Benchmark Phaseless Auxiliary-Field Quantum Monte Carlo Method for Small Molecules

Abstract

We report a scalable Fortran implementation of the phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) and demonstrate its excellent performance and beneficial scaling with respect to system size. Furthermore, we investigate modifications of the phaseless approximation that can help to reduce the overcorrelation problems common to the ph-AFQMC. We apply the method to the 26 molecules in the HEAT set, the benzene molecule, and water clusters. We observe a mean absolute deviation of the total energy of 1.15 kcal/mol for the molecules in the HEAT set, close to chemical accuracy. For the benzene molecule, the modified algorithm despite using a single-Slater-determinant trial wavefunction yields the same accuracy as the original phaseless scheme with 400 Slater determinants. Despite these improvements, we find systematic errors for the CN, CO 2 , and O 2 molecules that need to be addressed with more accurate trial wavefunctions. For water clusters, we find that the ph-AFQMC yields excellent binding energies that differ from CCSD(T) by typically less than 0.5 kcal/mol.

Toward Large-Scale AFQMC Calculations: Large Time Step Auxiliary-Field Quantum Monte Carlo

“…The total computational cost to perform k K -th order block Krylov expansion increases by the factor k K k T true( 1 + k normalK N e N + ( k K N e N ) 2 true) compared to the k T -th order Taylor expansion. This can add computational overhead for small basis sets, but for larger basis sets the computational overhead is usually negligible, especially when compared to other more expensive tasks in AFQMC …”

“…Although originally developed as the path integral MC algorithm, − it experienced a renaissance with the introduction of open-ended random walks and the phaseless approximation . Since then, numerous applications for molecules − and solids − have confirmed that ph-AFQMC is a promising alternative to CCSD(T) and FN-DMC. It has also been successfully applied to excited states , and at finite temperature. − ph-AFQMC exhibits low polynomial scaling ( O false( N 3 false) − O false( N 4 false) ) with system size that can even be reduced to cubic scaling by utilizing tensor hypercontractions, , stochastic resolution of identity, or employing a plane wave basis .…”

“…It has also been successfully applied to excited states , and at finite temperature. − ph-AFQMC exhibits low polynomial scaling ( O false( N 3 false) − O false( N 4 false) ) with system size that can even be reduced to cubic scaling by utilizing tensor hypercontractions, , stochastic resolution of identity, or employing a plane wave basis . The phaseless approximation controls the Fermionic sign problem in ph-AFQMC, however, it also introduces systematic errors that are challenging to control. , Mahajan et al developed a fast method for the evaluation of the local energies for multideterminantal trial wave functions and demonstrated that the phaseless approximation can be systematically improved using better trial wave functions. , Recently, there have been attempts to directly enhance the phaseless approximation. ,, …”

Expanding the Design Space of Constraints in Auxiliary-Field Quantum Monte Carlo

“…Two decades after the invention of ph-AFQMC, we now have sufficient computational power to rigorously test alternatives to the cosine projection on benchmark sets, and efforts to do so have already emerged in the literature. For example, Sukurma et al recently tested a reformulation of the cosine projection in which they delay projection until evaluating observables, instead removing the origin by killing walkers if their total phase grows beyond π/2. They found qualitatively similar yet statistically distinct results to ph-AFQMC on a small benchmark set of small main-group molecules.…”

Toward Linear Scaling Auxiliary-Field Quantum Monte Carlo with Local Natural Orbitals

“…The phaseless auxiliary-field quantum Monte Carlo (AFQMC) method has gained popularity due to its comparatively low N 4 computational scaling (i.e., the same as mean-field theory) and impressive accuracy even for strongly correlated systems. − AFQMC is a descendent of the determinantal QMC method , and was extensively developed for electronic structure by Zhang and co-workers . Recent algorithmic developments aimed at reducing the cost of AFQMC include the use of low-rank Coulomb integrals, − stochastic resolution of identity, and local trial states…”