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Chemistry in 1d via DMRG: Benchmarks for small atoms, ions, and molecules

Lucas O. Wagner, Edwin Miles Stoudenmire, Kieron Burke, Steven R. White

Published Feb 21, 2012• 2 citations
OpenAlex

Abstract

The density matrix renormalization group (DMRG) method, an efficient solver of lattice models in 1d, has been adapted to solve real-space problems. We use DMRG to study the behavior of soft-Coulomb interacting matter, reporting accurate benchmarks for small systems. We compare with Hartree-Fock and the local density approximation, noting similarities and differences with 3d. We study how correlation grows as bonds stretch.

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The density matrix renormalization group (DMRG) method, an efficient solver of lattice models in 1d, has been adapted to solve real-space problems. We use DMRG to study the behavior of soft-Coulomb interacting matter, reporting accurate benchmarks for small systems. We compare with Hartree-Fock and the local density approximation, noting similarities and differences with 3d. We study how correlation grows as bonds stretch.
The ab-initio density matrix renormalization group (DMRG) is a tool that can be applied to a wide variety of interesting problems in quantum chemistry. Here, we examine the density matrix renormalization group from the vantage point of the quantum chemistry user. What kinds of problems is the DMRG well-suited to? What are the largest systems that can be treated at practical cost? What sort of accuracies can be obtained, and how do we reason about the computational difficulty in different molecules? By examining a diverse benchmark set of molecules: π-electron systems, benchmark main-group and transition metal dimers, and the Mn-oxo-salen and Fe-porphine organometallic compounds, we provide some answers to these questions, and show how the density matrix renormalization group is used in practice.
An implementation of the single-site density matrix renormalization group (DMRG) approach for the quantum-chemical application to calculate approximate as well as exact electronic ground-state energies and wave functions of atoms and molecules is presented.The algorithm is formulated in terms of the variational and sizeconsistent class of wave functions, matrix product states, and their operator analogues, matrix product operators.A procedure to construct quantum-chemical Hamiltonian operators with long-range Coulomb interactions as matrix product operators is developed.The many-electron Hamiltonian operators are represented as sums and products of one-electron operators.For the efficient matrix product operator construction of quantum-chemical Hamiltonians, different basis spaces including discrete position and momentum space as well as molecular and atomic orbitals are investigated.Discrete position space yields the best matrix product operator size scaling, but introduces artificial long-range couplings in the kinetic energy term of the Hamiltonian which are a consequence of the DMRG algorithm in terms of matrix product states.The huge amount of discrete space points needed for an accurate description poses a problem because of the electron-electron interaction.Therefore, a momentum space or Fourier series expansion of the electron-electron interaction is shown which makes it possible to compress its matrix product operator representation.DMRG applied to quantum-chemical Hamiltonians in matrix product operator form can serve as means to directly (without mean-field treatment in advance) investigate quantum-chemical problems and find the analytically exact ground-state solution of small atoms and molecules.The next basis space analyzed in detail is the atomic orbital basis.The matrix product operator construction and the direct application of the DMRG method in this basis is examined.A physical compression scheme based on a singular value decomposition of the electron-electron interaction in the atomic orbital basis is demonstrated.Furthermore, DMRG calculations in the atomic orbital basis for certain systems where mean-field theory yields qualitatively wrong results and where higher-order approximative correction methods like truncated configuration interaction and coupled cluster cannot fully capture strong electron correlation effects are compared to results obtained in the molecular orbital basis.The comparison shows that atomic orbitals describe the physical situation well and thus lead to a tremendous reduction of the matrix product state dimension and the computational effort.i
This study integrates a Hermitian coupled-cluster downfolding technique with the Density Matrix Renormalization Group (DMRG) method to accurately treat both static and dynamic correlations in complex electronic systems. By calculating ground-state energies of active-space Hamiltonians via DMRG, we achieve efficient and accurate simulations of strongly correlated systems, demonstrated on molecular benchmarks including N 2, benzene, porphyrin, and tetramethyleneethane. This combined approach offers a promising advancement in computational chemistry for complex chemical processes.
We demonstrate how to parallelize the density matrix renormalization group (DMRG) algorithm in real space through a straightforward modification of serial DMRG. This makes it possible to apply at least an order of magnitude more computational power to challenging simulations, greatly accelerating investigations of two-dimensional systems and large parameter spaces. We discuss details of the algorithm and present benchmark results including a study of valence-bond-solid order within the square-lattice Q2{Q}_{2}Q2​ model and N\'eel order within the triangular lattice Heisenberg model. The parallel DMRG algorithm also motivates an alternative canonical form for matrix product states.
The recently proposed Clifford augmented density matrix renormalization group (CA-DMRG) method seamlessly integrates Clifford circuits with matrix product states and takes advantage of the expression power from both. CA-DMRG has been shown to be able to achieve higher accuracy than standard DMRG on commonly used lattice models, with only moderate computational overhead compared to the latter. In this work, we propose an efficient scheme in CA-DMRG to deal with ab initio quantum chemistry Hamiltonians and apply it to study several molecular systems. Our numerical results show that CA-DMRG can reach higher accuracy than DMRG using the same bond dimension, pointing out a promising route to push the boundary of solving ab initio quantum chemistry with strong static correlations.