Complete Active Space Method Research Articles

Exact-Two-Component Complete Active Space Method with Variational Treatment of Magnetic Field and Spin-Orbit Coupling: Application to X-ray Magnetic Circular Dichroism Spectroscopy.

We introduce an exact-two-component complete active space self-consistent-field (X2C-CASSCF) method formulated under the restricted-magnetic-balance condition. This framework allows for the nonperturbative treatment of static magnetic fields using gauge-including atomic orbitals (GIAOs). The GIAO-X2C-CASSCF methodology effectively captures all microstates within the same 2J + 1-degenerate manifold and their splitting in a static magnetic field, which are not accessible through single-reference-based methods. We also present mathematical recursive expressions for evaluating one-electron relativistic integrals by using GIAOs in the presence of a finite magnetic field. Benchmark studies include oxygen and nitrogen K-edge X-ray magnetic circular dichroism spectroscopy (XMCD) for closed-shell organic compounds, as well as L-edge XMCD spectroscopy for the high-spin open-shell transition metal ion Mn2+ and the tetrahedral Mn(II)O46- complex.

Error Control and Automatic Detection of Reference Active Spaces in Many-Body Expanded Full Configuration Interaction.

We present a wide-reaching revamp of the generalized many-body expanded full configuration interaction (MBE-FCI) method. First, we outline how to automatize the selection of reference active spaces, whereby the inherent bias introduced through a manual identification is reduced, also within the context of traditional complete active space methods. Second, we allow for the use of compact orbital clusters as expansion objects, which works to circumvent the unfavorable scaling with the number of orbitals included in the space complementary to the reference orbitals. Finally, we present a new algorithm for ensuring that many-body expansions can be efficiently terminated while conservatively accounting for resulting errors. These developments are all tested on a variety of molecular systems and different orbital representations to illustrate the abilities of our algorithm to produce correlation energies within predetermined error bounds, significantly broadening the overall applicability of the MBE-FCI method.

On the Fluorescence Properties and Nonradiative Transitions in Medium-Sized All-Trans Polyenes.

The nonadiabatic photodynamics of all-trans linear polyenes with N = 4-8 conjugated double bonds is studied from an electronic structure perspective. Excitation energies and stationary points for the 1Bu and 2Ag singlet states have been computed by using the state-average complete active space (SA-CASSCF) method and its second-order perturbation theory variant (MS-CASPT2). The dependence of the two low-lying excited states on the "chain length" N has been elucidated. In addition, the 1Bu-2Ag crossing seam has been mapped out in a suitable two-dimensional coordinate space and its minimum within the subspace has been determined. This minimum is found to increase substantially and monotonously in energy with increasing N. This increase is discussed and interpreted in relation to the fluorescence properties of these systems. In particular, it allows to understand the crossover from S1(2Ag) fluorescence for smaller N to S2(1Bu) (or dual) fluorescence for larger N.

Complete Active Space Methods for NISQ Devices: The Importance of Canonical Orbital Optimization for Accuracy and Noise Resilience.

To avoid the scaling of the number of qubits with the size of the basis set, one can divide the molecular space into active and inactive regions, which is also known as complete active space methods. However, selecting the active space alone is not enough to accurately describe quantum mechanical effects such as correlation. This study emphasizes the importance of optimizing the active space orbitals to describe correlation and improve the basis-dependent Hartree-Fock energies. We will explore classical and quantum computation methods for orbital optimization and compare the chemically inspired ansatz, UCCSD, with the classical full CI approach for describing the active space in both weakly and strongly correlated molecules. Finally, we will investigate the practical implementation of a quantum CASSCF, where hardware-efficient circuits must be used and noise can interfere with accuracy and convergence. Additionally, we will examine the impact of using canonical and noncanonical active orbitals on the convergence of the quantum CASSCF routine in the presence of noise.

Reimagining the Wave Function in Density Functional Theory: Exploring Strongly Correlated States in Pancake-Bonded Radical Dimers

Pancake bonding between π-conjugated radicals challenges conventional electronic structure approximations, due to the presence of both dispersion (van der Waals) interactions and "strong" electron correlation. Here we use a reimagined wave function-in-density functional theory (DFT) approach to model pancake bonds. Our generalized self-interaction correction extends DFT's reference system of noninteracting electrons, by introducing electron-electron interactions within an active space. We show that a small variation on our previous derivation recovers a DFT-corrected complete active space method proposed by Pijeau and Hohenstein. Comparison of the two approaches shows that the latter provides reasonable dissociation curves for single bonds and pancake bonds, including excited states inaccessible to conventional linear response time-dependent DFT. The results motivate broader adoption of wavefunction-in-DFT approaches for modeling pancake bonds.

SQMBox: Interfacing a semiempirical integral library to modular ab initio electronic structure enables new semiempirical methods.

Ab initio and semiempirical electronic structure methods are usually implemented in separate software packages or use entirely different code paths. As a result, it can be time-consuming to transfer an established ab initio electronic structure scheme to a semiempirical Hamiltonian. We present an approach to unify ab initio and semiempirical electronic structure code paths based on a separation of the wavefunction ansatz and the needed matrix representations of operators. With this separation, the Hamiltonian can refer to either an ab initio or semiempirical treatment of the resulting integrals. We built a semiempirical integral library and interfaced it to the GPU-accelerated electronic structure code TeraChem. Equivalency between ab initio and semiempirical tight-binding Hamiltonian terms is assigned according to their dependence on the one-electron density matrix. The new library provides semiempirical equivalents of the Hamiltonian matrix and gradient intermediates, corresponding to those provided by the ab initio integral library. This enables the straightforward combination of semiempirical Hamiltonians with the full pre-existing ground and excited state functionality of the ab initio electronic structure code. We demonstrate the capability of this approach by combining the extended tight-binding method GFN1-xTB with both spin-restricted ensemble-referenced Kohn-Sham and complete active space methods. We also present a highly efficient GPU implementation of the semiempirical Mulliken-approximated Fock exchange. The additional computational cost for this term becomes negligible even on consumer-grade GPUs, enabling Mulliken-approximated exchange in tight-binding methods for essentially no additional cost.

Nonorthogonal Active Space Decomposition of Wave Functions with Multiple Correlation Mechanisms

The nonorthogonal active space decomposition (NO-ASD) methodology is proposed for describing systems containing multiple correlation mechanisms. NO-ASD partitions the wave function by a correlation mechanism, such that the interactions between different correlation mechanisms are treated with an effective Hamiltonian approach, while interactions between correlated orbitals in the same correlation mechanism are treated explicitly. As a result, the determinant expansion scales polynomially with the number of correlation mechanisms rather than exponentially, which significantly reduces the factorial scaling associated with the size of the correlated orbital space. Despite the nonorthogonal framework of NO-ASD, the approach can take advantage of computational efficient matrix element evaluation when performing nonorthogonal coupling of orthogonal determinant expansions. In this work, we introduce and examine the NO-ASD approach in comparison to complete active space methods to establish how the NO-ASD approach reduces the problem dimensionality and the extent to which it affects the amount of correlation energy recovered. Calculations are performed on ozone, nickel-acetylene, and isomers of μ-oxo dicopper ammonia.

Computational Spectroscopy of the Cr-Cr Bond in Coordination Complexes.

We report the accurate computational vibrational analysis of the Cr-Cr bond in dichromium complexes using second-order multireference complete active space methods (CASPT2), allowing direct comparison with experimental spectroscopic data both to facilitate interpreting the low-energy region of the spectra and to provide insights into the nature of the bonds themselves. Recent technological development by the authors has realized such computation for the first time. Accurate simulation of the vibrational structure of these compounds has been hampered by their notorious multiconfigurational electronic structure that yields bond distances that do not correlate with bond order. Some measured Cr-Cr vibrational stretching modes, ν(Cr2), have suggested weaker bonding, even for so-called ultrashort Cr-Cr bonds, while others are in line with the bond distance. Here, we optimize geometries and compute ν(Cr2) with CASPT2 for three well-characterized complexes, Cr2(O2CCH3)4(H2O)2, Cr2(mhp)4, and Cr2(dmp)4. We obtain CASPT2 harmonic ν(Cr2) modes in good agreement with experiment at 282 cm-1 for Cr2(mhp)4 and 353 cm-1 for Cr2(dmp)4, compute 50Cr and 54Cr isotope shifts, and demonstrate that the use of the so-called IPEA shift leads to improved Cr-Cr distances. Additionally, normal mode sampling was used to estimate anharmonicity along ν(Cr2), leading to an anharmonic mode of 272 cm-1 for Cr2(mhp)4 and 333 cm-1 for Cr2(dmp)4.

Molecular Structure, Thermodynamic and Spectral Characteristics of Metal-Free and Nickel Complex of Tetrakis(1,2,5-thiadiazolo)porphyrazine.

The Knudsen effusion method with mass spectrometric control of the vapor composition was used to study the possibility of a congruent transition to the gas phase and to estimate the enthalpy of sublimation of metal-free tetrakis(1,2,5-thiadiazolo)porphyrazine and its nickel complex (H2TTDPz and NiTTDPz, respectively). The geometrical and electronic structure of H2TTDPz and NiTTDPz in ground and low-lying excited electronic states were determined by DFT calculations. The electronic structure of NiTTDPz was studied by the complete active space (CASSCF) method, following accounting dynamic correlation by multiconfigurational quasi-degenerate second-order perturbation theory (MCQDPT2). A geometrical structure of D2h and D4h symmetry was obtained for H2TTDPz and NiTTDPz, respectively. According to data obtained by the MCQDPT2 method, the nickel complex possesses the ground state 1A1g, and the wave function of the ground state has the form of a single determinant. Electronic absorption and vibrational (IR and resonance Raman) spectra of H2TTDPz and NiTTDPz were studied experimentally and simulated theoretically.

Combining density-based dynamical correlation with a reduced-density-matrix strong-correlation description

A combined density and first-order reduced-density-matrix (1RDM) functional method is proposed for the calculation of potential energy curves (PECs) of molecular multibond dissociation. Its 1RDM functional part, a pair density functional, efficiently approximates the ab initio pair density of the complete active space self-consistent-field (CASSCF) method. The corresponding approximate on top pair density {\Pi} is employed to correct for double counting a correlation functional of density functional theory (DFT). The proposed ELS-DM{\Pi}DFT method with the extended L\"owdin-Shull (ELS) 1RDM functional with dispersion and multibond (DM) corrections augmented with the {\Pi}DFT functional closely reproduces PECs of multibond dissociation in the paradigmatic N_2 , H_2O, and H_2CO molecules calculated with the recently proposed CAS{\Pi}DFT (CASSCF augmented with a {\Pi} based scaled DFT correlation correction) method. Furthermore, with the additional M-correction, ELS-DM{\Pi}DFT+M reproduces well the benchmark PEC of the N_2 molecule by Lie and Clementi.

Peculiarities of electronic structure and chemical bonding in iron and cobalt metal complexes of porphyrazine and tetra(1,2,5-thiadiazole)porphyrazine

The geometrical and electronic structures of iron and cobalt metal complexes of porphyrazine and tetra(1,2,5-thiadiazole)porphyrazine in ground and low-lying excited electronic states were determined by DFT calculations and the complete active space (CASSCF) method with following accounting dynamic correlation by multiconfigurational quasidegenerate second-order perturbation theory (MCQDPT2). A geometrical structure of D[Formula: see text] symmetry has been obtained for all four complexes. According to data obtained by the MCQDPT2 method, the complexes of cobalt and iron possess the ground states 2A[Formula: see text] and 3A[Formula: see text], respectively, and wave functions of the ground states have the form of a single determinant. It is shown that the crystal field theory (CFT) can be used to describe the sequence of electronic states of the investigated complexes. The nature of the bonds between metal atoms and nitrogen atoms has been described using the analysis of the electron density distribution in the frame of Bader’s quantum theory of atoms in molecule (QTAIM).

Electronic structure of hybrid pentaheptite carbon nanoflakes containing boron-nitrogen motifs.

The electronic structure of isomeric graphene nanoflakes (NFs) heavily doped with boron and nitrogen atoms has been explored. Dispersion-corrected B3LYP functional has been used for the geometry optimizations. A complete active space method has been used for the energy evaluations. Combined boron and nitrogen doping promotes polyradicalic antiferromagnetic ground states in the NFs and affects the nanoflake geometry. There is a charge transfer from boron to nitrogen atoms which increases with the doping level. This transfer does not involve carbon atoms. Combined doping reduces both the ionization potentials (IPs) and the electron affinities (EAs) of the NFs similar to nitrogen doping alone. Boron does not affect either IPs or EAs being neither n- nor p-type dopant for the isomeric graphene NFs. All hybrid NFs show a tendency to increase the band gaps with doping level, which is promoted by the increment of the bond length alternation with doping. Finally, the hole reorganization energies for the NFs were found to be lower than the electronic ones, positioning the hybrid NF as hole-transporting systems. Graphical Abstract Color coded natural charge differences between charged and neutral states. The excess of positive charge is green for cation radicals and the excess of negative charge is red in anion radicals.

Improvement of Ab Initio Ligand Field Theory by Means of Multistate Perturbation Theory.

Over the last few years, ab initio ligand field theory (AILFT) has evolved into an important tool for the extraction of ligand field models from ab initio calculations. The inclusion of dynamic correlation on top of complete active space self-consistent field (CASSCF) reference functions, which is important for accurate results, was so far realized at the level of second-order N-electron valence state perturbation theory (NEVPT2). In this work, we introduce two alternative methods for the inclusion of dynamic correlation into AILFT calculations, the second-order dynamic correlation dressed complete active space method (DCD-CAS(2)) and the Hermitian quasi-degenerate NEVPT2 (HQD-NEVPT2). These methods belong to the class of multistate perturbation theory approaches, which allow for the mixing of CASSCF states under the effect of dynamic correlation (state-mixing). The two new versions of AILFT were tested for a diverse set of transition-metal complexes. It was found that the multistate methods have, compared to NEVPT2, an AILFT fit with smaller root mean square deviations (rmsds) between ab initio and AILFT energies. A comparison of AILFT excitation energies with the experiment shows that for some systems, the agreement gets better at the multistate level because of the smaller rmsds. However, for some systems, the agreement gets worse, which could be attributed to a cancellation of errors at the NEVPT2 level that is partly removed at the multistate level. An investigation of trends in the extracted ligand field parameters shows that at the multistate level, the ligand field splitting Δ gets larger, whereas the Racah parameters B and C get smaller and larger, respectively. An investigation of the reasons for the observed improvement for octahedral CrIII halide complexes shows that the possibility of state-mixing relaxes constraints that are present at the NEVPT2 level and that keep Δ and B from following their individual preferences.

The combination of multipartitioning of the Hamiltonian with canonical Van Vleck perturbation theory leads to a Hermitian variant of quasidegenerate N-electron valence perturbation theory.

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.

Modeling of silicon- and aluminum-doped phosphorene nanoflakes.

The electronic structure of phosphorene nanoflakes (PNFs) doped with Al and Si has been explored using hybrid functional BHandHlyp/def2-SVP and complete active space (CASSCF) methods. Doping increases the bond length alternation and changes the overall PNF shape. Doping also decreases singlet-triplet splitting in the PNFs. This effect is most notable for Si doping where singlet and triplet states become virtually degenerated. Doping also reduces band gaps and changes the nature of the ground states for Si-doped systems. The ground state of Si-doped PNFs becomes polyradicalic. In general, dopants with even number of valence electrons promote polyradicalic ground state. Doped systems show increased electron affinities (EAs), while the ionization potentials are much less affected. Larger EAs are related with the delocalization of an extra electron over the empty or partially empty 3p orbitals of the dopants. Doping increases the reorganization energies in all cases. Al-doped PNFs are the hole transport materials while Si-doped nanoflakes tend to be electron transport systems. Graphical abstract.

Spin-dependent properties in the framework of the dynamic correlation dressed complete active space method.

We report an extension of the recently proposed 2nd order dynamic correlation dressed complete active space method [S. Pathak et al., J. Chem. Phys. 147, 234109 (2017)] to incorporate spin-dependent relativistic effects into the Hamiltonian. The result is an effective Hamiltonian that contains the effects of static correlation, dynamic correlation, and relativistic effects on an equal footing. All contributions necessary for the description of magnetic phenomena and electron paramagnetic resonance (EPR) spectroscopy, namely, spin-orbit coupling, magnetic hyperfine coupling, Zeeman interaction, and direct electronic spin-spin coupling, are incorporated. We also suggest a novel analysis of g-matrices and A-matrices based on the singular value decomposition, which can provide not only the magnitude but also the sign of the principal components and allows for a transparent decomposition into different physical contributions. The new method was tested for excitation energies of first-row transition metal ions as well as D-tensors and g-shifts of first-row transition metal complexes using minimal active spaces. It was observed that state-mixing effects are usually small in these cases and that the results are comparable to nondegenerate N-electron valence state perturbation theory (NEVPT2) in conjunction with quasi-degenerate perturbation theory (QDPT). Results on EPR parameters of pseudo-square-planar Cu(ii) complexes show that state-mixing with a ligand-to-metal-charge-transfer configuration greatly improves results compared with NEVPT2/QDPT but also demonstrate that future modifications of the 0th order Hamiltonian or more elaborate electron correlation treatments will be necessary in order to achieve better agreement with the experiment.

Magnetism of metallacrown single-molecule magnets: From a simplest model to realistic systems

Electronic and magnetic properties of molecular nanomagnets are determined by competing energy scales due to the crystal field splitting, the exchange interactions between transition metal atoms, and relativistic effects. We present a comprehensive theory embracing all these phenomena based on first-principles calculations. In order to achieve this goal, we start from the ${\mathrm{FeNi}}_{4}$ cluster as a paradigm. The system can be accurately described on the ab initio level yielding all expected electronic states in a range of multiplicities from 1 to 9, with a ferromagnetic ground state. By adding the spin-orbit coupling between them we obtain the zero-field splitting. This allows to introduce a spin Hamiltonian of a giant spin model, which operates on a smaller energy scale. We compare the computed parameters of this Hamiltonian with the experimental and theoretical magnetic anisotropy energies of the monolayer Ni/Cu(001). In line with them, we find that the anisotropy almost entirely originates from the second-order spin-orbit coupling, the spin-spin coupling constitutes only a small fraction. Finally, we include the ligand atoms in our consideration. This component has a decisive role for the stabilization of molecules in experimental synthesis and characterization, and also substantially complicates the theory by bringing the superexchange mechanisms into play. Since they are higher-order effects involving two hopping matrix elements, not every theory can describe them. Our generalization of the corresponding perturbation theory substantiates the use of complete active space methods for the description of superexchange. At the same time, our numerical results for the $\left\{{\mathrm{CuFe}}_{4}\right\}$ system demonstrate that the Goodenough-Kanamori rules, which are often used to determine the sign of these exchange interactions, cannot deliver quantitative predictions due to the interplay of other mechanisms, e. g., involving multicenter Coulomb integrals. We conclude by comparing ab initio values of the exchange interaction constants for the $\left\{{\mathrm{CuCu}}_{4}\right\}$ and $\left\{{\mathrm{CuFe}}_{4}\right\}$ metallacrown magnetic molecules with experimental values determined by fitting of the magnetic susceptibility curves ${\ensuremath{\chi}}_{M}T(T)$, and attribute the remaining discrepancy between them to the role of virtual electron excitations into and out of the active space (dynamical correlations).

A dynamic correlation dressed complete active space method: Theory, implementation, and preliminary applications.

Complete Active Space SCF (CASSCF) theory may provide poor 0th order descriptions due to the lack of dynamic correlation. The most popular post-CASSCF approaches for recovering dynamic correlation are methods which keep the configuration interaction coefficients fixed at the CASSCF level and use internal contraction. This may result in severe inaccuracies where the wavefunction changes considerably under the influence of dynamic correlation. In this paper, we propose and compare several variants of a straightforward method of the "perturb-then-diagonalize" type that is aimed at keeping this balance while remaining computationally tractable and numerically stable. The method is loosely based on the theory of intermediate Hamiltonians and has been given the acronym "dynamic correlation dressed CAS" (DCD-CAS), with the second-order treatment, DCD-CAS(2), being the most practically useful member of the family. The dynamic correlation energy is treated to second order with a 0th order Hamiltonian based on Dyall's Hamiltonian. The method is orbitally invariant with respect to unitary transformations in the occupied, active, and virtual subspaces. It yields the ground- and low-lying excited states at the same time. Detailed numerical evaluations show that DCD-CAS(2) is superior to NEVPT2 for the difficult situations mentioned above while being very close to it when CASSCF provides a good 0th order description.

Electronic structure of boron-doped finite graphene sheets: unrestricted DFT and complete active space calculations

B3LYP and complete active space methods were applied to study graphene nanoribbons (GNRs) doped with boron atoms. The restricted B3LYP solutions were found to be unstable in all but two cases, and the complete active space calculations prove the multiconfigurational character of the ground state contributing with two most important configurations. The exception is the structure c4 where the system has single reference ground state in spite of the instability of the restricted wavefunction.The distance between dopant atoms, their mutual position and their location within the nanoribbon impact the relative stability of doped nanoribbons. B doping does not modify the ionisation potentials of doped GNRs. However, it notably increases the electron affinity of the core-doped nanoribbons. The doping also has a notable impact on the reorganisation energy of the nanoribbons. The reorganisation energy of B-doped GNRs is higher than the corresponding reorganisation energy of pristine and nitrogen-doped GNRs.

On the calculation of complete dissociation curves of closed-shell pseudo-onedimensional systems via the complete active space method of increments.

The method of increments (MoI) has been employed using the complete active space formalism in order to calculate the dissociation curve of beryllium ring-shaped clusters Be(n) of different sizes. Benchmarks obtained through different quantum chemical methods including the ab initio density matrix renormalization group were used to verify the validity of the MoI truncation which showed a reliable behavior for the whole dissociation curve. Moreover we investigated the size dependence of the correlation energy at different interatomic distances in order to extrapolate the values for the periodic chain and to discuss the transition from a metal-like to an insulator-like behavior of the wave function through quantum chemical considerations.