Unity of Kohn-Sham Density Functional Theory and Reduced Density Matrix Functional Theory
UUnity of Kohn-Sham Density Functional Theory and Reduced Density MatrixFunctional Theory
Neil Qiang Su ∗ Department of Chemistry, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education) andRenewable Energy Conversion and Storage Center (RECAST), Nankai University, Tianjin 300071, China
This work presents a theory to unify the two independent theoretical frameworks of Kohn-Sham(KS) density functional theory (DFT) and reduced density matrix functional theory (RDMFT).The generalization of the KS orbitals to hypercomplex number systems leads to the hypercomplexKS (HCKS) theory, which extends the search space for the density in KS-DFT to a space thatis equivalent to natural spin orbitals with fractional occupations in RDMFT. Thereby, HCKS isable to capture the multi-reference nature of strong correlation by dynamically varying fractionaloccupations. Moreover, the potential of HCKS to overcome the fundamental limitations of KS isverified on systems with strong correlation, including atoms of transition metals. As a promisingalternative to the realization of DFT, HCKS opens up new possibilities for the development andapplication of DFT in the future.
Introduction. —Built upon the Hohenberg-Kohn the-orem [1, 2], Kohn-Sham (KS) density functional theory(DFT) [3–5] is a formally exact theoretical frameworktoward the many-electron problem. Due to the favor-able balance between accuracy and efficiency, KS-DFThas won enormous popularity that can manifest itself inthe countless applications across physics, materials sci-ence, chemistry, and biology [6–8]. Nonetheless, the greatsuccess of KS-DFT, along with commonly used densityfunctional approximations (DFAs), is clouded by the im-proper treatment of strong correlation [9]. Strong cor-relation represents the intractable electronic interactionstemming from the multi-reference nature of systems,which has long posed a major challenge to KS-DFT [10–13].It is generally recognized that intrinsic errors associ-ated with commonly used DFAs [9, 14–20] and the slowprogress in systematically eliminating these errors to bet-ter handle strong correlation have severely limited theapplicability of KS-DFT [9, 18, 21, 22]. The enlighteningwork by Lee, Bertels, Small, and Head-Gordon [23] showsthat commonly used DFAs can better treat the strongcorrelation in singlet biradicals by using complex spin-restricted orbitals in KS-DFT. This thus raises a deepquestion: In addition to the errors inherent in existingDFAs, is there still any limitation in the understandingand application of the KS-DFT framework? Apparently,in-depth insights into this question is important for fur-ther development and application of DFT.Beside KS-DFT, reduced density matrix functionaltheory (RDMFT) [24–32] provides an alternative ap-proach to the many-electron problem. The Gilbert’s the-orem [24] guarantees that the one-electron reduced den-sity matrix (1-RDM) instead of the density can be usedas the fundamental variable for the energy functional,which thus makes RDMFT an exact theoretical frame-work independent of KS-DFT. RDMFT has proved itsgreat potential to overcome the fundamental limitationsof KS-DFT through the successful application in predict- ing dissociation energy curves [33–35], and fundamentalgaps for finite systems and extended solids [36] as well asfor Mott insulators [28]. Therefore, establishing a con-nection with the theoretical framework of RDMFT willfurther improve our understanding of KS-DFT and theproblems of existing DFAs.This letter seeks to come up with a theory to unifythe two theoretical frameworks of KS-DFT and RDMFT.This is achieved by generalizing the conventional KS de-terminant to hypercomplex number systems. The result-ing hypercomplex KS (HCKS) theory extends the searchspace for the density in KS-DFT to a space that is equiv-alent to natural spin orbitals with fractional occupationsin RDMFT. The potential of HCKS in capturing thephysical essence of strong correlation is demonstrated onatoms of multi-reference character, including transitionmetals.
T heory. —The total energy E tot [ ρ σ ] in KS-DFT in-cludes the kinetic energy T s [ ρ σ ], the external energy E ext [ ρ σ ], the Coulomb energy E H [ ρ σ ], and the exchange-correlation (XC) energy E XC [ ρ σ ]. They are all function-als of the density that can be formed by the occupiedKS orbitals in the KS determinant ( N σ being the σ -spinelectron number), ρ σ ( r ) = N σ (cid:88) k =1 | ϕ σk ( r ) | , (1)while T s [ ρ σ ] can be explicitly formulated as − (cid:80) α,βσ (cid:80) N σ k =1 (cid:104) ϕ σk |∇ | ϕ σk (cid:105) [3]. Therefore, the min-imization of E tot [ ρ σ ] with respect to ρ σ is equivalent tothe minimization with respect to { ϕ σp } , subject to theorthonormalization condition (cid:104) ϕ σp | ϕ σq (cid:105) = δ pq . (2)In RDMFT, the four terms in the total energy areuniquely determined by 1-RDM ( γ σ ), which are T [ γ σ ], E ext [ γ σ ], E H [ γ σ ] and E XC [ γ σ ] respectively. In terms of a r X i v : . [ phy s i c s . c h e m - ph ] J a n the natural spin orbitals { ψ σp } and the natural occupationnumbers { n σp } , γ σ in the spectral representation reads (cid:80) Kp =1 | ψ σp (cid:105) n σp (cid:104) ψ σp | ( K being the dimension of the basisset), and the diagonal element (cid:104) r | γ σ | r (cid:105) is the density, ρ σ ( r ) = K (cid:88) p =1 n σp | ψ σp ( r ) | . (3) T [ γ σ ] has the form, − (cid:80) α,βσ (cid:80) Kp =1 n σp (cid:104) ψ σp |∇ | ψ σp (cid:105) . Theground-state energy can be obtained by the minimiza-tion of E tot [ γ σ ] with respect to γ σ , or equivalently, withrespect to both { ψ σp } and { n σp } , subject to the orthonor-malization condition, (cid:104) ψ σp | ψ σq (cid:105) = δ pq , and the Pauli ex-clusion principle and the N -representability constraint[37], 0 ≤ n σp ≤ , K (cid:88) p =1 n σp = N σ . (4)Unlike KS-DFT, RDMFT allows dynamically varyingfractional occupations to capture the multi-reference na-ture of strong correlation.The connection between KS-DFT and RDMFT is es-tablished by introducing hypercomplex. The conceptof hypercomplex [38], and the theory of Clifford alge-bra that generalizes real numbers, complex numbers toquanternions, octonions, and other hypercomplex num-bers have important applications in a variety of fields in-cluding theoretical physics [39]. Here, the HCKS orbitalsare formulated as ϕ σp ( r ) = φ σ, p ( r ) + n (cid:88) µ =1 φ σ,µp ( r ) e µ , (5)where { φ σ,µp } are a set of real functions, and { e , e , · · · , e n } are a basis of dimension n in a Cliffordalgebra, such that [39] e µ = − e µ e ν = − e ν e µ . (6)The conjugate hypercomplex of the HCKS orbitals are¯ ϕ σp ( r ) = φ σ, p ( r ) − (cid:80) nµ =1 φ σ,µp ( r ) e µ . Therefore, Eq 5 pro-vides a general set of high dimensional orbitals for theHCKS determinant, while complex KS orbitals are a spe-cial case for n = 1.Without loss of generality, { φ σ,µp } can be expanded ona set of orthonormal functions { χ p } and read φ σ,µp ( r ) = K (cid:88) q =1 χ q ( r ) V σ,µpq . (7)Here V σ,µ is a K × K matrix associated with the µ -thcomponent of the HCKS orbitals. The orthonormaliza-tion condition of Eq 2 becomes [40] n (cid:88) µ =0 V σ,µ V σ,µT = n (cid:88) µ =0 V σ,µT V σ,µ = I K , (8) and (cid:40) V σ,µ V σ,νT = V σ,ν V σ,µT ;V σ,µT V σ,ν = V σ,νT V σ,µ , (9)where the superscript T denotes the transpose, and I K isthe K × K identity matrix. The density correspondingto the HCKS determinant reads [40], ρ σ ( r ) = K (cid:88) p,q =1 χ p ( r ) D σpq χ q ( r ) , (10)and D σ is defined byD σ = n (cid:88) µ =0 V σ,µT I N σ K V σ,µ , (11)where I N σ K is a K × K diagonal matrix, with the first N σ diagonal elements being 1 and the rest being 0. D σ is symmetric and can be diagonalized by an orthogonalmatrix U σ , D σ = U σ Λ σ U σT , (12)where Λ σ is a diagonal matrix, diag( λ σ , λ σ , · · · , λ σK ), and { λ σp } are the eigenvalues of D σ , which satisfy [40]0 ≤ λ σp ≤ K (cid:88) p =1 λ σp = N σ . (13)By inserting Eq 12, the density of Eq 10 can be writtenas ρ σ ( r ) = K (cid:88) p =1 λ σp | χ σp ( r ) | , (14)where χ σp ( r ) = (cid:80) Kq =1 χ q ( r ) U σqp (thereby { χ σk } being or-thonormal). Similarly, the kinetic energy reads T s [ ρ σ ] = − α,β (cid:88) σ K (cid:88) p =1 λ σp (cid:104) χ σp |∇ | χ σp (cid:105) . (15)To further verify that { λ σp } in Eqs 14 and 15, associatedwith any given orthonormal functions { χ σk } , can take anyvalues subject to Eq 13 when K ≤ n +1, here construct aspecial set of HCKS orbitals. Given a set of orthonormal { χ σk } , { φ σ,µp } of Eq 7 are constructed with the followingform φ σ,µp = χ σµ +1 ( r ) V σ,µpµ +1 , (16)i.e. the components corresponding to the same e µ in allthe HCKS orbitals are formed by the same function χ σµ +1 ,thereby V σ,µpq = (cid:40) W σpq , q = µ + 10 , q (cid:54) = µ + 1 , (17)where W σ is a K × K matrix. The orthonormalizationcondition in terms of W σ readsW σ W σT = I K , (18)which thus requires W σ to be an orthogonal matrix. Sim-ilar derivations lead to Eqs 14 and 15 for ρ σ ( r ) and T [ ρ σ ]respectively, with λ σp = (cid:80) N σ k =1 ( W σkp ) . As W σ can be anyorthogonal matrix, { λ σp } can take any values subject toEq 13.Therefore, the extension of KS-DFT to hypercomplexorbitals leads to a density that has the same search spaceas the density of Eq 3 constructed by real orbitals inRDMFT, with the same form of kinetic energies as well.The resulting HCKS theory thus unifies KS-DFT andRDMFT, which extends the search space of KS-DFT tonatural spin orbitals with fractional occupations. When n = 0, the orbitals of Eq 5 are real and HCKS reduces theconventional KS method; when n = 1, the orbitals arecomplex, which are used in the complex, spin-restrictedKS (CRKS) method [23]. Results. —The numerical performance of HCKS andits potential in describing strong correlation are evalu-ated with the finite basis simulation. Atoms of multi-reference character, including transition metals, are cal-culated. While two XC functionals, PBE [41] and BLYP[42, 43], are examined here, similar results can be ob-tained with other commonly used functionals. All calcu-lations were performed using a local modified version ofthe NWChem package [44].The density for the lowest singlet state of C atom isfirst examined. The singlet-state C atom is of biradicalnature, with two electrons in the 2p sub-shell. Normally,the spin-restricted KS (RKS) would have one doubly oc-cupied p orbital and destroy the degeneracy of the porbitals. Fig 1a shows the density of RKS when p z isoccupied, which maintains the spin symmetry but losesthe space symmetry. Different from RKS, CRKS leads totwo half filled p orbitals for the singlet C atom [23, 40].Fig 1b shows that the CRKS density with both p x and p y half filled maintains the space symmetry along the z axis,but it cannot guarantee the apace symmetry along both xand y axes. In contrast, HCKS, or equivalently, RDMFT,automatically converges to the spin-restricted solution,with each p orbital 1/3 filled. Therefore, the density byHCKS can maintain both spin and space symmetry; seeFig 1c. Similar results can be obtained for other systemsof multi-reference character such as singlet states of O,S, and Si atoms, and even the d and f block transitionmetals.In addition, the energies of C atom under varying ex-ternal charged environments are tested. Fig 2 shows thatthe triplet-singlet energy gaps by KS seriously deviatefrom the results of CCSDT, especially when the pointcharges are far away from the C atom. By compari-son, HCKS significantly improves the performance of KS, x y (a)(b)(c) zx y z FIG. 1. Density on real-space grids for the lowest singlet stateof C atom. Views along the z, y and x axes are providedrespectively. (a) RKS density (purple), (b) CRKS density(green), and (c) HCKS (or equivalently, RDMFT) density(blue) are plotted at the isosurface value of 0.2 au. The XCfunctional PBE is applied for all the calculation. The basisset used is aug-cc-pVQZ [45, 46]. in use of the same XC functionals. The occupations ofthe three p orbitals change gradually from { , , } to { / , / , / } as the point charges move away from theC atom. Therefore, HCKS is able to capture the physi-cal essence of multi-reference nature through dynamicallyvarying occupations.Here test also some atoms of 3d transition metals,which are generally considered to be a big challengeto KS. Due to the partially filled 3d sub-shell and thenear degeneracy of 4s and 3d sub-shells, these atoms orsystems containing them often have a plethora of low-lying degenerate and near-degenerate states, which makethem much more difficult to correctly describe than main-group compounds. Fig 3 shows that KS with both PBEand BLYP XC functionals overestimates the energy gapsbetween high and low spin-states of Cr, Fe and Ni, dueto the lack of strong correlation in the low spin-states.In contrast, HCKS improves the performance in use ofthe same XC functionals. The fractional occupationsfor (nearly) degenerate orbitals render HCKS great po-tential for handling strong correlation. For example,PBE@HCKS converges to a set of spin-restricted orbitalsfor the singlet state of Ni, with occupations for both α and β spins being 0.916 for each 3d orbital and 0.420 for4s orbital. This further proves that HCKS can improvethe description of strong correlation while maintaining C CCC CC PBE@KS PBE@HCKS BLYP@KS BLYP@HCKS Ref
FIG. 2. Triplet-singlet energy gaps of C atom under varyingexternal charged environments. The energies are calculatedwith two point charges of 0.3 au each placed equidistantlyon two sides of the C atom, with the distance between thetwo point charges ranging from 4 to 12 ˚A. KS and HCKS(or equivalently, RDMFT) in use of the same XC functionals(PBE and BLYP) are examined, while CCSDT [47] resultsare used as reference. The basis set used is aug-cc-pVQZ. Allenergies are in kcal/mol. both spin and space symmetry.
Conclusions. —This letter presented a theory to unifythe two theoretical frameworks of KS-DFT and RDMFT.This is achieved by HCKS that generalizes the KS or-bitals to hypercomplex number systems. HCKS extendsthe search space for the density in KS-DFT to a spacethat is equivalent to natural spin orbitals with fractionaloccupations in RDMFT. The test on the singlet birad-ical C atom shows that HCKS can maintain both spinand space symmetry for the density with equally frac-tionally occupied p orbitals, which cannot be achievedby RKS. Besides, the calculation of energy gaps betweenhigh and low spin states, for both C atom under varyingexternal charged environments and atoms of transitionmetals, demonstrates that HCKS is able to capture themulti-reference nature of strong correlation by dynam-ically varying fractional occupations, while KS in useof the same XC functionals cannot. Therefore, HCKSshows great potential to overcome the fundamental lim-itations of KS, which thus provides an alternative to therealization of DFT, and opens up new channels for thedevelopment and evaluation of approximate functionals.The Supplemental Material for this work is availableon line [40]. Support from the National Natural Science ∆E( D- F) ∆E( G- D) ∆E( G- S) Cr Fe NiPBE@KS PBE@HCKS
BLYP@KS BLYP@HCKS
Ref
FIG. 3. Energy gaps between high and low spin-states of tran-sition metals. Septet-singlet, quintet-singlet, triplet-singletenergy gaps for Cr, Fe and Ni atoms respectively are calcu-lated. KS and HCKS (or equivalently, RDMFT) in use of thesame XC functionals (PBE and BLYP) are examined, whilethe experimental data [48] are used as reference. The basisset used is def2-TZVP [49]. All energies are in kcal/mol.
Foundation of China (Grant 22073049), the Natural Sci-ence Foundation of Tianjin City (20JCQNJC01760), andFundamental Research Funds for the Central Universities(Nankai University: No. 63206008) is appreciated. Ded-icated to the 100th anniversary of Chemistry at NankaiUniversity. ∗ [email protected][1] P. Hohenberg and W. Kohn, Phys. Rev. , B864(1964).[2] M. Levy, Proc. Natl. Acad. Sci. USA , 6062 (1979).[3] W. Kohn and L. J. Sham, Phys. Rev. , A1133 (1965).[4] R. G. Parr and W. Yang, Density-Functional Theoryof Atoms and Molecules , Oxford University Press: NewYork, 1989.[5] R. Dreizler and E. Gross,
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Unity of Kohn-Sham Density Functional Theory and Reduced Density MatrixFunctional Theory
Neil Qiang Su
Department of Chemistry, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education) and RenewableEnergy Conversion and Storage Center (RECAST), Nankai University, Tianjin 300071, China
Email: [email protected]
CONTENTS
References 4I. Reduced Density Matrix Functional Theory 6II. Kohn-Sham Density Functional Theory 7III. Hypercomplex numbers 8IV. Hypercomplex Kohn-Sham Theory 8
I. REDUCED DENSITY MATRIX FUNCTIONAL THEORY
In RDMFT, the electronic ground-state energy can be written as a functional of the one-body reduced densitymatrix (1-RDM) γ σ [2, 24] E DMFT [ γ σ ] = T [ γ σ ] + E ext [ γ σ ] + E H [ γ σ ] + E XC [ γ σ ] , (S1)where the kinetic energy is T [ γ σ ] = − α,β (cid:88) σ (cid:90) ∇ r γ σ ( r , r (cid:48) ) | r (cid:48) = r d r , (S2)the external energy is E ext [ γ σ ] = α,β (cid:88) σ (cid:90) v ext ( r ) γ σ ( r , r (cid:48) ) | r (cid:48) = r d r , (S3)the Coulomb energy is E H [ γ σ ] = 12 α,β (cid:88) σσ (cid:48) (cid:90) γ σ ( r , r (cid:48) ) γ σ (cid:48) ( r , r (cid:48) ) | r (cid:48) = r r (cid:48) = r r d r d r , (S4)and the exchange-correlation (XC) energy E XC [ γ σ ] is unknown. The total 1-RDM is γ ( r , r (cid:48) ) = γ α ( r , r (cid:48) ) + γ β ( r , r (cid:48) ) = N α,β (cid:88) σ , ··· ,σ N (cid:90) d r · · · d r N Ψ ∗ ( r (cid:48) σ , x , · · · , x N )Ψ( r σ , x , · · · , x N ) , (S5)where x p = r p σ p are combined spatial and spin coordinates. Diagonalization of 1-RDM γ σ generates the natural spinorbitals, { ψ σp } , and their occupation numbers, { n σp } , thus γ σ = K (cid:88) p =1 | ψ σp (cid:105) n σp (cid:104) ψ σp | ; γ σ ( r , r (cid:48) ) = (cid:104) r | γ σ | r (cid:48) (cid:105) , (S6)where its diagonal element is the density, ρ σ ( r ) = (cid:104) r | γ σ | r (cid:105) = K (cid:88) p =1 n σp | ψ σp ( r ) | . (S7)The ground-state energy can be obtained by minimizing Eq. S1 with respect to 1-RDM, or equivalently, by minimizingEq. S1 with respect to both { ψ σp } and { n σp } , subject to (cid:104) ψ σp | ψ σq (cid:105) = δ pq , ≤ n σp ≤ , K (cid:88) p =1 n σp = N σ . (S8)Here, σ is the spin index, it can be α or β ; N σ is the electron number of σ spin; K is the dimension of basicfunctions, which should be not smaller than N σ . II. KOHN-SHAM DENSITY FUNCTIONAL THEORY
In KS-DFT [1, 3], the total energy is E KS [ ρ σ ] = T s [ ρ σ ] + E ext [ ρ σ ] + E H [ ρ σ ] + E XC [ ρ σ ] , (S9)where the kinetic energy is T s [ ρ σ ] = − α,β (cid:88) σ (cid:90) ∇ r γ sσ ( r , r (cid:48) ) | r (cid:48) = r d r = − α,β (cid:88) σ N σ (cid:88) k =1 (cid:104) ϕ σk |∇ | ϕ σk (cid:105) , (S10)the external energy is E ext [ ρ σ ] = α,β (cid:88) σ (cid:90) v ext ( r ) ρ σ ( r ) d r , (S11)the Coulomb energy is E H [ ρ σ ] = 12 α,β (cid:88) σσ (cid:48) (cid:90) ρ σ ( r ) ρ σ (cid:48) ( r ) r d r d r , (S12)while the form of E XC [ ρ σ ] is unknown. γ sσ is 1-RDM of the noninteracting reference system, which can be established by occupied orbitals, γ sσ = N σ (cid:88) k =1 | ψ σk (cid:105)(cid:104) ψ σk | ; γ sσ ( r , r (cid:48) ) = (cid:104) r | γ sσ | r (cid:48) (cid:105) , (S13)and the density is ρ σ ( r ) = (cid:104) r | γ sσ | r (cid:105) = N σ (cid:88) k =1 | ϕ σk ( r ) | . (S14)Here { ϕ σp } are the KS orbitals, which are a set of orthonormal orbitals that span the space of dimension K , (cid:104) ϕ σp | ϕ σq (cid:105) = δ pq , (S15)thereby 1 K = K (cid:88) p =1 | ϕ σp (cid:105)(cid:104) ϕ σp | . (S16)Here 1 K represents the identity operator of the space. III. HYPERCOMPLEX NUMBERS
Hypercomplex [38] and the theory of Clifford algebra [39] that generalizes real numbers, complex numbers toquanternions, octonions, and other hypercomplex numbers are introduced here.Clifford algebra is a unital associative algebra generated by a vector space with a quadratic form, which hasimportant applications in a variety of fields including theoretical physics. In a Clifford algebra, given a basis ofdimension n , { e , e , · · · , e n } , such that [39] e µ = − e µ e ν = − e ν e µ , (S17)further imposing closure under multiplication generates a multivector space spanned by a basis of the 2 n distinctproducts of { e , e , · · · , e n } , i.e. { , e , e , · · · , e e , · · · , e e e , · · · } . These distinct products form the basis of ahypercomplex number system. Unlike { e , e , · · · , e n } , the remaining elements in this basis need not anti-commute,depending on how many simple exchanges must be carried out for the swap.Here we consider the following hypercomplex number C = n (cid:88) µ =0 c µ e µ = c · n (cid:88) µ =1 c µ e µ , (S18)where e = 1, and the coefficients { c µ } are real. The corresponding conjugate hypercomplex number of Eq. S18 is¯ C = c · − n (cid:88) µ =1 c µ e µ . (S19) IV. HYPERCOMPLEX KOHN-SHAM THEORY
Now, we generalize the KS orbitals to hypercomplex number systems, ϕ σp ( r ) = n (cid:88) µ =0 φ σ,µp ( r ) e µ . (S20)The density of Eq. S14 becomes ρ σ ( r ) = N σ (cid:88) k =1 n (cid:88) µ =0 [ φ σ,µk ( r )] , (S21)and the kinetic energy of Eq. S10 becomes T s [ ρ σ ] = − α,β (cid:88) σ N σ (cid:88) k =1 n (cid:88) µ =0 (cid:104) φ σ,µk |∇ | φ σ,µk (cid:105) . (S22)Without loss of generality, we expand { φ σ,µp } on a set of orthonormal functions { χ p } , i.e. φ σ,µp ( r ) = K (cid:88) q =1 χ q ( r ) V σ,µpq . (S23)Here { V σ,µ } are a set of K × K matrices. Eq. S15 becomes n (cid:88) µ =0 V σ,µ V σ,µT = I K , (S24)and V σ,µ V σ,νT = V σ,ν V σ,µT . (S25)Eq. S16 leads to (cid:104) χ p | χ q (cid:105) = (cid:80) Kr =1 (cid:104) χ p | ϕ σr (cid:105)(cid:104) ϕ σr | χ q (cid:105) = δ pq , such that n (cid:88) µ =0 V σ,µT V σ,µ = I K , (S26)and V σ,µT V σ,ν = V σ,νT V σ,µ . (S27)I K denotes the K × K identity matrix.Here we define two K × K symmetric matrices D σ and E σ ,D σ = n (cid:88) µ =0 V σ,µT I N σ K V σ,µ ; E σ = n (cid:88) µ =0 V σ,µT I K V σ,µT = n (cid:88) µ =0 V σ,µT V σ,µT = I K . (S28)I N σ K is a K × K diagonal matrix, with the first N σ diagonal elements being 1 and the rest being 0. D σ can bediagonalized by an orthogonal matrix U σ , thus D σ = U σ Λ σ U σT , (S29)where Λ σ is a diagonal matrix with diagonal elements { λ σp } being the eigenvalues of D σ . The positive semi-definitenessof D σ guarantees that all the eigenvalues are not negative, with the maximal eigenvalue smaller than that of E σ .Thereby, 0 ≤ λ σp ≤ K (cid:88) p =1 λ σp = Tr (D σ ) = Tr (cid:32) n (cid:88) µ =0 V σ,µT I N σ K V σ,µ (cid:33) = Tr (cid:32) I N σ K n (cid:88) µ =0 V σ,µ V σ,µT (cid:33) = N σ . (S30)The density and kinetic energy can be formulated as ρ σ ( r ) = K (cid:88) p,q =1 χ p ( r ) D σpq χ q ( r ) = K (cid:88) p =1 λ σp | χ σp ( r ) | , (S31)and T [ ρ σ ] = − α,β (cid:88) σ K (cid:88) p,q =1 D σpq (cid:104) χ p |∇ | χ q (cid:105) = − α,β (cid:88) σ K (cid:88) p =1 λ σp (cid:104) χ σp |∇ | χ σp (cid:105) , (S32)where χ σp ( r ) = (cid:80) Kq =1 χ q ( r ) U σqp .For K ≤ n + 1, to show that { λ σp } can be any values that satisfy Eq. S30, here we construct a special set of orbitals { ϕ σp } on any given set of orthonormal functions { χ σp } , which take the following form ϕ σp ( r ) = n (cid:88) µ =0 φ σ,µp e µ = K − (cid:88) µ =0 χ σµ +1 V σ,µpµ +1 e µ , (S33)i.e. the components corresponding to the same e µ in all the orbitals { ϕ σp } are formed by only the function χ σµ +1 ,thereby V σ,µpq = (cid:40) W σpq , q = µ + 10 , q (cid:54) = µ + 1 , (S34)where W σ is a K × K matrix. Now the conditions of Eqs. S24-S27 becomeW σ W σT = I K , (S35) W k,µ +1 W l,ν +1 δ µν = W k,ν +1 W l,µ +1 δ µν , (S36)0[W σT W σ ] µ +1 ,µ +1 = 1 , (S37)[W σT W σ ] µ +1 ,ν +1 = [W σT W σ ] ν +1 ,µ +1 . (S38)Eqs. S35-S38 can be satisfied once W σ is an orthogonal matrix. Now, the density and the kinetic energy becomes ρ σ ( r ) = K (cid:88) p =1 N σ (cid:88) k =1 ( W σkp ) | χ σp ( r ) | , (S39)and T [ ρ σ ] = − α,β (cid:88) σ K (cid:88) p =1 N σ (cid:88) k =1 ( W σkp ) (cid:104) χ σp |∇ | χ σp (cid:105) . (S40)Hence λ σp = (cid:80) N σ k =1 ( W σkp ) . Because W σ can be any unitary matrix, { λ σp } can be any values subject to0 ≤ λ σp ≤ K (cid:88) p =1 λ σp = N σ . (S41)When n = 0, the orbitals of Eq. S20 are real and the hypercomplex KS (HCKS) reduces the conventional KSmethod; when n = 1, the orbitals of Eq. S20 are complex, which are used in the complex, spin-restricted KS (CRKS)method [23]. Take the singlet state of C atom as an example to compare KS, CRKS and HCKS. The three methodsall lead to fully occupied 1s and 2s sub-shells that are lower in energy than the 2p sub-shell. For the 2p sub-shell,the spin-restricted KS (RKS) has one doubly occupied p orbital; CRKS leads to two half filled p orbitals as Eq. S20with nn