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CheMPS2: a free open-source spin-adapted implementation of the density matrix renormalization group for ab initio quantum chemistry
Sebastian Wouters, Ward Poelmans, Paul W. Ayers, Dimitri Van Neck
TL;DR
Quantum-chemistry DMRG needs efficient implementations that extend active spaces beyond exact diagonalization while preserving accurate variational calculations. CheMPS2 provides a free open-source spin-adapted implementation with DMRG-SCF, and applies it to carbon-dimer dissociation curves and 1s core correlation.
Problem
Exact diagonalization is restricted by exponential Hilbert-space growth, limiting quantum-chemistry calculations to small active spaces.
Method
CheMPS2 implements two-site spin-adapted DMRG and an augmented-Hessian Newton-Raphson DMRG-SCF method with exact Hessian.
Results
The study calculates the 12 lowest carbon-dimer bond dissociation curves and assesses 1s core correlation using specified DMRG and DMRG-SCF levels.
Takeaways & Limitations
CheMPS2 provides a freely available implementation for DMRG-based ab initio quantum chemistry calculations.
Takeaways & Limitations
Optimal orbital choice and ordering remain under debate, motivating future use of two-orbital mutual information.
Abstract
from arXiv · showhide
The density matrix renormalization group (DMRG) has become an indispensable numerical tool to find exact eigenstates of finite-size quantum systems with strong correlation. In the fields of condensed matter, nuclear structure and molecular electronic structure, it has significantly extended the system sizes that can be handled compared to full configuration interaction, without losing numerical accuracy. For quantum chemistry (QC), the most efficient implementations of DMRG require the incorporation of particle number, spin and point group symmetries in the underlying matrix product state (MPS) ansatz, as well as the use of so-called complementary operators. The symmetries introduce a sparse block structure in the MPS ansatz and in the intermediary contracted tensors. If a symmetry is non-abelian, the Wigner-Eckart theorem allows to factorize a tensor into a Clebsch-Gordan coefficient and a reduced tensor. In addition, the fermion signs have to be carefully tracked. Because of these challenges, implementing DMRG efficiently for QC is not straightforward. Efficient and freely available implementations are therefore highly desired. In this work we present CheMPS2, our free open-source spin-adapted implementation of DMRG for ab initio QC. Around CheMPS2, we have implemented the augmented Hessian Newton-Raphson complete active space self-consistent field method, with exact Hessian. The bond dissociation curves of the 12 lowest states of the carbon dimer were obtained at the DMRG(28 orbitals, 12 electrons, D$_{\mathsf{SU(2)}}$=2500)/cc-pVDZ level of theory. The contribution of $1s$ core correlation to the $X^1Σ_g^+$ bond dissociation curve of the carbon dimer was estimated by comparing energies at the DMRG(36o, 12e, D$_{\mathsf{SU(2)}}$=2500)/cc-pCVDZ and DMRG-SCF(34o, 8e, D$_{\mathsf{SU(2)}}$=2500)/cc-pCVDZ levels of theory.
PROGRAM SUMMARY
CheMPS2 is a free, open-source implementation of spin-adapted DMRG for ab initio quantum chemistry. It extends feasible active spaces beyond exact diagonalization while retaining a variational energy upper bound.
- CheMPS2 provides a free, open-source spin-adapted DMRG implementation for ab initio quantum chemistry.
- Exact diagonalization handles only systems up to 18 electrons in 18 orbitals because the many-body Hilbert space grows exponentially.
- DMRG extends numerically exact solutions to active spaces of about 40 electrons in 40 orbitals.
- A carbon-dimer dissociation point with 28 orbitals and D_SU(2)=2500 takes about 8 hours and 6 GB of RAM on a 16-core node, with energy 0.1 mEh above convergence.
1. Introduction
DMRG addresses multireference quantum-chemistry problems that exceed full configuration interaction by using an MPS-based variational approach. The introduction situates CheMPS2 among symmetry-aware, excited-state, and tensor-network methods for larger active spaces.
- Full configuration interaction is limited to about 18 electrons in 18 orbitals, whereas DMRG reaches about 40 electrons in 40 orbitals.
- DMRG optimization of an MPS provides an energy upper bound that can be systematically improved by increasing the virtual dimension.
- DMRG-SCF replaces the FCI active-space solver with DMRG and uses the active-space two-body reduced density matrix for orbital gradients and Hessians.
- State-specific, state-averaged, and linear-response DMRG approaches target ground and excited states in different variational representations.
- Tree and complete-graph tensor networks can require smaller virtual dimensions, but MPS optimization is currently more efficient for ab initio quantum chemistry.
2. DMRG for ab initio quantum chemistry
The paper formulates DMRG through an MPS ansatz whose virtual dimensions control the accessible Hilbert space. CheMPS2 optimizes neighboring sites in canonical form, using gauge freedom to organize the calculation.
- DMRG is formulated as variational optimization of an open-boundary MPS.
- Orbital occupations index physical states, while bond or virtual indices connect neighboring MPS tensors and control the accessible Hilbert-space portion.
- MPS gauge freedom permits equivalent tensor representations without changing the wavefunction.
- CheMPS2 uses a two-site DMRG algorithm that simultaneously optimizes neighboring tensors at each micro-iteration.
- Canonical forms left-normalize tensors before the optimized pair and right-normalize tensors after it.
the effective Hamiltonian equation.
CheMPS2 solves the local effective-Hamiltonian problem within two-site DMRG sweeps, then controls accuracy through truncation, orbital ordering, and extrapolation procedures.
- the effective Hamiltonian equation.: The local eigenvalue equation is derived by varying the Lagrangian with respect to the complex conjugate of the two-site tensor.
- the effective Hamiltonian equation.: Canonical choices remove the overlap matrix, after which Davidson’s algorithm finds the lowest eigenpair and singular-value decomposition truncates the solution.
- the effective Hamiltonian equation.: During sweeps, CheMPS2 constructs the effective problem, solves it, decomposes the solution, truncates singular values, and updates neighboring MPS tensors.
- the effective Hamiltonian equation.: Complementary operators and stored renormalized operators make effective-Hamiltonian actions feasible without fully constructing the Hamiltonian.
- 2.6. Convergence: Orbital choice and ordering affect local-minimum avoidance and convergence, while mutual information is proposed for future orbital ordering.
- 2.6. Convergence: Two extrapolation schemes assess convergence with increasing renormalized basis dimension, including energy versus discarded weight.
3. Symmetry-adapted DMRG
CheMPS2 enforces SU(2) spin, U(1) particle-number, and abelian point-group symmetries by representing MPS tensors as irreducible tensor operators and separating Clebsch-Gordan coefficients from reduced tensors.
- Global symmetries: CheMPS2 implements SU(2) total electronic spin, U(1) particle number, and abelian point-group symmetry for the MPS wavefunction.The supported real-valued point groups are C1, Ci, C2, Cs, D2, C2v, C2h, and D2h.
- Symmetry-adapted tensors: MPS site tensors are required to be irreducible tensor operators of the total symmetry group.Local and virtual basis states are represented using good spin, spin projection, particle number, and point-group irrep quantum numbers.
- Reduced tensors: Each irreducible tensor operator factorizes into a Clebsch-Gordan coefficient and a reduced tensor through the Wigner-Eckart theorem.This factorization enables contractions using reduced tensors rather than explicit spin projections.
- Block sparsity: Clebsch-Gordan coefficients encode local spin recoupling, particle conservation, and point-group symmetry conservation.Zero coefficients imply that the corresponding reduced-tensor blocks need not be stored.
- Global sector selection: The global target sector is selected by fixing the leftmost virtual irrep to the trivial sector and the rightmost virtual irrep to the desired spin, particle number, and point-group symmetry.This is the singlet-embedding strategy.
- Two-site optimization: The reduced two-site object is grouped by intermediate spin, particle number, and point-group irrep before singular-value decomposition.The largest retained singular values determine which reduced basis states survive truncation.
4. CheMPS2 library
The CheMPS2 library provides configurable DMRG calculations, excited-state targeting, sparse-tensor parallelization, and a state-specific DMRG-SCF implementation using an exact-Hessian augmented-Hessian Newton-Raphson method.
- Library inputs: CheMPS2 accepts molecular Hamiltonian matrix elements with eightfold permutation symmetry and preserves SU(2) total electronic spin.Users can generate and supply these matrix elements from their preferred molecular electronic structure program.
- Calculation setup: A CheMPS2 calculation combines a Hamiltonian, a symmetry-sector specification, and a convergence scheme to determine the DMRG computation.The selected sector includes total electronic spin, particle number, and point-group symmetry.
- Convergence scheme: Each convergence-scheme instruction specifies the retained basis dimension D, energy threshold Econv, maximum sweeps Nmax, and noise prefactor γnoise.Noise is added before reduced two-site tensor decomposition, while Schmidt spectra are truncated to D.
- Convergence outcome: CheMPS2 returns the minimal variational energy encountered during the executed micro-iterations.Macro-iterations stop when the energy difference falls below Econv or when Nmax is reached.
- Excited states: Excited states are computed with a state-specific algorithm that projects out lower-lying states within the chosen SU(2) ⊗ U(1) ⊗ P sector.The current MPS is stored before solving for the next excitation.
- Parallelization: The library parallelizes sparse tensor contractions and the construction of renormalized operators with OpenMP.These operations occur during effective-Hamiltonian application and between micro-iterations.
- DMRG-SCF: The implemented DMRG-SCF routine is a state-specific augmented-Hessian Newton-Raphson CASSCF method with exact Hessian.It returns a converged DMRG-SCF energy when the orbital-rotation gradient reaches a predefined threshold.
5. Carbon dimer
The carbon dimer calculations use symmetry-sector analysis, optimized orbital ordering, and extrapolation to characterize low-lying states and core correlation. The results reveal several avoided crossings and show that 1s core correlation requires an expanded orbital space and basis set.
- Motivation: The carbon dimer’s low-lying states are quasi-degenerate, with avoided crossings between states sharing spin and D∞h symmetry.These features make the system a demanding test case for electronic-structure methods.
- State identification: FCI coefficients were extracted to distinguish low-lying states and identify their symmetry labels, including 1Ag, 3B1u, and 3Δu states.CheMPS2 used D2h symmetry to obtain the 12 targeted states.
- Irrep ordering: Bonding and anti-bonding orbitals were grouped on the DMRG lattice because the standard D2h irrep ordering was not optimal for convergence.The grouped ordering was used for the reported calculations.
- Core correlation: The 1s core correlation is captured only at the DMRG(36o, 12e, D_SU(2)=2500)/cc-pCVDZ level of theory.Without the necessary orbital freedom it cannot be captured; the DMRG-SCF(34o, 8e, D_SU(2)=2500)/cc-pCVDZ curve differs by about 2mEh in non-parallelity.
- Core correlation: Extra basis-set freedom is required at small interatomic distances to describe the more complicated core dynamics in the united-atom limit.The cc-pCVDZ curves therefore behave differently from the cc-pVDZ curves at short distances.
6. Summary
The paper presents CheMPS2 as an efficient, symmetry-adapted DMRG implementation for ab initio quantum chemistry, with applications to carbon-dimer states and core correlation. It also identifies orbital ordering and future parallelization as ongoing practical considerations.
- DMRG in quantum chemistry: DMRG can act as a large-active-space FCI solver, extending numerically exact calculations beyond the small systems accessible to exact diagonalization.The paper distinguishes this role from using DMRG as an approximate MPS wavefunction for building excitations.
- Implementation: Symmetry adaptation improves DMRG performance through sparse block structure and data compression for non-abelian groups.The Wigner–Eckart theorem underlies both effects in the MPS ansatz.
- Implementation: CheMPS2 provides a high-level DMRG interface and a DMRG-SCF implementation intended to help users understand and adapt the provided tests.The DMRG-SCF algorithm is implemented in CheMPS2::CASSCF.
- Applications: The carbon-dimer application calculated the 12 lowest bond dissociation curves at the DMRG(28o, 12e, D_SU(2)=2500)/cc-pVDZ level.The supplied passage reports the state count and computational level, while the associated figure passages identify triplet-state curve sets in the cc-pVDZ basis.
- Applications: The study assessed 1s core correlation using DMRG(36o, 12e, D_SU(2)=2500)/cc-pCVDZ calculations and relative energies referenced to that level.The comparison includes DMRG-SCF(34o, 8e, D_SU(2)=2500)/cc-pCVDZ.
- Future directions: Future work includes incorporating two-orbital mutual information and its derivatives to optimize orbitals and ordering, alongside an MPI implementation distributing computations across processors.Orbital choice and ordering affect convergence and susceptibility to local minima.
Appendix A. Reduced tensors
The appendix describes how renormalized operators are factorized into spin-adapted reduced tensors using Clebsch–Gordan coefficients and SU(2) representation theory.
- Canonical structure: Renormalized partial Hamiltonian terms are considered during sweeps with left-normalized tensors to the left and right-normalized tensors to the right.This canonical structure is the setting for the appendix’s reduced-tensor construction.
- Reduced tensors: Equation (A.1) is factorized into Clebsch–Gordan coefficients and a reduced spin-1/2 L-tensor.The appendix identifies the reduced tensor as the object carrying the remaining spin-adapted information.
- SU(2) decomposition: For second-quantized operators on different sites, the renormalized operator decomposes into terms with spin-0 and spin-1 reduced tensors.This decomposition follows from SU(2) representation theory.