Lower Level Of Theory Research Articles

Learning Electronic Polarizations in Aqueous Systems.

The polarization of periodically repeating systems is a discontinuous function of the atomic positions, a fact which seems at first to stymie attempts at their statistical learning. Two approaches to build models for bulk polarizations are compared: one in which a simple point charge model is used to preprocess the raw polarization to give a learning target that is a smooth function of atomic positions and the total polarization is learned as a sum of atom-centered dipoles and one in which instead the average position of Wannier centers around atoms is predicted. For a range of bulk aqueous systems, both of these methods perform perform comparatively well, with the former being slightly better but often requiring an extra effort to find a suitable point charge model. As a challenging test, we also analyze the performance of the models at the air-water interface. In this case, while the Wannier center approach delivers accurate predictions without further modifications, the preprocessing method requires augmentation with information from isolated water molecules to reach similar accuracy. Finally, we present a simple protocol to preprocess the polarizations in a data-driven way using a small number of derivatives calculated at a much lower level of theory, thus overcoming the need to find point charge models without appreciably increasing the computation cost. We believe that the training strategies presented here help the construction of accurate polarization models required for the study of the dielectric properties of realistic complex bulk systems and interfaces with ab initio accuracy.

Ab Initio-Based Modeling of Thermodynamic Cyclic Voltammograms: A Benchmark Study on Ag(100) in Bromide Solutions.

Experimental cyclic voltammograms (CVs) measured in the slow scan rate limit can be entirely described in terms of the thermodynamic equilibrium quantities of the electrified solid-liquid interface. They correspondingly serve as an important benchmark for the quality of first-principles calculations of interfacial thermodynamics. Here, we investigate the partially drastic approximations made presently in computationally efficient calculations for the well-defined showcase of an Ag(100) model electrode in Br-containing electrolytes, where the nontrivial part of the CV stems from the electrosorption of Br ions. We specifically study the entanglement of common approximations in the treatment of solvation and field effects, as well as in the way macroscopic averages of the two key quantities, namely, the potential-dependent adsorbate coverage and electrosorption valency, are derived from the first-principles energetics. We demonstrate that the combination of energetics obtained within an implicit solvation model and a perturbative second order account of capacitive double layer effects with a constant-potential grand-canonical Monte Carlo sampling of the adsorbate layer provides an accurate description of the experimental CV. However, our analysis also shows that error cancellation at lower levels of theory may equally lead to good descriptions even though key underlying physics such as the disorder-order transition of the Br adlayer at increasing coverages is inadequately treated.

A Δ-learning strategy for interpretation of spectroscopic observables.

Accurate computations of experimental observables are essential for interpreting the high information content held within x-ray spectra. However, for complicated systems this can be difficult, a challenge compounded when dynamics becomes important owing to the large number of calculations required to capture the time-evolving observable. While machine learning architectures have been shown to represent a promising approach for rapidly predicting spectral lineshapes, achieving simultaneously accurate and sufficiently comprehensive training data is challenging. Herein, we introduce Δ-learning for x-ray spectroscopy. Instead of directly learning the structure-spectrum relationship, the Δ-model learns the structure dependent difference between a higher and lower level of theory. Consequently, once developed these models can be used to translate spectral shapes obtained from lower levels of theory to mimic those corresponding to higher levels of theory. Ultimately, this achieves accurate simulations with a much reduced computational burden as only the lower level of theory is computed, while the model can instantaneously transform this to a spectrum equivalent to a higher level of theory. Our present model, demonstrated herein, learns the difference between TDDFT(BLYP) and TDDFT(B3LYP) spectra. Its effectiveness is illustrated using simulations of Rh L3-edge spectra tracking the C-H activation of octane by a cyclopentadienyl rhodium carbonyl complex.

Accelerating Anharmonic Spectroscopy Simulations via Local-Mode, Multilevel Methods.

Ab initio computer simulations of anharmonic vibrational spectra provide nuanced insight into the vibrational behavior of molecules and complexes. The computational bottleneck in such simulations, particularly for ab initio potentials, is often the generation of mode-coupling potentials. Focusing specifically on two-mode couplings in this analysis, the combination of a local-mode representation and multilevel methods is demonstrated to be particularly symbiotic. In this approach, a low-level quantum chemistry method is employed to predict the pairwise couplings that should be included at the target level of theory in vibrational self-consistent field (and similar) calculations. Pairs that are excluded by this approach are "recycled" at the low level of theory. Furthermore, because this low-level pre-screening will eventually become the computational bottleneck for sufficiently large chemical systems, distance-based truncation is applied to these low-level predictions without substantive loss of accuracy. This combination is demonstrated to yield sub-wavenumber fidelity with reference vibrational transitions when including only a small fraction of target-level couplings; the overhead of predicting these couplings, particularly when employing distance-based, local-mode cutoffs, is a trivial added cost. This combined approach is assessed on a series of test cases, including ethylene, hexatriene, and the alanine dipeptide. Vibrational self-consistent field (VSCF) spectra were obtained with an RI-MP2/cc-pVTZ potential for the dipeptide, at approximately a 5-fold reduction in computational cost. Considerable optimism for increased accelerations for larger systems and higher-order couplings is also justified, based on this investigation.

Transfer-learned potential energy surfaces: Toward microsecond-scale molecular dynamics simulations in the gas phase at CCSD(T) quality.

The rise of machine learning has greatly influenced the field of computational chemistry and atomistic molecular dynamics simulations in particular. One of its most exciting prospects is the development of accurate, full-dimensional potential energy surfaces (PESs) for molecules and clusters, which, however, often require thousands to tens of thousands of abinitio data points restricting the community to medium sized molecules and/or lower levels of theory (e.g., density functional theory). Transfer learning, which improves a global PES from a lower to a higher level of theory, offers a data efficient alternative requiring only a fraction of the high-level data (on the order of 100 are found to be sufficient for malonaldehyde). This work demonstrates that even with Hartree-Fock theory and a double-zeta basis set as the lower level model, transfer learning yields coupled-cluster single double triple [CCSD(T)]-level quality for H-transfer barrier energies, harmonic frequencies, and H-transfer tunneling splittings. Most importantly, finite-temperature molecular dynamics simulations on the sub-μs time scale in the gas phase are possible and the infrared spectra determined from the transfer-learned PESs are in good agreement with the experiment. It is concluded that routine, long-time atomistic simulations on PESs fulfilling CCSD(T)-standards become possible.

On the Challenge of Obtaining an Accurate Solvation Energy Estimate in Simulations of Electrocatalysis

The effect of solvation on the free energy of reaction intermediates adsorbed on electrocatalyst surfaces can significantly change the thermochemical overpotential, but accurate calculations of this are challenging. Here, we present computational estimates of the solvation energy for reaction intermediates in oxygen reduction reaction (ORR) on a B-doped graphene (BG) model system where the overpotential is found to reduce by up to 0.6 V due to solvation. BG is experimentally reported to be an active ORR catalyst but recent computational estimates using state-of-the-art hybrid density functionals in the absence of solvation effects have indicated low activity. To test whether the inclusion of explicit solvation can bring the calculated activity estimates closer to the experimental reports, up to 4 layers of water molecules are included in the simulations reported here. The calculations are based on classical molecular dynamics and local minimization of energy using atomic forces evaluated from electron density functional theory. Data sets are obtained from regular and coarse-grained dynamics, as well as local minimization of structures resampled from dynamics simulations. The results differ greatly depending on the method used and the solvation energy estimates are deemed untrustworthy. It is concluded that a significantly larger number of water molecules is required to obtain converged results for the solvation energy. As the present system includes up to 139 atoms, it already strains the limits of computational feasibility, so this points to the need for a hybrid simulation approach where efficient simulations of much larger number of solvent molecules is carried out using a lower level of theory while retaining the higher level of theory for the reacting molecules as well as their near neighbors and the catalyst. The results reported here provide a word of caution to the computational catalysis community: activity predictions can be inaccurate if too few solvent molecules are included in the calculations.

Concordant Mode Approach for Molecular Vibrations.

The Concordant Mode Approach (CMA) is advanced as a novel hierarchy for increasing the system size and level of theory feasible for quantum chemical computations of harmonic vibrational frequencies. The key concept behind CMA is that transferrable, internal-coordinate normal modes computed at an appropriate lower level of theory (B) comprise a superb basis for converging to vibrational frequencies given by a higher level of theory (A). Accordingly, high-level harmonic frequencies can be evaluated via CMA from a collection of single-point energies that essentially scales linearly in the number of atoms, providing nearly order-of-magnitude CPU time speedups. The accuracy of CMA methods was established by comprehensive tests on over 120 molecules for target Level A = CCSD(T)/cc-pVTZ with auxiliary Level B choices of both CCSD(T)/cc-pVDZ and B3LYP/6-31G(2df,p). Remarkably, the frequency residuals given by the diagonal CMA-0A(nc) scheme exhibit mean absolute deviations (MADs) of only 0.2 cm-1 and standard deviations less than 0.5 cm-1; the corresponding zero-point vibrational energies (ZPVEs) have negligible errors in the vicinity of 0.3 cm-1.

Naturally-meaningful and efficient descriptors: machine learning of material properties based on robust one-shot ab initio descriptors

Establishing a data-driven pipeline for the discovery of novel materials requires the engineering of material features that can be feasibly calculated and can be applied to predict a material’s target properties. Here we propose a new class of descriptors for describing crystal structures, which we term Robust One-Shot Ab initio (ROSA) descriptors. ROSA is computationally cheap and is shown to accurately predict a range of material properties. These simple and intuitive class of descriptors are generated from the energetics of a material at a low level of theory using an incomplete ab initio calculation. We demonstrate how the incorporation of ROSA descriptors in ML-based property prediction leads to accurate predictions over a wide range of crystals, amorphized crystals, metal–organic frameworks and molecules. We believe that the low computational cost and ease of use of these descriptors will significantly improve ML-based predictions.

ONIOM Method with Charge Transfer Corrections (ONIOM-CT): Analytic Gradients and Benchmarking.

Hybrid methods such as ONIOM that treat different regions of a large molecule using different methods are widely used to investigate chemical reactions in a variety of materials and biological systems. However, there are inherent sources of significant errors due to the standard treatment of the boundary between the regions using hydrogen link atoms. In particular, an unbalanced charge distribution in the chemically important model region is a potential source of such problems. We have previously suggested ONIOM-CT (ONIOM with charge transfer corrections) which addresses this issue by applying a potential in the form of point charges to obtain a desired charge redistribution. The metric for charge redistribution relies on the type of population analysis used to obtain the charges. ONIOM-CT has been implemented using Mulliken and Löwdin population analyses and has been shown to improve computed reaction energies for illustrative chemical reactions. In this work, we derive and implement the analytic gradients for ONIOM-CT that requires solving two sets of coupled-perturbed self-consistent equations, one each for the model system and the full system. However, both are needed only at the low level of theory, allowing for an efficient formulation and implementation for both Mulliken and Löwdin population analyses. Benchmarking and illustrative geometry optimizations have been carried out for a previously studied set of reactions involving a single link atom between regions. Additionally, we have generalized our method for the treatment of model systems involving multiple link atoms to enable applications for a broader set of problems. The generalized methods are illustrated for both charge models. Furthermore, we have studied a set of three proton transfer reactions and demonstrate that significant improvement is achieved by ONIOM-CT over ONIOM using both Mulliken and Löwdin population analyses.

A Semiclassical Framework for Mixed Quantum Classical Dynamics.

Semiclassical (SC) approximations for quantum dynamic simulations in complex chemical systems range from rigorously accurate methods that are computationally expensive to methods that exhibit near-classical scaling with system size but are limited in their ability to describe quantum effects. In practical studies of high-dimensional reactions, neither extreme is the best choice: frequently a high-level quantum mechanical description is only required for a handful of modes, while the majority of environment modes that do not play a key role in the reactive event of interest are well served with a lower level of theory. In this feature, we introduce modified Filinov filtration as a powerful tool to construct mixed quantum-classical SC theories where different subsystems can be quantized to different extents without introducing ad hoc intersubsystem interaction terms. We demonstrate that these Filinov-based SC methods can systematically tune between quantum and classical limit SC behavior, offering a practical way forward to accurate and computationally efficient simulations of high-dimensional quantum processes.

Core-hole delocalization for modeling x-ray spectroscopies: A cautionary tale.

The influence of core-hole delocalization for x-ray photoelectron, x-ray absorption, and x-ray emission spectrum calculations is investigated in detail using approaches including response theory, transition-potential methods, and ground state schemes. The question of a localized/delocalized vacancy is relevant for systems with symmetrically equivalent atoms, as well as near-degeneracies that can distribute the core orbitals over several atoms. We show that the issues relating to core-hole delocalization are present for calculations considering explicit core-hole states, e.g., when using a core-excited or core-ionized reference state or for fractional occupation numbers. As electron correlation eventually alleviates the issues, but even when using coupled-cluster single-double and perturbative triple, there is a notable discrepancy between core-ionization energies obtained with localized and delocalized core-holes (0.5eV for the carbon K-edge). Within density functional theory, the discrepancy correlates with the exchange interaction involving the core orbitals of the same spin symmetry as the delocalized core-hole. The use of a localized core-hole allows for a reasonably good inclusion of relaxation at a lower level of theory, whereas the proper symmetry solution involving a delocalized core-hole requires higher levels of theory to account for the correlation effects involved in orbital relaxation. For linear response methods, we further show that if x-ray absorption spectra are modeled by considering symmetry-unique sets of atoms, care has to be taken such that there are no delocalizations of the core orbitals, which would otherwise introduce shifts in absolute energies and relative features.

Interpolating Moving Ridge Regression (IMRR): A machine learning algorithm to predict energy gradients for ab initio molecular dynamics simulations

Ab initio molecular dynamics (AIMD) simulations are a direct way to visualize chemical reactions and help elucidate non-statistical dynamics that does not follow the intrinsic reaction coordinate. However, due to the enormous amount of ab initio energy gradient calculations needed, it has been largely restrained to limited sampling and low level of theory (i.e., density functional theory with small basis sets). To overcome this issue, a number of machine learning (ML) methods have been developed to predict the energy gradient of the system of interest. In this manuscript, we outline the theoretical foundations of a novel ML method which trains from a varying set of atomic positions and their energy gradients, called “Interpolating Moving Ridge Regression” (IMRR), and directly predicts the energy gradient of a new set of atomic positions. Several key theoretical findings are presented regarding the inputs used to train IMRR and its predicted energy gradient. A hyperparameter used to guide IMRR is rigorously examined as well. IMRR is then applied to three bimolecular reactions studied with AIMD, including HBr+ + CO2, H2S + CH, and C4H2 + CH, to demonstrate its performance on different chemical systems of different sizes. This manuscript also compares the computational cost of the energy gradient calculation with IMRR vs. ab initio, and the results highlight IMRR as a viable option to greatly increase the efficiency of AIMD.

Two Wrongs Can Make a Right: A Transfer Learning Approach for Chemical Discovery with Chemical Accuracy

Appropriately identifying and treating molecules and materials with significant multi-reference (MR) character is crucial for achieving high data fidelity in virtual high throughput screening (VHTS). Nevertheless, most VHTS is carried out with approximate density functional theory (DFT) using a single functional. Despite development of numerous MR diagnostics, the extent to which a single value of such a diagnostic indicates MR effect on chemical property prediction is not well established. We evaluate MR diagnostics of over 10,000 transition metal complexes (TMCs) and compare to those in organic molecules. We reveal that only some MR diagnostics are transferable across these materials spaces. By studying the influence of MR character on chemical properties (i.e., MR effect) that involves multiple potential energy surfaces (i.e., adiabatic spin splitting, $\Delta E_\mathrm{H-L}$, and ionization potential, IP), we observe that cancellation in MR effect outweighs accumulation. Differences in MR character are more important than the total degree of MR character in predicting MR effect in property prediction. Motivated by this observation, we build transfer learning models to directly predict CCSD(T)-level adiabatic $\Delta E_\mathrm{H-L}$ and IP from lower levels of theory. By combining these models with uncertainty quantification and multi-level modeling, we introduce a multi-pronged strategy that accelerates data acquisition by at least a factor of three while achieving chemical accuracy (i.e., 1 kcal/mol) for robust VHTS.

Data-driven dynamical mean-field theory: An error-correction approach to solve the quantum many-body problem using machine learning

Machine learning opens new avenues for modelling correlated materials. Quantum embedding approaches, such as the dynamical mean-field theory (DMFT), provide corrections to first-principles calculations for strongly correlated materials, which are poorly described at lower levels of theory. Such embedding approaches are computationally demanding on classical computing architectures, and hence remain restricted to small systems, which limits the scope of applicability without exceptional computational resources. Here we outline a data-driven machine learning process for solving the Anderson Impurity Model (AIM) - the central component of DMFT calculations. The key advance is the use of an ensemble error-correction approach to generate fast and accurate solutions of AIM. An example calculation of the Mott transition using DMFT in the single band Hubbard model is given as an example of the technique, and is validated against the most accurate available method. This new approach is called data-driven dynamical mean-field theory (\textbf{d$^{3}$MFT}).

Maximally localized dynamical quantum embedding for solving many-body correlated systems.

Quantum computing opens new avenues for modeling correlated materials, which are notoriously challenging to solve due to the presence of large electronic correlations. Quantum embedding approaches, such as dynamical mean-field theory, provide corrections to first-principles calculations for strongly correlated materials, which are poorly described at lower levels of theory. Such embedding approaches are computationally demanding on classical computing architectures and hence remain restricted to small systems, limiting the scope of their applicability. Hitherto, implementations on quantum computers have been limited by hardware constraints. Here, we derive a compact representation, where the number of quantum states is reduced for a given system while retaining a high level of accuracy. We benchmark our method for archetypal quantum states of matter that emerge due to electronic correlations, such as Kondo and Mott physics, both at equilibrium and for quenched systems. We implement this approach on a quantum emulator, demonstrating a reduction of the required number of qubits.

Fragment-Based Local Coupled Cluster Embedding Approach for the Quantification and Analysis of Noncovalent Interactions: Exploring the Many-Body Expansion of the Local Coupled Cluster Energy.

Herein, we introduce a fragment-based local coupled cluster embedding approach for the accurate quantification and analysis of noncovalent interactions in molecular aggregates. Our scheme combines two different expansions of the domain-based local pair natural orbital coupled cluster (DLPNO-CCSD(T)) energy: the many-body expansion (MBE) and the local energy decomposition (LED). The low-order terms in the MBE are initially computed in the presence of an environment that is treated at a low level of theory. Then, LED is used to decompose the energy of each term in the embedded MBE into additive fragment and fragment-pairwise contributions. This information is used to quantify the total energy of the system while providing at the same time in-depth insights into the nature and cooperativity of noncovalent interactions. Two different approaches are introduced and tested, in which the environment is treated at different levels of theory: the local coupled cluster in the Hartree–Fock (LCC-in-HF) method, in which the environment is treated at the HF level; and the electrostatically embedded local coupled cluster method (LCC-in-EE), in which the environment is replaced by point charges. Both schemes are designed to preserve as much as possible the accuracy of the parent local coupled cluster method for total energies, while being embarrassingly parallel and less memory intensive. These schemes appear to be particularly promising for the study of large and complex molecular aggregates at the coupled cluster level, such as condensed phase systems and protein–ligand interactions.

Efficient and Accurate Approach To Estimate Hybrid Functional and Large Basis-Set Contributions to Condensed-Phase Systems and Molecule-Surface Interactions.

We present a graph theoretic approach to adaptively compute contributions from many-body approximations in an efficient manner and perform accurate hybrid density functional theory (DFT) electronic structure calculations for condensed-phase systems. The salient features of the approach are ONIOM-like. (a) The full-system calculation is performed at a lower level of theory (pure DFT) by enforcing periodic boundary conditions. (b) This treatment is then improved through a correction term that captures many-body interactions up to any given order within a higher (in this case, hybrid DFT) level of theory. (c) In the spirit of ONIOM, contributions from the many-body approximations that arise from the higher level of theory [i.e., from (b) above] are included through extrapolation from the lower level calculation. The approach is implemented in a general, system-independent, fashion using the graph-theoretic functionalities available within Python. For example, the individual one-body components within the unit cell are designated as "nodes" within a graph. The interactions between these nodes are captured through edges, faces, tetrahedrons, and so forth, thus building a many-body interaction hierarchy. Electronic energy extrapolation contributions from all of these geometric entities are included within the above-mentioned ONIOM paradigm. The implementation of the method simultaneously uses multiple electronic structure packages. Here, for example, we present results which use both the Gaussian suite of electronic structure programs and the Quantum ESPRESSO program within a single calculation. Thus, the method integrates both plane-wave basis functions and atom-centered basis functions within a single structure calculation. The method is demonstrated for a range of condensed-phase problems for computing (i) hybrid DFT energies for condensed-phase systems at pure DFT cost and (ii) large, triple-zeta, multiply polarized, and diffuse atom-centered basis-set energies at computational costs commensurate with much smaller sets of basis functions. The methods are demonstrated through calculations performed on (a) homogeneous water surfaces as well as heterogeneous surfaces that contain organic solutes studied using two-dimensional periodic boundary conditions and (b) bulk simulations of water enforced through three-dimensional periodic boundary conditions. A range of structures are considered, and in all cases, the results are in good agreement with those obtained using large atom-centered basis and hybrid DFT calculations on the full periodic systems at significantly lower cost.

Multistate QM/QM Extrapolation of UV/Vis Absorption Spectra with Point Charge Embedding.

The simulation of UV/vis absorption spectra of large chromophores is prohibitively expensive with accurate quantum mechanical (QM) methods. Thus, hybrid methods, which treat the core chromophoric region at a high level of theory while the substituent effects are treated with a more computationally efficient method, may provide the best compromise between cost and accuracy. The ONIOM (Our own N-layered Integrated molecular Orbital molecular Mechanics) method has proved successful at describing ground-state processes. However, for excited states, it suffers from difficulties in matching the correct excited states among the different levels of theory. We devised an approach, based on the ONIOM extrapolation formula, to combine two QM levels of theory to extrapolate entire excitation bands rather than individual states. In this contribution, we extend the same QM/QM hybrid scheme to include polarization effects on the core region through point charge embedding. The charges are computed to reproduce the electrostatic potential of the entire chromophore at the low level of theory, with proper constraints to avoid overpolarization issues at the boundary between layers. We test this approach on a variety of model compounds that show how the multistate QM/QM-embedding scheme is able to accurately reproduce the spectrum of the entire system at the high level of theory better than (i) the bare QM/QM hybrid scheme, (ii) the low-level calculation on the entire system, and (iii) the high-level calculation on the core region.

Embedded, graph‐theoretically defined many‐body approximations for wavefunction‐in‐DFT and DFT‐in‐DFT: Applications to gas‐ and condensed‐phase ab initio molecular dynamics, and potential surfaces for quantum nuclear effects

AbstractWe present a graph‐theoretic approach to adaptively compute many‐body approximations in an efficient manner to perform (a) accurate post‐Hartree–Fock (HF) ab initio molecular dynamics (AIMD) at density functional theory (DFT) cost for medium‐ to large‐sized molecular clusters, (b) hybrid DFT electronic structure calculations for condensed‐phase simulations at the cost of pure density functionals, (c) reduced‐cost on‐the‐fly basis extrapolation for gas‐phase AIMD and condensed phase studies, and (d) accurate post‐HF‐level potential energy surfaces at DFT cost for quantum nuclear effects. The salient features of our approach are ONIOM‐like in that (a) the full system (cluster or condensed phase) calculation is performed at a lower level of theory (pure DFT for condensed phase or hybrid DFT for molecular systems), and (b) this approximation is improved through a correction term that captures all many‐body interactions up to any given order within a higher level of theory (hybrid DFT for condensed phase; CCSD or MP2 for cluster), combined through graph‐theoretic methods. Specifically, a region of chemical interest is coarse‐grained into a set of nodes and these nodes are then connected to form edges based on a given definition of local envelope (or threshold) of interactions. The nodes and edges together define a graph, which forms the basis for developing the many‐body expansion. The methods are demonstrated through (a) ab initio dynamics studies on protonated water clusters and polypeptide fragments, (b) potential energy surface calculations on one‐dimensional water chains such as those found in ion channels, and (c) conformational stabilization and lattice energy studies on homogeneous and heterogeneous surfaces of water with organic adsorbates using two‐dimensional periodic boundary conditions.

Coupling Constants, High Spin, and Broken Symmetry States of Organic Radicals: an Assessment of the Molecules-in-Molecules Fragmentation-Based Method.

We extend the application of our multilayer molecules-in-molecules (MIM) fragmentation-based method to the study of open-shell systems, particularly organic radicals. A test set of organic mono-, di- and polyradicals with a wide range in size, containing up to 360 atoms, was investigated. Total energies computed with MIM using density functional theory (DFT) were compared with full, unfragmented energies to assess the performance of MIM and to develop a systematic protocol for the treatment of large radical systems. More specifically, a two-layer (MIM2) model with a fragmentation scheme along the backbone involving covalently bonded dimers, trimers, or tetramers was considered, with DFT at a smaller basis set serving as the low level of theory. The MIM method was evaluated on the high-spin state and several possible broken-symmetry (BS) states for di- and polyradicals. When relevant spin-spin interactions were considered, the errors in total energies were less than 1 kcal mol-1. In addition, the applicability of MIM2 was extended to predict the intersite magnetic exchange coupling constants (J), which were compared with reference values. Further, since the energy levels derived from Hamiltonian diagonalization are physically more meaningful, the calculated J values estimated from the BS-DFT methodology were used to obtain the lower spin state energies of the polyradicals. The difference in calculated total energies of the lower spin state between full and MIM2 lie within 1 kcal mol-1 in the majority of these cases. Our rigorous, quantum chemical study demonstrates that MIM can be successfully applied to the study of large organic radicals reliably and accurately within the framework of BS-DFT.