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

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.

New physical insights into the supporting subspace factorization of XMS-CASPT2 and generalization to multiple spin states via spin-free formulation

This paper introduces a spin-free formulation of the supporting subspace factorization [C. Song and T. J. Martínez, J. Chem. Phys. 149, 044108 (2018)], enabling a reduction in the computational scaling of the extended multi-state complete active space second-order perturbation (XMS-CASPT2) method for arbitrary spins. Compared to the original formulation that is defined in the spin orbitals and is limited to singlet states, the spin-free formulation in this work treats different spin states equivalently, thus naturally generalizing the idea beyond singlet states. In addition, we will present a new way of deriving the supporting subspace factorization with the purpose of understanding its physical interpretation. In this new derivation, we separate the sources that make CASPT2 difficult into the "same-site interactions" and "inter-site interactions." We will first show how the Kronecker sum can be used to remove the same-site interactions in the absence of inter-site interactions, leading to MP2 energy in dressed orbitals. We will then show how the inter-site interactions can be exactly recovered using Löwdin partition, where the supporting subspace concept will naturally arise. The new spin-free formulation maintains the main advantage of the supporting subspace factorization, i.e., allowing XMS-CASPT2 energies to be computed using highly optimized MP2 energy codes and Fock build codes, thus reducing the scaling of XMS-CASPT2 to the same scaling as MP2. We will present and discuss results that benchmark the accuracy and performance of the new method. To demonstrate how the new method can be useful in studying real photochemical systems, the supporting subspace XMS-CASPT2 is applied to a photoreaction sensitive to magnetic field effects. The new spin-free formulation makes it possible to calculate the doublet and quartet states required in this particular photoreaction mechanism.

Tensor Hypercontraction of Cluster Perturbation Theory: Quartic Scaling Perturbation Series for the Coupled Cluster Singles and Doubles Ground-State Energies

Even though cluster perturbation theory has been shown to be a robust noniterative alternative to coupled cluster theory, it is still plagued by high order polynomial computational scaling and the storage of higher order tensors. We present a proof-of-concept strategy for implementing a cluster perturbation theory ground-state energy series for the coupled cluster singles and doubles energy with N4 computational scaling using tensor hypercontraction (THC). The reduction in computational scaling by two orders is achieved by decomposing two electron repulsion integrals, doubles amplitudes and multipliers, as well as selected double intermediates to the THC format. Using the outlined strategy, we showcase that the THC pilot implementations retain numerical accuracy to within 1 kcal/mol relative to corresponding conventional and density fitting implementations, and we empirically verify the N4 scaling.

A local pair-natural orbital-based complete-active space perturbation theory using orthogonal localized virtual molecular orbitals

The multireference second-order perturbation theory (CASPT2) is known to deliver a quantitative description of various complex electronic states. Despite its near-size-consistent nature, the applicability of the CASPT2 method to large, real-life systems is mostly hindered by large computational and storage costs for the two-external tensors, such as two-electron integrals, amplitudes, and residuum. To this end, Menezes and co-workers developed a reduced-scaling CASPT2 scheme by incorporating the local pair-natural orbital (PNO) representation of the many-body wave functions using non-orthonormal projected atomic orbitals (PAOs) into the CASPT theory [F. Menezes et al., J. Chem. Phys. 145, 124115 (2016)]. Alternatively, in this paper, we develop a new PNO-based CASPT2 scheme using the orthonormal localized virtual molecular orbitals (LVMOs) and assess its performance and accuracy in comparison with the conventional PAO-based counterpart. Albeit the compactness, the LVMOs were considered to perform somewhat poorly compared to PAOs in the local correlation framework because they caused enormously large orbital domains. In this work, we show that the size of LVMO domains can be rendered comparable to or even smaller than that of PAOs by the use of the differential overlap integrals for domain construction. Optimality of the MOs from the CASSCF treatment is a key to reducing the LVMO domain size for the multireference case. Due to the augmented Hessian-based localization algorithm, an additional computational cost for obtaining the LVMOs is relatively minor. We demonstrate that the LVMO-based PNO-CASPT2 method is routinely applicable to large, real-life molecules such as Menshutkin SN2 reaction in a single-walled carbon nanotube reaction field.

A Theoretical Perspective on the Actinic Photochemistry of 2-Hydroperoxypropanal

The photochemical reactions triggered by the sunlight absorption of transient volatile organic compounds in the troposphere are notoriously difficult to characterize experimentally due to the unstable and short-lived nature of these organic molecules. Some members of this family of compounds are likely to exhibit a rich photochemistry given the diversity of functional groups they can bear. Even more interesting is the photochemical fate of volatile organic compounds bearing more than one functional group that can absorb light—this is the case, for example, of α-hydroperoxycarbonyls, which are formed during the oxidation of isoprene. Experimental observables characterizing the photochemistry of these molecules like photoabsorption cross-sections or photolysis quantum yields are currently missing, and we propose here to leverage a recently developed computational protocol to predict in silico the photochemical fate of 2-hydroperoxypropanal (2-HPP) in the actinic region. We combine different levels of electronic structure methods—SCS-ADC(2) and XMS-CASPT2—with the nuclear ensemble approach and trajectory surface hopping to understand the mechanistic details of the possible nonradiative processes of 2-HPP. In particular, we predict the photoabsorption cross-section and the wavelength-dependent quantum yields for the observed photolytic pathways and combine them to determine in silico photolysis rate constants. The limitations of our protocol and possible future improvements are discussed.

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.

Steric and Electronic Origins of Fluorescence in GFP and GFP-like Proteins

Fluorescent proteins have become routine tools for biological imaging. However, their nanosecond lifetimes on the excited state present computational hurdles to a full understanding of these photoactive proteins. In this work, we simulate approximately 0.5 nanoseconds of ab initio molecular dynamics to elucidate steric and electronic features responsible for fluorescent protein behavior. Using green fluorescent protein (GFP) and Dronpa2─widely used fluorescent proteins with contrasting functionality─as case studies, we leverage previous findings in the gas phase and solution to explore the deactivation mechanisms available to these proteins. Starting with ground-state analyses, we identify steric (the distribution of empty pockets near the chromophore) and electronic (electric fields exerted on chromophore moieties) factors that offer potential avenues for rational design. Picosecond timescale simulations on the excited state reveal that the chromophore can access twisted structures in Dronpa2, while the chromophore is largely confined to planarity in GFP. We couple ab initio multiple spawning (AIMS) and enhanced sampling simulations to discover and characterize conical intersection seams that facilitate internal conversion, which is a rare event in both systems. Our AIMS simulations correctly capture the relative fluorescence profiles of GFP and Dronpa2 within the first few picoseconds, and we attribute the diminished fluorescence intensity of Dronpa2, relative to GFP, to flexible chromophore intermediates on the excited state. Furthermore, we predict that twisted chromophore intermediates produce red-shifted intensities in the Dronpa2 fluorescence spectrum. If confirmed experimentally, this spectroscopic signature would provide valuable insights when screening and developing novel fluorescent proteins.

Excited state non-adiabatic dynamics of large photoswitchable molecules using a chemically transferable machine learning potential

Light-induced chemical processes are ubiquitous in nature and have widespread technological applications. For example, photoisomerization can allow a drug with a photo-switchable scaffold such as azobenzene to be activated with light. In principle, photoswitches with desired photophysical properties like high isomerization quantum yields can be identified through virtual screening with reactive simulations. In practice, these simulations are rarely used for screening, since they require hundreds of trajectories and expensive quantum chemical methods to account for non-adiabatic excited state effects. Here we introduce a diabatic artificial neural network (DANN), based on diabatic states, to accelerate such simulations for azobenzene derivatives. The network is six orders of magnitude faster than the quantum chemistry method used for training. DANN is transferable to azobenzene molecules outside the training set, predicting quantum yields for unseen species that are correlated with experiment. We use the model to virtually screen 3100 hypothetical molecules, and identify novel species with high predicted quantum yields. The model predictions are confirmed using high-accuracy non-adiabatic dynamics. Our results pave the way for fast and accurate virtual screening of photoactive compounds.

Resolving the ultrafast dynamics of the anionic green fluorescent protein chromophore in water

The chromophore of the green fluorescent protein (GFP) is critical for probing environmental influences on fluorescent protein behavior. Using the aqueous system as a bridge between the unconfined vacuum system and a constricting protein scaffold, we investigate the steric and electronic effects of the environment on the photodynamical behavior of the chromophore. Specifically, we apply ab initio multiple spawning to simulate five picoseconds of nonadiabatic dynamics after photoexcitation, resolving the excited-state pathways responsible for internal conversion in the aqueous chromophore. We identify an ultrafast pathway that proceeds through a short-lived (sub-picosecond) imidazolinone-twisted (I-twisted) species and a slower (several picoseconds) channel that proceeds through a long-lived phenolate-twisted (P-twisted) intermediate. The molecule navigates the non-equilibrium energy landscape via an aborted hula-twist-like motion toward the one-bond-flip dominated conical intersection seams, as opposed to following the pure one-bond-flip paths proposed by the excited-state equilibrium picture. We interpret our simulations in the context of time-resolved fluorescence experiments, which use short- and long-time components to describe the fluorescence decay of the aqueous GFP chromophore. Our results suggest that the longer time component is caused by an energetically uphill approach to the P-twisted intersection seam rather than an excited-state barrier to reach the twisted intramolecular charge-transfer species. Irrespective of the location of the nonadiabatic population events, the twisted intersection seams are inefficient at facilitating isomerization in aqueous solution. The disordered and homogeneous nature of the aqueous solvent environment facilitates non-selective stabilization with respect to I- and P-twisted species, offering an important foundation for understanding the consequences of selective stabilization in heterogeneous and rigid protein environments.

A diagrammatic approach for automatically deriving analytical gradients of tensor hyper-contracted electronic structure methods

We introduce a diagrammatic approach to facilitate the automatic derivation of analytical nuclear gradients for tensor hyper-contraction (THC) based electronic structure methods. The automatically derived gradients are guaranteed to have the same scaling in terms of both operation count and memory footprint as the underlying energy calculations, and the computation of a gradient is roughly three times as costly as the underlying energy. The new diagrammatic approach enables the first cubic scaling implementation of nuclear derivatives for THC tensors fitted in molecular orbital basis (MO-THC). Furthermore, application of this new approach to THC-MP2 analytical gradients leads to an implementation, which is at least four times faster than the previously reported, manually derived implementation. Finally, we apply the new approach to the 14 tensor contraction patterns appearing in the supporting subspace formulation of multireference perturbation theory, laying the foundation for developments of analytical nuclear gradients and nonadiabatic coupling vectors for multi-state CASPT2.

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.

Estimates of electron correlation based on density expansions

Two methods for estimating the correlation energy of molecules and other electronic systems are discussed based on the assumption that the correlation energy can be partitioned between atomic regions. In the first method, the electron density is expanded in terms of atomic contributions using rigorous electron repulsion bounds, and in the second method, correlation contributions are associated with basis function pairs. These methods do not consider the detailed nature of localized excitations but instead define a correlation energy per electron factor that is unique to a specific atom. The correlation factors are basis function dependent and are determined by configuration interaction (CI) calculations on diatomic and hydride molecules. The correlation energy estimates are compared with the results of high-level CI calculations for a test set of 27 molecules representing a wide range of bonding environments (average error of 2.6%). An extension based on truncated CI calculations in which d-type and hydrogen p-type functions are eliminated from the virtual space combined with estimates of dynamical correlation contributions using atomic correlation factors is discussed and applied to the dissociation of several molecules.

Putting Photomechanical Switches to Work: An Ab Initio Multiple Spawning Study of Donor-Acceptor Stenhouse Adducts

Photomechanical switches are light sensitive molecules capable of transducing the energy of a photon into mechanical work via photodynamics. In this Letter, we present the first atomistic investigation of the photodynamics of a novel class of photochromes called donor-acceptor Stenhouse adducts (DASA) using state-of-the-art ab initio multiple spawning interfaced with state-averaged complete active-space self-consistent field theory. Understanding the Z/E photoisomerization mechanism in DASAs at the molecular level is crucial in designing new derivatives with improved photoswitching capabilities. Our dynamics simulations show that the actinic step consists of competing nonradiative relaxation pathways that collectively contribute to DASAs' low (21% in toluene) photoisomerization quantum yield. Furthermore, we highlight the important role the intramolecular hydrogen bond plays in the selectivity of photoisomerization in DASAs, identifying it as a possible structural element to tune DASA properties. Our fully ab initio simulations reveal the key degrees of freedom involved in the actinic step, paving the way for the rational design of new generations of DASAs with improved quantum yield and efficiency.