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Piecuch and Kowalski [8]. An interesting attempt to interaction (FCI) space created by the entire set of spin- address the existence of a cluster operator and cluster orbitals B = {χ1, ..., χk, ..., χK }. We here assume the expansion in the open-shell case was done by Jeziorski spin-orbitals to be eigenfunctions of the system’s Fock op- and Paldus [9]. The first advances within the rigor- erator. Note that the following analysis can be applied ous realm of local functional analysis were performed by for any single-particle operator fulfilling the properties R.S. and Rohwedder, providing analyses of the closed- used in Ref. [16] – not only the Fock operator. shell CC method for non multi-configuration systems [10– Based on the single reference approach, the TCC 12]. Since then, the local mathematical analysis of CC method needs a large CAS to cover most of the static schemes was extended by A.L. and S.K. analyzing the ex- correlations. As the size of the CAS scales exponentially, tended CC method [13] and revisiting the bivariational an efficient approximation scheme for strongly correlated approach [14, 15], and F.M.F. et al. providing the first systems is indispensable for the TCC method to have local mathematical analysis of a multi-reference scheme, practical significance. One of the most efficient schemes namely the CC method tailored by tensor network states for static correlation is the DMRG method [21]. Going (TNS-TCC) [16]. As this mathematical branch of CC back to the physicists S. R. White and R. L. Martin [22], theory is very young, channeling abstract mathemati- it was introduced to as an alternative cal results into the quantum-chemistry community is a to the CI or CC approach. However, the major disad- highly interdisciplinary task merging both fields. A first vantage of the DMRG is that in order to compute dy- attempt in this direction was done by A.L. and F.M.F. namical correlation high bond dimensions (tensor ranks) linking the physical assumption of a HOMO-LUMO gap may be necessary, making the DMRG a potentially costly to the somewhat abstract G˚arding inequality and in that method [2, 21]. Nevertheless, as a TNS-TCC method, context presenting key aspects of Refs. [10–13] from a the DMRG-TCC approach is an efficient method since more quantum chemical point of view [17]. the CAS is supposed to cover the statically correlated With this article, we aim to bridge the mathemati- orbital of the system. This avoids the DMRG method’s cal results in Ref. [16] of the TNS-TCC method into the weak point and allows to employ a much larger CAS com- quantum-chemistry community. Furthermore, we derive pared to the traditional CI-TCC method. chemical properties of the TCC method and extend these A notable advantage of the TCC approach over the results with a numerical study on the complete active conventional MRCC methods is that excitation operators space (CAS) dependence of the DMRG-TCCSD error. commute. This is due to the fact that the Hartee–Fock method yields a single reference solution |ΨHFi, which implies that separating the cluster operator corresponds II. THE DMRG-TCC METHOD to a partition of excitation operators. Hence, Sˆ and Tˆ commute. This makes the DMRG-TCC method’s analy- As a post-Hartree–Fock method, the TCC approach sis much more accessible than conventional MRCC meth- was introduced by Kinoshita et al. in Ref. [18] as an ods and therewith facilitates establishing sound mathe- alternative to the expensive and knotty conventional matical results [16]. Note also that the dynamical corre- multi-reference methods. It divides the cluster opera- lation via the CCSD method distinguishes the DMRG- tor into a complete active space (CAS) part, denoted Sˆ, TCC method from standard CAS-SCF methods, which and an external (ext) part Tˆ, i.e., the wave function is suffer from an imbalance in the considered correlation parametrized as making these methods inapplicable for chemical phenom- ena like surface hopping. Consequently, due to its more |ΨTCCi = exp(Tˆ) exp(Sˆ)|ΨHFi . balanced correlation the DMRG-TCC approach is much better suited for these phenomena, which widens the Separating the cluster operator into several parts goes horizon of possible applications. We remark, however, back to Piecuch et al. [19, 20]. In this formulation the that the computationally most demanding step of the linked CC equations are given by DMRG-TCC calculation is the DMRG part, and its cost increases rapidly with k. Thus the above application re- (TCC) −Sˆ −Tˆ ˆ Tˆ Sˆ E = hΨHF|e e He e |ΨHFi , quires further investigations. Alternative to the dynam- (1) ˆ ˆ ˆ ˆ DMRG-MRCI ( 0= hΨ |e−Se−T Heˆ T eS|Ψ i . ical correction via the CC approach, the µ HF method in Ref. [23] utilizes an internally contracted CI Sˆ algorithm different from a conventional CI calculation. Computing |ΨCASi = e |ΨHFi first and keeping it fixed for the dynamical correction via the CCSD method re- stricts the above equations to |Ψµi not in the CAS, i.e., III. CHEMICAL PROPERTIES OF THE hΨ|Ψµi = 0 for all |Ψi in the CAS (we say that |Ψµi is in the L2-orthogonal complement of the CAS). We DMRG-TCC AB emphasize that this includes mixed states, e.g., |ΨIJ i A B where |ΨI i is an element of the CAS but |ΨJ i is not. It is desired that quantum-chemical computations pos- We consider a CAS created by spin-orbitals BCAS = sess certain features representing the system’s chemical {χ1, ..., χk}, forming a subspace of the full configuration and physical behavior. Despite their similarity, the CC 3 and TCC method have essentially different properties, The corresponding energies are given by which are here elaborated. A basic property of the CC (A) ¯ˆ (A) (B) ¯ˆ (B) method is the equivalence of linked and unlinked CC EA = hΨHF |HA|ΨHF i , EB = hΨHF |HB |ΨHF i , equations. We point out that this equivalence is in gen- eral not true for the DMRG-TCCSD scheme. This is and the amplitudes fulfill a consequence of the CAS ansatz since it yields mixed (A) ¯ˆ (A) (B) ¯ˆ (B) states, i.e., two particle excitations with one excitation 0= hΨµ |HA|ΨHF i , 0= hΨµ |HB|ΨHF i , into the CAS. The respective overlap integrals in the un- linked CC equations will then not vanish unless the single in terms of the effective, similarity-transformed Hamilto- excitation amplitudes are equal to zero. Generalizing this nians result for rank complete truncations of order n we find ˆ that all excitation amplitudes need to be zero but for the H¯A = exp(−SˆA − TˆA)HˆA exp(SˆA + TˆA) , n-th one. This is somewhat surprising as the equivalence H¯ˆ = exp(−Sˆ − Tˆ )Hˆ exp(Sˆ + Tˆ ) . of linked and unlinked CC equations holds for rank com- B B B B B B plete truncations of the single-reference CC method. The Hamiltonian of the compound system of the non- For the sake of simplicity we show this results for the interacting subsystems can be written as HˆAB = HˆA + DMRG-TCCSD method. The general case can be proven Hˆ . Since the TCC approach corresponds to a partition- in similar a fashion. We define the matrix representa- B ing of the cluster amplitude we note that H¯ˆ = H¯ˆ +H¯ˆ tion T with elements T = hΨ |eTˆ|Ψ i for µ,ν∈ / CAS. AB A B µ,ν µ ν for Note that, as T increases the excitation rank, T is an atomic lower triangular matrix and therefore not singu- ˆ H¯AB = exp(−SˆA − SˆB − TˆA − TˆB) lar. Assuming that the linked CC equations hold, the ˆ ˆ ˆ ˆ ˆ non-singularity of T yields × HAB exp(SA + TB + TA + TB) .

−Tˆ Tˆ Aµ := Tµ,ν hΨν |e Heˆ |ΨHFi (AB) (A) (B) With |ΨHF i = |ΨHF i ∧ |ΨHF i, the energy of the com- ν∈ /XCAS pound systems can be written as Tˆ −Tˆ Tˆ = hΨµ|e |Ψν ihΨν |e Heˆ |ΨHFi =0 . E = hΨ(AB)|H¯ˆ |Ψ(AB)i ν∈ /XCAS AB HF AB HF (A) (B) ¯ˆ ¯ˆ (A) (B) = hΨHF | ∧ hΨHF | HA + HB |ΨHF i ∧ |ΨHF i As the full projection manifold is complete under de- (A) ˆ (A) (B) ˆ (B) = hΨ |H¯A|Ψ i+ hΨ |H¯B|Ψ i = EA + EB . excitation, we obtain that HF HF HF HF

Tˆ Tˆ It remains to show that Aµ = hΨµ|Heˆ |ΨHFi− E0hΨµ|e |ΨHFi Tˆ ˆ Tˆ (AB) ˆ ˆ ˆ ˆ (AB) − hΨµ|e |Ψγ ihΨγ|He |ΨHFi . (2) |ΨDMRG−TCCi = exp(SA + SB + TA + TB)|ΨHF i γ∈CAS X (AB) Note that the first two terms on the r.h.s. in Eq. (2) solves the Schr¨odinger equation, i.e., for all hΨµ | holds (AB) ¯ˆ (AB) describe the unlinked CC equations. We analyze the last hΨµ |HAB|ΨHF i = 0. Splitting the argument into term on the r.h.s. in Eq. (2) by expanding the inner three case, we note that products, i.e., (A) (B) ¯ˆ (AB) (A) ¯ˆ (A) Tˆ hΨµ Ψ(HF)|HAB|ΨHF i = hΨµ |HA|ΨHF i =0 , hΨµ|e |Ψγi = hΨµ|Ψγi + hΨµ|Tˆ|Ψγi (A) (B) ¯ˆ (AB) (B) ¯ˆ (B) 1 2 hΨ(HF)Ψµ |HAB|ΨHF i = hΨµ |HB|ΨHF i =0 , + hΨµ|Tˆ |Ψγ i + ... . 2 (A) (B) ¯ˆ (AB) hΨµ Ψµ |HAB|Ψ i =0 , The first term in this expansion vanishes due to orthogo- HF ˆ nality. The same holds true for all terms where T enters where hΨ(A)Ψ(B)| = hΨ(A)| ∧ hΨ(B)|. This proves the en- to the power of two or higher. However, as the external ˆ ergy size extensivity for the untruncated TCC method. space contains mixed states, we find that hΨµ|T |Ψγi is From this we conclude the energy size extensivity for the not necessarily zero, namely, for hΨµ| = hΨα| ∧ hΨβ| and DMRG-TCCSD scheme, because the truncation only af- |Ψ i = |Ψ i with α ∈ ext and β ∈ CAS. This proves the γ β fects the product states hΨ(A)Ψ(B)| and these are zero in claim. µ µ the above projection. Subsequently, we elaborate the size extensivity of the Looking at TCC energy expression we observe that DMRG-TCCSD. Let two DMRG-TCCSD wave functions due to the Slater–Condon rules, these equations are in- for the individual subsystems A and B be dependent of CAS excitations higher than order three, (A) ˆ ˆ (A) ˆ |ΨDMRG−TCCi = exp(SA) exp(TA)|ΨHF i , i.e., amplitudes of Sn for n > 3. More precisely, due to the fact that in the TCCSD case external space ampli- (B) ˆ ˆ (B) |ΨDMRG−TCCi = exp(SB) exp(TB)|ΨHF i . tudes can at most contain one virtual orbital in the CAS, 4 the TCCSD amplitude expressions become independent quantity, which is in agreement with the chemical def- of Sˆ4, i.e., inition of correlation concepts [25]. We identify pairs

′ ′ ′ ′ ′ of spin-orbitals contributing to a high mutual informa- a a ˆ b c d e hφij |H|ψklmn i =0 , tion value as strongly correlated, the pairs contribut- ing to the plateau region as non-dynamically correlated where the primed variables a′,b′,c′, d′,e′ describe orbitals and the pairs contributing to the mutual information in the CAS, the non-primed variable a describes an or- tail as dynamically correlated (see Fig. 3). The mutual- bital in the external part and i,j,k,l,m,n are occupied information profile can be well approximated from a prior orbitals. Note, this does not imply that we can restrict DMRG computation on the full system. Due to the size the CAS computation to a manifold characterizing exci- of the full system we only compute a DMRG solution of tations with rank less or equal to three as for strongly cor- low bond dimension (also called tensor rank). These low- related systems these can still be relevant. However, it re- accuracy calculations, however, already provide a good duces the number of terms entering the DMRG-TCCSD qualitative entropy profile, i.e., the shapes of profiles ob- energy computations significantly. tained for low bond dimension, M, agree well with the the In contrast to the originally introduced CI-TCCSD ones obtained in the FCI limit. Here, we refer to Fig. 2 method taking only Sˆn for n =1, 2 into account [18], the and Fig. 3 showing the single orbital entropy and mutual additional consideration of Sˆ3 corresponds to an exact information profiles, respectively, for various M values treatment of the CAS contributions to the energy. We and for three different geometries of the N2 . emphasize that the additional terms do not change the The orbitals with large entropies can be identified from methods complexity. This is due to the fact that includ- the low-M calculations providing a black-box tool to form ing the CAS triple excitation amplitudes will not exceed the CAS space including the strongly correlated orbitals. the dominating complexities of the CCSD [24] nor of the A central observation is that for BCAS = {χ1, ..., χN } DMRG. However, the extraction of the CI-triples from (i.e. k = N), the DMRG-TCCSD becomes the CCSD the DMRG wave function is costly and a corresponding method and for BCAS = {χ1, ..., χK } (i.e. k = K), it is efficiency investigation is left for future work. the DMRG method. We recall that the CCSD method can not resolve static correlation and the DMRG method needs high tensor ranks for dynamically correlated sys- IV. ANALYSIS OF THE DMRG-TCC tems. This suggests that the error obtains a minimum for some k with N ≤ k ≤ K, i.e., there exists an optimal In the sequel we discuss and elaborate mathematical choice of k determining the basis splitting and therewith properties of the TCC approach and their influence on the choice of the CAS. Note that this feature becomes im- the DMRG-TCC method. The presentation here is held portant for large systems since high bond dimensions be- brief and the interested reader is referred to Ref. [16] and come simply impossible to compute with available meth- the references therein for further mathematical details. ods.

A. The Complete Active Space Choice B. Local Analysis of the DMRG-TCC Method

As pointed out in the previous section, the TCC The CC method can be formulated as non-linear method relies on a well-chosen CAS, i.e., a large enough Galerkin scheme [10], which is a well-established frame- CAS that covers the system’s static correlation. Conse- work in numerical analysis to convert the continuous quently, we require a quantitative measurement for the Schr¨odinger equation to a discrete problem. For the quality of the CAS, which presents the first obstacle for DMRG-TCC method a first local analysis was performed creating a non-empirical model since the chemical con- in Ref. [16]. There, a quantitative error estimate with cept of correlation is not well-defined [25]. In the DMRG- respect to the basis truncation was established. F.M.F. TCC method, we use a quantum information theory ap- et al. showed under certain assumptions (Assumption proach to classify the spin-orbital correlation. This clas- A and B in the sequel) that the DMRG-TCC method sification is based on the mutual-information possesses a locally unique and quasi-optimal solution (cf. Sec. 4.1 in Ref. [16]). In case of the DMRG-TCC method Ii|j = S(ρ{i})+ S(ρ{j}) − S(ρ{i,j}) . the latter means: For a fixed basis set the CC solution tailored by a DMRG solution on a fixed CAS is up to a This two particle entropy is defined via the von Neu- multiplicative constant the best possible solution in the mann entropy S(ρ)= −Tr(ρ ln ρ) of the reduced density approximation space defined by the basis set. In other operators ρ{X} [26]. Note that the mutual-information words, the CC method provides the best possible dynam- describes two-particle correlations, for a more general ical correction for a given CAS solution such as a DMRG connection between multi-particle correlations and ξ- solution. correlations we refer the reader to Szalay et. al [26]. Note that local uniqueness ensures that for a fixed ba- We emphasize that in practice this is a basis dependent sis set, the computed DMRG-TCC solution is unique in a 5 neighborhood around the exact solution. We emphasize with respect to the error of the wave function or clus- that this result is derived under the assumption that the ter amplitudes. Note that the TCC approach represents CAS solution is fixed. Consequently, for different CAS merely a partition of the cluster operator, however, its solutions we obtain in general different TCC solutions, error analysis is more delicate than the traditional CC i.e., different cluster amplitudes. method’s analysis. The TCC-energy error is measured Subsequently, parts of the results in Ref. [16] are ex- as difference to the FCI energy. Let |Ψ∗i describe the plained in a setting adapted to the FCI solution on the whole space, i.e., Hˆ |Ψ∗i = E|Ψ∗i. perspective. The TCC function is given by f(t; s) = Using the exponential parametrization and the above in- −Sˆ −Tˆ Tˆ Sˆ hΨµ|e e Heˆ e |ΨHFi, for |Ψµi not in the CAS. Note troduced separation of the cluster operator, we have that we use the convention where small letters s,t corre- ∗ ˆ∗ ˆ∗ spond to cluster amplitudes, whereas capital letters S,ˆ Tˆ |Ψ i = exp(T ) exp(S )|ΨHFi . (3) describe cluster operators. The corresponding TCC en- An important observation is that the TCC approach ig- ergy expression is given by nores the coupling from the external space into the CAS. It follows that the FCI solution on the CAS |ΨCASi = −Sˆ −Tˆ Tˆ Sˆ FCI E(t; s)= hΨHF|e e Heˆ e |ΨHFi . exp(SˆFCI)|ΨHFi is an approximation to the projection of |Ψ∗i onto the CAS Consequently, the linked TCC equations (1) then become CAS ˆ ∗ ˆ∗ |ΨFCI i≈ P |Ψ i = exp(S )|ΨHFi , E(TCC) = E(t; s) , where Pˆ = |Ψ ihΨ | + |Ψ ihΨ | is the L2- ( 0= f(t; s) . HF HF µ∈CAS µ µ orthogonal projection onto the CAS. For a reasonably P CAS Within this framework the locally unique and quasi- sized CAS the FCI solution |ΨFCI i is rarely computa- optimal solutions of the TCC method were obtained tionally accessible and we introduce the DMRG solution CAS under two assumptions (see Assumption A and B in on the CAS as an approximation of |ΨFCI i, Ref. [16]). |ΨCAS i = exp(Sˆ )|Ψ i ≈ |ΨCASi . First, Assumption A requires that the Fock operator DMRG DMRG HF FCI Fˆ is bounded and satisfies a so-called G˚arding inequality. Tailoring the CC method with these different CAS so- For a more detailed description of these properties in this lutions leads in general to different TCC solutions. In context we refer the reader to Ref. [17]. Second, it is as- CAS the case of |ΨFCI i, the TCC method yields the best sumed that there exists a CAS-ext gap in the spectrum possible solution with respect to the chosen CAS, i.e., of the Fock operator, i.e., there is a gap between the k- ∗ f(tCC; sFCI) = 0. This solution is in general differ- th and the k + 1-st orbital energies. Note, that spectral ent from tCC fulfilling f(tCC; sDMRG) = 0 and its trun- gap assumptions (cf. HOMO-LUMO gap) are standard cated version tCCSD satisfying PGalf(tCCSD; sDMRG) = 0, 2 in the analysis of dynamically correlated systems. To be where PGal denotes the l -orthogonal projection onto applicable to multi-configuration systems, the analysis in the corresponding Galerkin space. In the context of Ref. [16] rests on a CAS chosen such that the k-th and the DMRG-TCC theory, the Galerkin space represents the (k+1)-st spin orbital are non-degenerate. Intuitively, a truncation in the excitation rank of the cluster opera- this gap assumption means that the CAS captures the tor, e.g., DMRG-TCCD, DMRG-TCCSD, etc. static correlation of the system. For the following argument, suppose that an appropri- Assumption B is concerned with the fluctuation opera- ate CAS has been fixed. The total DMRG-TCC energy tor Wˆ = Hˆ − Fˆ. This operator describes the degeneracy error ∆E can be estimated as [16] of the system since it is the difference of the Hamiltonian ∗ ∗ and a single particle operator. Using the similarity trans- ∆E = |E(tCCSD; sDMRG) −E(t ; s )| ˆ ∗ (4) formed W and fixing the CAS amplitudes s, the map ≤ ∆ε + ∆εCAS + ∆εCAS ,

−Tˆ −Sˆ Sˆ Tˆ where each term of the r.h.s. in Eq. (4) is now discussed. t 7→ e e Weˆ e |ΨHFi As a technical remark, the norms on either the Hilbert is assumed to have a small enough Lipschitz-continuity space of cluster amplitudes or wave functions are here constant (see Eq. (20) in Ref. [16]). The physical inter- simply denoted k·k. These norms are not just the l2- pretation of this Lipschitz condition is at the moment or L2-norm, respectively, but also measure the kinetic unclear. energy. It should be clear from context which Hilbert space is in question and we refer to Ref. [11] for formal definitions. The first term is defined as C. Error Estimates for the DMRG-TCC Method ∆ε = |E(tCCSD; sDMRG) −E(tCC; sDMRG)| , A major difference between the CI and CC method is which describes the truncation error of the CCSD method CAS that the CC formalism is not variational. Hence, it is not tailored by |ΨDMRGi. We emphasize that the dynami- evident that the CC energy error decays quadratically cal corrections via the CCSD and the untruncated CC 6 method are here tailored by the same CAS solution. Note that the above error is caused by the assumed basis Hence, the energy error ∆ε corresponds to a single ref- splitting, namely, the correlation from the external part erence CC energy error, which suggests an analysis sim- into the CAS is ignored. Therefore, the best possible ∗ ilar to Refs. [10, 12]. Indeed, the Aubin–Nitsche duality solution for a given basis splitting (tCC,sFCI) differs in method [27–29] yields a quadratic a priori error estimate general from the FCI solution (t∗,s∗). in ktCCSD − tCCk (and in terms of the Lagrange mulitpli- Combining now the three quadratic bounds gives an ers, see Theorem 29 in Ref. [16]). overall quadratic a priori energy error estimate for the Second, we discuss the term DMRG-TCC method. The interested reader is referred to Ref. [16] for a more detailed treatment of the above ∆εCAS = |E(tCC; sDMRG) −E(tCC; sFCI)| . analysis.

Here, different CAS solutions with fixed external solu- tions are used to compute the energies. This suggests D. On the k-dependency of the Error Estimates that ∆εCAS is connected with the error The error estimate outlined above is for a fixed CAS, −SˆDMRG ˆ ˆ ˆ SˆDMRG ∆EDMRG = |hΨHF|e P HPe i.e., a particular basis splitting, and bounds the energy (5) −SˆFCI SˆFCI error in terms of truncated amplitudes. Because the TCC − e PˆHˆ Peˆ |ΨHFi| , solution depends strongly on the choice of the CAS, it is describing the approximation error of the DMRG solu- motivated to further investigate the k-dependence of the tion on the CAS (see Lemma 27 in Ref. [16]). Indeed, error ∆E. However, the above derived error bound has a highly complicated k dependence since not only the am- ∗ 2 plitudes but also the implicit constants (in .) and norms ∆εCAS . ∆EDMRG + ktCC − tCCk depend on k. Therefore, the analysis in Ref. [16] is not + k(Sˆ − Sˆ )|Ψ ik2 DMRG FCI HF (6) directly applicable to take the full k dependence into ac- ∗ 2 count. + εµ(t ) , CC µ In the limit where s → s we obtain that |µ|=1 DMRG FCI X ∗ tCC → tCC since the TCC method is numerically sta- A1...An n ble, i.e., a small perturbation in s corresponds to a small with εµ = ε = (λA − λI ), for 1 ≤ n ≤ k, I1...In j=1 j j perturbation in the solution t. Furthermore, if we assume where λi are the orbital energies. The εµ are the (trans- P that tCCSD ≈ tCC, which is reasonable for the equilibrium lated) Fock energies, more precisely, Fˆ|Ψ i = (Λ + µ 0 bond length of N2, the error can be bound as N εµ)|Ψµi, with Λ0 = i=1 λi. Note that the wave func- CAS ˆ ∗ 2 tion |ΨFCI i is in general not an eigenfunction of H, how- 2 tCCSD t P ∆Ek ≤ Ck (tCCSD)µ + − ∗ . ever, it is an eigenfunction of the projected Hamiltonian sDMRG s 2 |µ|=1     l PˆHˆ Pˆ. Eq. (5) involves the exponential parametrization.  X  (10) This can be estimated by the energy error of the DMRG wave function, denoted ∆EDMRG, namely, Here the subscript k on ∆Ek and Ck highlights the k- dependence. We remark that we here used the less ac- ∆EDMRG ≤ 2∆EDMRG (7) curate l2 structure on the amplitude space compared to ˆ CAS CAS 2 1 + kHk k |ΨDMRGi − |ΨFCI i kL . the H structure in Eq. (9). This yields k-independent vectors In Sec. V the energy error of the DMRG wave func- ∗ tion is controlled by the threshold value δεTr, i.e., tCCSD t , ∗ , ∆E (δε ). Hence, for well chosen CAS the differ- sDMRG s DMRG Tr     CAS CAS 2 ence k |ΨDMRGi − |ΨFCI i kL is sufficiently small such 2 that ∆EDMRG . 2∆EDMRG holds. This again shows the as well as an k-independent l norm. The k-depenence importance of a well-chosen CAS. Furthermore, the last of Ck will be investigated numerically in more detail in term in Eq. (6) can be eliminated via orbital rotations, Sec. V B 5. as it is a sum of single excitation amplitudes. Finally, we consider V. THE SPLITTING ERROR FOR N2 ∗ ∗ ∗ ∆εCAS = |E(tCC; sFCI) −E(t ; s )| . (8) Including the k dependence in the above performed ∗ ∗ Since (t ,s ) is a stationary point of E we have error analysis explicitly is a highly non-trivial task in- ∗ ∗ DE(t ; s ) = 0. A calculation involving Taylor expanding volving many mathematical obstacles and is part of our ∗ ∗ E around (t ,s ) (see Lemma 26 in Ref. [16]) yields current research. Therefore, we here extend the mathe-

∗ ∗ 2 ∗ 2 matical results from Sec. IV with a numerical investiga- ∆εCAS . ktCC − t k + ksFCI − s kl2 . (9) tion on this k dependence. Our study is presented for the 7

N2 molecule using the cc-pVDZ basis, which is a common tree, i.e., a one dimensional topology. Thus permuta- basis for benchmark computations developed by Dunning tions of orbitals along such an artificial chain effect the and coworkers [30]. We investigate three different geome- convergence for a given CAS choice [37, 38]. This orbital- tries of the Nitrogen dimer by stretching the molecule, ordering optimization can be carried out based on spec- thus the performance of DMRG-TCCSD method is as- tral graph theory [39, 40] by minimizing the entanglement 2 sessed against DMRG and single reference CC methods distance [41], defined as Idist = ij Ii|j |i − j| . In order for bond lengths r = 2.118 a0, 2.700 a0, and 3.600 a0. In to speed up the convergence of the DMRG procedure the the equilibrium geometry the system is weakly corre- configuration interaction based dynamicallyP extended ac- lated implying that single reference CC methods yield tive space (CI-DEAS) method is applied [32, 38]. In the reliable results. For increasing bond length r the system course of these optimization steps, the single orbital en- shows multi-reference character, i.e., static correlations tropy (Si = S(ρ{i})) and the two-orbital mutual infor- become more dominant. For r> 3.5 a0 this results in the mation (Ii|j ) are calculated iteratively until convergence variational breakdown of single reference CC methods is reached. The size of the active space is systematically [31]. This breakdown can be overcome with the DMRG- increased by including orbitals with the largest single site TCCSD method once a large and well chosen CAS is entropy values, which at the same time correspond to or- formed, we therefore refer to the DMRG-TCCSD method bitals contributing to the largest matrix elements of the as numerically stable with respect to the bond length mutual information. Thus, the decreasingly ordered val- along the potential energy surface (PES). ues of Si define the so-called CAS vector, CASvec, which As mentioned before, the DMRG method is in gen- provides a guide in what order to extend the CAS by eral less efficient to recover dynamic correlations since including additional orbitals. The bond dimensions M it requires large computational resources. However, due (tensor rank) in the DMRG method can be kept fixed to the specific CAS choice the computational resource or adapted dynamically (Dynamic Block State Selection for the DMRG part of the TCC scheme is expected to (DBSS) approach) in order to fulfill an a priori defined be significantly lower than a pure DMRG calculation for error margin [42, 43]. Accurate extrapolation to the trun- the same level of accuracy. cation free limit is possible as a function of the truncation error δεTr [42, 44]. In our DMRG implementation [45] we use a spatial A. Computational Details orbital basis, i.e., the local tensor space of a single or- bital is d = 4 dimensional. In this C4 representation an orbital can be empty, singly occupied with either a spin In practice, a routine application of the TCC method up or spin down electron, or doubly occupied with op- to strongly correlated molecular systems, i.e., to multi- posite spins. Note, in contrast to Sec. IV we need N/2 reference problems, became possible only recently since spatial orbitals to describe an N-electron wave function it requires a very accurate solution in a large CAS includ- and similar changes apply to the size of the basis set so ing all static correlations. Tensor network state methods that we use K ≡ K/2 from here on. The single orbital fulfill such a high accuracy criterion, but the efficiency entropy therefore varies between 0 and ln d = ln 4, while of the TNS-TCCSD method strongly depends on various the two-orbital mutual information varies between 0 and parameters of the involved algorithms. Some of these ln d2 = ln 16. are defined rigorously while others are more heuristic Next we provide a short description how to perform from the mathematical point of view. In this section we DMRG-TCCSD calculations in practice. Note that we present the optimization steps for the most important leave the discussion on the optimal choice of k for the parameters of the DMRG-TCCSD method and outline following sections. how the numerical error study in Sec. V B is performed. 1) First the CAS is formed from the full orbital space As elaborated in Sec. II and IVA, the CAS choice is es- by setting k = K. DMRG calculations are performed it- sential for the computational success of TNS-TCC meth- eratively with fixed low bond dimension (or with a large ods. In addition, the error of the TNS method used to error margin) in order to determine the optimal ordering approximate the CAS part depends on various approxi- and the CAS-vector as described above. Thus, the corre- mations. These include the proper choice of a finite di- sponding single-orbital entropy and mutual information mensional basis to describe the chemical compound, the are also calculated. These calculations already provide a tensor network structure, and the mapping of the molec- good qualitative description of the entropy profiles with ular orbitals onto the given network [32]. Fortunately, respect to the exact solution, i.e., strongly correlated or- all these can be optimized by utilizing concepts of quan- bitals can be identified. tum information theory, introduced in Sec. IV A (see also 2) Using a given N/2

CAS CAS the given CAS and perform a large-scale DMRG calcu- e.g., |ΨDMRGi or |ΨFCI i. Note that calculating all CI- lation with a low error threshold margin in order to get coefficients scales exponentially with the size of the CAS. CAS an accurate approximation of the |ΨFCI i. Note, that the However, since the system is dynamically correlated ze- DMRG method yields a normalized wave function, i.e., roth order, single and double excitation coefficients are CAS CAS 2 the overlap with the reference determinant |ΨHFi is not sufficient. Hence the error terms || |ΨDMRGi−|ΨFCI i||L necessarily equal to one. ˆ ˆ and ||(SFCI −SDMRG(δεTr))|ΨHFi|| in Eq. (6) and Eq. (7), 3) Using the matrix product state representation of respectively, can be well approximated. We remark that FCI |ΨDMRGi obtained by the DMRG method we determine this exponential scaling with the CAS size also effects the zero reference overlap, single and double CI coeffi- the computational costs of the CAS CI-triples, which cients of the full tensor representation of the wave func- are needed for an exact treatment of the TCCSD en- ˆ ˆ tion. Next, these are used to calculate the S1 and S2 am- ergy equation. However, investigations of the influence plitudes, which form the input of the forthcoming CCSD of the CAS CI-triples on the computed energies are left calculation. for future work. 4) Next, the cluster amplitudes for the external part, i.e., Tˆ1 and Tˆ2, are calculated in the course of the DMRG- TCCSD scheme. B. Results and Discussion 5) As we discus in the next section, finding the optimal CAS, i.e., k-splitting, is a highly non-trivial problem, and In this section, we investigate the overall error depen- at the present stage we can only present a solution that dence of DMRG-TCCSD as a function of k and as a is considered as a heuristic approach in terms of rigorous function of the DMRG-truncation error δεTr. For our mathematics. In practice, we repeat steps 2-4 for a large numerical error study we perform steps 1-4 discussed in DMRG-truncation error as a function of N/2

-3 4 10 10 10 1 2.118 M = 64 M = 256 0 M -4 10 = 512 10 M = 10000 j

| M = 64 i

3 I 10 10 -1 M = 256 -5 M = 512

10 ) M = 10000 M rel -2 E 10 M = 64 ∆ M = 256 -6 max( 10 M = 512 2 10 10 -3 M = 10000 Sorted values of 10 -7 2.118 10 -4 2.700 3.600 10 -8 10 1 10 -8 10 -6 10 -4 10 15 20 25 0 200 400 600 800 δεTr k

FIG. 3. Mutual information for r = 2.118 a0 (blue), 2.700 a0 FIG. 1. (a) Relative error of the ground-state energy as a (green), 3.600 a0 (red) obtained for the full orbital space (k = function of the DMRG-truncation error on a logarithmic scale 28) with DMRG for fixed M = 64, 256, 512 and for δεTr = obtained for the full orbital space (k = K) with r = 2.118 a0. −8 10 , Mmax = 10000. (b) Maximum number of block states as a function of k for −8 the a priori defined truncation error δεTr = 10 with r = 2.118 a (blue), 2.700 a (green), and 3.600 a (red). 0 0 0 2. Numerical Investigation of the Error’s k-dependence

In order to obtain |Ψ∗i in the FCI limit, we perform 1.2 −8 M = 64 high-accuracy DMRG calculations with δεTr = 10 . M = 256 The CAS was formed by including all Hartee–Fock or- 1 r = 3.600 M = 512 M = 10000 bitals and its size was increased systematically by in-

i M = 64

S cluding orbitals with the largest entropies according to 0.8 M = 256 M = 512 the CAS vector. Orbitals with degenerate single orbital M = 10000 entropies, due to symmetry considerations, are added 0.6 M = 64 M = 256 to the CAS at the same time. Thus there are some M = 512 missing k points in the following figures. For each re- 0.4 r = 2.700 M = 10000

Sorted values of stricted CAS we carry our the usual optimization steps of a DMRG scheme as discussed in Sec. V A, with low 0.2 r = 2.118 bond dimension followed by a high-accuracy calculation −8 with δεTr = 10 using eight sweeps [32]. Our DMRG 0 0 5 10 15 20 25 30 ground-state energies for 7

FIG. 2. Single orbital entropy for r = 2.118 a0 (blue), 2.700 a0 where k = N/2 = 7), and CCSDTQ reference energies, (green), 3.600 a0 (red) obtained for the full orbital space (k = are shown in Fig. 4 near the equilibrium bond length, 28) with DMRG for fixed M = 64, 256, 512 and for δεTr = r =2.118 a0. The single-reference calcu- −8 10 , Mmax = 10000. lations were performed in NWChem [48], we employed the cc-pVDZ basis set in the spherical representation. For k = K = 28 the CCSDTQPH energy was taken as a reference for the FCI energy [46]. tribution of static correlations for the stretched geome- The DMRG energy starts from the Hartree–Fock en- tries. Similarly the first 50-100 matrix elements of Ii|j ergy for k = 7 and decreases monotonically with increas- also take larger values for larger r while the exponen- ing k until the full orbital solution with k = 28 is reached. tial tail, corresponding to dynamic correlations, is less It is remarkable, however, that the DMRG-TCCSD en- effected. The gap between large and small values of the ergy is significantly below the CCSD energy for all CAS orbital entropies gets larger and its position shifts right- choices, even for a very small k = 9. The error, how- ward for larger r. Thus, for the stretched geometries ever, shows an irregular behavior taking small values for more orbitals must be included in the CAS during the several different k-s. This is due to the fact that the TCC scheme in order to determine the static correlations DMRG-TCCSD approach suffers from a methodological accurately. We remark here, that the orbitals contribut- error, i.e., certain fraction of the correlations are lost, ing to the high values of the single orbital entropy and since the CAS is frozen in the CCSD correction. This mutual information matrix elements change for the differ- supports the hypothesis of a k-dependent constant as dis- ent geometries according to chemical bond forming and cussed in Sec. IVD. Therefore, whether orbital k is part breaking processes [47]. of the CAS or external part provides a different method- 10

0.014 importance signaled by the increase in the single orbital entropy in Fig. 2. Thus the multi-reference character 0.012 of the wave function becomes apparent through the en- 1e-4 1e-5 0.01 tropy profiles. According to Fig. 5 the DMRG-TCCSD r = 2.118 1e-6

] 1e-7 energy for all k > 7 values is again below the CCSD h 1e-8 E 0.008 [ computation and for k> 15 it is even below the CCSDT 1e-5, CAS↑ GS CCSD E reference energy. For r = 3.600 a0 the CC computation 0.006 ∆ CCSDT CCSDTQ fluctuates with increasing excitation ranks and CCSDT 0.004 is even far below the FCI reference energy, revealing the variational breakdown of the single-reference CC method 0.002 for multi-reference problems. In contrast to this, the

0 DMRG-TCCSD energy is again below the CCSD energy 10 15 20 25 for all k > 7, but above the the CCSDT energy. The er- k ror furthermore shows a local minimum around k = 19.

0.035 FIG. 4. Ground-state energy of the N2 molecule near the equilibrium geometry, r = 2.118 a0, obtained with DMRG- TCCSD for 7 ≤ k ≤ 28 and for various DMRG truncation 0.03 errors δεTr. The CCSD, CCSDT and CCSDTQ reference en- 1e-4 0.025 1e-5 ergies are shown by dotted, dashed and dashed-dotted lines, r = 2.700 1e-6 −8 respectively. The DMRG energy with δεTr = 10 on the full ] 1e-7 h 0.02 1e-8 E space, i.e., k = 28, is taken as a reference for the FCI energy. [ 1e-5, CAS↑

−5 GS CCSD For δεTr = 10 the CAS was additionally formed by taking E 0.015

∆ CCSDT k orbitals according to increasing values of the single-orbital CCSDTQ entropy values, i.e., inverse to the other CAS extensions. This 0.01 is labeled by CAS↑. 0.005

0 ological error. This is clearly seen as the error increases 10 15 20 25 k between k = 10 and k = 15 although the CAS covers more of the system’s static correlation with increasing k. FIG. 5. Ground-state energy of the N molecule with This is investigated in more detail in Sec. V B 4. 2 bond length r = 2.7 a0, obtained with DMRG-TCCSD for Since several k-splits lead to small DMRG-TCCSD er- 7 ≤ k ≤ 28 and for various DMRG truncation errors δεTr. rors, the optimal k value from the computational point of The CCSD, CCSDT and CCSDTQ reference energies are view, is determined not only by the error minimum but shown by dotted, dashed and dashed-dotted lines, respec- −8 also by the minimal computational time, i.e., we need to tively. The DMRG energy with δεTr = 10 on the full space, take the computational requirements of the DMRG into i.e., k = 28, is taken as a reference for the FCI energy. For −5 account. Note that the size of the DMRG block states δεTr = 10 the CAS was additionally formed by taking k contributes significantly to the computational cost of the orbitals according to increasing values of the single-orbital DMRG calculation. The connection of the block size to entropy, i.e., inverse to the other CAS extensions. This is labeled by CAS↑. the CAS choice is shown in Fig. 1 (b), where the maxi- mal number of DMRG block states is depicted as a func- tion of k for the a priori defined truncation error margin For the stretched geometries static correlations are more −8 δεTr = 10 . Note that max(M) increases rapidly for pronounced, there are more orbitals with large entropies, 10

0.08 CAS split since ψDMRG differs only marginally from ψHF. 0.07 1e-4 0.06 1e-5 1e-6 4. Numerical Investigation on CAS-ext correlations 0.05 1e-7 1e-8 r = 3.600

] ↑

h 0.04 1e-5, CAS E

[ CCSD Taking another look at Fig. 2, we can confirm that al-

GS 0.03 CCSDT

E CCSDTQ ready for small k values the most important orbitals, i.e., ∆ 0.02 those with the largest entropies, are included in the CAS. 0.01 In Fig. 7 the sorted values of the mutual information ob-

0 10 0 0.3 -0.01 k < 28 k < 28 k = 28 -0.02 k = 28 10 15 20 25 0.25 k j | i r = 2.118 r = 2.118 I FIG. 6. Ground-state energy of the N2 molecule with 0.2 10 -2 bond length r = 3.6 a0, obtained with DMRG-TCCSD for 7 ≤ k ≤ 28 and for various DMRG truncation errors δεTr. 0.15 The CCSD, CCSDT and CCSDTQ reference energies are shown by dotted, dashed and dashed-dotted lines, respec- −8 tively. The DMRG energy with δεTr = 10 on the full space, 0.1 Sorted values of i.e., k = 28, is taken as a reference for the FCI energy. For 10 -4 −5 δεTr = 10 the CAS was additionally formed by taking k 0.05 orbitals according to increasing values of the single-orbital

k k k k entropy, i.e., inverse to the other CAS extensions. This is = 9 = 19 = 23 = 28 labeled by CAS↑. 0 0 500 1000 0 20 40

FIG. 7. (a) Sorted values of the mutual information obtained 6. For small k the DMRG solution basically provides the by DMRG(k) for 9 ≤ k ≤ 28 on a semi-logarithmic scale for Full-CI limit since the a priori set minimum number of N2 at r = 2.118 a0. (b) Sorted 40 largest matrix elements of ≤ ≤ block states Mmin ≃ 64 already leads to a very low trun- the mutual information obtained by DMRG(k) for 9 k 28 cation error. Therefore, the error of the DMRG-TCCSD on a lin-lin scale for N2 at r = 2.118 a0 is dominated by the methodological error. For k> 15 the effect of the DMRG truncation error becomes visible and tained by DMRG(k) for 9 ≤ k ≤ 28 is shown on a semi- for large k the overall error is basically determined by the logarithmic scale. It is apparent from the figure that the −4 −5 DMRG solution. For larger δεTr between 10 and 10 largest values of Ii|j change only slightly with increasing the DMRG-TCCSD error shows a minimum with respect k, thus static correlations are basically included for all re- to k. This is exactly the expected trend, since the CCSD stricted CAS. The exponential tail of Ii|j corresponding method fails to capture static correlation while DMRG to dynamic correlations, however, becomes more visible requires large bond dimension to recover dynamic corre- only for larger k values. We conclude, for a given k split lations, i.e., a low truncation error threshold. In addition, the DMRG method computes the static correlations ef- the error minima for different truncation error thresholds ficiently and the missing tail of the mutual information δεTr happen to be around the same k values. This has with respect to the full orbital space (k = 28) calculation an important practical consequence: the optimal k-split is captured by the TCC scheme. can be determined by performing cheap DMRG-TCCSD Correlations between the CAS and external parts can calculations using large DMRG truncation error thresh- also be simulated by a DMRG calculation on the full old as a function of k. orbital space using an orbital ordering according to the The figures furthermore indicate that ∆EGS has a high CAS-vector. In this case, the DMRG left block can be peak for 9 7, i.e, the correlations between the the equilibrium geometry is shown in Fig. 4 labeled by CAS and the external part vanish with increasing k. In CAS↑. As expected, the improvement of DMRG-TCCSD contrast to this, when an ordering according to CAS↑ is is marginal compared to CCSD up to a very large k ≃ 23 used the correlation between CAS and external part re- 12

mains always strong, i.e., some of the highly correlated to valid index-pairs) of sDMRG(k,δεTr ). orbitals are distributed among the CAS and the exter- In Fig. 8 (b) we show the error e(k,δεTr ) as a function of nal part. Nevertheless, both curves are smooth and they k of the Nitrogen dimer near the equilibrium bond length. cannot explain the error profile shown in Fig. 4. Note that the quantitative behavior is quite robust with respect to the bond dimension since the values only dif- fer marginally. We emphasize that the error contribution 5. Numerical Values for the Amplitude Error Analysis in Fig. 8 is dominated by second term in Eq. (11) since this is an order of magnitude larger than the contribution Since correlation analysis based on the entropy func- from the first and third terms in Eq. (11), respectively. tions cannot reveal the error profile shown in Fig. 4, The first term in Eq. (11) is furthermore related to the here we reinvestagte the error behavior as a function usual T1 diagnostic in CC [49], so it is not a surprise −3 of N/2 ≤ k ≤ K but in terms of the CC amlitudes. that a small value, ∼ 10 , was found. Comparing this Therefore, we also present a more detailed description of error profile to the one shown in Fig. 4 we can under- Eq. (10) in Sec. IV which includes the following terms: stand the irregular behavior and the peak in the error in ∆EGS between k = 9 and 17, and the other peaks 2 for k > 17 but the error minimum found for k = 19 re- e(k,δεTr )= tCCSD(k,δεTr ) µ µ: mains unexplained. Furthermore, we can conclude from |µX|=1  Fig. 8 (b) that the quotient ∆EGS(k)/e(k,δεTr ) is not ∗ 2 constant. This indicates that the constants involved in + t − t (k,δ Tr ) (11) k CCSD ε µ Sec. IV in particular the constant in Eq. (10) in Sec. IV D µ: |µX|=1,2 h  is indeed k dependent. ∗ 2 + sk − sDMRG(k,δεTr ) µ . 0.8 0.07 i 1e-4  1e-5 Here the valid index-pairs are µ = (i, a), with i = 0.7 0.06 n a 1e-6 (i1,...,in) ∈ {1,...,N/2} , and = (a1,...,an) ∈ 1e-7 n 0.6 1e-8 {N/2+1,...,K} . The excitation rank is given by 0.05 |µ| = n where n = 1 stands for singles, n = 2 for dou- 0.5

) r = 2.118 ) bles, and so on. The µ-s are the labels of excitation )

k 0.04 a † a1,...,an an a1 Tr ε

operatorsτ ˆ :=a ˆ aˆi, andτ ˆ :=τ ˆ ... τˆ . The δ i a i1,...,in in i1 0.4 r = 2.118 a1,...,an CAS( k, ̺ corresponding amplitudes are given as t . For in- ( i1,...,in ( e 0.03 valid index-pairs, i.e., index-pairs that are out of range, S 0.3 the amplitudes are always zero. The various amplitudes 0.02 appering in Eq. (11) are calculated according to the fol- 0.2 lowing rules: ∗ 0.1 0.01 1) The sk: amplitudes in the CAS(k) obtained by 1e-5, CAS ∗ −8 1e-5, CAS↑ DMRG(δεTr = 10 ) solution (represented by CI coef- ∗ 0 0 ficients c ) for CAS(K), 10 15 20 25 10 15 20 25 k k ∗a ∗ a ci (sk)i = ∗ , c FIG. 8. (a) Block entropy, S(ρ k ) as a function of k for 0 (12) CAS( ) c∗a1,a2 c∗a1 c∗a2 − c∗a2 c∗a1 r = 2.118 ordering orbitals along the DMRG chain according (s∗)a1,a2 = i1,i2 − i1 i2 i1 i2 to the same CAS and CAS↑ vectors as used in Fig. 4. (b) k i1,i2 c∗ c∗2 0 0 e(k, δεTr ) as a function of k of the Nitrogen dimer near the equilibrium bond length for DMRG truncation error thresh- where i,i ,i ∈ {1,...,N/2} and a,a ,a ∈ {N/2 + −4 −8 1 2 1 2 olds δεTr between 10 and 10 . 1,...,k}. ∗ 2) The tk: amplitudes not in the CAS(k) obtained from ∗ −8 the DMRG(δεTr = 10 ) solution (represented by CI co- efficients c∗) for CAS(K) projected onto CAS(k), i.e., the ∗ VI. CONCLUSION complement (with respect to valid index-pairs) of sk. 3) The sDMRG(k,δεTr ) amplitudes in the CAS(k) are obtained by the DMRG(δεTr ) solution (represented In this article we presented a fundamental study of the by CI coefficients c) for CAS(k). The amplitudes DMRG-TCCSD method. We showed that, unlike the tra- a a1,a2 sDMRG(k,δεTr )i ,sDMRG(k,δεTr )i1,i2 are the same as ditional single-reference CC method, the linked and un- ∗ Eq. (12), but with c → c, where i,i1,i2 ∈{1,...,N/2} linked DMRG-TCC equations are in general not equiv- and a,a1,a2 ∈{N/2+1,...,k}. alent. Furthermore, we showed energy size extensivity

4) The tCCSD(k,δεTr ): amplitudes not in the CAS(k) of the TCC, DMRG-TCC and DMRG-TCCSD method obtained by TCCSD, i.e., the complement (with respect and gave a proof that CAS excitations higher than order 13 three do not enter the TCC energy expression. method only if the treatment of large CAS becomes nec- In addition to these computational properties of the essary. DMRG-TCCSD method we presented the mathemati- In addition, the DMRG-TCCSD method avoids the cal error analysis performed in Ref. [16] from a quan- numerical breakdown of the CC approach even for multi- tum chemistry perspective. We showed local uniqueness reference (strongly correlated) systems and, using con- and quasi optimality of DMRG-TCC solutions, and high- cepts of quantum information theory, allows an efficient lighted the importance of the CAS-ext gap – a spectral black-box implementation. The numerical investigations gap assumption allowing to perform the analysis pre- showed also that the constants involved in the error es- sented here. Furthermore, we presented a quadratic a timation are most likely k dependent. This stresses the priori error estimate for the DMRG-TCC method, which importance of further mathematical work to include this aligns the error behavior of the DMRG-TCC method dependence explicitly in the analysis. with variational methods except for the upper bound con- Since the numerical error study showed a significant dition. We emphasize that the DMRG-TCC solution de- improvement for small CAS, we suspect the DMRG- pends strongly on the CAS choice. Throughout the anal- TCCSD method to be of great use for larger systems ysis we neglected this dependence as we assumed an op- with many strongly correlated orbitals as well as a many timal CAS choice as indicated in Sec. IV A. The explicit dynamically correlated orbitals [1, 2]. The oscillatory consideration of this dependence in the performed error behavior of the error, however, remains unexplained at analysis carries many mathematical challenges, which are this point. Despite the unknown reason of this behavior, part of our current research. Therefore, we extended this we note that the error minima are fairly robust with re- work with a numerical study of the k-dependence of the spect to the bond dimension. Hence, the DMRG-TCCSD DMRG-TCCSD error. method can be extended with a screening process using We presented computational data of the single-site en- low bond-dimension approximations to detect possible tropy and the mutual information that are used to choose error minima. In addition, our analysis is basis depen- the CAS. Our computations showed that these properties dent. Thus there is a need for further investigations based are qualitatively very robust, i.e., their qualitative behav- on, for example, fermionic mode transformation [50], and ior is well represent by means of a low-rank approxima- investigations of the influence of the CAS CI-triples on tion, which is a computational benefit. The numerical the computed energies. All these tasks, however, remain investigation of the k-dependence of the DMRG-TCCSD for future work. error revealed that the predicted trend in Sec. IVA is correct. We can clearly see that the error indeed first decays (for 7 ≤ k ≤ 9) and then increases again (for ACKNOWLEDGEMENTS 25 ≤ k ≤ 28) for low-rank approximations, i.e., 1e-4 respectively 1e-5. This aligns with the theoretical pre- diction based on the properties of the DMRG and single This work has received funding from the Research reference CC method. Additional to this general trend, Council of Norway (RCN) under CoE Grant No. 262695 the error shows oscillations. A first hypothesis is that (Hylleraas Centre for Quantum Molecular Sciences), this behavior is related to the ignored correlations in the from ERC-STG-2014 under grant No. 639508, from the transition k → k + 1. However, this was not able to be Hungarian National Research, Development and Inno- proven so far using entropy based measures but a similar vation Office (NKFIH) through Grant No. K120569, irregular behavior can be detected by a cluster ampli- from the Hungarian Quantum Technology National Ex- tude error analysis. The error minimum found for the cellence Program (Project No. 2017-1.2.1-NKP-2017- DMRG-TCCSD method, however, was not able to be 00001), from the Czech Science Foundation (grants no. proven within this article and is left for future work. An 16-12052S, 18-18940Y, and 18-24563S), and the Czech important feature that we would like to highlight here is Ministry of Education, Youth and Sports (project no. that a small CAS (k = 9) yields a significant improve- LTAUSA17033). FMF, AL, OL,¨ MAC are grateful for ment of the energy, and that the energies for all three ge- the mutual hospitality received during their visits at the ometries and all CAS choices outrun the single-reference Wigner Research Center for Physics in Budapest and the CC method. Although the computational costs of the Hylleraas Centre for Quantum Molecular Sciences. OL¨ DMRG-TCCSD method exceed the costs of the CCSD and JP acknowledge useful discussions with Marcel Nooi- method, this leads to a computational drawback of the jen.

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