We identify the dominant computational cost within the recently introduced stochastic and internally contracted FCIQMC-NEVPT2 method for large active space sizes. This arises from the contribution to the four-body intermediates arising from low-excitation level sampled determinant pairs. We develop an effective way to mitigate this cost via an additional stochastic step within the sampling of the required NEVPT2 intermediates. We find this systematically improvable additional sampling can reduce simulation time by 80% without introducing appreciable error. This saving is expected to increase for larger active spaces. We combine this enhanced sampling scheme with full stochastic orbital optimisation for the first time and apply it to find FCIQMC-NEVPT2 energies for spin states of an iron porphyrin system within (24,24) active spaces with relatively meagre computational resources. This active space size can now be considered as routine for NEVPT2 calculations of strongly correlated molecular systems within this improved stochastic methodology.
We identify the dominant computational cost within the recently introduced stochastic and internally contracted FCIQMC-NEVPT2 method for large active space sizes. This arises from the contribution to the four-body intermediates arising from low-excitation level sampled determinant pairs. We develop an effective way to mitigate this cost via an additional stochastic step within the sampling of the required NEVPT2 intermediates. We find this systematically improvable additional sampling can reduce simulation time by 80% without introducing appreciable error. This saving is expected to increase for larger active spaces. We combine this enhanced sampling scheme with full stochastic orbital optimisation for the first time and apply it to find FCIQMC-NEVPT2 energies for spin states of an iron porphyrin system within (24,24) active spaces with relatively meagre computational resources. This active space size can now be considered as routine for NEVPT2 calculations of strongly correlated molecular systems within this improved stochastic methodology.
Full Configuration Interaction Quantum Monte Carlo (FCIQMC) has been effectively applied to very large configuration interaction (CI) problems and was recently adapted for use as an active space solver and combined with orbital optimization. In this work, we detail an approach within FCIQMC to allow for efficient sampling of fully internally contracted multireference perturbation theories within the same stochastic framework. Schemes are described to allow for the close control over the resolution of stochastic sampling of the effective higher-body intermediates within the active space. It is found that while complete active space second-order perturbation theory seems less amenable to a stochastic reformulation, strongly contracted N-Electron Valence second-order Perturbation Theory (NEVPT2) is far more stable, requiring a similar number of walkers to converge the sc-NEVPT2 expectation values as to converge the underlying CI problem. We demonstrate the application of the stochastic approach to the computation of sc-NEVPT2 within a (24, 24) active space in a biologically relevant system and show that small numbers of walkers are sufficient for a faithful sampling of the sc-NEVPT2 energy to chemical accuracy, despite the active space already exceeding the limits of practicality for traditional approaches. This raises prospects of an efficient stochastic solver for multireference chemical problems requiring large active spaces, with an accurate treatment of external orbitals.
The multiconfigurational self-consistent field theory is considered the standard starting point for almost all multireference approaches required for strongly correlated molecular problems. The limitation of the approach is generally given by the number of strongly correlated orbitals in the molecule, since its cost will grow exponentially with this number. We present a new multiconfigurational self-consistent field approach, wherein linear determinant coefficients of a multiconfigurational wave function are optimized via the stochastic full configuration interaction quantum Monte Carlo technique at greatly reduced computational cost, with nonlinear orbital rotation parameters updated variationally based on this sampled wave function. This extends this approach to strongly correlated systems with far larger active spaces than it is possible to treat via conventional means. By comparison with this traditional approach, we demonstrate that the introduction of stochastic noise in both the determinant amplitudes and the gradient and Hessian of the orbital rotations does not preclude robust and reliable convergence of the orbital optimization. It can even improve the ability to avoid convergence to local minima in the orbital space, and therefore aid in finding variationally lower-energy solutions. We consider the effect on the convergence of the orbitals as the number of walkers and the sampling time within the active space increases, as well as the effect on the final energy and error. The scope of the new protocol is demonstrated with a study of the increasingly strongly correlated electronic structure in a series of polycyclic aromatic hydrocarbons, up to the large coronene molecule in a complete active space of 24 π electrons in 24 orbitals, requiring only modest computational resources
In this article we report a stochastic evaluation of the recently proposed multireference linearized coupled cluster theory [S. Sharma and A. Alavi, J. Chem. Phys. 143, 102815 (2015)]. In this method, both the zeroth-order and first-order wavefunctions are sampled stochastically by propagating simultaneously two populations of signed walkers. The sampling of the zeroth-order wavefunction follows a set of stochastic processes identical to the one used in the full configuration interaction quantum Monte Carlo (FCIQMC) method. To sample the first-order wavefunction, the usual FCIQMC algorithm is augmented with a source term that spawns walkers in the sampled first-order wavefunction from the zeroth-order wavefunction. The second-order energy is also computed stochastically but requires no additional overhead outside of the added cost of sampling the first-order wavefunction. This fully stochastic method opens up the possibility of simultaneously treating large active spaces to account for static correlation and recovering the dynamical correlation using perturbation theory. The method is used to study a few benchmark systems including the carbon dimer and aromatic molecules. We have computed the singlet-triplet gaps of benzene and m-xylylene. For m-xylylene, which has proved difficult for standard complete active space self consistent field theory with perturbative correction, we find the singlet-triplet gap to be in good agreement with the experimental values.
An algorithm to perform stochastic generalized active space calculations, Stochastic-GAS, is presented, that uses the Slater determinant based FCIQMC algorithm as configuration interaction eigensolver. Stochastic-GAS allows the construction and stochastic optimization of preselected truncated configuration interaction wave functions, either to reduce the computational costs of large active space wave function optimizations, or to probe the role of specific electron correlation pathways. As for the conventional GAS procedure, the preselection of the truncated wave function is based on the selection of multiple active subspaces while imposing restrictions on the interspace excitations. Both local and cumulative minimum and maximum occupation number constraints are supported by Stochastic-GAS. The occupation number constraints are efficiently encoded in precomputed probability distributions, using the precomputed heat bath algorithm, which removes nearly all runtime overheads of GAS. This strategy effectively allows the FCIQMC dynamics to a priori exclude electronic configurations that are not allowed by GAS restrictions. Stochastic-GAS reduced density matrices are stochastically sampled, allowing orbital relaxations via Stochastic-GASSCF, and direct evaluation of properties that can be extracted from density matrices, such as the spin expectation value. Three test case applications have been chosen to demonstrate the flexibility of Stochastic-GAS: (a) the Stochastic-GASSCF optimization of a stack of five benzene molecules, that shows the applicability of Stochastic-GAS towards fragment-based chemical systems; (b) an uncontracted stochastic MRCISD calculation that correlates 96 electrons and 159 molecular orbitals, and uses a large (32, 34) active space reference wave function for an Fe(II)-porphyrin model system, showing how GAS can be applied to systematically recover dynamic electron correlation, and how in the specific case of the Fe(II)-porphyrin dynamic correlation further differentially stabilizes the triplet over the quintet spin state; (c) the study of an Fe4S4 cluster's spin-ladder energetics via highly truncated stochastic-GAS wave functions, where we show how GAS can be applied to understand the competing spin-exchange and charge-transfer correlating mechanisms in stabilizing different spin-states.
We present two efficient and intruder-free methods for treating dynamic correlation on top of general multiconfiguration reference wave functions — including such as obtained by the density matrix renormalization group (DMRG) with large active spaces. The new methods are the second order variant of the recently proposed multireference linearized coupled cluster method (MRLCC) [Sharma, S.; Alavi, A. J. Chem. Phys. 2015, 143, 102815] and of N-electron valence perturbation theory (NEVPT2), with expected accuracies similar to MRCI+Q and (at least) CASPT2, respectively. Great efficiency gains are realized by representing the first order wave function with a combination of internal contraction (IC) and matrix product state perturbation theory (MPSPT). With this combination, only third order reduced density matrices (RDMs) are required. Thus, we obviate the need for calculating (or estimating) RDMs of fourth or higher order; these had so far posed a severe bottleneck for dynamic correlation treatments involving the large active spaces accessible to DMRG. Using several benchmark systems, including first and second row containing small molecules, Cr 2, pentacene, and oxo-Mn(Salen), we show that active spaces containing at least 30 orbitals can be treated using this method. On a single node, MRLCC2 and NEVPT2 calculations can be performed with over 550 and 1100 virtual orbitals, respectively. We also critically examine the errors incurred due to the three sources of errors introduced in the present implementation – calculating second order instead of third order energy corrections, use of internal contraction, and approximations made in the reference wave function due to DMRG.
An algorithm to perform stochastic generalized active space calculations, Stochastic-GAS, is presented, that uses the Slater determinant based FCIQMC algorithm as configuration interaction eigensolver. Stochastic-GAS allows the construction and stochastic optimization of preselected truncated configuration interaction wave functions, either to reduce the computational costs of large active space wave function optimizations, or to probe the role of specific electron correlation pathways. As for the conventional GAS procedure, the preselection of the truncated wave function is based on the selection of multiple active subspaces while imposing restrictions on the interspace excitations. Both local and cumulative minimum and maximum occupation number constraints are supported by Stochastic-GAS. The occupation number constraints are efficiently encoded in precomputed probability distributions, using the precomputed heat bath algorithm, which removes nearly all runtime overheads of GAS. This strategy effectively allows the FCIQMC dynamics to a priori exclude electronic configurations that are not allowed by GAS restrictions. Stochastic-GAS reduced density matrices are stochastically sampled, allowing orbital relaxations via Stochastic-GASSCF, and direct evaluation of properties that can be extracted from density matrices, such as the spin expectation value. Three test case applications have been chosen to demonstrate the flexibility of Stochastic-GAS: (a) the Stochastic-GASSCF optimization of a stack of five benzene molecules, that shows the applicability of Stochastic-GAS towards fragment-based chemical systems; (b) an uncontracted stochastic MRCISD calculation that correlates 96 electrons and 159 molecular orbitals, and uses a large (32, 34) active space reference wave function for an Fe(II)-porphyrin model system, showing how GAS can be applied to systematically recover dynamic electron correlation, and how in the specific case of the Fe(II)-porphyrin dynamic correlation further differentially stabilizes the triplet over the quintet spin state; (c) the study of an Fe4S4 cluster's spin-ladder energetics via highly truncated stochastic-GAS wave functions, where we show how GAS can be applied to understand the competing spin-exchange and charge-transfer correlating mechanisms in stabilizing different spin-states.