CheMPS2: A free open-source spin-adapted implementation of the density matrix renormalization group for ab initio quantum chemistry

Abstract

The density matrix renormalization group (DMRG) has become an indispensable numerical tool to find exact eigenstates of finitesize 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 SU(2) =2500)/cc-pVDZ level of theory. The contribution of 1s core correlation to the X 1 Σ + g bond dissociation curve of the carbon dimer was estimated by comparing energies at the DMRG(36o, 12e, D SU(2) =2500)/cc-pCVDZ and DMRG-SCF(34o, 8e, D SU(2) =2500)/cc-pCVDZ levels of theory.Keywords: density matrix renormalization group, matrix product state, SU(2) spin symmetry, abelian point group symmetry, ab initio quantum chemistry PROGRAM SUMMARYManuscript Title: CheMPS2: a free open-source spin-adapted implementation of the density matrix renormalization group for ab initio quantum chemistry

The ab-initio density matrix renormalization group in practice

“…As a practical note, the "noise" in the initial sweeps introduces random error into the DMRG wavefunction and can cause problems with extrapolation when using the data at small M. One can avoid this by carrying out a few sweeps at each M without noise before proceeding to higher M. 10 Alternatively, after the calculation is converged, we can carry out a "reverse schedule" where M starts at its maximum value and is then lowered to its smallest value, to obtain good data at small M (which is often biased by the warmup sweep). We call this the reverse schedule.…”

“…54,56 The entanglement structure of the MPS has even generated a niche in interpretative quantum chemistry. 52,[57][58][59][60] With the advent of publicly available quantum chemistry DMRG codes, 10,[20][21][22] the DMRG is transitioning into a method available not only to expert developers but also to the informed user. As with all complex methods, there is some degree of experience required to use it effectively.…”

“…[3][4][5][6][7][8][9][10][11][12][13][14][15] Developing from its roots in condensed matter, its earliest application to chemical problems used the semi-empirical π-electron Pariser-Parr-Pople Hamiltonian. [16][17][18] White and Martin introduced the first efficient formulation of the DMRG algorithm for ab-initio Hamiltonians (using some algorithmic contributions from Xiang 19 ) and the ab-initio density matrix renormalization group method was born.…”

Combining Internally Contracted States and Matrix Product States To Perform Multireference Perturbation Theory

“…We here focus on DMRG, 20,21 a variational method which minimizes the energy of a wavefunction parametrized as a matrix product state (MPS). 22,23 DMRG can handle active spaces of around [30][31][32][33][34][35][36][37][38][39][40] orbitals, and in some cases even up to 100 [24][25][26][27][28][29][30][31][32][33][34] . However, by themselves the mentioned methods are not efficient for obtaining quantitative accuracy and for this dynamic correlation must also be calculated.…”

Spin-adapted matrix product states and operators

“…To obtain the reduced elements of P , we follow the description by Wouters et al [18] and employ the formula in Eq. (6), which couples two tensor operators and reads…”