We describe the implementation of the frozen-orbital and downfolding approximations in the auxiliary-field quantum Monte Carlo (AFQMC) method. These approaches can provide significant computational savings, compared to fully correlating all of the electrons. While the many-body wave function is never explicit in AFQMC, its random walkers are Slater determinants, whose orbitals may be expressed in terms of any one-particle orbital basis. It is therefore straightforward to partition the full N-particle Hilbert space into active and inactive parts to implement the frozen-orbital method. In the frozen-core approximation, for example, the core electrons can be eliminated in the correlated part of the calculations, greatly increasing the computational efficiency, especially for heavy atoms. Scalar relativistic effects are easily included using the Douglas-Kroll-Hess theory. Using this method, we obtain a way to effectively eliminate the error due to single-projector, norm-conserving pseudopotentials in AFQMC. We also illustrate a generalization of the frozen-orbital approach that downfolds high-energy basis states to a physically relevant low-energy sector, which allows a systematic approach to produce realistic model Hamiltonians to further increase efficiency for extended systems.
The ground state electronic energy of benzene is the focus of a recent blind test by Eriksen and co-workers [arXiv:2008.02678]. In this note, we report the phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) results on the identical problem. We examine trial wavefunctions of the spin-restricted Hartree-Fock (RHF), and complete active space self-consistent field (CASSCF) with an active space of 6-electron and 6-orbital (i.e., CAS(6,6)) form for the cc-pVDZ basis set. ph-AFQMC+RHF deviates from the value where many methods agreed on (i.e., -863 m
Eh) by -3.1(3) m
Eh whereas ph-AFQMC+CAS(6,6) deviates from the same value by -1.3(4) m
Eh. In addition to this, we report the frozen core correlation energy of ph-AFQMC+RHF in the cc-pVTZ and cc-pVQZ basis sets as well as their complete basis set limit. Our findings highlight the accuracy, flexibility, and scalability of ph-AFQMC with simple trial wavefunctions.
A local embedding and effective downfolding scheme has been developed and implemented in the auxiliary-field quantum Monte Carlo (AFQMC) method. A local cluster in which electrons are fully correlated is defined, and the frozen orbital method is used on the remainder of the system to construct an effective Hamiltonian, which operates within the local cluster. Local embedding, which involves only the occupied sector, has previously been employed in the context of Co/graphene. Here, the methodology is extended in order to allow for effective downfolding of the virtual sector, thus allowing for significant reduction in the computational effort required for AFQMC calculations. The system size, which can be feasibly treated with AFQMC, is therefore greatly extended as only a single local cluster is explicitly correlated at the AFQMC level of theory. The approximation is controlled by the separate choice of the spatial size of the active occupied region ( R o ) and of the active virtual region ( R v ). The systematic dependence of the AFQMC energy on R o and R v is investigated, and it is found that relative AFQMC energies of physical and chemical interest converge rapidly to the full AFQMC treatment (i.e., using no embedding or downfolding).