Efficient and robust approximations to the full configuration interaction (full-CI) method such as the density matrix renormalization group (DMRG) and the full-CI quantum Monte Carlo (FCIQMC) algorithm allow for multiconfigurational self-consistent field (MC-SCF) calculations of molecules with many strongly correlated electrons. This opens up the possibility to treat large and complex systems that were previously untractable, but at the same time it calls for an efficient and reliable active space selection as the choice of how many electrons and orbitals enter the active space is critical for any multireference calculation. In this work we propose an Active Space Selection based on 1st order perturbation theory (ASS1ST) that follows a "bottom-up" strategy and utilizes a set of quasi-natural orbitals together with sensible thresholds for their occupation numbers. The required quasi-natural orbitals are generated by diagonalizing the virtual and internal part of the one-electron reduced density matrix that is obtained from strongly contracted n-electron valence perturbation theory (SC-NEVPT) on top of a minimal active space calculation. Self-consistent results can be obtained when the proposed selection scheme is applied iteratively. Initial applications on four chemically relevant benchmark systems indicate the capabilities of ASS1ST. Eventually, the strengths and limitations are critically discussed.
Efficient and robust approximations to the full configuration interaction (full-CI) method such as the density matrix renormalization group (DMRG) and the full-CI quantum Monte Carlo (FCIQMC) algorithm allow for multiconfigurational self-consistent field (MC-SCF) calculations of molecules with many strongly correlated electrons. This opens up the possibility to treat large and complex systems that were previously untractable, but at the same time it calls for an efficient and reliable active space selection as the choice of how many electrons and orbitals enter the active space is critical for any multireference calculation. In this work we propose an Active Space Selection based on 1st order perturbation theory (ASS1ST) that follows a "bottom-up" strategy and utilizes a set of quasi-natural orbitals together with sensible thresholds for their occupation numbers. The required quasi-natural orbitals are generated by diagonalizing the virtual and internal part of the one-electron reduced density matrix that is obtained from strongly contracted n-electron valence perturbation theory (SC-NEVPT) on top of a minimal active space calculation. Self-consistent results can be obtained when the proposed selection scheme is applied iteratively. Initial applications on four chemically relevant benchmark systems indicate the capabilities of ASS1ST. Eventually, the strengths and limitations are critically discussed.
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.
We present the density matrix renormalization group self-consistent field (DMRG-SCF) approach that is analogous to the complete active space self-consisted field (CASSCF) method but instead of using for the description of the active space the full configuration interaction (FCI) method, the DMRG-SCF uses the density matrix renormalization group (DMRG) method. The DMRG-SCF approach, similarly to CASSCF, properly describes the multiconfigurational character of the wave function but avoids the exponential scaling of the FCI method and replaces it with a polynomial scaling. Hence, calculations for a larger number of orbitals and electrons in the active space are possible since the DMRG method provides an efficient tool to automatically select from the full Hilbert space the many-body contracted basis states that are the most important for the description of the wave function.
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
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.
In multi-reference (MR) methods, addressing systems with large active spaces remains a challenge in the field. In Papers I and II of this series, we demonstrated that full rank N-electron valence state second-order perturbation theory (FR-NEVPT2) is a robust MR perturbation theory capable of computing strongly correlated systems with approximate density matrices. However, the previous FR-NEVPT2 implementation requires the computation and storage of fifth-order reduced density matrices (RDMs), limiting the usage of FR-NEVPT2 for systems with large active spaces. In the present work, as Paper III of the series, we report a new FR-NEVPT2 algorithm to handle systems with large active spaces. In the new algorithm, an approximate complete active space (CAS) self-consistent field (SCF) method, iterative configuration expansion (ICE) SCF, is employed to compute the reference wave functions for FR-NEVPT2. Then, the necessary Koopmans matrices of FR-NEVPT2 involving various RDMs are constructed using the intermediates designed by Kollmar et al. [J. Chem. Phys. 155, 234104 (2021)] to avoid storage bottlenecks. The performance of the new FR-NEVPT2 algorithm for systems with large active spaces is evaluated. Our results show that even with aggressive truncation parameters to truncate the ICE-SCF reference wave function, FR-NEVPT2 effectively recovers the missing static correlations of ICE-SCF. Several interesting systems with active spaces up to CAS(34,34) are studied using FR-NEVPT2 with ICE-SCF reference.
It is well-known that not only the orbital ordering but also the choice of the orbitals itself as the basis may significantly influence the computational efficiency of density-matrix renormalization group (DMRG) calculations. In this study, for assessing the efficiency of using various natural orbitals (NOs) as the DMRG basis, we performed benchmark DMRG calculations with different bases, which included the NOs obtained by various traditional electron correlation methods, as well as NOs acquired from preliminary moderate DMRG calculations (e.g., preserved states less than 500). The tested systems included N2, transition metal Cr2 systems, as well as 1D hydrogen polyradical chain systems under equilibrium and dissociation conditions and 2D hydrogen aggregates. The results indicate that a good compromise between the requirement for low computational costs of acquiring NOs and the demand for high efficiency of NOs as the basis of DMRG calculations may be very dependent on the studied systems' diverse electron correlation characteristics and the size of the active space. It is also shown that a DMRG-complete active space configuration interaction (DMRG-CASCI) calculation in a basis of carefully chosen NOs can provide a less expensive alternative to the standard DMRG-complete active space self-consistent field (DMRG-CASSCF) calculation and avoid the convergence difficulties of orbital optimization for large active spaces. The effect of different NO ordering schemes on DMRG-CASCI calculations is also discussed.