Identifying and removing intruder states in multireference Mo/ller–Plesset perturbation theory

Multiconfigurational Correlation at DFT + U Cost: On-Site Electron-Electron Interactions Yield a Block-Localized Configuration Interaction Hamiltonian

This work presents a first-principles wavefunction-in-DFT approach based on the Hubbard density functional theory (DFT) + U method. This approach begins with the standard DFT reference system of noninteracting electrons and introduces an electron-electron interaction projected onto DFT+U-type atomic states. The reference system's configuration interaction Hamiltonian is block-localized to these states and can be expressed in terms of state occupation numbers, state self-energies (which correspond to unscreened Hubbard U values), and the promotion energies of doubly excited Slater determinants. Simple approximations for the promotion energies provide multiconfigurational correlation energies without requiring explicit orbital localization/transform. Numerical results for fractionally occupied chromium atom, bonded chromium dimer, dissociating covalent bonds, and large active spaces show that the approach provides beyond-zero-sum accuracy at computational cost comparable to standard DFT+U.

Time-Dependent Density Functional Theory for X-ray Absorption Spectra: Comparing the Real-Time Approach to Linear Response

We simulate X-ray absorption spectra at elemental K-edges using time-dependent density functional theory (TDDFT) in both its conventional linear-response implementation and its explicitly time-dependent or "real-time" formulation. Real-time TDDFT simulations enable broadband spectra calculations without the need to invoke frozen occupied orbitals ("core/valence separation"), but we find that these spectra are often contaminated by transitions to the continuum that originate from lower-energy core and semicore orbitals. This problem becomes acute in triple-ζ basis sets, although it is sometimes sidestepped in double-ζ basis sets. Transitions to the continuum acquire surprisingly large dipole oscillator strengths, leading to spectra that are difficult to interpret. Meaningful spectra can be recovered by means of a filtering technique that decomposes the spectrum into contributions from individual occupied orbitals, and the same procedure can be used to separate L- and K-edge spectra arising from different elements within a given molecule. In contrast, conventional linear-response TDDFT requires core/valence separation but is free of these artifacts. It is also significantly more efficient than the real-time approach, even when hundreds of individual states are needed to reproduce near-edge absorption features and even when Padé approximants are used to reduce the real-time simulations to just 2-4 fs of time propagation. Despite the cost, the real-time approach may be useful to examine the validity of the core/valence separation approximation.

A Similarity Renormalization Group Approach to Green's Function Methods

The family of Green's function methods based on the GW approximation has gained popularity in the electronic structure theory thanks to its accuracy in weakly correlated systems combined with its cost-effectiveness. Despite this, self-consistent versions still pose challenges in terms of convergence. A recent study [Monino and Loos J. Chem. Phys. 2022, 156, 231101.] has linked these convergence issues to the intruder-state problem. In this work, a perturbative analysis of the similarity renormalization group (SRG) approach is performed on Green's function methods. The SRG formalism enables us to derive, from first-principles, the expression of a naturally static and Hermitian form of the self-energy that can be employed in quasiparticle self-consistent GW (qsGW) calculations. The resulting SRG-based regularized self-energy significantly accelerates the convergence of qsGW calculations, slightly improves the overall accuracy, and is straightforward to implement in existing code.

Approximations of density matrices in N-electron valence state second-order perturbation theory (NEVPT2). I. Revisiting the NEVPT2 construction

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.

Compressed-State Multistate Pair-Density Functional Theory

Multiconfiguration pair-density functional theory (MC-PDFT) is a multireference method that can be used to calculate excited states. However, MC-PDFT potential energy surfaces have the wrong topology at conical intersections because the last step of MC-PDFT is not a diagonalization of a model-space Hamiltonian matrix, as done in, for example, multistate second-order perturbation theory (MS-CASPT2). We have previously proposed methods that solve this problem by diagonalizing a model-space effective Hamiltonian matrix, where the diagonal elements are MC-PDFT energies for intermediate states, and the off-diagonal elements are evaluated by wave function theory. One previous method is called variational multistate PDFT (VMS-PDFT), whose intermediate states maximize the trace of the effective Hamiltonian, namely, the sum of the MC-PDFT energies of the model-space states; the VMS-PDFT is very robust but is more computationally expensive than another method, extended multistate PDFT (XMS-PDFT), in which the transformation to intermediate states is accomplished without needing any density functional evaluations. However, although VMS-PDFT was accurate in all cases tested, XMS-PDFT was accurate in only some of them. In the present paper, we propose a new method, called compressed-state multistate PDFT (CMS-PDFT), that is as efficient as XMS-PDFT and as accurate as VMS-PDFT. The new method maximizes the trace of the classical Coulomb energy of the intermediate states such that the electron densities of the intermediate states are compressed. We show that CMS-PDFT performs robustly even where XMS-PDFT fails.

How Much Dispersion Energy Is Included in the Multiconfigurational Interaction Energy?

We demonstrate how to quantify the amount of dispersion interaction recovered by supermolecular calculations with the multiconfigurational self-consistent field (MCSCF) wave functions. For this purpose, we present a rigorous derivation which connects the portion of dispersion interaction captured by the assumed wave function model-the residual dispersion interaction-with the size of the active space. Based on the obtained expression for the residual dispersion contribution, we propose a dispersion correction for the MCSCF that avoids correlation double counting. Numerical demonstration for model four-electron dimers in both ground and excited states described with the complete active space self-consistent field (CASSCF) reference serves as a proof-of-concept for the method. Accurate results, largely independent of the size of the active space, are obtained. For many-electron systems, routine CASSCF interaction energy calculations recover a tiny fraction of the full second-order dispersion energy. We found that the residual dispersion is non-negligible only for purely dispersion-bound complexes.

Ab Initio Probing of the Ground State of Tetraradicals: Breakdown of Hund's Multiplicity Rule

The electronic structure of organic σ-type polyradical including 2,4,6-tridehydropyridine radical cation (246-TDHP) and three isomers of tetradehydrobenzene (TDHB) have been studied using a computationally robust and cost-effective second-order multireference perturbative model which provides a balanced treatment of nondynamic and dynamic contributions to the electron correlation problem in the ground or excited electronic states which are imperative for predicting structural properties (e.g., ground state multiplicity, energy gaps between high-spin and low-spin states, etc.) of polyradicals. Energy gaps are useful to capture insight into the degree of interaction between the radical sites. An important finding of this study is that the tetraradicals considered here possess singlet ground states, contrary to Hund's rule. Present findings are in close agreement with the available high-level ab initio estimates at attainable cost implying that a perturbative description of the systems is adequate. The impact of N⁺ on the nature of ground state for the 246-TDHP have also analyzed. The singlet-triplet energy gaps for 1245- and 1234-TDHB are smaller than for o-benzyne mainly due to the ring strain. 1235-TDHB is 14.42 and 11.05 kcal/mol lower in energy than 1245- and 1234-isomers, respectively. IVO-SSMRPT predicts ¹A₁-³B₂ and ¹A₁-⁵B₂ gaps of 25.84 and 105.15 kcal/mol, respectively for the 246-TDHP cation.

Ab initio investigation of Br-3d core-excited states in HBr and HBr+ toward XUV probing of photochemical dynamics

Ultrafast X-ray/XUV transient absorption spectroscopy is a powerful tool for real-time probing of chemical dynamics. Interpretation of the transient absorption spectra requires knowledge of core-excited potentials, which necessitates assistance from high-level electronic-structure computations. In this study, we investigate Br-3d core-excited electronic structures of hydrogen bromide (HBr) using spin-orbit general multiconfigurational quasidegenerate perturbation theory (SO-GMC-QDPT). Potential energy curves and transition dipole moments are calculated from the Franck-Condon region to the asymptotic limit and used to construct core-to-valence absorption strengths for five electronic states of HBr (Σ 1 0 + ,   3 Π 1 ,   1 Π 1 ,   3 Π 0 + ,   3 Σ 1 ) and two electronic states of HBr+ (2Π3∕2, 2Σ1∕2). The results illustrate the capabilities of Br-3d edge probing to capture transitions of the electronic-state symmetry as well as nonadiabatic dissociation processes that evolve across avoided crossings. Furthermore, core-to-valence absorption spectra are simulated from the neutralΣ 1 0 + state and the ionicΠ 2 1 / 2 , 3 / 2 states by numerically solving the time-dependent Schrödinger equation and exhibit excellent agreement with the experimental spectrum. The comprehensive and quantitative picture of the core-excited states obtained in this work allows for transparent analysis of the core-to-valence absorption signals, filling gaps in the theoretical understanding of the Br-3d transient absorption spectra.

Theoretical Studies of Singlet Fission: Searching for Materials and Exploring Mechanisms

In this Review article, a survey is given for theoretical studies in the subject of singlet fission. Singlet fission converts one singlet exciton to two triplet excitons. With the doubled number of excitons and the longer lifetime of the triplets, singlet fission provides an avenue to improve the photoelectric conversion efficiency in organic photovoltaic devices. It has been a subject of intense research in the past decade. Theoretical studies play an essential role in understanding singlet fission. This article presents a Review of theoretical studies in singlet fission since 2006, the year when the research interest in this subject was reignited. Both electronic structure and dynamics studies are covered. Electronic structure studies provide guidelines for designing singlet fission chromophores and insights into the couplings between single- and multi-excitonic states. The latter provides fundamental knowledge for engineering interchromophore conformations to enhance the fission efficiency. Dynamics studies reveal the importance of vibronic couplings in singlet fission.

Four-Component Relativistic State-Specific Multireference Perturbation Theory with a Simplified Treatment of Static Correlation

The relativistic multireference (MR) perturbative approach is one of the most successful tools for the description of computationally demanding molecular systems of heavy elements. We present here the ground state dissociation energy surfaces, equilibrium bond lengths, harmonic frequencies, and dissociation energies of Ag₂, Cu₂, Au₂, and I₂ computed using the four-component (4c) relativistic spinors based state-specific MR perturbation theory (SSMRPT) with improved virtual orbital complete active space configuration interaction (IVO-CASCI) functions. The IVO-CASCI method is a simple, robust, useful and lower cost alternative to the complete active space self-consistent field approach for treating quasidegenerate situations. The redeeming features of the resulting method, termed as 4c-IVO-SSMRPT, lies in (i) manifestly size-extensivity, (ii) exemption from intruder problems, (iii) the freedom of convenient multipartitionings of the Hamiltonian, (iv) flexibility of the relaxed and unrelaxed descriptions of the reference coefficients, and (v) manageable cost/accuracy ratio. The present method delivers accurate descriptions of dissociation processes of heavy element systems. Close agreement with reference values has been found for the calculated molecular constants indicating that our 4c-IVOSSMRPT provides a robust and economic protocol for determining the structural properties for the ground state of heavy element molecules with eloquent MR character as it treats correlation and relativity on equal footing.

Behaviour of DFT-based approaches to the spin–orbit term of zero-field splitting tensors: a case study of metallocomplexes, MIII(acac)3 (M = V, Cr, Mn, Fe and Mo)

Zero-field splitting tensors of M^III(acac)₃ complexes are calculated using ab initio and DFT methods.

A simplified ab initio treatment of diradicaloid structures produced from stretching and breaking chemical bonds

With a proper choice of active spaces, the single root perturbation theory employing improved virtual orbitals can flawlessly describe the ground, excited, ionized, and dissociated states having varying degrees of degeneracy at the expense of low computational cost.

Tuning spin-states of carbynes and silylynes: a long jump with one leg

The challenge motivating this paper is to induce, by chemical substitution, a silylyne, SiR, or a congeneric carbyne, CR, to adopt the high-spin quartet rather than the low-spin doublet as its ground state. The difficulty is seen in the preference for the doublet of the parent SiH (doublet-quartet energy difference ∼39 kcal/mol, favoring the doublet) or CH (∼17 kcal/mol). Strategies for having high-spin ground state parallel those for silylenes and carbenes: greater electropositivity (σ-donation) and π-acceptance of the single substituent favor the high-spin state. The electronegativity trend can be understood from an ions in molecules way of thinking already present in the literature in the works of Boldyrev and Simons, and of Mavridis and Harrison; i.e., the quartet ground state spin of some CR/SiR species is largely determined by the ground state spin of C(-)/Si(-). In this study, we provide a diabatization analysis that solidly confirms the ions in molecules picture and explains the difference in the equilibrium internuclear distances for the two spin states. In general, electronegativity dominates the ordering of spin states. π-Acceptors also help to lower the quartet state energy of the many carbynes (silylynes) examined, whose range of doublet-quartet differences calculated is impressive, 120 (100) kcal/mol. The qualitative understanding gained leads to the prediction of some quartet-ground state carbynes (CMgH, CAlH2, CZnH, CSiH3, CSiF3, etc.) and a smaller number of silylynes (SiMgH, SiMgF, SiBeH, etc.). A beginning is made on the energetics of approach geometries of the fragments in the highly exoergic dimerization of CH to acetylene; it should proceed for the ground state doublet CH through C2h-like trajectories, with no activation energy.

The low-lying electronic states of pentacene and their roles in singlet fission

We present a detailed study of pentacene monomer and dimer that serves to reconcile extant views of its singlet fission. We obtain the correct ordering of singlet excited-state energy levels in a pentacene molecule (E (S1) < E (D)) from multireference calculations with an appropriate active orbital space and dynamical correlation being incorporated. In order to understand the mechanism of singlet fission in pentacene, we use a well-developed diabatization scheme to characterize the six low-lying singlet states of a pentacene dimer that approximates the unit cell structure of crystalline pentacene. The local, single-excitonic diabats are not directly coupled with the important multiexcitonic state but rather mix through their mutual couplings with one of the charge-transfer configurations. We analyze the mixing of diabats as a function of monomer separation and pentacene rotation. By defining an oscillator strength measure of the coherent population of the multiexcitonic diabat, essential to singlet fission, we find this population can, in principle, be increased by small compression along a specific crystal direction.

An ab initio MO study of heavy atom effects on the zero-field splitting tensors of high-spin nitrenes: how the spin–orbit contributions are affected

The CASSCF and hybrid CASSCF–MRMP2 methods reproduce the ZFS tensors of spin-septet 2,4,6-trinitrenopyridines, focusing on the heavy atom effects on the spin–orbit terms of the tensors.

Computational investigation of the photochemical deoxygenation of thiophene-S-oxide and selenophene-Se-oxide

A multireference CASSCF investigation of the dissociation of thiophene-S-oxide and selenophene-Se-oxide in their ground and accessible excited state demonstrates that dissociation can occur from T₂ with little or no barrier.

Multiconfiguration perturbation theory: Size consistency at second order

A modified version of a previously elaborated multiconfiguration perturbation theory (MCPT) [Rolik et al. J. Chem. Phys. 119, 1922 (2003)] is presented. In the modified formulation size consistency is ensured at second order in energy, by omitting projectors from the zero order Hamiltonian operator. This MCPT formulation is abbreviated as SC2-MCPT (size consistent at second order). To ensure proper separability, we also require that energy denominators are constructed as differences of some one-particle energies. A similar choice for energy denominators also renders the well-known multireference Moller-Plesset (MRMP) energy size consistent at second order. The same thing applies to the related multireference perturbation theory by Witek, Nakano, and Hirao.

Performance of Molecular Orbital Methods and Density Functional Theory in the Computation of Geometries and Energies of Metal Aqua Ions

Geometries and energies of various metal aqua ions have been computed with Hartree-Fock (HF), single-reference second-order perturbation theory according to Møller and Plesset (MP2), complete active space self-consistent field (CAS-SCF) methods, multi-configurational quasi-degenerate second-order perturbation theory (MCQDPT2), and density functional theory (DFT), whereby for the latter the most widely used functionals BLYP, B3LYP, LDA ("local density approximation"), and another functional (SOP) exhibiting Slater exchange, have been investigated. The geometries as well as the energies are sensitive to the computational method, results of which can lead to different conclusions. The geometries of certain reactants, products, intermediates, or transition states cannot be obtained with all of the above-mentioned computational methods: in some cases, the structures are very poor, exhibiting large errors in the bond parameters, and in others, the targeted species cannot be calculated at all. The different computational techniques do not always predict the same coordination number of a given aqua ion or the same preferred water exchange mechanism. Inaccuracies arising from inadequate basis sets are also discussed.

Intruder state avoidance multireference Møller-Plesset perturbation theory

A new perturbation approach is proposed that enhances the low-order, perturbative convergence by modifying the zeroth-order Hamiltonian in a manner that enlarges any small-energy denominators that may otherwise appear in the perturbative expansion. This intruder state avoidance (ISA) method can be used in conjunction with any perturbative approach, but is most applicable to cases where small energy denominators arise from orthogonal-space states-so-called intruder states-that should, under normal circumstances, make a negligible contribution to the target state of interests. This ISA method is used with multireference Møller-Plesset (MRMP) perturbation theory on potential energy curves that are otherwise plagued by singularities when treated with (conventional) MRMP; calculation are performed on the 1(3)Sigma(-)(u) state of O(2); and the 2(1)Delta, 3(1)Delta, 2(3)Delta, and 3(3)Delta states of AgH. This approach is also applied to other calculations where MRMP is influenced by intruder states; calculations are performed on the (3)Pi(u) state of N(2), the (3)Pi state of CO, and the 2(1)A' state of formamide. A number of calculations are also performed to illustrate that this approach has little or no effect on MRMP when intruder states are not present in perturbative calculations; vertical excitation energies are computed for the low-lying states of N(2), C(2), CO, formamide, and benzene; the adiabatic (1)A(1)-(3)B(1) energy separation in CH(2), and the spectroscopic parameters of O(2) are also calculated. Vertical excitation energies are also performed on the Q and B bands states of free-base, chlorin, and zinc-chlorin porphyrin, where somewhat larger couplings exists, and-as anticipated-a larger deviation is found between MRMP and ISA-MRMP.