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An improved version of ORCA's automated generator environment (ORCA-AGE II) is presented. The algorithmic improvements and the move to C++ as the programming language lead to a performance gain of up to two orders of magnitude compared to the previously developed P YTHON toolchain. Additionally, the restructured modular design allows for far more complex code engines to be implemented readily. Importantly, we have realised an extremely tight integration with the ORCA host program. This allows for a workflow in which only the wavefunction Ansatz is part of the source code repository while all actual high-level code is generated automatically, inserted at the appropriate place in the host program before it is compiled and linked together with the hand written code parts. This construction ensures longevity and uniform code quality. Furthermore the new developments allow ORCA-AGE II to generate parallelised production-level code for highly complex theories, such as fully internally contracted multireference coupled-cluster theory (fic-MRCC) with an enormous number of contributing tensor contractions. We also discuss the automated implementation of nuclear gradients for arbitrary theories. All these improvements enable the implementation of theories that are too complex for the human mind and also reduce development times by orders of magnitude. We hope that this work enables researchers to concentrate on the intellectual content of the theories they develop rather than be concerned with technical details of the implementation. [ABSTRACT FROM AUTHOR]

A factorization of the matrix elements of the Dyall Hamiltonian in N-electron valence state perturbation theory allowing their evaluation with a computational effort comparable to the one needed for the construction of the third-order reduced density matrix at the most is presented. Thus, the computational bottleneck arising from explicit evaluation of the fourth-order density matrix is avoided. It is also shown that the residual terms arising in the case of an approximate complete active space configuration interaction solution and containing even the fifth-order density matrix for two excitation classes can be evaluated with little additional effort by choosing again a favorable factorization of the corresponding matrix elements. An analogous argument is also provided for avoiding the fourth-order density matrix in complete active space second-order perturbation theory. Practical calculations indicate that such an approach leads to a considerable gain in computational efficiency without any compromise in numerical accuracy or stability. [ABSTRACT FROM AUTHOR]

Over the last decade, the second-order N-electron valence state perturbation theory (NEVPT2) has developed into a widely used multireference perturbation method. To apply NEVPT2 to systems with large active spaces, the computational bottleneck is the construction of the fourth-order reduced density matrix. Both its generation and storage become quickly problematic beyond the usual maximum active space of about 15 active orbitals. To reduce the computational cost of handling fourth-order density matrices, the cumulant approximation (CU) has been proposed in several studies. A more conventional strategy to address the higher-order density matrices is the pre-screening approximation (PS), which is the default one in the ORCA program package since 2010. In the present work, the performance of the CU, PS, and extended PS (EPS) approximations for the fourth-order density matrices is compared. Following a pedagogical introduction to NEVPT2, contraction schemes, as well as the approximations to density matrices, and the intruder state problem are discussed. The CU approximation, while potentially leading to large computational savings, virtually always leads to intruder states. With the PS approximation, the computational savings are more modest. However, in conjunction with conservative cutoffs, it produces stable results. The EPS approximation to the fourth-order density matrices can reproduce very accurate NEVPT2 results without any intruder states. However, its computational cost is not much lower than that of the canonical algorithm. Moreover, we found that a good indicator of intrude states problems in any approximation to high order density matrices is the eigenspectra of the Koopmans matrices. [ABSTRACT FROM AUTHOR]

In Paper I, the performances of pre-screening (PS), extended PS (EPS), and cumulant (CU) approximations to the fourth-order density matrix were examined in the context of second-order N-electron valence state perturbation theory (NEVPT2). It has been found that the CU, PS, and even EPS approximations with loose thresholds may introduce intruder states. In the present work, the origin of these "false intruder" states introduced by approximated density matrices is discussed. Canonical NEVPT2 implementations employ a rank reduction trick. By analyzing its residual error, we find that the omission of the rank reduction leads to a more stable multireference perturbation theory for incomplete active space reference wave functions. Such a full rank (FR)-NEVPT2 formulation is equivalent to the conventional NEVPT2 method for the complete active space self-consistent field/complete active space configuration interaction reference wave function. A major drawback of the FR-NEVPT2 formulation is the necessity of the fifth-order density matrix. To avoid the construction of the high-order density matrices, the combination of the FR-NEVPT2 with the CU approximation is studied. However, we find that the CU approximation remains problematic as it still introduces intruder states. The question of how to robustly and efficiently perform internally contracted multireference perturbation theories with approximate densities remains a challenging field of investigation. [ABSTRACT FROM AUTHOR]

A zeroth-order Hamiltonian based on Koopmans matrices for complete active space second-order perturbation theory (CASPT2) is presented. This Hamiltonian involves three types of Fock matrices. The original CASPT2 Fock matrix is retained for excitation classes where the excitation does not change the number of electrons in the complete active space (CAS). For excitation classes involving a change in the number of electrons in the CAS, two alternative Fock matrices corresponding to either positive or negative ionization of the CAS are introduced. These are constructed such that they exactly reproduce the Koopmans matrices for a singly ionized CAS. Test calculations indicate that the method gives better excitation energies than CASPT2 without using empirical parameters, for example, the ionization potential-electron affinity shift, which is also designed to improve excitation energies. The method is also less prone to intruder states than conventional CASPT2. Moreover, the dissociation curve for the chromium dimer looks much more reasonable than the one obtained with conventional CASPT2. [ABSTRACT FROM AUTHOR]

Many recent developments in the area of multistate multireference perturbation theories focused on methods that use a state-averaged 0th order Hamiltonian. We recently found that the dynamic correlation dressed complete active space method fails in describing ligand field and charge transfer states in a balanced way precisely because it uses a state-averaged 0th order Hamiltonian [L. Lang and F. Neese, J. Chem. Phys. 150, 104104 (2019)]. The multipartitioning idea allows the use of state-specific 0th order Hamiltonians in a multistate framework and could therefore alleviate the mentioned problem. However, the effective Hamiltonian is non-Hermitian in the traditional formulation of multipartitioning, which can lead to unphysical behavior, especially for nearly degenerate states. In order to achieve a more balanced treatment of states with different physical character and at the same time have a Hermitian effective Hamiltonian, we combine in this work multipartitioning with canonical Van Vleck perturbation theory. At the 2nd order, the result is a Hermitian variant of multipartitioning quasidegenerate N-electron valence state perturbation theory. The effect of model space noninvariance of the method is discussed and the benefit of a Hermitian formulation is highlighted with numerical examples. The method is shown to give good results for the calculation of electronic transitions of the [CuCl4]2−complex and for the calculation of electron paramagnetic resonance parameters, which are two examples where the balance between ligand field and charge transfer configurations is of utmost importance. [ABSTRACT FROM AUTHOR]

In thiswork, explicitly correlated second order N-electron valence state perturbation theory (NEVPT2-F12) has been derived and implemented for the first time. The NEVPT2-F12 algorithm presented here is based on a fully internally contracted wave function and includes the correction of semi-internal excitation subspaces. The algorithm exploits the resolution of identity (RI) approximation to improve the computational efficiency. The overall O(N5) scaling of the computational effort is documented. In Sec. III, the dissociation processes of diatomic molecules and the singlet-triplet gap of several systems are studied. For all relative energies studied in this work, the errors with respect to the complete basis set (CBS) limit for the NEVPT2-F12 method are within 1 kcal/mol. For moderately sized active spaces, the computational cost of a RI-NEVPT2-F12 correlation energy calculation for each root is comparable to a closed-shell RI-MP2-F12 calculation on the same system. [ABSTRACT FROM AUTHOR]

Multireference (MR) methods occupy an important class of approaches in quantum chemistry. In many instances, for example, in studying complex magnetic properties of transition metal complexes, they are actually the only physically satisfactory choice. In traditional MR approaches, single and double excitations are performed with respect to all reference configurations (or configuration state functions, CSFs), which leads to an explosive increase of computational cost for larger reference spaces. This can be avoided by the internal contraction scheme proposed by Meyer and Siegbahn, which effectively reduces the number of wavefunction parameters to their single-reference counterpart. The "fully internally contracted" scheme (FIC) is well known from the popular CASPT2 approach. An even shorter expansion of the wavefunction is possible with the "strong contraction" (SC) scheme proposed by Angeli and Malrieu in their NEVPT2 approach. Promising multireference configuration interaction formulations (MRCI) employing internal contraction and strong contraction have been reported by several authors. In this work, we report on the implementation of the FIC-MRCI and SC-MRCI methodologies, using a computer assisted implementation strategy. The methods are benchmarked against the traditional uncontracted MRCI approach for ground and excited states of small molecules (N2, O2, CO, CO+, OH, CH, and CN). For ground states, the comparison includes the "partially internally contracted" MRCI based on the Celani-Werner ansatz (PC-MRCI). For the three contraction schemes, the average errors range from 2% to 6% of the uncontracted MRCI correlation energies. Excitation energies are reproduced with ∼0.2 eV accuracy. In most cases, the agreement is better than 0.2 eV, even in cases with very large differential correlation contributions as exemplified for the d-d and ligand-to-metal charge transfer transitions of a Cu[NH3]2+ 4 model complex. The benchmark is supplemented with the investigation of typical potential energy surfaces (i.e., N2, HF, LiF, BeH2, ethane C-C bond stretching, and the ethylene double bond torsion). Our results indicate that the SC-scheme, which is successful in the context of second- and third-order perturbation theory, does not offer computational advantages and at the same time leads to much larger errors than the PC and FIC schemes. We discuss the advantages and disadvantages of the PC and FIC schemes, which are of comparable accuracy and, for the systems tested, also of comparable efficiency. [ABSTRACT FROM AUTHOR]

Multi-reference (MR) electronic structure methods, such as MR configuration interaction or MR perturbation theory, can provide reliable energies and properties for many molecular phenomena like bond breaking, excited states, transition states or magnetic properties of transition metal complexes and clusters. However, owing to their inherent complexity, most MR methods are still too computationally expensive for large systems. Therefore the development of more computationally attractive MR approaches is necessary to enable routine application for large-scale chemical systems. Among the state-of-the-art MR methods, second-order N-electron valence state perturbation theory (NEVPT2) is an efficient, size-consistent, and intruder-state-free method. However, there are still two important bottlenecks in practical applications of NEVPT2 to large systems: (a) the high computational cost of NEVPT2 for large molecules, even with moderate active spaces and (b) the prohibitive cost for treating large active spaces. In this work, we address problem (a) by developing a linear scaling "partially contracted" NEVPT2 method. This development uses the idea of domain-based local pair natural orbitals (DLPNOs) to form a highly efficient algorithm. As shown previously in the framework of single-reference methods, the DLPNO concept leads to an enormous reduction in computational effort while at the same time providing high accuracy (approaching 99.9% of the correlation energy), robustness, and black-box character. In the DLPNO approach, the virtual space is spanned by pair natural orbitals that are expanded in terms of projected atomic orbitals in large orbital domains, while the inactive space is spanned by localized orbitals. The active orbitals are left untouched. Our implementation features a highly efficient "electron pair prescreening" that skips the negligible inactive pairs. The surviving pairs are treated using the partially contracted NEVPT2 formalism. A detailed comparison between the partial and strong contraction schemes is made, with conclusions that discourage the strong contraction scheme as a basis for local correlation methods due to its non-invariance with respect to rotations in the inactive and external subspaces. A minimal set of conservatively chosen truncation thresholds controls the accuracy of the method. With the default thresholds, about 99.9% of the canonical partially contracted NEVPT2 correlation energy is recovered while the crossover of the computational cost with the already very efficient canonical method occurs reasonably early; in linear chain type compounds at a chain length of around 80 atoms. Calculations are reported for systems with more than 300 atoms and 5400 basis functions. [ABSTRACT FROM AUTHOR]