Extended Second-Order Multireference Algebraic Diagrammatic Construction Theory for Charged Excitations

Forte: A suite of advanced multireference quantum chemistry methods

Forte is an open-source library specialized in multireference electronic structure theories for molecular systems and the rapid prototyping of new methods. This paper gives an overview of the capabilities of Forte, its software architecture, and examples of applications enabled by the methods it implements.

Electron Binding Energies of Open-Shell Species from Diagonal Electron-Propagator Self-Energies with Unrestricted Hartree-Fock Spin-Orbitals

For closed-shell molecules, valence electron binding energies may be calculated accurately and efficiently with ab initio electron-propagator methods that have surpassed their predecessors. Advantageous combinations of accuracy and efficiency range from cubically scaling methods with mean errors of 0.2 eV to quintically scaling methods with mean errors of 0.1 eV or less. The diagonal self-energy approximation in the canonical Hartree-Fock basis is responsible for the enhanced efficiency of several methods. This work examines the predictive capabilities of diagonal self-energy approximations when they are generalized to the canonical spin-orbital basis of unrestricted Hartree-Fock (UHF) theory. Experimental data on atomic electron binding energies and high-level, correlated calculations in a fixed basis for a set of open-shell molecules constitute standards of comparison. A review of the underlying theory and analysis of numerical errors lead to several recommendations for the calculation of electron binding energies: (1) In calculations that employ the diagonal self-energy approximation, Koopmans's identity for UHF must be qualitatively correct. (2) Closed-shell reference states are preferable to open-shell reference states in calculations of molecular ionization energies and electron affinities. (3) For molecular electron binding energies between doublets and triplets, calculations of electron detachment energies are more accurate and efficient than calculations of electron attachment energies. When these recommendations are followed, mean absolute errors increase by approximately 0.05 eV with respect to their counterparts obtained with closed-shell reference orbitals.

Efficient Spin-Adapted Implementation of Multireference Algebraic Diagrammatic Construction Theory. I. Core-Ionized States and X-ray Photoelectron Spectra

We present an efficient implementation of multireference algebraic diagrammatic construction theory (MR-ADC) for simulating core-ionized states and X-ray photoelectron spectra (XPS). Taking advantage of spin adaptation, automatic code generation, and density fitting, our implementation can perform calculations for molecules with more than 1500 molecular orbitals, incorporating static and dynamic correlation in the ground and excited electronic states. We demonstrate the capabilities of MR-ADC methods by simulating the XPS spectra of substituted ferrocene complexes and azobenzene isomers. For the ground electronic states of these molecules, the XPS spectra computed using the extended second-order MR-ADC method (MR-ADC(2)-X) are in a very good agreement with available experimental results. We further show that MR-ADC can be used as a tool for interpreting or predicting the results of time-resolved XPS measurements by simulating the core ionization spectra of azobenzene along its photoisomerization, including the XPS signatures of excited states and the minimum energy conical intersection. This work is the first in a series of publications reporting the efficient implementations of MR-ADC methods.

Reference Energies for Valence Ionizations and Satellite Transitions

Upon ionization of an atom or a molecule, another electron (or more) can be simultaneously excited. These concurrently generated states are called "satellites" (or shakeup transitions) as they appear in ionization spectra as higher-energy peaks with weaker intensity and larger width than the main peaks associated with single-particle ionizations. Satellites, which correspond to electronically excited states of the cationic species, are notoriously challenging to model using conventional single-reference methods due to their high excitation degree compared to the neutral reference state. This work reports 42 satellite transition energies and 58 valence ionization potentials (IPs) of full configuration interaction quality computed in small molecular systems. Following the protocol developed for the quest database [Véril, M.; Scemama, A.; Caffarel, M.; Lipparini, F.; Boggio-Pasqua, M.; Jacquemin, D.; and Loos, P.-F. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2021, 11, e1517], these reference energies are computed using the configuration interaction using a perturbative selection made iteratively (CIPSI) method. In addition, the accuracy of the well-known coupled-cluster (CC) hierarchy (CC2, CCSD, CC3, CCSDT, CC4, and CCSDTQ) is gauged against these new accurate references. The performances of various approximations based on many-body Green's functions (GW, GF2, and T-matrix) for IPs are also analyzed. Their limitations in correctly modeling satellite transitions are discussed.

Simulating transient X-ray photoelectron spectra of Fe(CO)5 and its photodissociation products with multireference algebraic diagrammatic construction theory

Accurate simulations of transient X-ray photoelectron spectra (XPS) provide unique opportunities to bridge the gap between theory and experiment in understanding the photoactivated dynamics in molecules and materials. However, simulating X-ray photoelectron spectra along a photochemical reaction pathway is challenging as it requires accurate description of electronic structure incorporating core-hole screening, orbital relaxation, electron correlation, and spin-orbit coupling in excited states or at nonequilibrium ground-state geometries. In this work, we employ the recently developed multireference algebraic diagrammatic construction theory (MR-ADC) to investigate the core-ionized states and X-ray photoelectron spectra of Fe(CO)5 and its photodissociation products (Fe(CO)4, Fe(CO)3) following excitation with 266 nm light. The simulated transient Fe 3p and CO 3σ XPS spectra incorporating spin-orbit coupling and high-order electron correlation effects are shown to be in a good agreement with the experimental measurements by Leitner et al. [J. Chem. Phys., 2018, 149, 044307]. Our calculations suggest that core-hole screening, spin-orbit coupling, and ligand-field splitting effects are similarly important in reproducing the experimentally observed chemical shifts in transient Fe 3p XPS spectra of iron carbonyl complexes. Our results also demonstrate that the MR-ADC methods can be very useful in interpreting the transient XPS spectra of transition metal compounds.

New-Generation Electron-Propagator Methods for Calculations of Electron Affinities and Ionization Energies: Tests on Organic Photovoltaic Molecules

A new generation of ab initio electron-propagator self-energies recently superseded its antecedents' accuracy and computational efficiency in calculating vertical ionization energies (VIEs) of closed-shell molecules. (See J. Chem. Phys. 2021, 155, 204107, J. Chem. Theory Comput. 2022, 18, 4927, J. Chem. Phys. 2023, 159, 124109.) No adjustable parameters were introduced in the generation of reference orbitals or in the construction of self-energies. The same approach has been extended in this work to vertical electron affinities (VEAs). Calculations were performed on 24 conjugated, organic photovoltaic molecules with diverse functional groups. These molecules are considerably larger than those studied in previous tests on VIEs. Several new-generation self-energies produce mean absolute errors (MAEs) below 0.1 eV versus ΔCCSD(T) (i.e., total energy differences from the coupled-cluster singles, doubles, and perturbative triples method) VIEs and VEAs obtained with identical basis sets. A composite model employs cubically and quintically scaling algorithms and power-law basis-set extrapolations based on augmented double-triple or triple-quadruple ζ data. Its MAEs are near 0.05 eV versus benchmark values, with 0.03 eV error bars for the lowest VIE and the highest VEA of each molecule. A more efficient and equally accurate composite model for calculating VIEs avoids full transformations of electron repulsion integrals to the molecular orbital basis. High probability factors support the diagonal self-energy approximation, wherein Dyson orbitals are proportional to canonical, Hartree-Fock orbitals.

Performance Tests of the Second-Order Approximate Internally Contracted Multireference Coupled-Cluster Singles and Doubles Method icMRCC2

Benchmark results are presented for the second-order approximation of the internally contracted multireference coupled-cluster method with single and double excitations, icMRCC2 [Köhn, Bargholz, J. Chem. Phys. 2019, 151, 041106], which was designed as a multireference analogue of the single-reference second-order approximate coupled-cluster method CC2 [Christiansen, Koch, Jørgensen, Chem. Phys. Lett. 1995, 243, 409-418]. Vertical excitation energies of various small to medium-sized organic molecules are investigated based on established test sets from the literature. Additionally, the spectroscopic constants of ground and excited states of diatomics and the geometric parameters of excited triatomic molecules were determined and compared to the experimental data. The results show that the method clearly extends the applicability of single-reference CC2, including doubly excited states, and also artifacts of CC2 like too low Rydberg excitations and too weak multiple bonds are eliminated. The method is computationally more demanding than standard multireference second-order perturbation theories but improves significantly in accuracy, as shown by the benchmark results. In addition, it is demonstrated that small active spaces are often sufficient to obtain accurate energies with icMRCC2. Example applications like the automerization of cyclobutadiene, the deactivation pathway of ethylene, and the excited states of an iron complex with a noninnocent nitrosyl ligand demonstrate the potential of icMRCC2 in cases with strong multireference character.

Intruder-free cumulant-truncated driven similarity renormalization group second-order multireference perturbation theory

Accurate multireference electronic structure calculations are important for constructing potential energy surfaces. Still, even in the case of low-scaling methods, their routine use is limited by the steep growth of the computational and storage costs as the active space grows. This is primarily due to the occurrence of three- and higher-body density matrices or, equivalently, their cumulants. This work examines the effect of various cumulant truncation schemes on the accuracy of the driven similarity renormalization group second-order multireference perturbation theory. We test four different levels of three-body reduced density cumulant truncations that set different classes of cumulant elements to zero. Our test cases include the singlet-triplet gap of CH2, the potential energy curves of the XΣg+1 and AΣu+3 states of N2, and the singlet-triplet splittings of oligoacenes. Our results show that both relative and absolute errors introduced by these cumulant truncations can be as small as 0.5 kcal mol-1 or less. At the same time, the amount of memory required is reduced from O(NA6) to O(NA5), where NA is the number of active orbitals. No additional regularization is needed to prevent the intruder state problem in the cumulant-truncated second-order driven similarity renormalization group multireference perturbation theory methods.

Algebraic Diagrammatic Construction Schemes Employing the Intermediate State Formalism: Theory, Capabilities, and Interpretation

Algebraic diagrammatic construction (ADC) schemes represent a family of ab initio methods for the calculation of excited electronic states and electron-detached and -attached states. All ADC methods have been demonstrated to possess great potential for molecular applications, e.g., for the calculation of absorption or photoelectron spectra or electron attachment processes. ADC originates from Green's function or propagator theory; however, most recent ADC developments heavily rely on the intermediate state representation or effective Liouvillian formalisms, which comprise new ADC methods and computational schemes for high-order properties. The different approaches for the calculation of excitation energies, ionization potentials, and electron affinities are intimately related, and they provide a coherent description of these quantities at equivalent levels of theory and with comparable errors. Most quantum chemical program packages contain ADC methods; however, the most complete ADC suite of methods can be found in the recent release of Q-Chem.

Algebraic Diagrammatic Construction Theory for Simulating Charged Excited States and Photoelectron Spectra

Charged excitations are electronic transitions that involve a change in the total charge of a molecule or material. Understanding the properties and reactivity of charged species requires insights from theoretical calculations that can accurately describe orbital relaxation and electron correlation effects in open-shell electronic states. In this Review, we describe the current state of algebraic diagrammatic construction (ADC) theory for simulating charged excitations and its recent developments. We start with a short overview of ADC formalism for the one-particle Green's function, including its single- and multireference formulations and extension to periodic systems. Next, we focus on the capabilities of ADC methods and discuss recent findings about their accuracy for calculating a wide range of excited-state properties. We conclude our Review by outlining possible directions for future developments of this theoretical approach.

Simulating Spin-Orbit Coupling with Quasidegenerate N-Electron Valence Perturbation Theory

We present the first implementation of spin-orbit coupling effects in fully internally contracted second-order quasidegenerate N-electron valence perturbation theory (SO-QDNEVPT2). The SO-QDNEVPT2 approach enables the computations of ground- and excited-state energies and oscillator strengths combining the description of static electron correlation with an efficient treatment of dynamic correlation and spin-orbit coupling. In addition to SO-QDNEVPT2 with the full description of one- and two-body spin-orbit interactions at the level of two-component Breit-Pauli Hamiltonian, our implementation also features a simplified approach that takes advantage of spin-orbit mean-field approximation (SOMF-QDNEVPT2). The accuracy of these methods is tested for the group 14 and 16 hydrides, 3d and 4d transition metal ions, and two actinide dioxides (neptunyl and plutonyl dications). The zero-field splittings of group 14 and 16 molecules computed using SO-QDNEVPT2 and SOMF-QDNEVPT2 are in good agreement with the available experimental data. For the 3d transition metal ions, the SO-QDNEVPT2 method is significantly more accurate than SOMF-QDNEVPT2, while no substantial difference in the performance of two methods is observed for the 4d ions. Finally, we demonstrate that for the actinide dioxides the results of SO-QDNEVPT2 and SOMF-QDNEVPT2 are in good agreement with the data from previous theoretical studies of these systems. Overall, our results demonstrate that SO-QDNEVPT2 and SOMF-QDNEVPT2 are promising multireference methods for treating spin-orbit coupling with a relatively low computational cost.

Assessment of State-Averaged Driven Similarity Renormalization Group on Vertical Excitation Energies: Optimal Flow Parameters and Applications to Nucleobases

We present a comprehensive excited-state benchmark for the state-averaged (SA) driven similarity renormalization group (DSRG) [Li, C.; Evangelista, F. A. J. Chem. Phys. 2018, 148, 124106]. Following the QUEST database [Véril, M.; Scemama, A.; Caffarel, M.; Lipparini, F.; Boggio-Pasqua, M.; Jacquemin, D.; Loos, P.-F. Wiley Interdiscip. Rev. Comput. Mol. Sci. 2021, 11, e1517], 280 vertical transition energies of 35 medium-sized molecules are computed using the SA-DSRG derived second- and third-order perturbation theories (PT2/PT3) along with a nonperturbative approach [sq-LDSRG(2)]. Comparing to the theoretical best estimates, the optimal flow parameter is found to be 0.35 and 2.0 Eh-2 for SA-DSRG-PT2 and SA-DSRG-PT3, respectively. For SA-sq-LDSRG(2), a flow parameter of 1.5 Eh-2 provides converged equations without compromising the accuracy. We then assess the accuracy of the SA-DSRG hierarchy using these parameters. The SA-DSRG-PT2 scheme outperforms the level-shifted CASPT2 by 0.10 eV in mean absolute error (MAE), yet this accuracy is slightly inferior than that of CASPT2 with the ionization-potential-electron-affinity shift. Both SA-DSRG-PT3 and SA-sq-LDSRG(2) yield a MAE of 0.10 eV, which is comparable to that of CASPT3 (0.09 eV). Finally, we compute vertical excitation energies of several low-lying singlet states of nucleobases. The SA-sq-LDSRG(2) approach provides highly accurate results for π → π* excitations, while n → π* transitions are better described by SA-DSRG-PT3.

Large-Scale Benchmarking of Multireference Vertical-Excitation Calculations via Automated Active-Space Selection

We have calculated state-averaged complete-active-space self-consistent-field (SA-CASSCF), multiconfiguration pair-density functional theory (MC-PDFT), hybrid MC-PDFT (HMC-PDFT), and n-electron valence state second-order perturbation theory (NEVPT2) excitation energies with the approximate pair coefficient (APC) automated active-space selection scheme for the QUESTDB benchmark database of 542 vertical excitation energies. We eliminated poor active spaces (20–40% of calculations) by applying a threshold to the SA-CASSCF absolute error. With the remaining calculations, we find that NEVPT2 performance is significantly impacted by the size of the basis set the wave functions are converged in, regardless of the quality of their description, which is a problem absent in MC-PDFT. Additionally, we find that HMC-PDFT is a significant improvement over MC-PDFT with the translated PBE (tPBE) density functional and that it performs about as well as NEVPT2 and second-order coupled cluster on a set of 373 excitations in the QUESTDB database. We optimized the percentage of SA-CASSCF energy to include in HMC-PDFT when using the tPBE on-top functional, and we find the 25% value used in tPBE0 to be optimal. This work is by far the largest benchmarking of MC-PDFT and HMC-PDFT to date, and the data produced in this work are useful as a validation of HMC-PDFT and of the APC active-space selection scheme. We have made all the wave functions produced in this work (orbitals and CI vectors) available to the public and encourage the community to utilize this data as a tool in the development of further multireference model chemistries.

Simulating X-ray photoelectron spectra with strong electron correlation using multireference algebraic diagrammatic construction theory

A new theoretical approach for the simulations of X-ray photoelectron spectra of strongly correlated molecular systems that combines multireference algebraic diagrammatic construction theory (MR-ADC) with a core–valence separation (CVS) technique.

An efficient implementation of the NEVPT2 and CASPT2 methods avoiding higher-order density matrices

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.

Analytic Energy Gradients for the Driven Similarity Renormalization Group Multireference Second-Order Perturbation Theory

We derive analytic energy gradients of the driven similarity renormalization group (DSRG) multireference second-order perturbation theory (MRPT2) using the method of Lagrange multipliers. In the Lagrangian, we impose constraints for a complete-active-space self-consistent-field reference wave function and the semicanonical orthonormal molecular orbitals. Solving for the associated Lagrange multipliers is found to share the same asymptotic scaling of a single DSRG-MRPT2 energy computation. A pilot implementation of the DSRG-MRPT2 analytic gradients is used to optimize the geometry of the singlet and triplet states of p-benzyne. The equilibrium bond lengths and angles are similar to those computed via other MRPT2s and Mukherjee's multireference coupled cluster theory. An approximate DSRG-MRPT2 method that neglects the contributions of the three-body density cumulant is found to introduce negligible errors in the geometry of p-benzyne, lending itself to a promising low-cost approach for molecular geometry optimizations using large active spaces.

Density Matrix Renormalization Group with Dynamical Correlation via Adiabatic Connection

The quantum chemical version of the density matrix renormalization group (DMRG) method has established itself as one of the methods of choice for calculations of strongly correlated molecular systems. Despite its great ability to capture strong electronic correlation in large active spaces, it is not suitable for computations of dynamical electron correlation. In this work, we present a new approach to the electronic structure problem of strongly correlated molecules, in which DMRG is responsible for a proper description of the strong correlation, whereas dynamical correlation is computed via the recently developed adiabatic connection (AC) technique which requires only up to two-body active space reduced density matrices. We report the encouraging results of this approach on typical candidates for DMRG computations, namely, n-acenes (n = 2 → 7), Fe(II)-porphyrin, and the Fe₃S₄ cluster.

Multireference Algebraic Diagrammatic Construction Theory for Excited States: Extended Second-Order Implementation and Benchmark

We present an implementation and benchmark of new approximations in multireference algebraic diagrammatic construction theory for simulations of neutral electronic excitations and UV/vis spectra of strongly correlated molecular systems (MR-ADC). Following our work on the first-order MR-ADC approximation [J. Chem. Phys. 2018, 149, 204113], we report the strict and extended second-order MR-ADC methods (MR-ADC(2) and MR-ADC(2)-X) that combine the description of static and dynamic electron correlation in the ground and excited electronic states without relying on state-averaged reference wave functions. We present an extensive benchmark of the new MR-ADC methods for excited states in several small molecules, including the carbon dimer, ethylene, and butadiene. Our results demonstrate that, for weakly correlated electronic states, the MR-ADC(2) and MR-ADC(2)-X methods outperform the third-order single-reference ADC approximation and are competitive with the results from equation-of-motion coupled cluster theory. For states with multireference character, the performance of the MR-ADC methods is similar to that of an N-electron valence perturbation theory. In contrast to conventional multireference perturbation theories, the MR-ADC methods have many attractive features, such as a straightforward and efficient calculation of excited-state properties and a direct access to excitations outside of the frontier (active) orbitals.

Efficient implementation of the single-reference algebraic diagrammatic construction theory for charged excitations: Applications to the TEMPO radical and DNA base pairs

We present an efficient implementation of the second- and third-order single-reference algebraic diagrammatic construction (ADC) theory for electron attachment and ionization energies and spectra [EA/IP-ADC(n), n = 2, 3]. Our new EA/IP-ADC program features spin adaptation for closed-shell systems, density fitting for efficient handling of the two-electron integral tensors, and vectorized and parallel implementation of tensor contractions. We demonstrate capabilities of our efficient implementation by applying the EA/IP-ADC(n) (n = 2, 3) methods to compute the photoelectron spectrum of the (2,2,6,6-tetramethylpiperidin-1-yl)oxyl (TEMPO) radical, as well as the vertical and adiabatic electron affinities of TEMPO and two DNA base pairs (guanine-cytosine and adenine-thymine). The spectra and electron affinities computed using large diffuse basis sets with up to 1028 molecular orbitals are found to be in good agreement with the best available results from the experiment and theoretical simulations.