Multireference Driven Similarity Renormalization Group: A Second-Order Perturbative Analysis

Frozen Natural Orbitals for the State-Averaged Driven Similarity Renormalization Group

We present a reduced-cost implementation of the state-averaged driven similarity renormalization group (SA-DSRG) based on the frozen natural orbital (FNO) approach. The natural orbitals (NOs) are obtained by diagonalizing the one-body reduced density matrix from SA-DSRG second-order perturbation theory (SA-DSRG-PT2). We consider three criteria to truncate the virtual NOs for the subsequent electron correlation treatment beyond SA-DSRG-PT2. An additive second-order correction is applied to the SA-DSRG Hamiltonian to reintroduce correlation effects from the discarded orbitals. The FNO SA-DSRG method is benchmarked on 35 small organic molecules in the QUEST database. When keeping 98-99% of the cumulative occupation numbers, the mean absolute error in the vertical transition energies due to FNO is less than 0.01 eV. Using the same FNO threshold, we observe a speedup of 9 times compared to the conventional SA-DSRG implementation for nickel carbonyl with a quadruple-ζ basis set. The FNO approach enables nonperturbative SA-DSRG computations on chloroiron corrole [FeCl(C19H11N4)] with more than 1000 basis functions, surpassing the current limit of a conventional implementation.

Modular Approach to Selected Configuration Interaction in an Arbitrary Spin Basis: Implementation and Comparison of Approaches

A modular selected configuration interaction (SCI) code has been developed that is based on the existing Monte-Carlo configuration interaction code (MCCI). The modularity allows various selection protocols to be implemented with ease and allows for fair comparison between wave functions built via different criteria. We have initially implemented adaptations of existing SCI theories, which are based on either energy- or coefficient-driven selection schemes. These codes have been implemented not only in the basis of Slater determinants (SDs) but also in the basis of configuration state functions (CSFs) and extended to state-averaged regimes. This allows one to take advantage of the reduced dimensionality of the wave function in the CSF basis and also the guarantee of pure spin states. All SCI methods were found to be able to predict potential energy surfaces to high accuracy, producing compact wave functions, when compared to full configuration interaction (FCI) for a variety of bond-breaking potential energy surfaces. The compactness of the error-controlled adaptive configuration interaction approach, particularly in the CSF basis, was apparent with nonparallelity errors within chemical accuracy while containing as little as 0.02% of the FCI CSF space. The size-to-accuracy was also extended to FCI spaces approaching one billion configurations.

Dynamic Correlation on the Adaptive Sampling Configuration Interaction (ASCI) Reference Function: ASCI-DSRG-MRPT2

A balanced description of static and dynamic electron correlations is at the heart of quantum chemical methods. To obtain accurate results in strongly correlated systems using wave-function-based methods, a large active space is necessary to ensure correct descriptions of static correlations. Correcting the results for dynamic correlations is also necessary. In this work, we present implementations of second-order perturbation theory for dynamic correlations based on the adaptive sampling configuration interaction self-consistent field (ASCI-SCF) method. In particular, we implemented spin-free driven similarity renormalization group second-order multireference perturbation theory (DSRG-MRPT2). The extrapolation of the ASCI + PT2 energy based on the relaxed Hamiltonian in DSRG-MRPT2 gives a reasonable approximation of DSRG-MRPT2 based on CASSCF. We demonstrate the application of the ASCI-DSRG-MRPT2 method in evaluations of the spin-state energy gaps in iron porphyrins, polyacenes, and periacenes along with the reaction energies of methane oxidation by FeO+ and electrocyclic ring formation in cethrene.

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.

Unraveling the Spin-State Energetics of FeN4 Complexes with Ab Initio Methods

A systematic analysis was conducted to explore the spin-state energetics of a series of 19 FeN4 complexes. The performance of a large number of multireference methods was assessed, highlighting the significant challenges associated with accurately describing the spin-state energetics of FeN4 complexes. Most multireference methods were found to be susceptible to errors originating from the reference CASSCF wavefunction, leading to an overstabilization of high-spin states. Nonetheless, a few multireference methods, namely, CASPT2/CC, DSRG-MRPT3, and LDSRG(2), demonstrated promising performance compared to the benchmark CCSD(T) method. Furthermore, our study revealed that FeN4 complexes having a quintet ground state are exceedingly rare. Accordingly, only one specific model (Fe(L2)) and one synthesized complex (Fe(OTBP)) have the quintet ground state among the studied complexes. This scarcity of quintet FeN4 complexes highlights the unique nature of these systems and raises intriguing questions regarding the factors influencing spin states, such as the size of the macrocycle cavity, the introduction of substituents, or the induction of out-of-plane deformation.

A Regularized Second-Order Correlation Method from Green's Function Theory

We present a scalable single-particle framework to treat electronic correlation in molecules and materials motivated by Green's function theory. We derive a size-extensive Brillouin-Wigner perturbation theory from the single-particle Green's function by introducing the Goldstone self-energy. This new ground state correlation energy, referred to as Quasi-Particle MP2 theory (QPMP2), avoids the characteristic divergences present in both second-order Møller-Plesset perturbation theory and Coupled Cluster Singles and Doubles within the strongly correlated regime. We show that the exact ground state energy and properties of the Hubbard dimer are reproduced by QPMP2 and demonstrate the advantages of the approach for larger Hubbard models where the metal-to-insulator transition is qualitatively reproduced, contrasting with the complete failure of traditional methods. We apply this formalism to characteristic strongly correlated molecular systems and show that QPMP2 provides an efficient, size-consistent regularization of MP2.

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.

A study of core-excited states of organic molecules computed with the generalized active space driven similarity renormalization group

This work examines the accuracy and precision of x-ray absorption spectra computed with a multireference approach that combines generalized active space (GAS) references with the driven similarity renormalization group (DSRG). We employ the x-ray absorption benchmark of organic molecule (XABOOM) set, consisting of 116 transitions from mostly organic molecules [Fransson et al., J. Chem. Theory Comput. 17, 1618 (2021)]. Several approximations to a full-valence active space are examined and benchmarked. Absolute excitation energies and intensities computed with the GAS-DSRG truncated to second-order in perturbation theory are found to systematically underestimate experimental and reference theoretical values. Third-order perturbative corrections significantly improve the accuracy of GAS-DSRG absolute excitation energies, bringing the mean absolute deviation from experimental values down to 0.32 eV. The ozone molecule and glyoxylic acid are particularly challenging for second-order perturbation theory and are examined in detail to assess the importance of active space truncation and intruder states.

CO Inversion on a NaCl(100) Surface: A Multireference Quantum Embedding Study

We develop a multireference quantum embedding model to investigate a recent experimental observation of the isomerization of vibrationally excited CO molecules on a NaCl(100) surface [Science 2020, 367, 175-178]. To explore this mechanism, we built a reduced potential energy surface of CO interacting with NaCl(100) using a second-order multireference perturbation theory, modeling the adsorbate-surface interaction with our previously developed active space embedding theory (ASET). We considered an isolated CO molecule on NaCl(100) and a high-coverage CO monolayer (1/1), and for both we generated potential energy surfaces parametrized by the CO stretching, adsorption, and inversion coordinates. These surfaces are used to determine stationary points and adsorption energies and to perform a vibrational analysis of the states relevant to the inversion mechanism. We found that for near-equilibrium bond lengths, CO adsorbed in the C-down configuration is lower in energy than in the O-down configuration. Stretching of the C-O bond reverses the energetic order of these configurations, supporting the accepted isomerization mechanism. The vibrational constants obtained from these potential energy surfaces show a small (< 10 cm^-1) blue- and red-shift for the C-down and O-down configurations, respectively, in agreement with experimental assignments and previous theoretical studies. Our vibrational analysis of the monolayer case suggests that the O-down configuration is energetically more stable than the C-down one beyond the 16th vibrational excited state of CO, a value slightly smaller than the one from quasi-classical trajectory simulations (22nd) and consistent with the experiment. Our analysis suggests that CO-CO interactions in the monolayer play an important role in stabilizing highly vibrationally excited states in the O-down configuration and reducing the barrier between the C-down and O-down geometries, therefore playing a crucial role in the inversion mechanism.

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.

Twenty Years of Auxiliary-Field Quantum Monte Carlo in Quantum Chemistry: An Overview and Assessment on Main Group Chemistry and Bond-Breaking

In this work, we present an overview of the phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) approach from a computational quantum chemistry perspective and present a numerical assessment of its performance on main group chemistry and bond-breaking problems with a total of 1004 relative energies. While our benchmark study is somewhat limited, we make recommendations for the use of ph-AFQMC for general main-group chemistry applications. For systems where single determinant wave functions are qualitatively accurate, we expect the accuracy of ph-AFQMC in conjunction with a single-determinant trial wave function to be between that of coupled-cluster with singles and doubles (CCSD) and CCSD with perturbative triples (CCSD(T)). For these applications, ph-AFQMC should be a method of choice when canonical CCSD(T) is too expensive to run. For systems where multireference (MR) wave functions are needed for qualitative accuracy, ph-AFQMC is far more accurate than MR perturbation theory methods and competitive with MR configuration interaction (MRCI) methods. Due to the computational efficiency of ph-AFQMC compared to MRCI, we recommended ph-AFQMC as a method of choice for handling dynamic correlation in MR problems. We conclude with a discussion of important directions for future development of the ph-AFQMC approach.

Automatic derivation of many-body theories based on general Fermi vacua

This paper describes Wick&d, an implementation of the algebra of second-quantized operators normal ordered with respect to general correlated references and the corresponding Wick theorem [D. Mukherjee, Chem. Phys. Lett. 274, 561 (1997) and W. Kutzelnigg and D. Mukherjee, J. Chem. Phys. 107, 432 (1997)]. Wick&d employs a compact representation of operators and a backtracking algorithm to efficiently evaluate Wick contractions. Since Wick&d can handle both fully and partially contracted terms, it can be applied to both projective and Fock-space many-body formalisms. To demonstrate the usefulness of  Wick&d, we use it to evaluate the single-reference coupled cluster equations up to octuple excitations and report an automated derivation and implementation of the second-order driven similarity renormalization group multireference perturbation theory.

Regularized CASPT2: an Intruder-State-Free Approach

In this work we present a new approach to fix the intruder-state problem (ISP) in CASPT2 based on σ^p regularization. The resulting σ^p-CASPT2 method is compared to previous techniques, namely, the real and imaginary level shifts, on a theoretical basis and by performing a series of systematic calculations. The analysis is focused on two aspects, the effectiveness of σ^p-CASPT2 in removing the ISP and the sensitivity of the approach with respect to the input parameter. We found that σ^p-CASPT2 compares favorably with respect to previous approaches and that different versions, σ¹-CASPT2 and σ²-CASPT2, have different potential application domains. This analysis also reveals the unsuitability of the real level shift technique as a general way to avoid the intruder-state problem.

Analytical Gradient Theory for Spin-Free State-Averaged Second-Order Driven Similarity Renormalization Group Perturbation Theory (SA-DSRG-MRPT2) and Its Applications for Conical Intersection Optimizations

Second-order multireference-driven similarity renormalization group perturbation theory (DSRG-MRPT2) provides an efficient means of correcting the dynamical correlation with the multiconfiguration reference function. The state-averaged DSRG-MRPT2 (SA-DSRG-MRPT2) method is the simplest means of treating the excited states with DSRG-MRPT2. In this method, the Hamiltonian dressed with dynamical correlation is diagonalized in the CASCI state subspace (SA-DSRG-MRPT2c) or the configuration subspace (SA-DSRG-MRPT2). This work develops analytical gradient theory for spin-free SA-DSRG-MRPT2(c) with the density-fitting approximation. We check the accuracy of the analytical gradients against the numerical gradients. We present applications for optimizing minimum energy conical intersections (MECI) of ethylene and retinal model chromophores (PSB3 and RPSB6). We investigate the dependence of the optimized geometries and energies on the flow parameters and reference relaxations. The smoothness of the SA-DSRG-MRPT2(c) potential energy surfaces near the reference (complete active space self-consistent field) MECI is comparable to the XMCQDPT2 one. These results render SA-DSRG-MRPT2(c) theory a promising approach for studies of conical intersections.

Second-Order Active-Space Embedding Theory

Quantum embedding schemes are a promising way to extend multireference computations to large molecules with strong correlation effects localized on a small number of atoms. This work introduces a second-order active-space embedding theory [ASET(2)] which improves upon mean-field frozen embedding by treating fragment-environment interactions via an approximate canonical transformation. The canonical transformation employed in ASET(2) is formulated using the driven similarity renormalization group. The ASET(2) scheme is benchmarked on the N═N bond dissociation in pentyldiazene, the S₀ to S₁ excitation in 1-octene, and the interaction energy of the O₂-benzene complex. The ASET(2) explicit treatment of fragment-environment interactions beyond the mean-field level generally improves the accuracy of embedded computations, and it becomes necessary to achieve an accurate description of excitation energies of 1-octene and the singlet-triplet gap of the O₂-benzene complex.

Theoretical Calculation of Core-Excited States along Dissociative Pathways beyond Second-Order Perturbation Theory

We extend the multireference driven similarity renormalization (MR-DSRG) method to compute core-excited states by combining it with a GASSCF treatment of orbital relaxation and static electron correlation effects. We consider MR-DSRG treatments of dynamical correlation truncated at the level of perturbation theory (DSRG-MRPT2/3) and iterative linearized approximations with one- and two-body operators [MR-LDSRG(2)] in combination with a spin-free exact-two-component (X2C) one-electron treatment of scalar relativistic effects. This approach is calibrated and tested on a series of 16 core-excited states of five closed- and open-shell diatomic molecules containing first-row elements (C, N, and O). All GASSCF-MR-DSRG theories show excellent agreement with experimental adiabatic transitions energies, with mean absolute errors ranging between 0.17 and 0.35 eV, even for the challenging partially doubly excited states of the N₂⁺ molecule. The vibrational structure of all these transitions, obtained from using a full potential energy scan, shows a mean absolute error as low as 25 meV for DSRG-MRPT2 and 12/13 meV for DSRG-MRPT3 and MR-LDSRG(2). We generally find that a treatment of dynamical correlation that goes beyond the second-order level in perturbation theory improves the accuracy of the potential energy surface, especially in the bond-dissociation region.

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.

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.

Spin-free formulation of the multireference driven similarity renormalization group: A benchmark study of first-row diatomic molecules and spin-crossover energetics

We report a spin-free formulation of the multireference (MR) driven similarity renormalization group (DSRG) based on the ensemble normal ordering of Mukherjee and Kutzelnigg [J. Chem. Phys. 107, 432 (1997)]. This ensemble averages over all microstates of a given total spin quantum number, and therefore, it is invariant with respect to SU(2) transformations. As such, all equations may be reformulated in terms of spin-free quantities and they closely resemble those of spin-adapted closed-shell coupled cluster (CC) theory. The current implementation is used to assess the accuracy of various truncated MR-DSRG methods (perturbation theory up to third order and iterative methods with single and double excitations) in computing the constants of 33 first-row diatomic molecules. The accuracy trends for these first-row diatomics are consistent with our previous benchmark on a small subset of closed-shell diatomic molecules. We then present the first MR-DSRG application on transition-metal complexes by computing the spin splittings of the [Fe(H₂O)₆]²⁺ and [Fe(NH₃)₆]²⁺ molecules. A focal point analysis (FPA) shows that third-order perturbative corrections are essential to achieve reasonably converged energetics. The FPA based on the linearized MR-DSRG theory with one- and two-body operators and up to a quintuple-ζ basis set predicts the spin splittings of [Fe(H₂O)₆]²⁺ and [Fe(NH₃)₆]²⁺ to be -35.7 and -17.1 kcal mol^-1, respectively, showing good agreement with the results of local CC theory with singles, doubles, and perturbative triples.

Excited States of Xanthophylls Revisited: Toward the Simulation of Biologically Relevant Systems

Xanthophylls are a class of oxygen-containing carotenoids, which play a fundamental role in light-harvesting pigment-protein complexes and in many photoresponsive proteins. The complexity of the manifold of the electronic states and the large sensitivity to the environment still prevent a clear and coherent interpretation of their photophysics and photochemistry. In this Letter, we compare cutting-edge ab initio methods (CC3 and DMRG/NEVPT2) with time-dependent DFT and semiempirical CI (SECI) on model keto-carotenoids and show that SECI represents the right compromise between accuracy and computational cost to be applied to real xanthophylls in their biological environment. As an example, we investigate canthaxanthin in the orange carotenoid protein and show that the conical intersections between excited states and excited-ground states are mostly determined by the effective bond length alternation coordinate, which is significantly tuned by the protein through geometrical constraints and electrostatic effects.

Toward an Accurate Ab Initio Description of Low-Lying Singlet Excited States of Polyenes

The low-lying excited states of carotenoids play a crucial role in many important biophysical processes such as photosynthesis. Most of these excited states are strongly correlated, which makes them both challenging for a qualitative ab initio description and an engaging model system for trying out emerging multireference methods. Among these methods, driven similarity renormalization group (DSRG) and its perturbative version (DSRG-MRPT2) are especially attractive in terms of both accuracy and moderate numerical complexity. In this paper, we applied density matrix renormalization group (DMRG) followed by DSRG-MRPT2 for the calculation of vertical and adiabatic excitation energies into the 2A_g^-, 1B_u^-, and 1B_u⁺ electronic states of polyenes containing from 8 to 13 conjugating double bonds acting as a model for natural carotenoids. It was shown that the DSRG flow parameter should be adjusted to ensure both the energy convergence with respect to it and the agreement with the experimental data. With the increased flow parameter, the proposed combination of methods provides a reasonable agreement with the experiment. The deviations of the adiabatic excitation energies are less than 1000 cm^-1 for the 2A_g^- and less than 3000 cm^-1 for the excited states of the B_u symmetry, which in terms of accuracy significantly outperforms the N-electron valence state perturbation theory. At the same time, DSRG-MRPT2 is shown to be robust with respect to variation of quality of the DMRG reference wave function such as the orbital optimization or the number of electronic states in the averaging.

Exploration of interlacing and avoided crossings in a manifold of potential energy curves by a unitary group adapted state specific multi-reference perturbation theory (UGA-SSMRPT)

The Unitary Group Adapted State-Specific Multi-Reference Perturbation Theory (UGA-SSMRPT2) developed by Mukherjee et al. [J. Comput. Chem. 36, 670 (2015)] has successfully realized the goal of studying bond dissociation in a numerically stable, spin-preserving, and size-consistent manner. We explore and analyze here the efficacy of the UGA-SSMRPT2 theory in the description of the avoided crossings and interlacings between a manifold of potential energy curves for states belonging to the same space-spin symmetry. Three different aspects of UGA-SSMRPT2 have been studied: (a) We introduce and develop the most rigorous version of UGA-SSMRPT2 that emerges from the rigorous version of UGA-SSMRCC utilizing a linearly independent virtual manifold; we call this the "projection" version of UGA-SSMRPT2 (UGA-SSMRPT2 scheme P). We compare and contrast this approach with our earlier formulation that used extra sufficiency conditions via amplitude equations (UGA-SSMRPT2 scheme A). (b) We present the results for a variety of electronic states of a set of molecules, which display the striking accuracy of both the two versions of UGA-SSMRPT2 with respect to three different situations involving weakly avoided crossings, moderate/strongly avoided crossings, and interlacing in a manifold of potential energy curves (PECs) of the same symmetry. Accuracy of our results has been benchmarked against IC-MRCISD + Q.

Comparison of many-particle representations for selected-CI I: A tree based approach

The full configuration interaction (FCI) method is only applicable to small molecules with few electrons in moderate size basis sets. One of the main alternatives to obtain approximate FCI energies for bigger molecules and larger basis sets is selected CI. However, due to: (a) the lack of a well-defined structure in a selected CI Hamiltonian, (b) the potentially large number of electrons together with c) potentially large orbital spaces, a computationally and memory efficient algorithm is difficult to construct. In the present series of papers, we describe our attempts to address these issues by exploring tree-based approaches. At the same time, we devote special attention to the issue of obtaining eigenfunctions of the total spin squared operator since this is of particular importance in tackling magnetic properties of complex open shell systems. Dedicated algorithms are designed to tackle the CI problem in terms of determinant, configuration (CFG) and configuration state function many-particle bases by effective use of the tree representation. In this paper we describe the underlying logic of our algorithm design and discuss the advantages and disadvantages of the different many particle bases. We demonstrate by the use of small examples how the use of the tree simplifies many key algorithms required for the design of an efficient selected CI program. Our selected CI algorithm, called the iterative configuration expansion, is presented in the penultimate part. Finally, we discuss the limitations and scaling characteristics of the present approach.

Single-Root Multireference Brillouin-Wigner Perturbative Approach to Excitation Energies

The state-specific Brillouin-Wigner multireference perturbation theory [which employs Jeziorski-Monkhorst parametrization of the wave function] using improved virtual orbitals, denoted as IVO-BWMRPT, is applied to calculate excitation energies (EEs) for methylene, ethylene, trimethylenemethane, and benzyne systems exhibiting various degrees of diradical character. In IVO-BWMRPT, all of the parameters appearing in the wave function ansatz are optimized for a specific electronic state. For these systems, the IVO-BWMRPT method provides EEs that are in close agreement with the benchmark results and experiments, where available, indicating that the method does not introduce imbalance in the target-specific treatment of closed- and open-shell states involved. The good performance of the present methodology is primarily related to structural compactness of the formalism. Overall, present findings are encouraging for both further development of the approach and chemical applications on the energy differences of strongly correlated systems.

Reduced Density Matrix Cumulants: The Combinatorics of Size-Consistency and Generalized Normal Ordering

Reduced density matrix cumulants play key roles in the theory of both reduced density matrices and multiconfigurational normal ordering. We present a new, simpler generating function for reduced density matrix cumulants that is formally identical with equating the coupled cluster and configuration interaction ansätze. This is shown to be a general mechanism to convert between a multiplicatively separable quantity and an additively separable quantity, as defined by a set of axioms. It is shown that both the cumulants of probability theory and the reduced density matrices are entirely combinatorial constructions, where the differences can be associated with changes in the notion of "multiplicative separability" for expectation values of random variables compared to reduced density matrices. We compare our generating function to that of previous works and criticize previous claims of probabilistic significance of the reduced density matrix cumulants. Finally, we present a simple proof of the generalized normal ordering formalism to explore the role of reduced density matrix cumulants therein. While the formalism can be used without cumulants, the combinatorial structure of expressing RDMs in terms of cumulants is the same combinatorial structure on cumulants that allows for a simple extended generalized Wick's theorem.

Reduced Density Matrix-Driven Complete Active Apace Self-Consistent Field Corrected for Dynamic Correlation from the Adiabatic Connection

The recently proposed multireference adiabatic connection (AC) formalism [Pernal, Phys. Rev. Lett. 120, 013001 (2018)] is applied to recover dynamic electron correlation effects lacking in variational two-electron reduced density matrix (v2RDM)-driven complete active space self-consistent field theory (CASSCF). The AC approach is validated by computing potential energy curves for the dissociation of molecular nitrogen and the symmetric double dissociation of H₂O while enforcing two sets of approximate N-representability conditions in the underlying v2RDM-driven CASSCF calculations (either two-particle or two-particle plus partial three-particle conditions). The AC yields smaller absolute errors than second-order N-electron perturbation theory (NEVPT2) at all molecular geometries for both sets of the N-representability conditions considered. The efficacy of the approach for thermochemistry is also assessed for a set of 31 small-molecule reactions. When imposing partial three-particle N-representability conditions, mean and maximum unsigned errors in reaction energies from the AC are superior to those from NEVPT2.

Reduced scaling extended multi-state CASPT2 (XMS-CASPT2) using supporting subspaces and tensor hyper-contraction

We present a reduced scaling formulation of the extended multi-state CASPT2 (XMS-CASPT2) method, which is based on our recently developed state-specific CASPT2 (SS-CASPT2) formulation using supporting subspaces and tensor hyper-contraction. By using these two techniques, the off-diagonal elements of the effective Hamiltonian can be computed with only O(N³) operations and O(N²) memory, where N is the number of basis functions. This limits the overall computational scaling to O(N⁴) operations and O(N²) memory. Thus, excited states can now be obtained at the same reduced (relative to previous algorithms) scaling we achieved for SS-CASPT2. In addition, we also investigate how the energy denominators can be factorized with the Laplace quadrature when some of the denominators are negative, which is critical for excited state calculations. An efficient implementation of the method has been developed using graphical processing units while also exploiting spatial sparsity in tensor operations. We benchmark the accuracy of the new method by comparison to non-THC formulated XMS-CASPT2 for the excited states of various molecules. In our tests, the THC approximation introduces negligible errors (≈0.01 eV) compared to the non-THC reference method. Scaling behavior and computational timings are presented to demonstrate performance. The new method is also interfaced with quantum mechanics/molecular mechanics (QM/MM). In an example study of green fluorescent protein, we show how the XMS-CASPT2 potential energy surfaces and excitation energies are affected by increasing the size of the QM region up to 278 QM atoms with more than 2300 basis functions.

A zeroth-order active-space frozen-orbital embedding scheme for multireference calculations

Multireference computations of large-scale chemical systems are typically limited by the computational cost of quantum chemistry methods. In this work, we develop a zeroth-order active space embedding theory [ASET(0)], a simple and automatic approach for embedding any multireference dynamical correlation method based on a frozen-orbital treatment of the environment. ASET(0) is combined with the second-order multireference driven similarity renormalization group and tested on several benchmark problems, including the excitation energy of 1-octene and bond-breaking in ethane and pentyldiazene. Finally, we apply ASET(0) to study the singlet-triplet gap of p-benzyne and 9,10-anthracyne diradicals adsorbed on a NaCl surface. Our results show that despite its simplicity, ASET(0) is a powerful and sufficiently accurate embedding scheme applicable when the coupling between the fragment and the environment is in the weak to medium regime.

An adiabatic connection for doubly-occupied configuration interaction wave functions

An adiabatic connection (AC) is developed as an electron correlation correction for doubly occupied configuration interaction (DOCI) wave functions. Following the work of Pernal [Phys. Rev. Lett. 120, 013001 (2018)], the working equations of the approach, termed AC-DOCI, are rooted in the extended random phase approximation (ERPA) and require knowledge of only the ground-state two-electron reduced density matrix (2RDM) from the DOCI. As such, the AC is naturally suited to modeling electron correlation in variational 2RDM (v2RDM)-based approximations to the DOCI. The v2RDM-driven AC-DOCI is applied to the dissociation of molecular nitrogen and the double dissociation of water; the approach yields energies that are similar in quality to those from second-order multireference perturbation theory near equilibrium, but the quality of the AC-DOCI energy degrades at stretched geometries. The exact adiabatic connection path suggests the assumption that the one-electron reduced-density matrix is constant along the AC path is invalid at stretched geometries, but this deficiency alone cannot explain the observed behavior. Rather, it appears that the ERPA's single-particle-transition ansatz cannot, in general, provide good approximations to the 2RDM along the AC path. The AC-DOCI is also applied to a set of 45 reaction energies; for these systems, the approach has an average accuracy that is comparable to that of single-reference second-order many-body perturbation theory.

Improving the Efficiency of the Multireference Driven Similarity Renormalization Group via Sequential Transformation, Density Fitting, and the Noninteracting Virtual Orbital Approximation

This study examines several techniques to improve the efficiency of the linearized multireference driven similarity renormalization group truncated to one- and two-body operators [MR-LDSRG(2)]. We propose a sequential MR-LDSRG(2) [sq-MR-LDSRG(2)] scheme, in which one-body substitutions are folded exactly into the Hamiltonian. This new approach is combined with density fitting (DF) to reduce the storage cost of two-electron integrals. To further avoid storage of large four-index intermediates, we propose a noninteracting virtual orbital (NIVO) approximation of the Baker-Campbell-Hausdorff series that neglects commutators terms with three and four virtual indices. The NIVO approximation reduces the computational prefactor of the MR-LDSRG(2), bringing it closer to that of coupled cluster with singles and doubles (CCSD). We test the effect of the DF and NIVO approximations on the MR-LDSRG(2) and sq-MR-LDSRG(2) methods by computing properties of eight diatomic molecules. The diatomic constants obtained by DF-sq-MR-LDSRG(2)+NIVO are found to be as accurate as those from the original MR-LDSRG(2) and coupled cluster theory with singles, doubles, and perturbative triples. Finally, we demonstrate that the DF-sq-MR-LDSRG(2)+NIVO scheme can be applied to chemical systems with more than 550 basis functions by computing the automerization energy of cyclobutadiene with a quintuple-ζ basis set. The predicted automerization energy is found to be similar to the value computed with Mukherjee's state-specific multireference coupled cluster theory with singles and doubles.

Analytic gradients for the single-reference driven similarity renormalization group second-order perturbation theory

We derive and implement analytic energy gradients for the single-reference driven similarity renormalization group second-order perturbation theory (DSRG-PT2). The resulting equations possess an asymptotic scaling that is identical to that of the second-order Møller-Plesset perturbation theory (MP2), indicating that the exponential regularizer in the DSRG equations does not introduce formal difficulties in the gradient theory. We apply the DSRG-PT2 method to optimizing the geometries of 15 small molecules. The equilibrium bond lengths computed with DSRG-PT2 are found similar to those of MP2, yielding a mean absolute error of 0.0033 Å and a standard deviation of 0.0045 Å when compared with coupled cluster with singles, doubles, and perturbative triples.

Multireference Theories of Electron Correlation Based on the Driven Similarity Renormalization Group

The driven similarity renormalization group (DSRG) provides an alternative way to address the intruder state problem in quantum chemistry. In this review, we discuss recent developments of multireference methods based on the DSRG. We provide a pedagogical introduction to the DSRG and its various extensions and discuss its formal properties in great detail. In addition, we report several illustrative applications of the DSRG to molecular systems.

Constructing Molecular π-Orbital Active Spaces for Multireference Calculations of Conjugated Systems

Molecules with conjugated π systems often feature strong electron correlation and therefore require multireference methods for a reliable computational description. A key prerequisite for the successful application of such methods is the choice of a suitable active space. Herein the automated π-orbital space (PiOS) method for selecting active spaces for multireference calculations of conjugated π systems is presented. This approach allows the construction of small but effective active spaces based on Hückel theory. To demonstrate its performance, π → π* excitations for benzene, octatetraene, and free-base porphine are computed. In addition, this technique can be combined with the automated atomic valence active space method to compute excitations in complex systems with multiple conjugated fragments. This combined approach was used to generate two-dimensional potential energy surfaces for multiple electronic states associated with photoinduced electron-coupled double proton transfer in the blue-light-using flavin photoreceptor protein. These types of methods for the automated selection of active space orbitals are important for ensuring consistency and reproducibility of multireference approaches for a wide range of chemical and biological systems.

Electronic Excited States from the Adiabatic-Connection Formalism with Complete Active Space Wave Functions

It is demonstrated how the recently proposed multireference adiabatic-connection (AC) approximation for electron correlation energy ( Pernal , K. Electron Correlation from the Adiabatic Connection for Multireference Wave Functions . Phys. Rev. Lett. 2018 , 120 , 013001 ) can be extended to predicting correlation energy in excited states of molecules. It is the first successful application of the AC approach to computing excited-states energies of molecules using a complete active space (CAS) wave function as a reference. The unique feature of the AC-CAS approach with respect to popular methods such as CASPT2 and NEVPT2 is that it requires only one- and two-particle reduced density matrices, making it possible to efficiently treat large spaces of active electrons. Application of the simpler variant of AC, the AC0, which is based on the first-order expansion of the AC integrand at the uncorrelated system limit, to excited states yields excitation energies with accuracy rivaling that of the NEVPT2 method but at greatly reduced computational cost.