[Paper Review] Accurate variational electronic structure calculations with the density matrix renormalization group

Sebastian Wouters|arXiv (Cornell University)|May 6, 2014

Quantum many-body systems2 references3 citations

TL;DR

This paper presents an accurate variational approach to electronic structure calculations using the density matrix renormalization group (DMRG) with matrix product state (MPS) wavefunctions. By exploiting symmetries such as SU(2) spin and U(1) particle number, and introducing non-abelian symmetry adaptations, the method achieves high accuracy and efficiency, enabling precise treatment of strong correlation in systems like hydrogen chains and the carbon dimer.

ABSTRACT

During the past 15 years, the density matrix renormalization group (DMRG) has become increasingly important for ab initio quantum chemistry. The underlying matrix product state (MPS) ansatz is a low-rank decomposition of the full configuration interaction tensor. The virtual dimension of the MPS controls the size of the corner of the many-body Hilbert space that can be reached. Whereas the MPS ansatz will only yield an efficient description for noncritical one-dimensional systems, it can still be used as a variational ansatz for other finite-size systems. Rather large virtual dimensions are then required. The two most important aspects to reduce the corresponding computational cost are a proper choice and ordering of the active space orbitals, and the exploitation of the symmetry group of the Hamiltonian. By taking care of both aspects, DMRG becomes an efficient replacement for exact diagonalization in quantum chemistry. DMRG and Hartree-Fock theory have an analogous structure. The former can be interpreted as a self-consistent mean-field theory in the DMRG lattice sites, and the latter in the particles. It is possible to build upon this analogy to introduce post-DMRG methods. Based on an approximate MPS, these methods provide improved ansätze for the ground state, as well as for excitations. Exponentiation of the single-particle (single-site) excitations for a Slater determinant (an MPS with open boundary conditions) leads to the Thouless theorem for Hartree-Fock theory (DMRG), an explicit nonredundant parameterization of the entire manifold of Slater determinants (MPS wavefunctions). This gives rise to the configuration interaction expansion for DMRG. The Hubbard-Stratonovich transformation lies at the basis of auxiliary field quantum Monte Carlo for Slater determinants. An analogous transformation for spin-lattice Hamiltonians allows to formulate a promising variant for MPSs.

Motivation & Objective

• To develop a systematically improvable, variational electronic structure method for strongly correlated systems using the DMRG algorithm.
• To enhance computational efficiency and accuracy by incorporating exact symmetries (SU(2), U(1), abelian point groups) into the MPS ansatz.
• To enable accurate treatment of multireference systems with strong static correlation, such as the carbon dimer and hydrogen chains.
• To extend DMRG beyond ground-state energy by introducing post-DMRG methods for excitations and improved ansätze.
• To explore the feasibility of uniform MPS in the thermodynamic limit for one-dimensional molecular systems.

Proposed method

• Utilizes matrix product state (MPS) wavefunctions as a low-rank tensor decomposition of the full CI wavefunction, with the virtual dimension D controlling correlation capture.
• Employs symmetry-adapted MPS by constructing wavefunctions that are eigenstates of the Hamiltonian's symmetry group, using Clebsch-Gordan coefficients and reduced tensors.
• Applies the Wigner-Eckart theorem to factorize the MPS, introducing block-sparsity and reducing memory and computational cost.
• Implements the CheMPS2 DMRG code with support for SU(2) spin, U(1) particle number, and abelian point group symmetries.
• Introduces a Thouless-type parameterization for DMRG via exponentiation of single-site excitations, enabling nonredundant parameterization of the MPS manifold.
• Explores auxiliary-field quantum Monte Carlo variants for MPS using Hubbard-Stratonovich transformations on spin-lattice Hamiltonians.

Experimental results

Research questions

• RQ1 How can symmetry exploitation in the MPS ansatz reduce computational cost while preserving accuracy in DMRG calculations?
• RQ2 What is the optimal orbital ordering and choice for achieving fast convergence in QC-DMRG for molecular systems?
• RQ3 Can post-DMRG methods based on configuration interaction expansions improve DMRG ground-state and excited-state descriptions?
• RQ4 How does the systematic bias in DMRG energies behave with increasing virtual dimension D, and can it be extrapolated reliably?
• RQ5 Is a uniform MPS ansatz in the thermodynamic limit viable for one-dimensional ab initio systems with exponentially decaying correlations?

Key findings

• The CheMPS2 implementation successfully resolves low-lying states of the carbon dimer with multireference character by exploiting symmetry sectors.
• For hydrogen chains, QC-DMRG with small virtual dimensions (D) achieves numerical convergence due to effective electron screening, enabling accurate longitudinal response properties.
• The use of symmetry leads to significant memory and time savings through block-sparsity and information compression, especially for non-abelian groups.
• Auxiliary-field quantum Monte Carlo with MPS walkers shows promise as a post-DMRG method for improved energy and excitation energy estimates.
• Systematic bias in DMRG energies with increasing D can be estimated and potentially extrapolated, supporting reliable convergence monitoring.
• A uniform MPS ansatz is feasible for one-dimensional systems like polyenes due to exponential decay of correlations, enabling thermodynamic limit studies.