two sides, arising from the strong electron correlations which traditional LDA and GGA fail to capture. The main features of the spectral functions given by our DFT+DMFT calculations are in good agreement with previous theoretical results except for some small details such as the exact peak positions and intensities [24-27, 46, 66, 76, 78, 79]. Our results also correctly describe the strong particle-hole asymmetry of the lower and upper Hubbard band as the intensity of the occupied lower Hubbard band is much smaller than the empty upper Hubbard band. This is the consequence that the V 4+ , with only one 3d electron, is far away from half-filling.
Similar to previous DFT+DMFT studies [24-27, 46, 66, 76, 79], our spectral function results are in excellent agreement with the experimental photoemission spec-trum [78]. The first main feature lies in the reproduction of the lower Hubbard band at around −2.0 eV, which is the manifestation of the strong correlation among 3d electrons. The second feature, in quantitative agreement with experiment, is that the lower Hubbard band vanishes at ∼ −1.0 eV, and then the quasiparticle band starts to rise sharply.

MnO and NiO
Next we test our DFT+DMFT scheme on the late transition metal oxides MnO and NiO. Both MnO and NiO crystallize in face-centered cubic (FCC) NaCl-type structures where 3d orbitals are split into threefold degenerate t 2g and twofold degenerate e g orbitals as in the SrVO 3 case. In the MnO system, we adopt the parameters of U =5.0 eV, J =1.0 eV, and the temperature T  Hubbard model and ab-initio DFT, static DFT+U [6,20,21] and the more advanced dynamical DFT+DMFT [84,85] successfully open the gaps in those prototypical transitional metal oxides. Then obtained band gaps are in quantitative agreement with experiments, if proper parameters are used.
We then apply our DFT+DMFT implementation to both MnO and NiO, and the obtained spectral functions are presented in Fig. 3 and Fig. 4, respectively. Again in each figure, we present six panels containing results obtained by combining two NAO-based DFT codes and three DMFT impurity solvers. The most prominent feature in the DFT+DMFT spectra of both MnO and NiO is that a sizeable gap is opened up, arising apparently from the strong Coulomb interactions whose effect is properly captured by DMFT. One can further see from the fact that the number of 3d electrons of Mn given by n = − m G imp mm (β) is 5. Similar to the SrVO 3 case, the results presented in all six panels are very close, except that the peaks given by the Rutgers impurity solver is slightly more pronounced.
For NiO, the number of nominal 3d electrons is approximately 8, and DFT+DMFT calculations lead to a splitting of the 3d bands, with fully occupied t 2g orbitals and half-filled e g orbitals. Our results in general agree well with previous DFT+DMFT results reported in the literature [84,85]. However, Fig. 4 also reveals that both the DFT codes and the impurity solvers have certain influence on the obtained DFT+DMFT energy spectra. Although there are no qualitative differences, the shape and width of the left peak, and the depth of the dip between the first and middle peaks show noticeable quantitative differences. It is not entirely clear to us yet what factors caused such differences. We note that, however, NiO is a prototypical charge transfer insulator [86], and the competition between the strong Coulomb interactions among the Ni 3d electrons and the hybridization between the Ni 3d and O 2p electrons governs its underlying physics. Thus a complete DFT+DMFT treatment of NiO should also include O 2p electrons in the game. With the O 2p excluded, the DFT+DMFT calculations are probably more sensitive to the details in the band structures and numerical techniques behind the impurity solvers. Figure 3 and 4 only contain the spectral information of 3d electrons. Our DFT+DMFT implementation also allows for calculating the k-resolved spectral function in the KS orbital space, as determined by where the real-frequency self-energy is evaluated by analytical continuation of the imaginary frequency-selfenergy through the maximum entropy formalism in Ref. [87]. The corresponding results for MnO and NiO are presented in Figs. 5 and 6, respectively. Now for simplicity only results obtained using the two DFT codes combined with the Rutgers impurity solver are presented. The agreement between our theoretical gaps and experimental gaps [88][89][90] is rather satisfactory. The two k-resolved spectral functions of MnO is nearly iden-tical whereas again noticeable differences, in particular regarding the spectral weights of certain bands, exist for NiO. Nevertheless the main features, e.g., the energy positions and dispersion of the bands, are reasonably similar in Fig. 6(a) and 6(b).
NiO has been extensively studied experimentally, and reliable experimental photoemission spectra are available. In Fig. 7, we directly compare our DFT+DMFT spectrum of NiO with the experimental spectrum of Sawatzky and Allen [89] measured by x-rayphotoemission (XPS) and bremsstrahlung-isochromatspectroscopy (BIS) techniques. Again, the two sets of theoretical spectra are obtained using FHI-aims and ABACUS codes interfaced with the Rutgers impurity solver, respectively. For the spectrum below the Fermi level, corresponding to the (negative of) energy cost for removing a particle from the occupied levels, we reproduce the sharp peak around −2.0 eV and a local minimum at −3.0 eV given by XPS. For the spectrum above the Fermi level, corresponding to the (negative of) energy cost for adding a particle to the system, our curves match perfectly with the BIS result. Although small deviations of the shoulder peak at approximately −3.5 eV are visible, our DFT+DMFT spectrum, in general, agrees well with previous theoretical work [84] and the experiment data [89]. In ABACUS, the pseudopotential method is used and only the valence electrons are explicitly included. Furthermore, as given in the beginning of this section, the spatial cutoffs of NAOs used in ABACUS are much smaller than those used in FHI-aims. By contrast, in FHI-aims full potential is used with all electrons being included in DFT calculations. Although these two NAO schemes are rather different, we get nearly identical DFT+DMFT results of SrVO 3 and MnO and reasonably similar results for NiO. It proves the efficacy and robustness of our DFT+DMFT formalism and implementation. We expect that the DFT+DMFT formalism presented in this paper should work for other NAO-based DFT codes as well.

B. f -electron systems
The f -electron systems, including 4f lanthanides and 5f actinides, are an important class of strongly correlated materials characterized by partially filled f -type orbitals. These systems exhibit a variety of exotic phenomena, such as the heavy-fermion behavior, metal-insulator transition, Kondo physics, volume collapses accompanying phase transitions, etc. It has been generally accepted that traditional DFT calculations based on static meanfield-type approximations such as LDA and GGAs do not provide adequate accuracy for describing these physical scenarios. DFT+DMFT has proved to be a powerful approach to tackle these systems and achieved remarkable successes in the last two decades. In this section we test our DFT+DMFT implementation on several typical f -electron systems, including 4f systems like α and γ cerium (Ce) metal and Ce 2 O 3 , and 5f systems like PuO 2 and Pu 2 O 3 .
DFT calculations are carried out using FHI-aims with the default tight tier1 basis sets. For these basis sets, the cutoff radii of Ce, Pu, and O elements are 7 A, 6.0 A, and 6.0 A, respectively. Analogous to the case of 3delectron systems, the GGA-PBE functional [70] is used in the DFT calculations and 11×11×11 Monkhorst-Pack k-point mesh [83] is used for Brillouin-zone integration, which is deemed dense enough to obtain accurate DFT band structures. Unlike the practice in previous sections where results obtained using three DMFT impurity solvers are shown for comparison, here for simplicity only the results obtained using the Rutgers solver will be presented.

Ce metal
Despite its simple FCC structure and the fact that only one 4f valence electron is present per Ce atom, Ce exhibits spectacular physical properties, which has attracted considerable research interest over the last forty decades [66,[91][92][93][94][95][96][97][98][99]. At low temperature and ambient pressure, Ce crystalizes in the α phase (smaller volume), where the system is paramagnetic and shows Pauli-like magnetic susceptibility without forming local magnetic moments. At high temperature, Ce transforms into the γ phase (larger volume), which instead carries local magnetic moments and exhibits Curie-Weiss behavior of magnetic susceptibility. When increasing pressure or decreasing temperature, Ce undergoes the famous isostructural γ-α phase transition accompanied by a 14% volume collapse and a drastic change of magnetic properties [66,[93][94][95][96][97][98][99]. A theoretical understanding of this behavior poses a great challenge to condensed matter physics and has motivated lots of experimental and theoretical investigations. From the perspective of first-principles computations, standard DFT calculations are unable to give a proper description of this phase transition, and the approaches going beyond conventional DFT are required.
In this paper, the crystal structures of α and γ Ce phases are discerned by different volumes, i.e., 29 A 3 for α and 34 A 3 for γ phase, respectively, which covers the volume ranges during the α-γ transition [94]. The chosen energy window for KS subset C is [−5.0 eV, 5.0 eV]. The Coulomb interaction parameter U given by constrained DFT is 6.0 eV [66,98]. Here we used the value of U of 6.0 eV and Hund parameter J of 0.5 eV [96,98]. DFT+DMFT calculations are done at the temperature of 800 K.
Our DFT+DMFT spectra for Ce are shown in Fig. 8. First of all, the energy spectra of both phases show the characteristic three-peak structures of strongly correlated metals. One can also see that, going from the α phase to the γ phase, the central quasiparticle peak gets substantially suppressed, and its spectral weights transfer to the lower and upper Hubbard bands . Such results are in agreement with experimental observations [91,92] and previous DFT+DMFT calculations for Ce [94][95][96]98]. The relative strengths of the quasi-particle peak and the upper Hubbard peak of both α and γ phases are also captured qualitatively. However, a direct comparison of the calculated spectra with the experimental spectra reveals that the energy separation between the quasi-particle peak and upper Hubbard band is underestimated in our calculations, while that between the lower Hubbard band and quasi-particle peak is overestimated. Compared to previous DFT+DMFT calculations, the location of the lower Hubbard peak of our results is close to that reported in the literature [94,98] where the maximum of lower Hubbard peaks is around -4.0∼−3.0 eV. Concerning the energy positions of the upper Hubbard band, we found that the results reported in the literature also vary quite a bit, with those reported in Ref. [98] agreeing best with experiment. Our results are close to those reported in Ref. [96], where the upper Hubbard peaks are located at around 2.5 eV. In general, our DFT+DMFT spectrums are in quantitative agreement with experimental and previous theoretical results. The discrepancies between our results and experimental and previous theoretical results indicate that details of the DFT calculations, the definition of the projector, as well as the numerical implementation of the impurity solvers will still have appreciable influence on the outcomes of DFT+DMFT calculations. Further investigations along these lines are still needed. The resonant inverse photoemission spectroscopies (RIPES) are taken from Ref. [91], and the photoemission (PE) spectra are taken from Ref. [92].

Ce2O3
Ce 2 O 3 is a Mott insulator with a gap of about 2.4 eV [22,100] arising from the strong correlation among 4f electrons. This system has been calculated using a variety of approaches such as DFT+U, GW [101], and DFT+DMFT [22,28]. Ce 2 O 3 crystalizes in the hexagonal lattice with space group P3m1. The experimental lattice parameters a = b = 3.891 A, c = 6.059 A [102] are used in our calculations. The energy window for KS subset C is [−3.0 eV, 2.0 eV]. Constrained DFT calculation predicted the parameter U varying from 5.5 to 8.0 eV, and we adopt U as 6.5 eV and J as 0.5 eV. The temperature is set to be 300 K.
Here we calculate the total density of states (TDOS) of Ce 2 O 3 through where A(k, ω) is defined in Eq. (31). The result is shown in Fig. 9, our DFT+DMFT calculation gives a band gap of about 2.5 eV, which agrees well with the experimental and previous theoretical results [22,28,100,101]. For example, all predict that the lower Hubbard peaks are sharp and narrow whereas the upper Hubbard peaks are strong and wide. The interesting feature about the upper Hubbard part is that it emerges as a low plateau between 1 and 3 eV above the Fermi level, continued by a pronounced peak between 3 and 4 eV. Note that the peaks mainly composed of O p bands sitting below the lower Hubbard peaks does not show up here due to the fact that those O p characteristic bands are not included in the subset C in the DFT+DMFT calculations whereas they are retained in Refs. [22,28,101].

Pu2O3 and PuO2
Plutonium (Pu)-based oxides such as Pu 2 O 3 and PuO 2 are essential for the fuel components in current nuclear reactors, as well as transmutation of the minor actinides from spent nuclear fuels [103,104]. A clear understanding of the physio-chemical properties of plutoniumbased oxides is of key importance for the safe operation and development of nuclear reactor systems and nuclear waste reprocessing [105,106], and the correct description of the oxidation and reduction processes [107,108]. The physical, chemical, and thermodynamical properties of Pu-based oxides such as the chemical bonding and electronic structure are intimately related to the states of strongly correlated 5f electrons. Conversely, the relative tendency of delocalization versus localization of strongly correlated 5f electrons is extremely sensitive to the physical and chemical environment of the Pu atom [109]. The description of complicated behaviors of 5f electrons, e.g., whether they are settled in the delocalized or localized states, or in the intermediate regime, is out of the reach of standard DFT. In contrast, DFT+DMFT is becoming a promising approach that facilitates an indepth understanding of this type of materials.
In this paper, we look at two plutonium oxides, i.e., Pu 2 O 3 in its β phase [110] (simply called Pu 2 O 3 below) and PuO 2 [111] with our DFT+DMFT implementation. We use the Hubbard parameter U We calculated DFT+DMFT TDOS of Pu 2 O 3 through the method introduced in the case of Ce 2 O 3 . The TDOS here contains contributions from both the correlated orbitals and the rest (here spd) orbitals. The projected density of states (PDOS) belonging to correlated (5f ) orbitals, is evaluated by whereas the PDOS for the spd orbitals is obtained by taking the difference between the TDOS and ρ f (ω). The results are depicted in Fig. 10. Figure 10 indicates that the band gap of Pu 2 O 3 as determined by DFT+DMFT is about 1.7 eV, which is in good agreement with the experimental bands gap of 1.8 eV [28,112]. Additionally, the occupation analysis yields an average occupation number n f =5.0 for Pu 5f electrons through n = − m G imp mm (β), which is consistent with the chemical environment of Pu 3+ in Pu 2 O 3 .
In Fig. 11, the DFT+DMFT TDOS of PuO 2 is plotted. The bands gap is predicted to be about 2.5 eV, which is in good agreement with previous theoretical work [113] and the experimental band gap of 2.8 eV [114]. In PuO 2 , the peak of occupied Pu 5f electrons is noticeably sharper than that of Pu 2 O 3 , which indicates that the Pu 5f electrons in PuO 2 are likely more localized than the case of Pu 2 O 3 . This localization picture is in agreement with the bigger gaps in PuO 2 . The occupation analysis gives an average occupation number of n f =4.0 for Pu 5f electrons, which agrees with the chemical state of Pu 4+ in PuO 2 .

IV. SUMMARY
In summary, we developed and implemented a formalism that allows us to carry out DFT+DMFT calculations within the NAO basis set framework. For transition  metal compounds and f -electron systems, the most localized d or f -type NAO is used to define the local correlated subspace. Such a choice is physically intuitive and implementationally convenient for NAO basis sets.
Following what is usually done in the literature, only a subset of the KS bands C out of the full KS space around the Fermi energy, which hosts the majority of electrons with strong correlations, are enclosed in the definition of the projector. Our projector scheme is mathematically equivalent to the projective-Wannierfunction approach adopted in Ref. [24]. We implemented such a DFT+DMFT formalism by interfacing two NAObased DFT codes, i.e., FHI-aims [37] and ABACUS [38] with three DMFT quantum impurity solvers, i.e., PACS, iQIST [72][73][74], and the Rutgers impurity solver [30,62,75]. In particular, interfacing with the allelectron FHI-aims code allows one to study all types of strongly correlated materials over the periodic table.
Our DFT+DMFT formalism and implementation are testified for three typical series of strongly correlated materials, namely the 3d transitional metal compounds SrVO 3 , MnO and NiO, 4f materials including the Ce metal and Ce 2 O 3 , as well as the 5f actinides Pu 2 O 3 and PuO 2 . For SrVO 3 and MnO, our calculated one-electron removal and addition spectra are in good agreement with previously reported DFT+DMFT results and experimental data. Furthermore, the calculated results are rather robust against the use of different DFT codes or different impurity solvers. For NiO, the obtained DFT+DMFT results show noticeable dependence on the chosen DFT codes and/or impurity solvers, although the key spectroscopic features are captured by all calculations. For felectron systems, the characteristic three-peak structures are obtained for the Ce metal, whereas for the correlated insulators, the obtained DFT+DMFT band gaps are in good agreement with experiments. However, there remain issues calling for further investigations, like the energy separation of the quasiparticle peak and the upper Hubbard band for the Ce metal.
Admittedly, our scheme is still in its infancy, and at the quantitative level there are still issue to be sorted out. Further improvement would be necessary for more reliable descriptions of charge-transfer-type Mott insulators, and the intricate lanthanides and actinides. However, we consider that our attempt of developing an infrastructure that merge NAO-based DFT codes and DMFT-based techniques is rather rewarding. This will not only enable standard DFT+DMFT calculations within the NAO basis set framework, but also paves the way for developing more advanced schemes by combining beyond-DFT approaches like hybrid functionals [40,41,115] or GW [42], recently available in NAO-based DFT codes, with the DMFT machinery.


The Historical Formation and Academic Characteristics of the Tao-Ge School in Daoism

: The Tao-Ge School refers to the Taoists Ge Hong and Tao Hongjing as the main representatives in the Six dynasties period. It also includes numerous alchemists, physicians, and craftsmen who constituted a Daoist science and technology school in the Middle Ages. This school comprised an academic community characterized by scattered, small master–apprentice or family inheritance groups. It was founded and became popular between the 4th and 12th centuries. With alchemy, medicine, and technology as its main practice, it reﬂected the scientiﬁc and technological academic characteristics of cognitive positivism and rationalization, which emphasized an active understanding of nature, commitment to technological innovation, and a theoretical construction of knowledge. Historically, the Tao-Ge School had a profound impact on the practical methods and theoretical perspective of ancient Chinese science and technology, standing as an example of a school of science and technology with traditional Chinese cultural characteristics.

Introduction
It is difficult to imagine that the brilliant scientific and technological achievements of ancient China were merely caused by scattered individual experiences or practical craftsmanship, without the support of a rational thought paradigm and academic tradition. Setting aside its fortuitous history, behind the rich scientific and technological achievements that emerged are thought and cognitive traditions accumulated by countless thinkers and practitioners. In fact, just as the Milesian and Pythagoras Schools advocated rational cognition around nature in ancient Greece, the Tao-Ge School, based on Chinese culture, inclined toward understanding nature and technological inventions by emphasizing rational cognitive thought.
However, in the history of ancient Chinese religious culture, the Tao-Ge School, which was popular between the 4th and 12th centuries, has rarely been discussed by scholars (Cai 2007). This is likely because it is not like mainline Confucianism, the tradition of Lao Zi and Zhuang Zi in Daoism, or Zen Buddhism, which are humanistic and spiritual disciplines devoted to ethical systems and moral character. In contrast, the Tao-Ge School involved the study of shu shu 數術 (including astrology, numerology, divination, and talismans) and fang ji 方技 (medical technology and knowledge), focusing on nature, medicine, and smelting. It has made far-reaching contributions to the practice and theoretical development of ancient Chinese science and technology from a unique perspective. This article discusses the Tao-Ge School's connotations and value for Daoist science and technology history.

Historical Formation of the Tao-Ge School
The Tao-Ge School was mentioned in the manuscript Daode zhenjing cangshi zuan weikai tike wensu 道德真經藏室纂微開題科文疏, written by Xue Zhixuan 薛致玄 during the Northern Song dynasty. Volume 5 of the work states: In 1088, the famous Taoist Chen Jingyuan 陳景元 went to Dongjing (the capital of the Northern Song Dynasty) to work and meet Su Song 蘇頌. Su Song said to Chen Jingyuan, "You should recruit more disciples and teach them your academic knowledge. Otherwise, the Tao-Ge School may be lost" 元佑三年, 陳景元因過京 師 . . . . . . 朝廷複還右街道職. 右僕射蘇頌謂曰: 真靖當以所業授門弟子, 不爾, 則 恐陶葛之學不傳於來世. (Xue 1988, p. 371) Su Song 蘇頌 was one of the greatest scientists in ancient China (Yan 1986, p. 158). The Tao-Ge School originated as a result of Su Song's advice given above. Su Song highly valued the Tao-Ge School-which included astronomy, geography, medicine, alchemy, and other intellectual traditions-and learned a great deal from it. In fact, Su Song's scientific and technological thought and knowledge is deeply rooted in Daoism (Cai 2011, pp. 30-34). During this time, the Taoist academic tradition was undergoing major changes, which have been called the "Taoist School sudden change" 道家之學翕然一變 (Xue 1988, p. 731); the Lao-Zhuang School and the inner elixir (nei dan 內丹) began to flourish, while the Tao-Ge School and the outer elixir (wai dan 外丹) were gradually declining. The dialogue between Su Song 蘇頌 and Chen Jingyuan 陳景元 thus revealed a major transition in the history of ancient science and technology. How did the Tao-Ge School come into existence and spread?
Historical material confirms that the Tao-Ge School was created by Ge Hong 葛洪 of the Eastern Jin dynasty and Tao Hongjing 陶弘景of Southern Liang. Ge Hong was not only the most famous alchemist in Chinese history but also one of the most outstanding medical scientists. He believed that "golden elixirs" were the most important medicine to enable Daoist cultivators to obtain longevity and immortality-"I have collected and read a lot of books on regimen" 餘考覽養性之書, 鳩集久視之方, and "These Books show that elixir is the decisive factor for longevity" 莫不以還丹金液為大要者焉 (Ge 1988a, p. 181). Similarly, "medicine" is the key factor to maintaining good health-"health preservation is based on not harming the body" 養生以不傷為本 (Ge 1988a, p. 221); "while cultivating Daoism, we should also learn medical skills" 是故古之初為道者, 莫不兼修醫術, 以救近 禍焉 (Ge 1988a, p. 229). To this end, Ge Hong wrote far-reaching medical masterpieces, including Yuhanfang 玉函方 and Zhouhoubeijifang 肘後備急方. Tao Hongjing agreed with and followed Ge Hong's perspective on attaching importance to alchemy and medicine and offered that cultivating Daoism and becoming an immortal "must take medicine first" 莫不以藥道為先 (Tang 1986, p. 16), and he (like Ge Hong) wrote numerous innovative and important medical books, such as Bencaojingjizhu 本草經集注, Buquezhouhoubaiyifang 補闕肘後百一方, and Yao zong jue 藥總訣. Their shared emphasis was on the importance of alchemy (Wai dan 外丹) to facilitate longevity and medical assistance in the practice of Daoism; further, they advocated for technological innovation and a curious spirit aimed at acquiring knowledge. At the same time, they attached great importance to positivization and rationalization of knowledge. Joseph Needham conducted a systematic and in-depth study of the Taoist scientific and technological thought and achievements in the Ge and Tao age and spoke highly of them (Needham 1976, pp. 75-117). He pointed out, "Ko Hong (葛洪) was the greatest alchemist of his age, and the greatest Chinese alchemical writer of any age"; "the most celebrated in 5th-and 6th-century China was undoubtedly Thao Hung-Ching (陶弘景), who was also a great physician pharmaceutical naturalist like Ko Hong" (Needham 1976, pp. 79, 119).
During the Jin and Liang dynasties (4th-6th centuries), Ge and Tao became famous as Taoist science and technology masters, with alchemy and medicine as their main associated fields. They were widely respected and became known as "Ge-Tao" or "Tao-Ge". At that time, Yu Yuanwei's 庾元威 Lun shu 論書had already mentioned "Tao and Ge" (Zhang 1986, p. 127). During the Tang and Song dynasties, Ge and Tao gained even more reputation in this period. Literati often praised Ge and Tao in their poems and regarded them as outstanding alchemists and medical experts. For example, "Tao and Ge's prescription for immortality is a secret and has not been taught to others, so it has remained unknown for thousands of years" 陶葛金丹訣, 千年秘不傳 (Lu 1986, p. 670); "From Qi-Suo to Tao-Ge, many immortals are good doctors" 請看岐叟至陶葛, 神仙多是良醫成 (Xie n.d.,p. 29). At the same time, they were also regarded as the most learned Taoists: "Ge and Tao, the world knows them as Taoists, who are individuals with all-round natural knowledge and technology" 葛陶二君, 世共知為有道之士, 於學無所不貫, 於術無所不通 (Ge 1988b, p. 1). Thus, the Tao-Ge School was a traditional school generally recognized by the Jin, Liang, and Song dynasties. Its philosophical and academic characteristics were erudition proficiency in divination and advocacy of science and technology (see Table 1). He offered the concept that "you don't understand the things, and you deeply think it's a shame" 一事不知, 深以為 恥; specifically, we should understand concepts, such as "Yin-Yang and the five elements (metal, wood, water, fire, and earth), practicing divination, geomantic omens, and medical skills" 尤明陰陽五行, 風角星算, 山川地理, 方圖產 物, 醫術本草 (Li 1975(Li , p. 1898).
It should be noted that this "science and technology school" does not refer to a strictly fixed school of inheritance, for example, a formal association, but was composed of many small-scale Taoist groups, mainly working in Taoist alchemy, medicine, and theurgy. These groups often had teachers, pupils, or families constituting their lineage and practiced secretly. The succession of teachings from Ge Hong to his disciples and their group composition are extremely complex. Joseph Needham's research pointed out that there are two lines of Ge Hong's teaching: One is to learn from Zheng Yin 鄭隱, which can be traced back to Ge Xuan 葛玄, Zuo Ci 左慈, Li Zhongfu 李仲甫, or Wang Sizhen 王思真, Yin Changsheng 陰長生, . . . Ma Mingsheng 馬鳴生, An Qisheng 安期生, and Heshang Zhangren 河上丈人. The other is from Bao Liang 鮑靚, which can be traced back to Lu Zihua 呂子華, Yin Changsheng 陰長生, . . . Ma Mingsheng 馬鳴生, An Qisheng 安期生, and Heshang Zhangren 河上丈人. It can be seen that this path is interrelated, and there are branch groups such as Wei Boyang 魏伯陽, Xu Zongshi 徐從事, Chunyu Shutong 淳 于叔通, Hu Gangzi 狐剛子, Xu Mai 許邁, Yang Xi 楊曦, Ding Yi 丁義, and Wu Meng 吳猛. (Needham 1976, pp. 76-78) It is worth noting that Ge Hong also taught the Dao of alchemy (dan dao丹道) and medical prescriptions to his disciples, such as Huang Yeren 黃野人 (Zhao 1988, p. 237) and Tengsheng 滕升 (Ma 1986, p. 653). They learned many kinds of danjing daoshu 丹經道術 (including Taoist magic arts and alchemy), the medicine technology of immortality and alchemy, which had a great influence. Although Tao Hongjing 陶弘景 was not a disciple of Ge Hong's lineage, he inherited Ge Hong's alchemy, medical thought, and knowledge, and can be viewed as a successor to Ge Hong's thought. According to historical data, Tao Hongjing's family was a medical family, and his grandfather "(Tao Long 陶隆) was good at using medicine" 陶隆善解性 (Zhang 1988, p. 730). Under the family's influence, Tao Hongjing learned dan dao丹道 and medicine skills from Sun Youyue 孫游嶽 who is a famous Taoist, and Tao "traveled through famous mountains, looking for immortal medicine" 遍歷名山, 尋訪仙藥 (Li 1975(Li , p. 1897. In Mount Maoshan 茅山, Tao Hongjing and his disciples (Zhou Ziliang 周子良, Huan Kai 桓闓, and Sun Wentao 孫文韜) formed an academic Taoist group that advocated for knowledge and craftsmanship-the Maoshan sect of the Shangqing School 上清派茅山宗. Thus, the ideas foundation of the Tao-Ge School was established. From the 6th to the 8th century, the Maoshan sect was successively led by Taoist Wang Yuanzhi 王遠知, Pan Shizheng 潘師正, Sima Chengzhen 司馬承禎, and Li Hanguang 李含光 (Needham 1976, p. 121).
Since Ge and Tao, this form of Taoist group inheritance, which is a mixture of masterapprentice and family relations, has continued and developed in secret. For example, in the Tang dynasty, Sun Simiao 孫思邈 and his disciples Meng Shen 孟詵, Zhang Ding 張鼎, and Lu Zhaolin 盧照鄰 formed a medicine group (Ouyang 1975, p. 5597); Huang Gong黃公 and Cao Yuan曹元 formed a medicine group originated from Hua Tuo華佗 (Wang 1986, p. 106); Zhang Wenzhong 張文仲, Li Qianzong 李虔縱, and Wei Cizang 韋慈藏 composed a famous doctors' group (Liu 1975, p. 5099); Chen Zangqi 陳藏器 and Ri Huazi 日華子 followed Tao Hongjing's herbal medicine knowledge (Xu 1986, p. 661); a famous doctor, Wang Bing (Qi Xuanzi) 王冰 (啟玄子, learned from Meng Shen 孟詵and Xuanzhu 玄珠先生; and Hu Yin 胡愔, a famous female Taoist doctor, originated from the lineage of Wei Huacun 魏華存, the founder of a sect of the Shangqing School (Gai 1999, pp. 22-24). In the Song dynasty, the official medical research group was composed of Taoist Ma Zhi 馬志and Hanlin medical officials Zhai Xu 翟煦, Zhang Su 張素, Wu Fugui 吳複圭, Wang Guang 王光, Chen Zhaoyu 陳昭遇, and An Ziliang 安自良 (Tang 1986, p. 24); the medical research group established by Taoist Wang Huaiyin 王懷隱, Wang You 王佑, Zheng Qi 鄭奇, and Chen Zhaoyu 陳 昭遇 (Tang 1986, p. 24), and Su Song 蘇頌 and Su Yi's 蘇繹 family, who followed Ge, Tao, Sun's dan dao丹道and medical knowledge (Cai 2011, pp. 30-34). Furthermore, there were numerous Taoist groups who practiced the outer elixir (wai dan, 外丹) and medicine in this period, such as the waidan jinsha School 外丹金砂派 and liugong School 硫汞派. , and numerous unrecorded alchemists and medical groups. It can be said that from the 4th century to the 12th century, this knowledge group and its inheritance from the Tao-Ge School constituted the development path of ancient Chinese science and technology.
From the perspective of sociology, the Tao-Ge School maintained homology and continuity of knowledge through community members, including teachers, pupils, and families, which was very important for the integrity of the knowledge system. Furthermore, based on the renewal of knowledge and theory, the Tao-Ge School adopted the academic inheritance method of revising, annotating, and expanding classic books, especially medical books, such as Shennongbencao 神農本草, Huangdi neijing 黃帝內經, and Zhouhoubeijifang 肘 後備急方. This process of knowledge inheritance and promotion was the main theoretical method of the Tao-Ge School. Its philosophy emphasized the importance of medicine in knowledge and the social function of medical assistance, which established the "Tao-Ge School" as a simple, practical, and standardized philosophical system (see Table 2).
In short, the Tao-Ge School was an academic tradition based on the practice of Daoist belief, which attaches importance to natural knowledge, technological invention, and knowledge rationalization. It comprises three basic academic characteristics: first, it highlights the social value of medicine; second, it emphasizes the accumulation and expansion of knowledge; and third, it focuses on the standardization and development of theory. He advocated that "everyone can understand" medicine (Sun 1955, p. 6), contributed the slogan that "all things in the world are miraculous drugs," and developed a method of "classifying according to the properties of the drug" 方證同條, 比類相 附 to organize medical records (Sun 1955, p. 97).

Knowledge Demarcation of the Tao-Ge School
In the 4th century, the Tao-Ge School maintained the unique perspective: "Daoism has the scientific spirit of conquering nature" (Feng 1996, p. 3). In its epistemology, the Tao-Ge School had a strong tendency to understand and transform nature but was also ideologically conscious of its "demarcation" with traditional Chinese humanistic thought. This "knowledge demarcation" demonstrates the scientific and technological directions of the Tao-Ge School.

Naturalness: Demarcation with Confucian Humanism
Before Ge Hong, the Taoist knowledge tradition had not recognized the legitimacy of the official orthodox ethical ideology. Confucianism believed that the Daoist focus on developing nature knowledge and immortality technology often led to a practical way of life that contradicted Confucian ethics, so "the scholar-officials were ashamed to be involved in agricultural and commerce, and ashamed to work of craftsmanship" 士大夫 恥涉農商, 羞務工技 (Yan n.d., p. 26). Ge and Tao proposed that Daoism attaches great importance to natural knowledge and craftsmanship technology as "necessary for Daoist practice" 修道所須 (Zhang 1988, p. 733), that is, Daoism has social value that Confucian ethics does not have.
Ge Hong defined the epistemological goal and direction of naturalization and scientization as "understanding the motion laws of all things in nature" 欲求物理in Baopuzi neipian 抱樸子內篇. He stated: "I just wanted to fully explain the truth of things. When the truth was understood, the things will become clear" 但欲盡物理耳, 理盡事窮 (Ge 1988a, p. 216).
The wu li 物理 can be interpreted as the law of all things in nature, that is, the essence and motion laws of nature and its entities. The so-called yu jin wu li 欲盡物理 means wanting to fully explain the truth of things.
Therefore, Ge Hong criticized Confucianism's limitations in knowledge as he believed, "Confucianism lacks exploration of natural phenomena and natural mysteries, but focuses on how to govern society" 不肯長奇怪, 開異途, 務於禮教 (Ge 1988a, p. 219). Further, Confucianism ignored the rational cognition of all things in nature and their motion laws and was often "limited to the knowledge of Confucian classics and the superficial understanding of things" 玩華藻於木末, 而不識所生之有本也 (Ge 1988a, p. 196) but did not acknowledge that "there is an infinite wealth of knowledge beyond Confucian classics" 夫五經所不載者無限矣, 周孔所不言者不少矣 (Ge 1988a, p. 201).
Like Ge Hong, Tao Hongjing also preferred to grasp the knowledge of astronomy, geography, biology, medicine, and technology from an epistemological perspective and was not interested in exploring Confucian classics. While he touched upon them, he did not research them in depth (Zhang 1988, p. 732). Regarding the style of empty debate in metaphysics that was popular among the scholars of the Southern dynasty at that time, he even called it san dan 散誕 (absurd) and tan kong 談空 (empty talk and no action) (Zhang 1988, p. 734). In fact, Tao Hongjing also perceived a demarcation of knowledge between Daoism and Confucianism: he believed that medical knowledge is related to human life, which is more important than Confucian ethics-"The Four Books and Five Classics, the etiquette and costumes of the army and the country, if there are any deviations in the detailed application, it can be understood that the harm is only reflected in the inappropriate deeds. But decoctions and medicines, once a mistake is made, it will involve life, it's so important, why not think carefully and proceed cautiously" 五經四部, 軍國禮 服, 若詳用乖越者, 猶可矣, 止於事蹟非宜爾. 至於湯藥, 一物有謬, 便性命及之. 千乘之君, 百 金之長, 何不深思戒慎邪 (Tang 1986, p. 17).
In the Tao-Ge School tradition, the development of Ge and Tao's epistemological thought can be traced along a continuous line. In the 8th century, Sun Simiao developed the philosophy of the Tao-Ge School to a new stage. He regarded medicine as the most important knowledge and offered the concept that "the top doctors should have extensive knowledge and technology" 大醫習業 (Sun 1955, p. 1). He emphasized that, in addition to medical expertise, they also needed to learn traditional knowledge, including Confucianism, Daoism, Buddhism, astronomy, geography, and divination, to perfect the practice of medicine (Sun 1955, p. 1). In the 9th century, female Taoist Hu Yin 胡愔 wrote medical books such as Huangtingneijingwuzangliufupuxietu 黃庭內景五臟六腑補瀉 圖, Huangtingneijingtu 黃庭內景圖, and Huangtingwaijingtu 黃庭外景圖. She transformed the mysterious cultivation technology of Daoism into practical medical and health care knowledge and actively promoted the progress of ancient Chinese medicine (Wang and Hao 1993, pp. 28-34). This demonstrates that under the influence of the Tao-Ge School, female Taoists could freely acquire knowledge, conduct in-depth research, and write books on medicine. Their remarkable work has influenced the studies in the field to a great extent. In the 10th century, Taoist Wang Huaiyin 王懷隱 followed the methods of Ge and Tao-he carefully collected and organized various traditional medical books, from which he compiled the important medical classic Taipingshenghuifang 太平聖惠方, which comprised various medical prescription books and numerous folk medical books in an omnibus that reflected the philosophy of the Tao-Ge School. In the 11th century, Su Song 蘇頌, who emphasized the social value of medicine, epitomized the thought of the Tao-Ge School. He believed that "medicine is the most important technology to save and help people" 救 恤之惠, 無先醫術, and that "The most important thing to get rid of diseases and ensure health is the medicine of Dao" 驅沴淫, 救昏亂, 保壽命, 躋康寧, 無先於此道, which can "let the people live in a harmonious and clean environment, so that they can live a long and healthy life" 納斯民于壽康, 召和氣於穹壤. 太平之致, 茲有助焉 (Su 1986, p. 699). He was responsible for the compilation of Bencaotujing 本草圖經 and acquired significant medical knowledge and achievements from his predecessors.
Thus, the Tao-Ge School criticized and demarcated the academic ideology of Confucianism, which emphasized humanistic knowledge and ignored natural knowledge, whereas the Daoism tradition of science, technology, and craft knowledge reflected a natural and empirical epistemological orientation, which has been fully affirmed and has persevered.

Positive Action: Demarcation with Daoist Metaphysical Thought
As is well known, Daoist beliefs and practical concepts have a rather complex relationship with Daoist philosophy. Daoism's cosmology, life theory, and epistemological thought are closely related to Lao-Zhuang's. However, the scientific and technological practices of Ge and Tao demonstrated practical characteristics with interests that differed from the metaphysical thought and spiritual temperament of Daoist philosophy.

Advocating the Spirit of Proactive Acquiring of Knowledge
The original meaning of wu wei 無為 (refraining from activity contrary to nature) advocated by Daoism does not refer to passive inaction but conforms to nature. However, the concept of wu wei tends to discourage the active understanding and transformation of nature. The metaphysics of the Wei and Jin dynasties have developed into features of clear, sparse, and empty thinness. In this regard, Ge and Tao criticized the metaphysical speculation at that time and offered positive, promising principles of craftsmanship practice.
Ge Hong advocated that Taoist beginners should spare no effort, dare to take risks, collect and refine medicine in the mountains, and actively use farming, grazing, and business to obtain the funds and wealth needed for alchemy and pharmacy (Ge 1988a, p. 244). Tao Hongjing pointed out that the metaphysical speculative style of the literati and officials in the Southern dynasty was san dan 散誕 (absurd) and tan kong 談空 (empty talk and no action); instead, he emphasized that cultivators should actively acquire "all kinds of astronomy, calendar, arithmetic, and geography, natural history, medical knowledge, and technology" (Zhao 1988, p. 242). Later generations of Daoist alchemists also insisted that "a Daoist alchemist must acquire the knowledge and technology of astronomy, geography, climate, and chemical smelting" 修丹之士, 須要上知天文, 下明地理, 洞達陰陽, 窮通爻象; 並節氣休王, 日時升降, 火候進退, 鼎爐法則 (Meng 1988, p. 220) and that they "must know the motion law of all things and learn lots of technical books" 必須精祥物理, 廣博方書 (Jin 1988, p. 112). They even shouted the slogan, "the true method of actively transform nature" 有為之真法: Laozi taught the Dao De Jing, and Sakyamuni founded Buddhism. They all only taught the truth of showing the true nature of human beings through inaction. Life and death go hand in hand. They only preserve their true nature, keep their minds quiet and empty, and ultimately cannot make their bodies live forever. Competing to make alchemy, so that people can live forever, the material in the body is constantly alive, and become an immortal. This is the true method of actively transform nature. 老子述五千文, 釋氏演三乘教, 只說無為見性之理, 生而有死, 唯全其性, 作虛明之 神, 終不能堅固其形.爭似還丹, 生前不死, 不息內質而獲非飛仙, 乃是有為之真法. (Meng 1988, p. 218) The slogan "the true method of actively transform nature" posed by Daoist alchemists was the opposite of the spirit of inaction and liberation of Lao-Zhuang and Buddhism. It represented the spiritual embodiment of the Tao-Ge School that actively recognized nature.

Affirmation of the Knowledge Inheritance Function of Words
To some extent, both Lao-Zhuang and Buddhism's philosophies ignored the cognitive function of words regarding the external world and specific things. The metaphysics of the Wei and Jin dynasties drew on the cognitive methodology of Lao-Zhuang and Buddhism, proposed that "words are not expressing the meaning of things" 言不意 (B. Wang 1986, p. 544), and tended to despise and deny the cognitive function of words. In this regard, the representatives of the Tao-Ge School raised objections. They insisted that "words" functioned to recognize all things in nature, accumulating and disseminating knowledge, and should not be discarded.
Ge Hong believed that words are indispensable tools for human beings to understand and grasp the actual world. He pointed out that "The bamboo trap can be discarded, but if the fish is not obtained, there cannot be without the trap; words can be discarded, but if Dao has not been practiced, there cannot be without words" 荃可以棄而魚未獲, 則不得無荃; 文可以廢而道未行, 則不可無文 (Ge 1988c, p. 326). Thus, Ge Hong affirmed the cognitive function and value of words. Tao Hongjing was more deeply aware of the function of words for knowledge generation and dissemination. He believed that words were not only the main tool for human beings to describe "knowledge" but also an indispensable carrier for recording, accumulating, and creating "knowledge".
In the Shennong era, there were no words; during the era of the Yellow Emperor, words were invented, so there was the Shennong bencao jing; since then, a large number of knowledge classics and ideological theories have appeared, which are inherited only by words 神農之時, 未有文字, 至於黃帝書記乃興, 於是神農本草, 列為四經.三家之說, 虢有損益, 豈非隨時適變, 殊途同歸者乎. (Tao 1958, p. 6437) It is evident that Ge Hong and Tao's knowledge generation and inheritance functions of words were different from the traditional concepts of Lao-Zhuang and Buddhism in that they actively affirmed and praised the role of words.

Advocating the Spirit of Technology's Craftsmanship and Creativity
Contrary to the belief that metaphysics, Confucianism, Buddhism, and other humanistic traditions despised or even belittled technology, the Tao-Ge School reflected a strong spirit of advocating technology. Ge Hong affirmed and respected the value of shu 術 (technology), and his attitude was very clear. He stressed that in Daoism, whether the goal is to cultivate and become immortals, or to cultivate the body with drugs, in daily life or production activities, we must rely on shu shu 數術 or fang shu 方術 (technology of immortality and alchemy) (Ge 1988a, p. 231), and proposed the slogan "the various alchemy and medicine technologies can promote immortality" 藉眾術之共成長生 (Ge 1988a, p. 216). From an ethical perspective, Ge Hong also represented "technology" (shu 術) positively: "sage is not only the exclusive name of Confucian sages, but also those who grasp all kinds of knowledge and skills can be called sage" 聖者人事之極號, 也不獨于文學 而已矣 (Ge 1988a, p. 215). For example, those who are good at medicine can be called sages of treating diseases 治疾之聖, those who are good at mechanical manufacturing can be described as sages of machinery 機械之聖, those who are good at woodworking technology can be described as sages of woodworking 木工之聖, those who are good at painting can be called painting sages 畫聖, and so on. Ge Hong's view of sages was very different from that of traditional Confucian and metaphysical values, which fully reflected his admiration for the value of technology.
Tao Hongjing affirmed practice activities that promoted human technology. He praised Shennong 神農 and regarded him as the founder of the ancient Chinese technological civilization: Shennong governs the world with Dao. He invented Bagua 八卦 and Yixue 易學 to communicate the relationship between Cosmos-man; he taught the people to engage in farming to eliminate the drawbacks of killing; medicines are used to treat diseases and save the people. These three achievements of Shennong have been passed down for a long time and will be famous for future generations 神農 氏之王天下也, 畫八卦, 以通鬼神之情; 造耕種, 以省殺生之弊; 宣藥療疾, 以拯天傷 之命.此三道者, 曆眾聖而滋彰. (Tang 1986, p. 11) Therefore, in Tao Hongjing's view, Shennong 神農 was the original inventor of farming and medicine; thus, his historical achievements were equivalent to those of the ancient sages Huangdi 黃帝 and Zhuanxu 顓頊. Notably, Tao Hongjing compared the human body during the practice of cultivation to utensils used during pottery casting, a quintessential practical activity of technology. "Cultivation is the most superb smelting technique, reflecting the ultimate principle of material change" 鑄煉之事極, 感變之理通 (Gao 1986, p. 337). In addition, in relation to medical practice, he proposed that "everything in the universe can be used by people" 蓋天地間物, 莫不為天地間用 and believed that "all things in nature can be used to make medicines" (Tang 1986, p. 16). In the Tang and Song dynasties, Daoist alchemists also inherited Ge and Tao's spirit of technology practice. They were self-confident and believed that they were "technicians who have Dao's knowledge" 有道 術士 (Jin 1988, p. 113). With superb craft technology, they could control all things and the law of motion in nature, so as to refine them into elixirs and medicine to save the people.
In summary, the Tao-Ge School demonstrated its independent scientific and technological spiritual characteristics through "knowledge demarcation" with Confucianism, abstruse Dao, and other humanistic thought (i.e., pursuing natural knowledge and emphasizing technological practices). It can be said, against the background of politics, ethics, and humanistic thought in the Chinese tradition, that the Tao-Ge School demonstrated a unique scientific spirit in its pursuit of natural knowledge and technological creation.

Theoretical Thought of the Tao-Ge School
Historically, Ge and Tao recognized the positivity and accuracy of knowledge and pursued its overall principles. Historical data indicate that the representatives of the Tao-Ge School consciously speculated about and practiced the theorization of knowledge and rational cognitive methods. It can be said that rationalized cognition is also one of the main academic characteristics of the Tao-Ge school.
As mentioned, Ge Hong was the first to propose the concept of yu jin wu li 欲盡物理 (understanding the motion laws of all things in the world). Wu li 物理 means the essence of nature and motion laws of all things (li理). In Baopuzi zhili 抱樸子至理, Ge Hong describes the concept of zhi yin 知隱 (know secret knowledge), "people only see the surface of things, but often do not know the secret essence of things" 世人不能知其隱者, 而但見其顯者, but that the Taoists will "recognize the law and essence of things" 為乎無為, 以全天理爾 (Ge 1988a, p. 189). This "yin" means the Dao 道 and li 理 (motion laws of all things) hidden behind the appearance of things. Although Ge Hong attached importance to experience and technology, he also began to pay attention to inquiry and exploration of wu 物 (all things in the world) and li 理 (motion laws of all things). He believed people should not only explore "the law of daily life" 尋常咫尺之近理, 人間取捨之細事 (Ge 1988a, p. 192) but also know the motion laws of all things in the universe: "As long as you grasp the law of the universe, you can grasp the motion laws of all things at the same time." 苟得其要, 則八 極之外, 如在指掌; 百代之遠, 有若同時. 不必在乎庭宇之左右, 俟乎瞻視之所及, 然後知之也 (Ge 1988a, p. 178).
It is evident that Ge Hong thought deeply about the cognitive method and its value. Consequently, he criticized the phenomenon that "people believed in divination, sacrifice, and witchcraft to cure diseases and do not believe in the medical skills of good doctors". 寧 煞生請福, 分蓍問祟, 不肯信良醫之攻病, 反用巫史之紛若 (Ge 1988a, p. 190). Tao Hongjing considered and explored rational cognition and the theorization of in-depth knowledge and even constructed an initial set of rational cognition methodologies.
First, it is necessary to grasp the experience and knowledge fully and accurately. He believed that to "understand and grasp the truth of nature, we must possess more comprehensive and sufficient knowledge; we cannot draw conclusions based on fragmentary and partial knowledge. More importantly, we must test the detailed empirical or experimental knowledge, rather than relying on subjective imagination and arbitrary speculation" 天地 間事理, 乃不可限以胸臆尋之 (Tao 1988a, p. 584).
Second, original knowledge is required to analyze empirical knowledge. Tao Hongjing believed that if we want to form logical and systematic knowledge, we must analyze the relevant knowledge materials by "observing the similarities and differences, distinguishing between right and wrong, seeing inside and outside, and knowing whether there is or not" 察同異, 別是非, 見內外, 知有無 (Gui 1988, p. 677). Thus, we can reasonably classify them on this basis. To this end, Tao Hongjing created a set of scientific classification methodologies: First is that the historical view of classification changes at any time and that new knowledge classification is based on changes in historical times and cognitive levels. Second is the dialectical classification methodology, which posits that classification should follow the principle of combining universality and particularity and analyze each specific matter. Third is the simple and practical view of the classification function, which states that knowledge classification should be based on the principles of practicality and simplicity. On this basis, people can establish a knowledge structure system of "consistent logic and standard knowledge style" 通立定格, 共為成准 (Ge 1988b, p. 2).
Finally, it is necessary to grasp knowledge materials logically to create a theoretical knowledge system. Tao Hongjing used a vivid cognition metaphor to subtly illustrate how to upgrade from perceptual cognition to rational cognition: The construction of the classical theory and knowledge systems can be compared to spinning and weaving. The warp and weft must be evenly distributed, connected and integrated into one; otherwise, the work of trying to theorize knowledge will end up like a failed weaving process, which cannot form true theoretical knowledge 夫經之為言, 徑也.經者, 常也, 通也.謂常通而無滯, 亦猶布 帛之有經矣, 必須銓綜緯緒, 僅乃成功.若機關疏越, 杼軸乖謬, 安能斐然成文. (Tao 1988b, p. 646) Tao Hongjing's metaphor of "weaving cloth" dovetails with Francis Bacon's metaphor of "bees", who represent the "true ancestor of the whole experimental science" in modern times. "Real philosophical work is similar to bees; it is neither wholly nor primarily dependent on the power of the mind, nor everything in memory that has been learned from natural history or experimentation with mechanisms, but is processed and transformed in understanding" (Francis Bacon 1955, p. 126). Marx commented in this regard: "Science is the science of experiments, and the scientific method is to use rational methods to organize perceptual materials. Induction, analysis, comparison, observation, and experiment are rational methods and important conditions" (Marx and Engels 1958, p. 175). Therefore, the scientific epistemological significance of Tao Hongjing's metaphor of "weaving cloth" lies in refining and transforming perceptual-experiential knowledge into a universal and holistic "theory" and knowledge system. It is worth mentioning that Tao Hongjing's cognitive metaphor predates Bacon's "bee" metaphor by more than 1100 years.
Historically, Ge and Tao's rational cognitive methods and philosophy have been inherited and perpetuated over generations of the Tao-Ge School. For example, Sun Simiao's medical research embodied this characteristic of theoretical thought. In his books Beijiqianjinyaofang 備急千金要方 and Qianjinyifang 千金翼方, although medical prescriptions occupy most of the pages, nearly all prescriptions are organized per medical theory. In medical treatment, Sun Simiao also astutely summarized medical theories. He summarized the methods of "zangfu bianzheng fa" 臟腑辨證法, "twelve lessers" 十二少, "twelve mores" 十二多 for health preservation, "classified according to the properties of the drug" 方證同 條, 比類相附, and developed the method of "nourishing the fetus" 養胎 (Ren et al. 2020, pp. 7-10). Although the Tao-Ge School was already declining in the 12th century, Wuwu 吳 (a Taoist of the Outer Elixir 外丹 School in the Southern Song dynasty) still proposed the concepts of "the law of nature" 自然之理: "by understanding the law of nature one can know the essence of things" 悟理得道 (Wu 1988, p. 285). He advocated that people should actively explore the essence of nature-"The universe contains Dao of nature, all things contain the law of nature. If you don't know Dao and the motion laws of all things, you can't know the essence of all things" 天下有自然之道, 萬物有自然之理.不得於理, 物 且不通, 而況道乎; "Why the classics written by FuXi and ShenNong can be spread in the world, because their books contain the ultimate law of the universe. People can understand this truth only by practice" 伏羲神農之書, 既傳於世, 豈固複有隱乎? 惟心契於道, 行合於理 者, 見之自然悟理而得道. Scholars "need to learn all kinds of alchemy, cultivating Daoism knowledge, and skills, then they can obtain the Dao" 悟理而得道, 予當披尋丹經百數十 家, 言雖不同而理實一貫 (Wu 1988, pp. 282-83). Thus, advocacy for rational cognition and the continual development of knowledge and theory is a consistent tradition of the Tao-Ge School.
In summary, the Tao-Ge School contains a valuable spiritual tradition of scientific rational cognition. Scholars of the past were not only satisfied with the inheritance of empirical knowledge and technology but also committed to the overall theoretical construction of knowledge. In the history of ancient Chinese science and technology, the Tao-Ge School embodied the spirit of traditional Chinese rational cognition and was undoubtedly integral to their development.

Conclusions
The research has demonstrated that the Tao-Ge School (4th-12th centuries AD), which was in existence for more than 800 years, first established a scientific research paradigm with a Chinese cultural temperament and was an iconic school of scientific and technological activity in the Middle Ages. The academic concepts and methodological characteristics of the Tao-Ge School can be roughly summarized into three points: (1) the primary goal was to understand and transform the world of nature, reflecting a different epistemological perspective than that of the traditional school of ancient Chinese humanities, (2) it reflected a practical spirit that advocated for and affirmed the value of technology, which was contrary to the perspective held by the traditional humanities school and depreciated its value, and (3) it promoted practical action with rational cognitive thinking and continual theorization of knowledge and was not simply limited to the inheritance of empirical knowledge and practical skills.
Historically, the Tao-Ge School exhibited a complex relationship and ideological tension between religion and technology in ancient China. Unfortunately, the scientific research program initiated by the Tao-Ge School was not embraced by subsequent generations. This may have been due to the extensive gap in the cognitive path between empirical research and theoretical research in Chinese mainstream traditional culture or, perhaps, to barriers in the dissemination of knowledge achievements or interruptions in the inheritance of knowledge. It remains unclear. However, if we systematically analyze existing knowledge of the Tao-Ge School, we can identify the obstacles and factors in the development of ancient Chinese science and technology that have been obscured by the sands of time. Such exploration opens a new prospect for understanding the relationship between religion and science in ancient China.
Author Contributions: Conceptualization, Methodology and writing original draft were implemented by R.Y. and Conceptualization, Writing-review and editing were by X.L. All authors have read and agreed to the published version of the manuscript.