Wavefunction optimization at the complete basis set limit with Multiwavelets and DMRG

TL;DR

The paper develops a novel integration of the density matrix renormalization group (DMRG) with multiresolution analysis and multiwavelets (MW) to reach the complete basis set limit in ab initio electronic structure calculations. By replacing CI steps with DMRG and deriving energy gradients directly from the DMRG tensor network within a Lagrangian formulation, the authors enable self-consistent orbital optimization in the MW/MRA framework. Numerical results on small molecules (e.g., H2, N2) show that MRA+DMRG consistently yields lower energies for a fixed number of orbitals and can outperform FCI with larger atomic bases, illustrating the potential to accurately treat multireference correlation at CBS accuracy with adaptive basis representations. This approach offers a path toward scalable, high-accuracy electronic structure calculations that converge to the CBS limit while maintaining manageable orbital counts.

Abstract

The density matrix renormalization group (DMRG) is a powerful numerical technique to solve strongly correlated quantum systems: it deals well with systems which are not dominated by a single configuration (unlike Coupled Cluster) and it converges rapidly to the Full Configuration Interaction (FCI) limit (unlike truncated Configuration Interaction (CI) expansions). In this work, we develop an algorithm integrating DMRG within the multiwavelet-based multiresolution analysis (MRA). Unlike fixed basis sets, multiwavelets offer an adaptive and hierarchical representation of functions, approaching the complete basis set limit to a specified precision. As a result, this combined technique leverages the multireference capability of DMRG and the complete basis set limit of MRA and multiwavelets. More specifically, we adopt a pre-existing Lagrangian optimization algorithm for orbitals represented in the MRA domain and improve its computational efficiency by replacing the original CI calculations with DMRG. Additionally, we substitute the reduced density matrices computation with the direct extraction of energy gradients from the DMRG tensors. We apply our method to small systems such H2, He, HeH2, BeH2 and N2. The results demonstrate that our approach reduces the final energy while keeping the number of orbitals low compared to FCI calculations on an atomic orbital basis set.

Figures (8)

• Figure 1: (\ref{['fig:scaling_functions']}) Scaling functions in $V_0$ for $k=3$. (\ref{['fig:mra_figure']}) Multi-reference grid in $3$D space. (\ref{['fig:multiwavelets_functions']}) MW in $W_0$ for $k=3$. Reproduced from figures_mra.
• Figure 2: MRA+DMRG flowchart. The steps colored in gray belong to the "MRA domain". The white blocks are orbital-agnostic.
• Figure 3: Final energies obtained from Hartree Fock, DMRG, FCI, and our method combining DMRG and MRA. When using MRA, a precision of $10^{-5}$ has been considered with polynomial order $k=9$. The MRA+DMRG method works in principle with any number of orbitals. However, the current pilot implementation is limited to 15 orbitals. For AO calculations, the number of orbitals is dictated by the choice of basis. For these reasons, the tables present empty entries.
• Figure 4: Dissociation paths of H2 and N2 with ground state energies obtained via HF and FCI with a minimal basis set and the MRA+DMRG algorithm with the same number of orbitals, respectively.
• Figure 5: State diagram for constructing the MPO representation of a molecular Hamiltonian \ref{['eq_def_second_quantization_Hamiltonian_mod']}. $c^{\dagger}_{\sigma}$ and $c_{\sigma}$ for $\sigma \in \{ \uparrow, \downarrow \}$ are the local bosonic creation and annihilation operators, $Z_{\sigma}$ is the Pauli-$Z$ gate acting on spin sector $\sigma \in \{ \uparrow, \downarrow \}$, and $I$ is the identity operation. The "'scaffolding" is highlighted in blue. The figure only shows a few operator strings and Hamiltonian terms for visual clarity.