The density matrix renormalization group for ab initio quantum chemistry
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The density matrix renormalization group for ab initio quantumchemistry
Sebastian Wouters and Dimitri Van Neck
Center for Molecular Modelling, Ghent University, Technologiepark 903, 9052 Zwijnaarde, BelgiumReceived: date / Revised version: date
Abstract.
During the past 15 years, the density matrix renormalization group (DMRG) has become in-creasingly important for ab initio quantum chemistry. Its underlying wavefunction ansatz, the matrixproduct state (MPS), is a low-rank decomposition of the full configuration interaction tensor. The virtualdimension of the MPS, the rank of the decomposition, controls the size of the corner of the many-bodyHilbert space that can be reached with the ansatz. This parameter can be systematically increased untilnumerical convergence is reached. The MPS ansatz naturally captures exponentially decaying correlationfunctions. Therefore DMRG works extremely well for noncritical one-dimensional systems. The active or-bital spaces in quantum chemistry are however often far from one-dimensional, and relatively large virtualdimensions are required to use DMRG for ab initio quantum chemistry (QC-DMRG). The QC-DMRGalgorithm, its computational cost, and its properties are discussed. Two important aspects to reduce thecomputational cost are given special attention: the orbital choice and ordering, and the exploitation of thesymmetry group of the Hamiltonian. With these considerations, the QC-DMRG algorithm allows to findnumerically exact solutions in active spaces of up to 40 electrons in 40 orbitals.
PACS.
At the basis of ab initio quantum chemistry lies Hartree-Fock (HF) theory [1–3]. In HF theory, a single Slater deter-minant (SD) is optimized by finding the set orbitals whichminimize its energy expectation value. The occupancy ofthe HF orbitals is definite: occupied orbitals are filled withprobability 1, and virtual orbitals are empty with prob-ability 1. The exact ground state is a linear combinationover all possible Slater determinants. The difference in en-ergy between the HF solution and the exact ground stateis the correlation energy. This energy is often (somewhatambiguously) divided into two contributions: static anddynamic correlation [4]. When near-degeneracies betweendeterminants occur, and more than one determinant isneeded to describe the qualitative behaviour of a molecule,it is said to have static correlation. This type of correlationoften arises in transition metal complexes or π -conjugatedsystems, as well as for geometries far from equilibrium. Itis typically resolved with only a few determinants. TheCoulomb repulsion results in a small nonzero occupancyof many virtual HF orbitals in the true ground state. Thiseffect is called dynamic correlation, and it constitutes theremainder of the energy gap.All static and dynamic correlation can in principlebe retrieved at HF cost with density functional theory(DFT). Hohenberg and Kohn have shown that the elec- tron density provides sufficient information to determineall ground state properties, and that there exists a uniqueuniversal functional of the electron density which can beused to obtain the exact ground state density [5]. Kohnand Sham rewrote the universal functional as the sumof the kinetic energy of a noninteracting system and anexchange-correlation functional [6]. This allows to rep-resent the electron density by means of the Kohn-ShamSlater determinant, which immediately ensures correct N-representability. Unfortunately, the universal functionalis unknown. Many approximate semi-empirical exchange-correlation functionals of various complexity have beenproposed. Because the exact exchange-correlation func-tional is unknown, not all correlation is retrieved withDFT. For single-reference systems, for which the exactsolution is dominated by a single SD, DFT is good incapturing dynamic correlation. For multireference (MR)systems, DFT fails to retrieve static correlation [7].Dynamic correlation can also be captured with ab ini-tio post-HF methods. These start from the optimized HForbitals and the corresponding SD, and build in dynamiccorrelation on top of the single SD reference. Commonlyknown are Møller-Plesset (Rayleigh-Schr¨odinger) pertur-bation theory [8], the configuration interaction (CI) ex-pansion [9, 10], and coupled cluster (CC) theory [11–13].These methods are truncated in their perturbation or ex-pansion order. An important property of wavefunctions Sebastian Wouters, Dimitri Van Neck: The density matrix renormalization group for ab initio quantum chemistry is size-consistency: the fact that for two noninteractingsubsystems, the compound wavefunction should be multi-plicatively separable and the total energy additively sep-arable. CI with N excitations is not size-consistent if thereare more than N electrons in the compound system, whereasCC is always size-consistent because of its exponentialwavefunction ansatz [4]. Because these post-HF methodsstart from a single SD reference, they have difficulty build-ing in static correlation. Mostly, very large expansion or-ders are required to retrieve static correlation.It is therefore better to resort to MR methods forsystems with pronounced static correlation. For such sys-tems, the subset of important orbitals (the active space),in which the occupation changes over the dominant de-terminants, is often rather small. This allows for a par-ticular MR solution method: the complete active space(CAS) self-consistent field (SCF) method [14–16]. Fromthe HF solution, a subset of occupied and virtual orbitalsis selected as active space. While the remaining occupiedand virtual orbitals are kept frozen at HF level, the elec-tronic structure in the active space is solved exactly (theCAS-part). Subsequently, the occupied, active, and virtualspaces are rotated to further minimize the energy. Thistwo-step cycle, which is sometimes implemented together,is repeated until convergence is reached (the SCF-part).CASSCF resolves the static correlation in the system. Dy-namic correlation can be built in on top of the CASSCFreference wavefunction by perturbation theory (CASPT2)[17, 18], a CI expansion (MRCI or CASCI) [19–23], or CCtheory (MRCC or CASCC) [24, 25]. For the latter, ap-proximate schemes such as canonical transformation (CT)theory [26] are often used.Because the many-body Hilbert space grows exponen-tially with the number of single-particle states, only smallactive spaces, of up to 18 electrons in 18 orbitals, can betreated in the CAS-part. In 1999, the density matrix re-normalization group (DMRG) was introduced in ab initio quantum chemistry (QC) [27]. This MR method allowsto find numerically exact solutions in significantly largeractive spaces, of up to 40 electrons in 40 orbitals. The electronic Hamiltonian can be written in second quan-tization as ˆ H = E + X ij t ij X σ ˆ a † iσ ˆ a jσ + 12 X ijkl v ij ; kl X στ ˆ a † iσ ˆ a † jτ ˆ a lτ ˆ a kσ . (1)The Latin letters denote spatial orbitals and the Greekletters electron spin projections. The t ij and v ij ; kl are theone- and two-electron integrals, respectively. In the oc-cupation number representation, the basis states of themany-body Hilbert space are | n ↑ n ↓ ...n L ↑ n L ↓ i = (cid:16) ˆ a † ↑ (cid:17) n ↑ (cid:16) ˆ a † ↓ (cid:17) n ↓ ... (cid:16) ˆ a † L ↑ (cid:17) n L ↑ (cid:16) ˆ a † L ↓ (cid:17) n L ↓ |−i . (2) The symmetry group of the Hamiltonian (1) is SU ( ) ⊗ U ( ) ⊗ P , or total electronic spin, particle-number, andmolecular point group symmetry. By defining the opera-tors ˆ S + = X i ˆ a † i ↑ ˆ a i ↓ , (3)ˆ S − = (cid:16) ˆ S + (cid:17) † = X i ˆ a † i ↓ ˆ a i ↑ , (4)ˆ S z = 12 X i (cid:16) ˆ a † i ↑ ˆ a i ↑ − ˆ a † i ↓ ˆ a i ↓ (cid:17) , (5)ˆ N = X i (cid:16) ˆ a † i ↑ ˆ a i ↑ + ˆ a † i ↓ ˆ a i ↓ (cid:17) , (6)ˆ S = ˆ S + ˆ S − + ˆ S − ˆ S + S z ˆ S z , (7)it can be easily checked that ˆ H , ˆ S , ˆ S z , and ˆ N form a set ofcommuting observables. This constitutes the SU ( ) totalelectronic spin and U ( ) particle-number symmetries. Forfixed particle number N , Eq. (1) can also be written asˆ H = E + 12 X ijkl h ij ; kl X στ ˆ a † iσ ˆ a † jτ ˆ a lτ ˆ a kσ , (8)with h ij ; kl = v ij ; kl + 1 N − t ik δ j,l + t jl δ i,k ) . (9)The molecular point group symmetry P consists of therotations, reflections, and inversions which leave the exter-nal potential due to the nuclei invariant. These symmetryoperations map nuclei with equal charges onto each other.The point group symmetry has implications for the spatialorbitals. Linear combinations of the single-particle basisfunctions can be constructed which transform accordingto a particular row of a particular irreducible represen-tation (irrep) of P [28]. As the Hamiltonian transformsaccording to the trivial irrep I of P , h ij ; kl can only benonzero if the reductions of I i ⊗ I j and I k ⊗ I l have at leastone irrep in common. Most molecular electronic structureprograms make use of the abelian point groups with real-valued character tables.An eigenstate of the Hamiltonian (8) can be writtenas | Ψ i = X { n jσ } C n ↑ n ↓ n ↑ n ↓ ...n L ↑ n L ↓ | n ↑ n ↓ n ↑ n ↓ ...n L ↑ n L ↓ i . (10)The size of the full CI (FCI) tensor grows as 4 L , exponen-tially fast with L . This tensor can be exactly decomposedby a singular value decomposition (SVD) as follows: C n ↑ n ↓ n ↑ n ↓ ...n L ↑ n L ↓ = C ( n ↑ n ↓ );( n ↑ n ↓ ...n L ↑ n L ↓ ) = P α U [1] ( n ↑ n ↓ ); α s [1] α V [1] α ;( n ↑ n ↓ ...n L ↑ n L ↓ ) . (11) ebastian Wouters, Dimitri Van Neck: The density matrix renormalization group for ab initio quantum chemistry 3 Define A [1] n ↑ n ↓ α = U [1] ( n ↑ n ↓ ); α s [1] α , (12)and decompose the right unitary V [1] again with an SVDas follows: V [1] α ;( n ↑ n ↓ n ↑ n ↓ ...n L ↑ n L ↓ ) = V [1] ( α n ↑ n ↓ );( n ↑ n ↓ ...n L ↑ n L ↓ ) = P α U [2] ( α n ↑ n ↓ ); α s [2] α V [2] α ;( n ↑ n ↓ ...n L ↑ n L ↓ ) . (13)Define A [2] n ↑ n ↓ α ; α = U [2] ( α n ↑ n ↓ ); α s [2] α . (14)Continue by successively decomposing the right unitaries V [ k ]. In this way, the FCI tensor can be exactly rewrittenas the following contracted matrix product: C n ↑ n ↓ n ↑ n ↓ n ↑ n ↓ ...n L ↑ n L ↓ = P { α k } A [1] n ↑ n ↓ α A [2] n ↑ n ↓ α ; α A [3] n ↑ n ↓ α ; α ...A [ L ] n L ↑ n L ↓ α L − , (15)which is graphically represented in Fig. 1. Except for thefirst and last orbital (or site ), Eq. (15) introduces a rank-3 tensor per site. One of its indices corresponds to thephysical index n i ↑ n i ↓ , the other two to the virtual or bond indices α i − and α i . In Fig. 1, tensors are represented bycircles, physical indices by open lines, and virtual indicesby connected lines. The graph hence represents how thecontracted matrix product decomposes the FCI tensor.Since no assumptions are made about the FCI tensor, thedimension of the indices { α k } has to grow exponentiallytowards the middle of this contracted product:dim ( α j ) = min (cid:0) j , L − j (cid:1) . (16)This is solely due to the increasing matrix dimensions inthe successive SVDs. Instead of variationally optimizingover the FCI tensor, one may as well optimize over thetensors of its decomposition (15). To make Eq. (15) ofpractical use, its dimensions can be truncated:dim ( α j ) = min (cid:0) j , L − j , D (cid:1) . (17)The corresponding ansatz is called a matrix product state(MPS) with open boundary conditions. The truncationdimension D is called the bond or virtual dimension. TheMPS ansatz can be optimized by the DMRG algorithm[27, 29, 30], yielding a variational upper bound for theground state energy.DMRG was invented in 1992 by White in the field ofcondensed matter theory [29]. ¨Ostlund and Rommer dis-covered in 1995 its underlying variational ansatz, the MPS[31, 32]. The discovery of the MPS ansatz allowed to un-derstand DMRG by means of quantum information the-ory. The area law for one-dimensional quantum systems,see section 3, was proven by Hastings in 2007 [33], and con-stitutes a hard proof that an MPS is very efficient in rep-resenting the ground state of noncritical one-dimensionalquantum systems. Fig. 1.
Tensors are represented by circles, physical indices byopen lines, and virtual indices by connected lines. The MPSgraph hence represents how the contracted matrix product de-composes the FCI tensor.
The MPS ansatz was in fact discovered earlier, un-der various names. Nishino found that they were used instatistical physics as a variational optimization technique[34]: in 1941 by Kramers and Wannier [35] and in 1968by Baxter [36]. Nightingale and Bl¨ote recycled Baxter’sansatz in 1986 to approximate quantum eigenstates [37].In 1987, Affleck, Kennedy, Lieb and Tasaki constructedthe exact valence-bond ground state of a particular next-nearest-neighbour spin chain [38]. They obtained an MPSwith bond dimension 2. In mathematics, the translation-ally invariant valence-bond state is known as a finitelycorrelated state [39, 40], and in the context of informationcompression, an MPS is known as a tensor train [41, 42].The concept of a renormalization group was first usedin quantum electrodynamics. The coarse-grained view ofa point-like electron breaks down at small distance scales(or large energy scales). The electron itself consists of elec-trons, positrons, and photons. The mass and charge con-tributions from this fine structure lead to infinities. Thesewere successfully resolved by Tomonaga, Schwinger, andFeynman [43–47]. Later, Wilson used a numerical renor-malization group (NRG) to solve the long-standing Kondoproblem [48]. He turned the coupling of the impurity to theconduction band into a half-infinite lattice problem by dis-cretizing the conduction band in momentum space. For in-creasing lattice sizes, only the lowest energy states are keptat each renormalization step. These are sufficient to studythe low-temperature thermodynamics of the impurity sys-tem. Although very successful for impurity systems, NRGfails for real-space lattice systems such as the discretizedparticle-in-a-box, spin-lattice, and Hubbard models. Forthese systems, the low energy states of a small subsystemare often irrelevant for the ground state of the total sys-tem [49]. Consider for example the ground state of theparticle-in-a-box problem. By concatenating the solutionof two smaller sized boxes, an unphysical node is intro-duced in the approximation of the ground state of thelarger problem. It was White who pointed out this prob-lem and resolved it with his DMRG method [29]. Insteadof selecting the degrees of freedom with lowest energy, themost relevant degrees of freedom should be selected.
Sebastian Wouters, Dimitri Van Neck: The density matrix renormalization group for ab initio quantum chemistry
Fig. 2.
Bipartition of the L single-particle states. This section attempts to clarify the broader context ofDMRG. A brief introduction to quantum entanglement,the von Neumann entropy, and the area law is given.Consider the bipartition of L orthonormal single-particlestates into two subsystems A and B in Fig. 2. Suppose {| A i i} and {| B j i} are the orthonormal basis states of themany-body Hilbert spaces of resp. subsystem A and B .The Hilbert space of the composite system is spanned bythe product space {| A i i} ⊗ {| B j i} , and a general quan-tum many-body state | Ψ i of the composite system can bewritten as | Ψ i = X ij C ij | A i i | B j i . (18)The Schmidt decomposition of | Ψ i is obtained by perform-ing an SVD on C ij and by rotating the orthonormal bases {| A i i} and {| B j i} with the unitary matrices: | Ψ i = X ij C ij | A i i | B j i = X ijk U ik σ k V kj | A i i | B j i = X k σ k | e A k i | e B k i . (19)For normalized | Ψ i : h Ψ | Ψ i = X k σ k = 1 . (20)For the given bipartition, one is sometimes interested inthe optimal approximation | e Ψ i of | Ψ i in a least squaressense k | e Ψ i − | Ψ i k . It can be shown that the optimal ap-proximation, with a smaller number of terms in the sum-mation (18), is obtained by keeping the states with thelargest Schmidt numbers σ k in Eq. (19). This fact will beof key importance for the DMRG algorithm (see section4.3).In classical theories, the sum over k can contain onlyone nonzero value σ k . A measurement in subsystem A thendoes not influence the outcome in subsystem B , and thetwo subsystems are not entangled. In quantum theories,the sum over k can contain many nonzero values σ k . State | e A k i in subsytem A occurs with probability σ k , as canbe observed from the reduced density matrix (RDM) ofsubsystem A :ˆ ρ A = Tr B | Ψ i h Ψ | = X j h B j | Ψ i h Ψ | B j i = X ijl | A i i C ij C † jl h A l | = X k | e A k i σ k h e A k | . (21) Analogously the RDM of subsystem B can be constructed:ˆ ρ B = X k | e B k i σ k h e B k | . (22)From (19), it follows that the measurement of | e A k i insubsystem A implies the measurement of | e B k i in subsys-tem B with probability 1. Measurements in A and B arehence not independent, and the two subsystems are saidto be entangled.Consider for example two singly occupied orbitals A and B in the spin-0 singlet state: | Ψ i = |↑ A ↓ B i − |↓ A ↑ B i√ . (23)The measurements of the spin projections of the electronsare not independent. Each possible spin projection of theelectron in A can be measured with probability , but thesimultaneous measurement of both spin projections willalways yield h Ψ | ˆ S zA ˆ S zB | Ψ i = −
14 (24)with probability 1.The RDMs ˆ ρ A and ˆ ρ B allow to define the von Neu-mann entanglement entropy [50]: S A | B = − Tr A ˆ ρ A ln ˆ ρ A = − Tr B ˆ ρ B ln ˆ ρ B = − X k σ k ln σ k . (25)This quantum analogue of the Shannon entropy is a mea-sure of how entangled subsystems A and B are. If they arenot entangled, σ = 1 and σ k = 0 for k ≥
2, which implies S A | B = 0. If they are maximally entangled, σ k = σ l for all k and l , which implies S A | B = ln( Z ), with Z the minimumof the sizes of the many-body Hilbert spaces of A and B .A Hamiltonian which acts on a K -dimensional quan-tum lattice system in the thermodynamic limit is calledlocal if there exists a distance cutoff beyond which the in-teraction terms decay at least exponentially. Consider the ground state | Ψ i of a gapped K -dimensional quantumsystem in the thermodynamic limit, and select as subsys-tem a hypercube with side L and volume L K . The vonNeumann entropy is believed to obey an area law [51–53]: S hypercube ∝ L K − . (26)This is the result of a finite correlation length, as onlylattice sites in the immediate vicinity of the hypercube’sboundary are then correlated with lattice sites on theother side of the boundary. This is a theorem for one-dimensional systems [33] and a conjecture in higher di-mensions [52], supported by numerical examples and theo-retical arguments [53]. For critical quantum systems, witha closed excitation gap, there can be logarithmic correc-tions to the area law [52, 54].For gapped one-dimensional systems, consider as sub-system a line segment of length L . Its boundary consists ebastian Wouters, Dimitri Van Neck: The density matrix renormalization group for ab initio quantum chemistry 5 Fig. 3.
Several tensor network states. Tensors are representedby circles, physical indices by open lines, and virtual indicesby connected lines. The graph hence represents how the ansatzdecomposes the FCI tensor. of two points. Due to the finite correlation length in theground state, the entanglement of the subsystem does notincrease with L , if L is significantly larger than the correla-tion length. The von Neumann entropy is then a constantindependent of L , and the ground state | Ψ i can be wellrepresented by retaining only a finite number of states D in the Schmidt decomposition of any bipartition of the lat-tice in two semi-infinite line segments. This is the reasonwhy the MPS ansatz and the corresponding DMRG algo-rithm work very well to study the ground states of gappedone-dimensional systems.The MPS ansatz | Ψ i = X { n jσ }{ α k } A [1] n ↑ n ↓ α A [2] n ↑ n ↓ α ; α ...A [ L ] n L ↑ n L ↓ α L − | n ↑ n ↓ n ↑ n ↓ ...n L ↑ n L ↓ i , (27)is shown graphically in Fig. 3. Except for the first andlast orbital (or site), the MPS ansatz introduces a rank-3tensor per site. One of its indices corresponds to the phys-ical index n i ↑ n i ↓ , the other two to the virtual indices α i − and α i . Similar to Fig. 1, tensors are represented by cir-cles, physical indices by open lines, and virtual indices byconnected lines in Fig. 3. The graph hence represents howthe ansatz decomposes the FCI tensor. The finite size D of the virtual indices can capture finite-length correlationsalong the one-dimensional chain. Stated more rigorously:for a system in the thermodynamic limit, all correlationfunctions C MPS ( ∆x ) measured in an MPS ansatz with fi-nite D decay exponentially with increasing site distance ∆x [40, 55]: C MPS ( ∆x ) ∝ e − α∆x . (28) Unless the lattice size is reasonably small [56], an MPSis not efficient to represent the ground state of higher di-mensional or critical systems. Fortunately, efficient tensornetwork states (TNS) for higher dimensional and criticallattice systems, which do obey the correct entanglementscaling laws, have been developed [55]. There even exists acontinuous MPS ansatz for one-dimensional quantum fieldtheories [57].The ansatz for two-dimensional systems is called theprojected entangled pair state (PEPS) [58], see Fig. 3. In-stead of two virtual indices, each tensor now has four vir-tual indices, which allows to arrange the sites in a squarelattice. A finite virtual dimension D still introduces a fi-nite correlation length, but due to the topology of thePEPS, this is sufficient for two-dimensional systems, evenin the thermodynamic limit. Analogous extensions existfor other lattice topologies.The ansatz for critical one-dimensional systems is calledthe multi-scale entanglement renormalization ansatz (MERA)[59], see Fig. 3. This ansatz has two axes: x along thephysical one-dimensional lattice and z along the renor-malization direction . Consider two sites separated by ∆x along x . The number of virtual bonds between those sitesis only of order ∆z ∝ ln ∆x . With finite D , all correlationfunctions C MERA ( ∆x ) measured in a MERA decay expo-nentially with increasing renormalization distance ∆z : C MERA ( ∆x ) ∝ e − α∆z ∝ e − β ln ∆x = ( ∆x ) − β , (29)and therefore only algebraically with increasing lattice dis-tance ∆x [55, 59].An inconvenient property of the PEPS, MERA, andMPS with periodic boundary conditions [60], is the in-troduction of loops in the network. This results in theinability to exploit the TNS gauge invariance to workwith orthonormal renormalized environment states, seesections 4.2 and 4.3. One particular network which avoidssuch loops, but which is still able to capture polynomiallydecaying correlation functions, is the tree TNS (TTNS)[61, 62], see Fig. 3. From a central tensor with z virtualbonds, Y consecutive onion-like layers are built of tensorswith also z virtual bonds. The last layer consists of tensorswith only 1 virtual bond. An MPS is hence a TTNS with z = 2. The number of sites L increases as [63, 64]: L = 1 + z Y X k =1 ( z − k − = z ( z − Y − z − . (30)Hence Y ∝ ln( L ) for z ≥
3. The maximum number ofvirtual bonds between any two sites is 2 Y . The correlationfunctions in a TTNS with finite D and z ≥ Y : C TTNS ( L ) ∝ e − αY ∝ e − β ln L = L − β , (31)and therefore only algebraically with increasing numberof sites L [61, 62].For higher-dimensional or critical systems, DMRG canstill be useful [56]. The virtual dimension D then has to beincreased to a rather large size to obtain numerical con-vergence. In the case of multiple dimensions, the question Sebastian Wouters, Dimitri Van Neck: The density matrix renormalization group for ab initio quantum chemistry arises if one should work in real or momentum space, andhow the corresponding single-particle degrees of freedomshould be mapped to the one-dimensional lattice [65].
Abinitio quantum chemistry can be considered as a higher-dimensional system, due to the full-rank two-body interac-tion in the Hamiltonian (8), and the often compact spatialextent of molecules. Nevertheless, DMRG turned out to bevery useful for ab initio quantum chemistry (QC-DMRG)[27, 63, 64, 66–135].An excellent description of QC-DMRG in terms of re-normalization transformations is given in Chan and Head-Gordon [68]. Section 4 contains a description in terms ofthe underlying MPS ansatz, because this approach will beused in section 9 to introduce SU ( ) ⊗ U ( ) ⊗ P symmetryin the DMRG algorithm. The properties of the DMRGalgorithm are discussed in section 5. Several convergencestrategies are listed in section 6. An overview of the strate-gies to choose and order orbitals is given in section 7. Aconverged DMRG calculation can be the starting point ofother methods. These methods are summarized in section8. Section 10 gives an overview of the currently existingQC-DMRG codes, and the systems which have been stud-ied with them. DMRG can be formulated as the variational optimizationof an MPS ansatz [31, 32]. The MPS ansatz (27) has openboundary conditions, because sites 1 and L only have onevirtual index. The sites are assumed to be orbitals, whichhave 4 possible occupancies |−i , |↑i , |↓i , and |↑↓i . Hence-forth | n i i will be used as a shorthand for | n i ↑ n i ↓ i . To beof practical use, the virtual dimensions α j are truncatedto D : dim( α j ) = min(4 j , L − j , D ). With increasing D , theMPS ansatz spans a larger region of the full Hilbert space,but it is of course not useful to make D larger than 4 ⌊ L ⌋ as the MPS ansatz then spans the whole Hilbert space.A Slater determinant has gauge freedom: a rotationin the occupied orbital space alone, or a rotation in thevirtual orbital space alone, does not change the physicalwavefunction. Only occupied-virtual rotations change thewavefunction. An MPS has gauge freedom as well. If fortwo neighbouring sites i and i + 1, the left MPS tensorsare right-multiplied with the non-singular matrix G :˜ A [ i ] n i α i − ; α i = X β i A [ i ] n i α i − ; β i G β i ; α i , (32)and the right MPS tensors are left-multiplied with theinverse of G :˜ A [ i + 1] n i +1 α i ; α i +1 = X β i G − α i ; β i A [ i + 1] n i +1 β i ; α i +1 , (33)the wavefunction does not change, i.e. ∀ n i , n i +1 , α i − , α i +1 : X α i ˜ A [ i ] n i α i − ; α i ˜ A [ i +1] n i +1 α i ; α i +1 = X α i A [ i ] n i α i − ; α i A [ i +1] n i +1 α i ; α i +1 . (34) The two-site DMRG algorithm consists of consecutive sweeps or macro-iterations , where at each sweep step the MPStensors of two neighbouring sites are optimized in the micro-iteration . Suppose these sites are i and i + 1. Thegauge freedom of the MPS is used to bring it in a partic-ular canonical form. For all sites to the left of i , the MPStensors are left-normalized: X α k − ,n k ( A [ k ] n k ) † α k ; α k − A [ k ] n k α k − ; β k = δ α k ,β k , (35)and for all sites to the right of i + 1, the MPS tensors areright-normalized: X α k ,n k A [ k ] n k α k − ; α k ( A [ k ] n k ) † α k ; β k − = δ α k − ,β k − . (36)Left-normalization can be performed with consecutive QR-decompositions: A [ k ] n k α k − ; α k = A [ k ] ( α k − n k ); α k = P β k Q [ k ] ( α k − n k ); β k R β k ; α k = P β k Q [ k ] n k α k − ; β k R β k ; α k . (37)The MPS tensor Q [ k ] is now left-normalized. The R -matrixis multiplied into A [ k + 1]. From site 1 to i −
1, the MPStensors are left-normalized this way, without changing thewavefunction. Right-normalization occurs analogously withLQ-decompositions. In section 4.4, it will become clearthat this normalization procedure only needs to occur atthe start of the DMRG algorithm.At this point, it is instructive to make the analogyto the renormalization group formulation of the DMRGalgorithm. Define the following vectors: | α Li − i = X { n j }{ α ...α i − } A [1] n α ...A [ i − n i − α i − ; α i − | n ...n i − i , (38) | α Ri +1 i = X { n j }{ α i +2 ...α L − } A [ i + 2] n i +2 α i +1 ; α i +2 ...A [ L ] n L α L − | n i +2 ...n L i . (39)Due to the left- and right-normalization described above,these vectors are orthonormal: h α Li − | β Li − i = δ α i − ,β i − , (40) h α Ri +1 | β Ri +1 i = δ α i +1 ,β i +1 . (41) {| α Li − i} and {| α Ri +1 i} are renormalized bases of the many-body Hilbert spaces spanned by resp. orbitals 1 to i − i + 2 to L . Consider for example the left side. Forsite k from 1 to i −
2, the many-body basis is augmented byone orbital and subsequently truncated again to at most D renormalized basis states: {| α Lk − i} ⊗ {| n k i} →| α Lk i = P α k − ,n k A [ k ] n k α k − ; α k | α Lk − i | n k i . (42)DMRG is hence a renormalization group for increasingmany-body Hilbert spaces. The next section addresses howthis renormalization transformation is chosen. ebastian Wouters, Dimitri Van Neck: The density matrix renormalization group for ab initio quantum chemistry 7 Fig. 4.
Optimization of the MPS tensors at sites i and i +1 inthe two-site DMRG algorithm. The effective Hamiltonian equa-tion (45), obtained by variation of the Lagrangian (44), can beinterpreted as the approximate diagonalization of the exactHamiltonian ˆ H in the orthonormal basis {| α Li − i} ⊗ {| n i i} ⊗{| n i +1 i} ⊗ {| α Ri +1 i} . Combine the MPS tensors of the two sites under consid-eration into a single two-site tensor: X α i A [ i ] n i α i − ; α i A [ i + 1] n i +1 α i ; α i +1 = B [ i ] n i ; n i +1 α i − ; α i +1 . (43)At the current micro-iteration of the DMRG algorithm, B [ i ] (the flattened column form of the tensor B [ i ]) is usedas an initial guess for the effective Hamiltonian equation.This equation is obtained by variation of the Lagrangian[91] L = h Ψ ( B [ i ]) | ˆ H | Ψ ( B [ i ]) i − E i h Ψ ( B [ i ]) | Ψ ( B [ i ]) i (44)with respect to the complex conjugate of B [ i ]: H [ i ] eff B [ i ] = E i B [ i ] . (45)The canonical form in Eqs. (35)-(36) ensured that no over-lap matrix is present in this effective Hamiltonian equa-tion. In the DMRG language, this equation can be in-terpreted as the approximate diagonalization of the ex-act Hamiltonian ˆ H in the orthonormal basis {| α Li − i} ⊗{| n i i} ⊗ {| n i +1 i} ⊗ {| α Ri +1 i} , see Fig. 4. Because of the un-derlying MPS ansatz, DMRG is variational: E i is alwaysan upper bound to the energy of the true ground state.The lowest eigenvalue and corresponding eigenvectorof the effective Hamiltonian are searched with iterativesparse eigensolvers. Typical choices are the Lanczos orDavidson algorithms [136, 137]. Once B [ i ] is found, it isdecomposed with an SVD: B [ i ] ( α i − n i );( n i +1 α i +1 ) = X β i U [ i ] ( α i − n i ); β i κ [ i ] β i V [ i ] β i ;( n i +1 α i +1 ) . (46)Note that U [ i ] is hence left-normalized and V [ i ] right-normalized. The sum over β i is truncated if there are morethan D nonzero Schmidt values κ [ i ] β i , thereby keeping the D largest ones. This is the optimal approximation for thebipartition of {| α Li − i} ⊗ {| n i i} ⊗ {| n i +1 i} ⊗ {| α Ri +1 i} into A = {| α Li − i} ⊗ {| n i i} and B = {| n i +1 i} ⊗ | α Ri +1 i} . In theoriginal DMRG algorithm, U [ i ] and V [ i ] were obtained asthe eigenvectors of resp. ˆ ρ A and ˆ ρ B .A discarded weight can be associated with the trunca-tion of the sum over β i : w [ i ] disc D = X β i >D κ [ i ] β i . (47) This is the probability to measure one of the discardedstates in the subsystems A or B . The approximation intro-duced by the truncation becomes better with increasinglysmall discarded weight. Instead of working with a fixed D , one could also choose D dynamically in order to keep w [ i ] disc D below a preset threshold, as is done in Legeza’sdynamic block state selection approach [69]. So far, we have looked at a micro-iteration of the DMRGalgorithm. This micro-iteration happens during left or rightsweeps. During a left sweep, B [ i ] is constructed, the cor-responding effective Hamiltonian equation solved, the so-lution B [ i ] decomposed, the Schmidt spectrum truncated, κ [ i ] is contracted into U [ i ], A [ i ] is set to this contraction U [ i ] × κ [ i ], A [ i + 1] is set to V [ i ], and i is decreased by 1.Note that A [ i + 1] is right-normalized for the next micro-iteration as required. This stepping to the left occurs until i = 1, and then the sweep direction is reversed from left toright. Based on energy differences, or wavefunction over-laps, between consecutive sweeps, a convergence criteriumis triggered, and the sweeping stops.DMRG can be regarded as a self-consistent field method:at convergence the neighbours of an MPS tensor generatethe field which yields the local solution, and this local so-lution generates the field for its neighbours [68, 81, 91]. The effective Hamiltonian in Eq. (45) is too large to befully constructed as a matrix. Only its action on a par-ticular guess B [ i ] is available as a function. In order toconstruct H [ i ] eff B [ i ] efficiently for general quantum chem-istry Hamiltonians, several tricks are used. Suppose thata right sweep is performed and that the MPS tensors ofsites i and i + 1 are about to be optimized.Renormalized operators such as h α Li − | ˆ a † kσ ˆ a lτ | β Li − i with k, l ≤ i − k, l ≤ i − h α Li − | ˆ a † kσ ˆ a lτ | β Li − i = P α i − ,β i − ,n i − ( A [ i − n i − ) † α i − ; α i − h α Li − | ˆ a † kσ ˆ a lτ | β Li − i A [ i − n i − β i − ; β i − . (48)Note that no phases appear because an even number ofsecond-quantized operators was transformed. For an oddnumber, there should be an additional phase ( − n ( i − ↑ + n ( i − ↓ at the right-hand side (RHS) due to the Jordan-Wignertransformation [138]. Renormalized operators to the rightof B [ i ] can be loaded from disk, as they have been savedduring the previous left sweep.Once three second-quantized operators are on one sideof B [ i ], they are multiplied with the matrix elements h kl ; mn ,and a summation is performed over the common indices Sebastian Wouters, Dimitri Van Neck: The density matrix renormalization group for ab initio quantum chemistry to construct complementary renormalized operators [27,65, 68, 94]: h α Li − | ˆ Q nτ | β Li − i = X σ X k,l,m
Computational requirements per macro-iteration or sweep of the QC-DMRG algorithm. O (task) time memory disk H [ i ] eff { B [ i ] } ( a ) N vec L D N vec D -SVD and basis truncation LD D LD Renormalized operators L D L D L D Complementary renormalized operators L D + L D L D L D Total L D + N vec L D ( N vec + L ) D L D a ) The memory for the (complementary) renormalized operators is mentioned separately.
Fig. 5.
The geometries of all-trans polyenes C n H n +2 wereoptimized at the B3LYP/6-31G** level of theory for n = 12,14, 16, 18, 20, 22 and 24. The σ -orbitals were kept frozen atthe RHF/6-31G level of theory. The π -orbitals in the 6-31Gbasis were localized by means of the Edmiston-Ruedenberg lo-calization procedure [140], which maximizes P i v ii ; ii . The lo-calized π -orbitals belong to the A ′′ irrep of the C s point group,and were ordered according to the one-dimensional topologyof the polyene. For all polyenes, the average CPU time perDMRG sweep was determined with CheMPS2 [121], for two reduced virtual dimensions D . For the values of D shown here,the energies are converged to µE h accuracy due to the one-dimensional topology of the localized and ordered π -orbitals.Due to the imposed SU ( ) ⊗ U ( ) ⊗ C s symmetry, all tensorsbecome block-sparse, see section 9, which causes the scaling tobe below O ( L ). With increasing virtual dimension D , the MPS ansatzspans an increasing part of the many-body Hilbert space.In the following, E D denotes the minimum energy encoun-tered in Eq. (45) during the micro-iterations for a givenvirtual dimension D . Several calculations with increasing D can be performed, in order to assess the convergence.This even allows to make an extrapolation of the energyto the FCI limit. Several extrapolation schemes have beensuggested. Note that E FCI and { C i , p j , q k } below are pa-rameters to be fitted. The maximum discarded weight en-countered during the last sweep before convergence is ab-breviated as: w disc D = max i (cid:8) w [ i ] disc D (cid:9) . (53) The initial assumption of exponential convergence [27]ln ( E D − E FCI ) ∝ C + C D (54)was rapidly abandoned for the relation [68, 69, 141] E D − E FCI = C w disc D , (55)because the energy is a linear function of the RDM [68].An example of an extrapolation with Eq. (55) is shownin Fig. 6. The tail of the distribution of RDM eigenvaluesscales as [68, 142] κ [ i ] β i ∝ exp n − C (ln β i ) o . (56)Substituting this relation in Eq. (55) yields an improvedversion of Eq. (54) [68]:ln ( E D − E FCI ) ∝ C − C (ln D ) . (57)An example of an extrapolation with Eq. (57) is shown inFig. 10. Eqs. (55) and (57) are the most widely used ex-trapolation schemes in QC-DMRG. Three other relationshave been proposed as well. A relation for incrementalenergies ∆E D = E D − E D has been suggested [72]: ∆E D = C + C E D √ L D + 2 L D , (58)but the extrapolated E FCI often violates the variationalprinciple. An alternative relation based on the discardedweight has also been proposed [72]:ln ( E D − E FCI ) = C − C (cid:0) w disc D (cid:1) − , (59)as well as a Richardson-type extrapolation scheme, basedon the assumption that the energy is an analytic functionof w disc D [97]: E ( µν ) ( w disc D ) = p + p w disc D + ... + p µ (cid:0) w disc D (cid:1) µ q + q w disc D + ... + q ν (cid:0) w disc D (cid:1) ν . (60) To analyze the MPS wavefunction (27), suppose that the L orthonormal orbitals are the HF orbitals. An importantdifference with traditional post-HF methods such as CIexpansions, is that no FCI coefficients are a priori zero. Fig. 6.
Extrapolation of the variational DMRG ground-stateenergy E D with the discarded weight w disc D , for N in the cc-pVDZ basis near equilibrium (nuclear separation 2.118 a.u.).The calculation was performed with CheMPS2 [121] with SU ( ) ⊗ U ( ) ⊗ D symmetry, see section 9. D denotes the num-ber of reduced virtual basis states. The irrep ordering in theDMRG calculation was [A g B u B u B g B u B g B g A u ] in orderto place bonding and antibonding orbitals close to each otheron the one-dimensional DMRG lattice, see section 7.3 and Fig.8. An MPS hence captures CI coefficients of any particle-excitation rank relative to HF [75, 81]. A small virtualdimension implies little information content in the FCIcoefficient tensor, or equivalently that the many nonzeroFCI coefficients are in fact highly correlated. This has tobe contrasted with CI expansions, which are truncated intheir particle-excitation rank and therefore set many FCIcoefficients a priori to zero. The nonzero FCI coefficientsare however not a priori correlated in a CI expansion: theyare entirely free to be variationally optimized.
For a method to be size-consistent, the compound wave-function should be multiplicatively separable | Ψ i = | A i | B i and the energy additively separable E = E A + E B for non-interacting subsystems A and B . From the discussion ofthe Schmidt decomposition above, it follows immediatelythat an MPS is size-consistent if the orbitals of subsystems A and B do not overlap, and if they are separated into twogroups on the one-dimensional DMRG lattice [68, 132].The latter is for example realized if orbitals 1 to k corre-spond to subsystem A and orbitals k + 1 to L correspondto subsystem B . DMRG will then automatically retrievea product wavefunction, in which only one Schmidt valueis nonzero at the corresponding boundary. A good variational energy does not necessarily imply thatthe wavefunction is accurate. Suppose we have an or- thonormal MPS | Ψ MPS i with virtual dimension D whichhas been variationally optimized to approximate the trueground state | Ψ i with energy E . Suppose that | Ψ MPS i = p − ǫ | Ψ i + ǫ | e Ψ i (61)with h Ψ | e Ψ i = 0. Then k | Ψ MPS i − | Ψ i k = r(cid:16)p − ǫ − (cid:17) + ǫ = ǫ + O ( ǫ ) (62)and h Ψ MPS | ˆ H | Ψ MPS i − E = ǫ (cid:16) h e Ψ | ˆ H | e Ψ i − E (cid:17) . (63)The energy converges quadratically in the wavefunctionerror. Most DMRG convergence criteria rely on energyconvergence ( ǫ ≈ D where ǫ ≈
0, the MPS wavefunction is not invariant toorbital rotations. The orbital choice and their ordering ona one-dimensional lattice also influence the convergencerate with D . Strategies to choose and order orbitals arediscussed in section 7. Sparse iterative FCI eigensolversconverge the FCI tensor to a predefined threshold insteadof the energy. An FCI solution can therefore be consideredinvariant to orbital rotations. The DMRG algorithm can get stuck in a local minimumor a limit cycle, if D is insufficiently large [68]. The chanceof occurrence is larger for inconvenient orbital choices andorderings. Because the virtual dimension D cannot be in-creased indefinitely in practice, it is important to choosethe set of orbitals and their ordering well, see section 7.Additional considerations to enhance convergence are de-scribed here. It is better to use the two-site DMRG algorithm than theone-site version [143]. In the one-site version, the Hamil-tonian ˆ H is diagonalized during the micro-iterations inthe basis {| α Li − i} ⊗ {| n i i} ⊗ {| α Ri i} instead of {| α Li − i} ⊗{| n i i} ⊗ {| n i +1 i} ⊗ {| α Ri +1 i} . Because of the larger vari-ational freedom in the two-site DMRG algorithm, lowerenergy solutions are obtained, and the algorithm is lesslikely to get stuck [88]. It might therefore be worthwhileto optimize three or more MPS tensors simultaneously ina micro-iteration, or to group several orbitals into a singleDMRG lattice site [27].The two-site algorithm has another important advan-tage, when the symmetry group of the Hamiltonian is ex-ploited. The virtual dimension D is then distributed over ebastian Wouters, Dimitri Van Neck: The density matrix renormalization group for ab initio quantum chemistry 11 several symmetry sectors, see section 9. In the one-site al-gorithm, the virtual dimension of a symmetry sector hasto be changed manually during the sweeps [88], while theSVD (46) in the two-site algorithm automatically picksthe best distribution. White suggested to add perturbative corrections to theRDM in order to enhance convergence [143]. Instead ofusing perturbative corrections, one can also add noise tothe RDM prior to diagonalization or to B [ i ] prior to SVD[68]. The corrections or noise help to reintroduce lost sym-metry sectors (lost quantum numbers) in the renormalizedbasis, which are important for the true ground state. In-stead of adding noise or perturbative corrections, one canalso reserve a certain percentage of the virtual dimension D to be distributed equally over all symmetry sectors [74]. The wavefunction from which the QC-DMRG algorithmstarts has an influence on the converged energy (by gettingstuck in a local minimum) and on the rate of convergence[69, 73, 80]. The effect of the starting guess is estimatedto be an order of magnitude smaller than the effect of thechoice and ordering of the orbitals [80]. Nevertheless, itdeserves attention.One possibility is to choose a small active space tostart from, and subsequently augment this active spacestepwise with previously frozen orbitals [67], in analogy tothe infinite-system DMRG algorithm [29]. Natural orbitalsfrom a small CASSCF calculation or HF orbitals can beused to this end [80]. An alternative is to make an a prioriguess of how correlated the orbitals are. This can be donewith a DMRG calculation with small virtual dimension D , from which the approximate single-orbital entropiescan be obtained. The subsystem A is then chosen to bea single orbital in Eq. (25). The larger the single-orbitalentropy, the more it is correlated. The active space canthen be chosen and dynamically extended based on thesingle-orbital entropies [102].One can also decompose the wavefunction from a cheapCI calculation with single and double excitations into anMPS to start from [68, 80]. Another possibility is to dis-tribute D equally over the symmetry sectors, and to fillthe MPS with noise. This retrieves energies below the HFenergy well within the first macro-iteration [121, 139].To achieve a very accurate MPS quickly, it is also bestto start from calculations with relatively small virtual di-mension D , and to enlarge it stepwise [68, 80, 144]. There are many ways to set up a renormalization groupflow, and the specific setup influences the outcome. Oneconsideration of key importance in QC-DMRG is the choice
Fig. 7.
The computational details were discussed in thecaption of Fig. 5. The active space of C H , which con-sists of 28 π -orbitals, is studied both with ordered localizedorbitals (Edmiston-Ruedenberg) and canonical orbitals (re-stricted HF). The energy converges significantly faster withthe number of reduced virtual basis states D when orderedlocalized orbitals are used. and ordering of orbitals. Most molecules or active spacesare far from one-dimensional. By placing the orbitals on aone-dimensional lattice, and by assuming an MPS ansatzwith modest D , an artifical correlation length is intro-duced in the system, which can be a bad approximation.Over time, several rules of thumb have been establishedto choose and order the orbitals. Quantum information theory learns that locality is animportant concept, see section 3. The Coulomb interac-tion is however long-ranged. On the other hand, the mu-tual screening of electrons and nuclei can result in an ef-fectively local interaction. For elongated molecules suchas hydrogen chains [68, 81, 105, 107, 112, 115], polyenes[68, 78, 81, 90, 95], or acenes [84, 85, 111], which are moreor less one-dimensional, choosing a spatially local basishas turned out to be very beneficial. There are roughlythree ways to choose a local basis: symmetric orthogo-nalization as it lies closest to the original gaussian basisfunctions [84, 85, 105, 107, 115, 145], explicit localizationprocedures such as Pipek-Mezey or Edmiston-Ruedenberg[90, 111, 140, 146], and working in a biorthogonal basis[78, 105]. For the latter, the effective Hamiltonian is nothermitian anymore. The DMRG algorithm should then becorrespondingly adapted [78, 105, 147]. The adapted algo-rithm is slower and prone to convergence issues, and it istherefore better to use one of the other two localized bases[78, 105]. Fig. 7 illustrates the speed-up in energy conver-gence by using a localized basis for all-trans polyenes.
If the topology of the molecule does not provide hints forchoosing and ordering orbitals, it was investigated whetherthe Hamiltonian (1) can be of use. Several integral mea-sures have been proposed, for which a minimal bandwidthis believed to yield a good orbital order. Chan and Head-Gordon proposed to minimize the bandwidth of the one-electron integral matrix t ij of the HF orbitals [68]. Inquantum chemistry, it is often stated that the one-electronintegrals are an order of magnitude larger than the two-electron integrals, and that quantum chemistry thereforecorresponds to the small- U limit of the Hubbard model[69, 102, 148]. On the other hand, there are many two-electron integrals, and they may become important due totheir number. When other orbitals than the HF orbitalsare used, it may therefore be interesting to minimize thebandwidth of the Fock matrix [71]: F ij = t ij + X k ∈ occ (4 v ik ; jk − v ik ; kj ) . (64)Other proposed integral measures are the MP2-inspiredmatrix [72]: G ij = v ii ; jj | ǫ i − ǫ j | (65)where { ǫ i } are the HF single-particle energies, as well asseveral measures in Ref. [77]. These are the Coulomb ma-trix J ij = v ij ; ij , the exchange matrix K ij = v ij ; ji , themean-field matrix M ij = (2 J ij − K ij ), and two derivedquantities: J ′ ij = e − J ij (66) M ′ ij = e − M ij . (67)While the one-electron integrals t ij vanish when orbitals i and j belong to different molecular point group irreps, J ij and K ij do not. Ref. [77] used a genetic algorithm tofind the optimal HF orbital ordering, in order to assessthe proposed integral measures. This genetic algorithmwas expensive, which limited its usage to small test sys-tems. It favoured K ij bandwidth minimization, althoughno definite conclusions were drawn [77]. The exchange ma-trix K ij was recently used in two DMRG studies [111, 112]in conjunction with localized orbitals, because it then di-rectly reflects their overlaps and distances. DMRG can be analyzed by means of the underlying MPSansatz and quantum information theory. The latter cantell us something more than locality. Legeza and S´olyomproposed to use the single-orbital entropies to find an op-timal ordering [73]. Subsystem A is then chosen to be asingle orbital k in Eq. (25), and its entropy is denoted by S ( k ). It can be efficiently calculated in the DMRG al-gorithm, because the corresponding RDM ˆ ρ k can be builtfrom the expectation values h (1 − ˆ n k ↑ )(1 − ˆ n k ↓ ) i , h ˆ n k ↑ ˆ n k ↓ i , h ˆ n k ↑ (1 − ˆ n k ↓ ) i , and h (1 − ˆ n k ↑ )ˆ n k ↓ i , in which ˆ n kσ = ˆ a † kσ ˆ a kσ [82]. This procedure hence does not require to reorderany orbitals. The larger the single-orbital entropy S ( k ),the more orbital k is correlated. Legeza and S´olyom pro-posed to perform a small- D DMRG calculation to esti-mate S ( k ), and to place the orbitals with large S ( k ) inthe center of the chain, and the ones with small S ( k ) nearthe edges. They reasoned that orbitals close to the Fermisurface are more entangled and therefore have a largersingle-orbital entropy. Because DMRG only captures localcorrelations, these orbitals should lie close to each other.Rissler, Noack and White proposed to use the two-orbital mutual information I k,l to order the orbitals [82].In addition to the single-orbital entropies S ( k ) and S ( l ),the two-orbital entropy S ( k, l ) is also needed to calculate I k,l . It can be obtained by choosing for subsystem A thetwo orbitals k and l . S ( k, l ) can again be efficiently cal-culated in the DMRG algorithm, as its RDM can be builtfrom expectation values of operators acting on at mosttwo sites [82]. The so-called subadditivity property of theentanglement entropy dictates that: S ( k, l ) ≤ S ( k ) + S ( l ) . (68)Any entanglement between orbitals k and l reduces S ( k, l )with respect to S ( k ) + S ( l ). The two-orbital mutual in-formation is defined by: I k,l = 12 ( S ( k ) + S ( l ) − S ( k, l )) (1 − δ k,l ) ≥ , (69)and is thus a symmetric measure of the correlation be-tween orbitals k and l . Its bandwidth can be minimized,for example based on cost functions such as I = X k,l I k,l | k − l | η . (70)Rissler, Noack and White found no clear correspondencebetween I k,l and the integral measures of section 7.2. Theyobserved that I k,l is large between orbitals which belongto the same molecular point group irrep, as well as be-tween corresponding bonding and anti-bonding orbitalswith large partial occupations (far from empty or doublyoccupied) [82]. Later studies of various groups supportedthis finding and corresponding ordering [94, 95, 102, 110,115, 121]. For small molecules such as dimers, it is bestto group orbitals of the same molecular point group irrepinto blocks, and place irrep blocks of bonding and anti-bonding type next to each other. If in addition naturalorbitals (NO) are used, the orbitals within an irrep blockshould be reordered so that the ones with NO occupationnumber (NOON) closest to one, are nearest to the block oftheir bonding or anti-bonding colleagues [115]. Fig. 8 illus-trates the speed-up in energy convergence by reorderingthe point group irreps. ebastian Wouters, Dimitri Van Neck: The density matrix renormalization group for ab initio quantum chemistry 13 Fig. 8.
The computational details for N were discussed inthe caption of Fig. 6. The energy converges significantly fasterwith the number of reduced virtual basis states D when theirrep blocks of bonding and anti-bonding molecular orbitalsare placed next to each other. For elongated molecules, when the active space is studiedin a localized basis, v ij ; kl = Z d r d r φ ∗ i ( r ) φ k ( r ) φ ∗ j ( r ) φ l ( r ) | r − r | (71)vanishes faster than exponential with the separation oforbitals i and k , and the separation of orbitals j and l . By defining a threshold, below which these two-bodymatrix elements can be neglected, one can reduce thecost of the QC-DMRG algorithm in Tab. 1 to O ( L D )computational time, O ( LD ) memory, and O ( L D ) disk[27, 81, 84]. Quadratic scaling DMRG (QS-DMRG) is notvariational anymore because the Hamiltonian is altered,but the error can be controlled with the threshold. Atpresent, QC-DMRG can achieve FCI energy accuracy forabout 40 electrons in 40 highly correlated orbitals (incompact molecules) [106, 121]. With QS-DMRG, one canachieve FCI energy accuracy for 100 electrons in 100 or-bitals [81], and maybe more. It should however be re-peated, that this method relies on the topology of themolecule, and exploits the fact that DMRG works verywell for one-dimensional systems. QC-DMRG can at present achieve FCI energy accuracyfor about 40 electrons in 40 orbitals. The static correla-tion in active spaces up to this size can hence be resolved,while dynamic correlation has to be treated a posteri-ori. Luckily, QC-DMRG allows for an efficient extraction of the two-body RDM (2-RDM) [88, 90]. The 2-RDM isnot only required to calculate analytic nuclear gradients[68, 117], but also to compute the gradient and the Hes-sian in CASSCF [16]. It is therefore natural to introducea CASSCF variant with DMRG as active space solver,DMRG-CASSCF or DMRG-SCF [89, 90, 92]. Static cor-relation can be treated with DMRG-SCF. To add dynamiccorrelation as well, three methods have been introduced.With more effort, the 3-RDM and some specific con-tracted 4-RDMs can be extracted from DMRG as well.These are required to apply second-order perturbationtheory to a CASSCF wavefunction, called CASPT2, ininternally contracted form. The DMRG variant is calledDMRG-CASPT2 [104, 115, 117].Based on a CASSCF wavefunction, a configuration in-teraction expansion can be introduced, called MRCI. Re-cently, an internally contracted MRCI variant was pro-posed, which only requires the 4-RDM [116]. By approxi-mating the 4-RDM with a cumulant reconstruction fromlower-rank RDMs, DMRG-MRCI was made possible [116].Yet another way is to perform a canonical transforma-tion (CT) on top of an MR wavefunction, in internallycontracted form. When an MPS is used as MR wavefunc-tion, the method is called DMRG-CT [95, 96, 109].
In addition to ground states, DMRG can also find excitedstates. By projecting out lower-lying eigenstates [121], orby targeting a specific energy with the harmonic Davidsonalgorithm [84], DMRG solves for a particular excited state.In these state-specific algorithms, the whole renormalizedbasis is used to represent one single eigenstate. In state-averaged DMRG, several eigenstates are targeted at onceto prevent root-flipping. Their RDMs are weighted andsummed to perform the DMRG renormalization step [149].The renormalized basis then represents several eigenstatessimultaneously.DMRG linear response theory (DMRG-LRT) [93] al-lows to calculate response properties, as well as excitedstates. Once the ground state has been found, the MPStangent vectors to this optimized point can be used asan (incomplete) variational basis to approximate excitedstates [93, 119, 150–153]. As the tangent vectors to an op-timized Slater determinant yield the configuration inter-action with singles (CIS), also called the Tamm-Dancoffapproximation (TDA), for HF theory [4], the same namesare used for DMRG: DMRG-CIS or DMRG-TDA. Thevariational optimization in an (incomplete) basis of MPStangent vectors can be extended to higher-order tangentspaces as well. DMRG-CISD, or DMRG configuration in-teraction with singles and doubles, is a variational approx-imation to target both ground and excited states in thespace spanned by the MPS reference and its single anddouble tangent spaces [152].By linearizing the time-dependent variational principlefor MPS [154], the DMRG random phase approximation(DMRG-RPA) is found [119, 152, 153, 155], again in com-plete analogy with RPA for HF theory.
Two other related ansatzes have been employed in quan-tum chemistry: the TTNS [63, 64, 112] and the complete-graph TNS (CGTNS) [100, 101]: | Ψ i = X { n k } Y i Imposing SU ( ), U ( ), and P symmetry. The physical basis states of orbital k correspond to thefollowing symmetry eigenstates: |−i → | s = 0; s z = 0; N = 0; I = I i (89) |↑i → | s = 12 ; s z = 12 ; N = 1; I = I k i (90) |↓i → | s = 12 ; s z = − 12 ; N = 1; I = I k i (91) |↑↓i → | s = 0; s z = 0; N = 2; I = I i . (92)The virtual basis states are also labeled by the quantumnumbers of SU ( ) ⊗ U ( ) ⊗ P : | α i → | jj z N Iα i . (93)The equivalent of Eq. (84) is then A [ i ] ( ss z NI )( j L j zL N L I L α L );( j R j zR N R I R α R ) = h j L j zL ss z | j R j zR i δ N L + N,N R δ I L ⊗ I,I R T [ i ] ( sNI )( j L N L I L α L );( j R N R I R α R ) . (94)The SU ( ), U ( ), and P symmetries are locally imposed bytheir Clebsch-Gordan coefficients. These express nothingelse than resp. local allowed spin recoupling, local parti-cle number conservation, and local point group symmetryconservation. The index α keeps track of the number of re-duced renormalized basis states with symmetry ( j, N, I ).This equation again encompasses block-sparsity and in-formation compression.The desired global symmetry ( S G , N G , I G ) can be im-posed with the singlet-embedding strategy, see Fig. 9. As-sume that the MPS is part of a larger DMRG chain, towhich it is connected on its left and right ends. On theleft end, there is only one irrep ( j L , N L , I L ) = (0 , , I )in the virtual bond, which has virtual dimension 1. Onthe right end, there is also only one irrep ( j R , N R , I R ) =( S G , N G , I G ) in the virtual bond, which also has reduced virtual dimension 1. Eq. (94) and Fig. 9 imply that theaddition of an extra orbital to the left renormalized basisis repeated from symmetry sector (0 , , I ) at boundary 0to symmetry sector ( S G , N G , I G ) at boundary L .Towards the middle of this embedded MPS chain, thereduced virtual dimension has to grow exponentially forthe MPS to represent a general symmetry-adapted FCIstate. To make the MPS ansatz in Eq. (94) of practi-cal use, the total reduced virtual dimension per bond hasto be truncated. The extrapolation scheme (57) is shownfor the one-dimensional Hubbard model [148] with openboundary conditionsˆ H = − L − X i =1 X σ (cid:16) ˆ a † iσ ˆ a i +1 σ + ˆ a † i +1 σ ˆ a iσ (cid:17) + U L X i =1 ˆ a † i ↑ ˆ a i ↑ ˆ a † i ↓ ˆ a i ↓ (95)in Fig. 10. The SU ( ) ⊗ U ( ) ⊗ C symmetry introducesblock-sparsity and information compression. The latter Fig. 10. Convergence of the one-dimensional Hubbard modelwith open boundary conditions, L = 36 sites, N = 22 electrons, U = 6, in the spin singlet state. The convergence scheme (57) istested for a DMRG code without any imposed symmetries andfor CheMPS2 [121] with imposed SU ( ) ⊗ U ( ) ⊗ C symmetry. κ is the parameter C of Eq. (57), and D denotes the totalnumber of renormalized basis states at each virtual bond. For CheMPS2 , these are the reduced ones. can be seen in the faster energy convergence with the num-ber of reduced virtual basis states.Due to the abelian point group symmetry P , the matrixelements h ij ; kl of the Hamiltonian (8) are only nonzero if I i ⊗ I j = I k ⊗ I l . If P is nontrivial, this considerably reducesthe number of terms in the construction of the complemen-tary renormalized operators, and in the multiplication ofthe effective Hamiltonian with a trial vector.The operators ˆ b † cγ = ˆ a † cγ (96)ˆ b cγ = ( − − γ ˆ a c − γ (97)for orbital c correspond to resp. the ( s = , s z = γ, N =1 , I c ) row of irrep ( s = , N = 1 , I c ) and the ( s = , s z = γ, N = − , I c ) row of irrep ( s = , N = − , I c ) [172]. ˆ b † and ˆ b are hence both doublet irreducible tensor operators.As described in section 9.2, this fact permits exploitationof the Wigner-Eckart theorem for operators and (com-plementary) renormalized operators. Contracting termsof the type (94) and (96)-(97) can be done by implic-itly summing over the common multiplets and recouplingthe local, virtual and operator spins. As is shown in Refs.[107, 139], (complementary) renormalized operators thenformally consist of terms containing Clebsch-Gordan co-efficients and reduced tensors. In an actual implementa-tion such as Block [106] or CheMPS2 [107, 121], onlythe reduced tensors need to be calculated, and Wigner 3-jsymbols or Clebsch-Gordan coefficients are never used. ebastian Wouters, Dimitri Van Neck: The density matrix renormalization group for ab initio quantum chemistry 17 Table 2. Overview of QC-DMRG codes.Name AuthorsWhite [27, 82]Mitrushenkov [67, 105] Block ( a ) Chan & Sharma [68, 106] Qc-Dmrg-Budapest Legeza [69, 113] Qc-Dmrg-Eth Reiher [97, 108]Zgid [87, 89]Xiang [98] Rego Kurashige & Yanai [94, 114] CheMPS2 ( b ) Wouters [107, 121] Qc-Maquis Keller & Reiher [135] ( a ) Block is freely available from [174]. ( b ) CheMPS2 is freely available from [175]. 10 QC-DMRG codes and studied systems Table 2 gives an overview of the currently existing QC-DMRG codes. Two of them are freely available, Block and CheMPS2 . Four codes have SU ( ) spin symmetry:Zgid’s code, Rego , Block , and CheMPS2 . The formertwo explicitly retain entire multiplets at each virtual bond,while the latter two exploit the Wigner-Eckart theoremto work with a reduced renormalized basis and reducedrenormalized operators, see section 9.Two parallellization strategies are currently used: pro-cesses can become responsible of certain site indices of the(complementary) renormalized operators [74], or of certainsymmetry blocks in the virtual bonds [94]. For condensed-matter Hamiltonians, a real-space parallellization strategyhas appeared recently [173], which might also be useful forQC-DMRG.Many properties of many systems have been studied.QC-DMRG is of course able to calculate the ground stateenergy, but also excited state energies [69, 71, 79, 84, 90,117, 119, 121, 125], avoided crossings [64, 71, 79, 121], spinsplittings [85–87, 96, 99, 100, 106, 107, 118, 121, 122, 127],polyradical character by means of the NOON spectrum[85, 92, 111], static and dynamic polarizabilities [93, 107],static second hyperpolarizabilities [107], particle-particle,spin-spin, and singlet diradical correlation functions [85,106, 111, 116], as well as expectation values based on the1- or 2-RDM such as spin densities [108, 124] and dipolemoments [71].The systems which have been studied range from atomsand first-row dimers to large transition metal clusters and π -conjugated hydrocarbons. Several of them have repeat-edly received attention in the QC-DMRG community: – H O [27, 68–70, 73, 74, 76, 97, 98, 109, 112, 119] wasalready the subject of several FCI studies, due to itsnatural abundance and small number of electrons. – Hydrogen chains [68, 81, 88, 89, 105, 107, 112, 115]:these one-dimensional systems exhibit large static cor-relation at stretched geometries. They are optimal test-cases for QC-DMRG. – All-trans polyenes [68, 78, 81, 90, 95, 116, 119]: theyare also one-dimensional, with a large MR character. – N [67, 68, 72, 73, 75, 82, 83, 109, 112, 113, 115, 116]was already the subject of several FCI studies, dueto its MR character at stretched bond lengths and itssmall number of electrons. – Cr [67, 77, 80, 94, 104, 106, 112, 115] is only foundto be bonding at the CASPT2 level. A complete basisset extrapolation of DMRG-CASPT2 calculations inthe cc-pwCV(T,Q,5)Z basis, correlating 12 electronsin 28 orbitals, was needed to retrieve an acceptabledissociation energy [104]. – [Cu O ] [86, 94, 95, 102] requires accurate descrip-tions of both static and dynamic correlation along itsisomerization coordinate. DMRG-CT, correlating 28electrons in 32 orbitals, showed that the bis( µ -oxo) iso-mer is more stable than the µ − η : η peroxo isomer[95].Other QC-DMRG studies treat – the avoided crossings in LiF [64, 71], CsH [79, 113],and C [121] – the static correlation due to π -conjugation in acenes[84, 85, 111], poly(phenyl) carbenes [92, 99], perylene[109], graphene nanoribbons [111], free base porphyrin[96, 116], and spiropyran [117] – transition metal clusters such as [Fe S (SCH ) ] − [106,119], [Fe(NO)] [108, 110], Mn CaO in photosystemII [114], the dinuclear oxo-bridged complexes [Fe OCl ] − and [Cr O(NH ) ] [122], diferrate [H Fe O ] [126],and oxo-Mn(Salen) [127] – molecules with heavy elements, for which relativisticeffects become important, such as CsH [79, 113], thecomplexation of CUO with four Ne or Ar atoms [118],and the binding energy of TlH [120]For transition metal clusters, QC-DMRG is currently theonly viable choice due to the large active spaces whichhave to be handled. 11 Conclusion The DMRG algorithm is well understood by means ofthe underlying MPS wavefunction. This allows to assessDMRG with concepts from quantum information theory.Accurate extrapolation schemes are known for the evolu-tion of the variational energy with increasing virtual di-mension D, or with decreasing discarded weight. The useof symmetry to reduce the computational cost is also wellunderstood. Most progress can still be made in the orbitalchoice and ordering for nontrivial orbital topologies.The 2-RDM can be extracted efficiently from QC-DMRG,and is required to calculate the gradient and the Hessianin CASSCF. QC-DMRG is therefore an ideal candidateto replace the FCI solver in CASSCF. DMRG-SCF, asthe method is called, can resolve the static correlationin active spaces of up to 40 electrons in 40 orbitals. Sev-eral dynamical correlation theories for CASSCF have beenused with DMRG-SCF as well: DMRG-CASPT2, DMRG-MRCI, and DMRG-CT. 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