Ab Initio Finite Temperature Auxiliary Field Quantum Monte Carlo

Abstract

We present an ab initio auxiliary field quantum Monte Carlo method for studying the electronic structure of molecules, solids, and model Hamiltonians at finite temperature. The algorithm marries the ab initio phaseless auxiliary field quantum Monte Carlo algorithm known to produce high accuracy ground state energies of molecules and solids with its finite temperature variant, long used by condensed matter physicists for studying model Hamiltonian phase diagrams, to yield a phaseless, ab initio finite temperature method. We demonstrate that the method produces internal energies within chemical accuracy of exact diagonalization results across a wide range of temperatures for HO (STO-3G), C (STO-6G), the one-dimensional hydrogen chain (STO-6G), and the multiorbital Hubbard model. Our method effectively controls the phase problem through importance sampling, often even without invoking the phaseless approximation, down to temperatures at which the systems studied approach their ground states and may therefore be viewed as exact over wide temperature ranges. This technique embodies a versatile tool for studying the finite temperature phase diagrams of a plethora of systems whose properties cannot be captured by a Hubbard U term alone. Our results moreover illustrate that the severity of the phase problem for model Hamiltonians far exceeds that for many molecules at all of the temperatures studied.

Utilizing Essential Symmetry Breaking in Auxiliary-Field Quantum Monte Carlo: Application to the Spin Gaps of the C36 Fullerene and an Iron Porphyrin Model Complex

“…All of these methods require a specification of active space which can be as small as (8e, 11o) and as large as (44e, 44o). As shown in Table 3, This then gives a spectrum of the quintet-triplet gap (∆E Q-T (= E T − E Q )) from -13 kcal/mol (DMRG-CI) to 19.27(7) kcal/mol (ASCISCF). Perhaps, the most surprising result in this broad spectrum is that two selected CI methods (SHCISCF and ACISCF) show a discrepancy on the order of 20 kcal/mol.…”

“…In order to obtain a triplet ground state, a reasonable correlation model is supposed to decrease this gap to a negative value. In this sense, 19.27(7) kcal/mol of ACISCF is surprising because the correlation out of the active space is so significant that it seems to achieve not much of the cancellation of dynamic correlation. By contrast, all other previous MR studies achieved either a triplet ground state or at least small enough gaps (less than 5 kcal/mol) by benefiting from the cancellation of dynamic correlation.…”

Using Density Matrix Quantum Monte Carlo for Calculating Exact-on-Average Energies for ab Initio Hamiltonians in a Finite Basis Set

“…Due to the success of the recently developed finite-temperature auxiliary field quantum Monte Carlo (FT-AFQMC) method for small molecules, atoms, solids and models, we choose to directly compare our results for an equilibrium and stretched H 10 molecule (STO-6G) and H 2 O (STO-3G) to Rubenstein and coworkers. 43 One significant difference between the Figure 5: The energy results of three different methods:i−DMQMC (black), i−DMQMC , sampled over all symmetries and spin polarizations (grey, dotted); GC-FCI (orange crosses); FT-FCI (magenta circles) and FT-AFQMC(blue squares). The same H 2 O geometry was used for these calculations as above, but the integral was produced in the STO-3G basis set for comparison with Rubenstein and coworkers.…”

Reaching the Continuum Limit in Finite-Temperature Ab Initio Field-Theory Computations in Many-Fermion Systems

“…This framework, even with simple trial wave functions to impose an approximate constraint, has been shown to be very accurate and has been applied widely in ground-state calculations [23][24][25][26][27][28][29][30][31]. Finitetemperature generalization of the approach has also been developed [32][33][34][35].These finite temperature calculations all have computational complexity of O(N 3 s ) [36], as they are formulated in the grand-canonical ensemble to analytically evaluate the fermion trace along each path in auxiliary-field space, leading to determinants with dimension N s . In the majority of applications, for example dilute Fermi gas and all ab initio real material simulations, it is necessary to reach the continuum (large lattice or complete basis set) limit in order to obtain realistic results.…”

“…The low-rank factorizations can be applied in exactly the same way. For electronic Hamiltonians with long-range interactions as in molecules and solids, a phase problem arises and the phaseless approximation [27,34] is needed. The low-rank factorizations and the rest of the algorithm are identical to the sign problem case.…”