Initiator Full Configuration Interaction Quantum Research Articles

Optimizing Jastrow factors for the transcorrelated method.

We investigate the optimization of flexible tailored real-space Jastrow factors for use in the transcorrelated (TC) method in combination with highly accurate quantum chemistry methods, such as initiator full configuration interaction quantum Monte Carlo (FCIQMC). Jastrow factors obtained by minimizing the variance of the TC reference energy are found to yield better, more consistent results than those obtained by minimizing the variational energy. We compute all-electron atomization energies for the challenging first-row molecules C2, CN, N2, and O2 and find that the TC method yields chemically accurate results using only the cc-pVTZ basis set, roughly matching the accuracy of non-TC calculations with the much larger cc-pV5Z basis set. We also investigate an approximation in which pure three-body excitations are neglected from the TC-FCIQMC dynamics, saving storage and computational costs, and show that it affects relative energies negligibly. Our results demonstrate that the combination of tailored real-space Jastrow factors with the multi-configurational TC-FCIQMC method provides a route to obtaining chemical accuracy using modest basis sets, obviating the need for basis-set extrapolation and composite techniques.

Accelerating convergence to the thermodynamic limit with twist angle selection applied to methods beyond many-body perturbation theory.

We recently developed a scheme to use low-cost calculations to find a single twist angle where the coupled cluster doubles energy of a single calculation matches the twist-averaged coupled cluster doubles energy in a finite unit cell. We used initiator full configuration interaction quantum Monte Carlo as an example of an exact method beyond coupled cluster doubles theory to show that this selected twist angle approach had comparable accuracy in methods beyond coupled cluster. Furthermore, at least for small system sizes, we show that the same twist angle can also be found by comparing the energy directly (at the level of second-order Moller-Plesset theory), suggesting a route toward twist angle selection, which requires minimal modification to existing codes that can perform twist averaging.

A hybrid approach to extending selected configuration interaction and full configuration interaction quantum Monte Carlo.

We present an approach to combining selected configuration interaction (SCI) and initiator full configuration interaction quantum Monte Carlo (i-FCIQMC). In the current i-FCIQMC scheme, the space of initiators is chosen dynamically by a population threshold. Here, we instead choose initiators as the selected space (V) from a prior SCI calculation, allowing substantially larger initiator spaces for a given walker population. While SCI+PT2 adds a perturbative correction in the first-order interacting space beyond V, the approach presented here allows a variational calculation in the same space and a perturbative correction in the second-order interacting space. The use of a fixed initiator space reintroduces population plateaus into FCIQMC, but it is shown that the plateau height is typically only a small multiple of the size of V. Thus, for a comparable fundamental memory cost to SCI+PT2, a substantially larger space can be sampled. The resulting method can be seen as a complementary approach to SCI+PT2, which is more accurate but slower for a common selected/initiator space. More generally, our results show that approaches exist to significantly improve initiator energies in i-FCIQMC while still ameliorating the fermion sign problem relative to the original FCIQMC method.

Communication: An efficient and accurate perturbative correction to initiator full configuration interaction quantum Monte Carlo.

We present a perturbative correction within initiator full configuration interaction quantum Monte Carlo (i-FCIQMC). In the existing i-FCIQMC algorithm, a significant number of spawned walkers are discarded due to the initiator criteria. Here we show that these discarded walkers have a form that allows the calculation of a second-order Epstein-Nesbet correction, which may be accumulated in a trivial and inexpensive manner, yet substantially improves i-FCIQMC results. The correction is applied to the Hubbard model and the uniform electron gas and molecular systems.

A general non-Abelian density matrix renormalization group algorithm with application to the C2 dimer.

We extend our previous work [S. Sharma and G. K.-L. Chan, J. Chem. Phys. 136, 124121 (2012)], which described a spin-adapted (SU(2) symmetry) density matrix renormalization group algorithm, to additionally utilize general non-Abelian point group symmetries. A key strength of the present formulation is that the requisite tensor operators are not hard-coded for each symmetry group, but are instead generated on the fly using the appropriate Clebsch-Gordan coefficients. This allows our single implementation to easily enable (or disable) any non-Abelian point group symmetry (including SU(2) spin symmetry). We use our implementation to compute the ground state potential energy curve of the C2 dimer in the cc-pVQZ basis set (with a frozen-core), corresponding to a Hilbert space dimension of 10(12) many-body states. While our calculated energy lies within the 0.3 mEh error bound of previous initiator full configuration interaction quantum Monte Carlo and correlation energy extrapolation by intrinsic scaling calculations, our estimated residual error is only 0.01 mEh, much more accurate than these previous estimates. Due to the additional efficiency afforded by the algorithm, the excitation energies (Te) of eight lowest lying excited states: a(3)Πu, b(3)Σg (-), A(1)Πu, c(3)Σu (+), B(1)Δg, B(') (1)Σg (+), d(3)Πg, and C(1)Πg are calculated, which agree with experimentally derived values to better than 0.06 eV. In addition, we also compute the potential energy curves of twelve states: the three lowest levels for each of the irreducible representations (1)Σg (+), (1)Σu (+), (1)Σg (-), and (1)Σu (-), to an estimated accuracy of 0.1 mEh of the exact result in this basis.

Symmetry Breaking and Broken Ergodicity in Full Configuration Interaction Quantum Monte Carlo.

The initiator full configuration interaction quantum Monte Carlo method (i-FCIQMC) is applied to the binding curve of N2 in Slater-determinant Hilbert spaces formed of both canonical restricted Hartree-Fock (RHF) and symmetry-broken unrestricted Hartree-Fock (UHF) orbitals. By explicit calculation, we demonstrate that the technique yields the same total energy for both types of orbital but that as the bond is stretched, FCI expansions expressed in unrestricted orbitals are substantially more compact than their restricted counterparts and more compact than those expressed in split-localized orbitals. These unrestricted Hilbert spaces, however, become nonergodic toward the dissociation limit, and the total wave function may be thought of as the sum of two weakly coupled, spin-impure, functions whose energies are nonetheless very close to the exact energy. In this limit, it is a challenge for i-FCIQMC to resolve a spin-pure wave function. The use of unrestricted natural orbitals is a promising remedy for this problem, as their expansions are more strongly weighted toward lower excitations of the reference, and they provide stronger coupling to higher excitations than do UHF orbitals.

Spectroscopic accuracy directly from quantum chemistry: Application to ground and excited states of beryllium dimer

We combine explicit correlation via the canonical transcorrelation approach with the density matrix renormalization group and initiator full configuration interaction quantum Monte Carlo methods to compute a near-exact beryllium dimer curve, without the use of composite methods. In particular, our direct density matrix renormalization group calculations produce a well-depth of D(e) = 931.2 cm(-1) which agrees very well with recent experimentally derived estimates D(e) = 929.7±2 cm(-1) [J. M. Merritt, V. E. Bondybey, and M. C. Heaven, Science 324, 1548 (2009)] and D(e) = 934.6 cm(-1) [K. Patkowski, V. Špirko, and K. Szalewicz, Science 326, 1382 (2009)], as well the best composite theoretical estimates, D(e) = 938±15 cm(-1) [K. Patkowski, R. Podeszwa, and K. Szalewicz, J. Phys. Chem. A 111, 12822 (2007)] and D(e) = 935.1±10 cm(-1) [J. Koput, Phys. Chem. Chem. Phys. 13, 20311 (2011)]. Our results suggest possible inaccuracies in the functional form of the potential used at shorter bond lengths to fit the experimental data [J. M. Merritt, V. E. Bondybey, and M. C. Heaven, Science 324, 1548 (2009)]. With the density matrix renormalization group we also compute near-exact vertical excitation energies at the equilibrium geometry. These provide non-trivial benchmarks for quantum chemical methods for excited states, and illustrate the surprisingly large error that remains for 1 ¹Σ(g)⁻ state with approximate multi-reference configuration interaction and equation-of-motion coupled cluster methods. Overall, we demonstrate that explicitly correlated density matrix renormalization group and initiator full configuration interaction quantum Monte Carlo methods allow us to fully converge to the basis set and correlation limit of the non-relativistic Schrödinger equation in small molecules.

Taming the First-Row Diatomics: A Full Configuration Interaction Quantum Monte Carlo Study.

The initiator full configuration interaction quantum Monte Carlo (i-FCIQMC) method has recently been developed as a highly accurate stochastic electronic structure technique. It has been shown to calculate the exact basis-set ground state energy of small molecules, to within modest stochastic error bars, using tractable computational cost. Here, we use this technique to elucidate an often troublesome series of first-row diatomics consisting of Be2, C2, CN, CO, N2, NO, O2, and F2. Using i-FCIQMC, the dissociation energies of these molecules are obtained almost entirely to within chemical accuracy of experimental results. Furthermore, the i-FCIQMC calculations are performed in a relatively black-box manner, without any a priori knowledge or specification of the wave function. The size consistency of i-FCIQMC is also demonstrated with regards to these diatomics at their more multiconfigurational stretched geometries. The clear and simple i-FCIQMC wave functions obtained for these systems are then compared and investigated to demonstrate the dynamic identification of the dominant determinants contributing to significant static correlation. The appearance and nature of such determinants is shown to provide insight into both the i-FCIQMC algorithm and the diatomics themselves.