Ab Initio Density Matrix Renormalization Research Articles

Parallel Implementation of the Density Matrix Renormalization Group Method Achieving a Quarter petaFLOPS Performance on a Single DGX-H100 GPU Node.

We report cutting edge performance results on a single node hybrid CPU-multi-GPU implementation of the spin adapted ab initio Density Matrix Renormalization Group (DMRG) method on current state-of-the-art NVIDIA DGX-H100 architectures. We evaluate the performance of the DMRG electronic structure calculations for the active compounds of the FeMoco, the primary cofactor of nitrogenase, and cytochrome P450 (CYP) enzymes with complete active space (CAS) sizes of up to 113 electrons in 76 orbitals [CAS(113, 76)] and 63 electrons in 58 orbitals [CAS(63, 58)], respectively. We achieve 246 teraFLOPS of sustained performance, an improvement of more than 2.5× compared to the performance achieved on the DGX-A100 architectures and an 80× acceleration compared to an OpenMP parallelized implementation on a 128-core CPU architecture. Our work highlights the ability of tensor network algorithms to efficiently utilize high-performance multi-GPU hardware and shows that the combination of tensor networks with modern large-scale GPU accelerators can pave the way toward solving some of the most challenging problems in quantum chemistry and beyond.

Distributed Multi-GPU Ab Initio Density Matrix Renormalization Group Algorithm with Applications to the P-Cluster of Nitrogenase.

The presence of many degenerate d/f orbitals makes polynuclear transition-metal compounds, such as iron-sulfur clusters in nitrogenase, challenging for state-of-the-art quantum chemistry methods. To address this challenge, we present the first distributed multi-graphics processing unit (GPU) ab initio density matrix renormalization group (DMRG) algorithm suitable for modern high-performance computing (HPC) infrastructures. The central idea is to parallelize the most computationally intensive part─the multiplication of O(K2) operators with a trial wave function, where K is the number of spatial orbitals, by combining operator parallelism for distributing the workload with a batched algorithm for performing contractions on GPU. With this new implementation, we are able to reach an unprecedentedly large bond dimension D = 14,000 on 48 GPUs (NVIDIA A100 80 GB SXM) for an active space model (114 electrons in 73 active orbitals) of the P-cluster, which is nearly 3 times larger than the bond dimensions reported in previous DMRG calculations for the same system using only central processing units (CPUs).

A comparison between the one- and two-step spin-orbit coupling approaches based on the ab initio density matrix renormalization group.

The efficient and reliable treatment of both spin-orbit coupling (SOC) and electron correlation is essential for understanding f-element chemistry. We analyze two approaches to the problem: the one-step approach, where both effects are treated simultaneously, and the two-step state interaction approach. We report an implementation of the ab initio density matrix renormalization group with a one-step treatment of the SOC effect, which can be compared to prior two-step treatments on an equal footing. Using a dysprosium octahedral complex and bridged dimer as benchmark systems, we identify characteristics of problems where the one-step approach is beneficial for obtaining the low-energy spectrum.

Exploring optimal multimode vibronic pathways in singlet fission of azaborine analogues of perylene.

The development of new singlet fission chromophores is a vibrant area of research to explore the possibility of efficient photovoltaic devices. Using high-level ab-initio density matrix renormalization group calculations, we present a systematic analysis of BN-doped perylenes for their potential application as singlet fission candidates. Four singlet fission chromophores are identified considering the monomer-based properties and their excitonic characters are further analyzed in a dimer configuration optimized in a six-dimensional space for local maxima of fission rates. Furthermore, a multistate multimode vibronic Hamiltonian is employed to identify intra- and interstate vibrational pathways for excitation energy modulation. Several photophysical properties such as Davydov splitting, activation energy and vibronic admixture of multiexcitonic and charge-transfer states are calculated for physically accessible dimers. The optimal dimer packing results in appropriate vibrational relaxation of singlet fission states and promotes significant population transfer which would be more attenuated without such couplings. This work not only identifies potential singlet fission systems with favorable electronic properties but also highlights the sensitivity of dimer packings with respect to the substitution patterns in singlet fission chromophores.

On the highest oxidation states of the actinoids in AnO4 molecules (An=Ac - Cm): A DMRG-CASSCF study.

Actinoid tetroxide molecules AnO4 (An=Ac - Cm) are investigated with the ab initio density matrix renormalization group (DMRG) approach. Natural orbital shapes are used to read out the oxidation state (OS) of the f-elements, and the atomic orbital energies and radii are used to explain the trends. The highest OSs reveal a "volcano"-type variation: For An=Ac - Np, the OSs are equal to the number of available valence electrons, that is, AcIII , ThIV , PaV , UVI , and NpVII . Starting with plutonium as the turning point, the highest OSs in the most stable AnO4 isomers then decrease as PuV , AmV , and CmIII , indicating that the 5f-electrons are hard to be fully oxidized off from Pu onward. The variations are related to the actinoid contraction and to the 5f-covalency characteristics. Combined with previous work on OSs, we review their general trends throughout the periodic table, providing fundamental understanding of OS-relevant phenomena.

Post-Density Matrix Renormalization Group Methods for Describing Dynamic Electron Correlation with Large Active Spaces.

The ab initio density matrix renormalization group (DMRG) method has been well-established and has become one of the most accurate numerical methods for the precise electronic structure solution of large active spaces. In the past few years, to capture the missing dynamic correlation, various post-DMRG approaches have been proposed through the combination of DMRG and multireference quantum chemical methods or density functional theory. With this in mind, this work provides a brief overview of ab initio DMRG principles and the new developments within post-DMRG methods. For clarity, post-DMRG methods are classified into two main categories depending on whether high-order n-electron reduced density matrices are used, and their merits and disadvantages are properly discussed. Finally, we conclude by discussing unsolved bottlenecks and giving development perspectives of post-DMRG approaches, which are expected to yield quantitative descriptions of complex electronic structures in large strongly correlated molecules and materials.

Multireference Perturbation Theory Combined with PCM and RISM Solvation Models: A Benchmark Study for Chemical Energetics.

The polarizable continuum model (PCM) has been one of the most widely used approaches to take into account the solvation effect in quantum chemical calculations. In this paper, we performed a series of benchmark calculations to assess the accuracy of the PCM scheme combined with the second-order complete-active-space perturbation theory (CASPT2) for molecular systems in polar solvents. For solute molecules with extensive conjugated π orbitals, exemplified by elongated conjugated arylcarbenes, we have incorporated the ab initio density matrix renormalization group algorithm into the PCM-CASPT2 method. In the previous work, we presented a combination of the DMRG-CASPT2 method with the reference interaction site model (RISM) theory for describing the solvation effect using the radial distribution function and compared its performance to the widely used density-functional approaches (PCM-TD-DFT). The work here allows us to further show a more thorough assessment of the RISM model compared to the PCM with an equal level of the wave function treatment, the (DMRG-)CASPT2 theory, toward a high-accuracy electronic structure calculations for solvated chemical systems. With the exception that the PCM models are not capable of properly describing the hydrogen bondings, accuracy of the PCM-CASPT2 model is in most cases quite comparable to the RISM counterpart.

Low communication high performance ab initio density matrix renormalization group algorithms.

There has been recent interest in the deployment of ab initio density matrix renormalization group (DMRG) computations on high performance computing platforms. Here, we introduce a reformulation of the conventional distributed memory ab initio DMRG algorithm that connects it to the conceptually simpler and advantageous sum of the sub-Hamiltonian approach. Starting from this framework, we further explore a hierarchy of parallelism strategies that includes (i) parallelism over the sum of sub-Hamiltonians, (ii) parallelism over sites, (iii) parallelism over normal and complementary operators, (iv) parallelism over symmetry sectors, and (v) parallelism within dense matrix multiplications. We describe how to reduce processor load imbalance and the communication cost of the algorithm to achieve higher efficiencies. We illustrate the performance of our new open-source implementation on a recent benchmark ground-state calculation of benzene in an orbital space of 108 orbitals and 30 electrons, with a bond dimension of up to 6000, and a model of the FeMo cofactor with 76 orbitals and 113 electrons. The observed parallel scaling from 448 to 2800 central processing unit cores is nearly ideal.

Towards multistate multimode landscapes in singlet fission of pentacene: the dual role of charge-transfer states.

Singlet fission duplicates triplet excitons for improving light harvesting efficiency. The presence of the interaction between electronic and nuclear degrees of freedom complicates the interpretation of correlated triplet pairs. We report a quantum chemistry study on the significance and subtleties of multistate and multimode pathways in forming triplet pair states of the pentacene dimer through a six-state vibronic-coupling Hamiltonian derived from many-electron adiabatic wavefunctions of an ab initio density matrix renormalization group. The resulting spin values of the singlet manifolds on each pentacene center are computed, and the varying spin nature can be distinguished clearly with respect to dimer stacking and vibronic progression. Our monomer spin assignments reveal the coexistence of both lower-lying weak and higher-lying strong charge transfer states which interact vibronically with the triplet pair state, providing important implications for its generation and separation occurring in vibronic regions. This work conveys the importance of the many-electron process requiring close low-lying singlet manifolds to determine the subtle fission details, and represents an important step for understanding vibronically resolved spin states and conversions underlying efficient singlet fission.

Stimulated X-ray Raman and Absorption Spectroscopy of Iron-Sulfur Dimers.

Iron-sulfur complexes play an important role in biological processes such as metabolic electron transport. A detailed understanding of the mechanism of long-range electron transfer requires knowledge of the electronic structure of the complexes, which has traditionally been challenging to obtain, either by theory or by experiment, but the situation has begun to change with advances in quantum chemical methods and intense free electron laser light sources. We compute the spectra for stimulated X-ray Raman spectroscopy (SXRS) and absorption spectroscopy of homovalent and mixed-valence [2Fe-2S] complexes, using the ab initio density matrix renormalization group algorithm. The simulated spectra show clear signatures of the theoretically predicted dense low-lying excited states within the d-d manifold. Furthermore, the difference in spectral intensity between the absorption-active and Raman-active states provides a potential mechanism to selectively excite states by a proper tuning of the excitation pump, to access the electronic dynamics within this manifold.

The df-d Dative Bonding in a Uranium-Cobalt Heterobimetallic Complex for Efficient Nitrogen Fixation.

Single transition-metal site catalysts with s-, p-, or d-block atom anchor for nitrogen fixation have been extensively studied, and yet the studies of the f-block atom anchor are rarely reported. Thus, we investigate the feasibility of using a newly synthesized U-Co complex featuring a single CoI site coordinated by tetrakis(phophinoamide) and an UIV anchor for N2-to-NH3 conversion by theoretical modeling. We characterize the evolution of oxidation states of U and Co along the reaction pathways from ab initio density matrix renormalization group (DMRG) calculations, and we find that the variation of the Co → U dative bond is correlated with the changes of oxidation states. Both uranium and cobalt can serve as electron reservoirs to facilitate breaking the N-N bond. Our study demonstrates the viability of metal → metal dative bonds, particularly the df-d one, for the reduction of N2 to NH3, and thus, this opens up a new avenue to the rational design of efficient catalyst for nitrogen fixation.

Peridinin Torsional Distortion and Bond-Length Alternation Introduce Intramolecular Charge-Transfer and Correlated Triplet Pair Intermediate Excited States.

The nature of intramolecular charge transfer (ICT) and the mechanism of intramolecular singlet fission (SF) in peridinin remain open research questions. Obtaining an understanding of the population evolution from the bright state to dark state following a photoinduced electronic transition is critical. Unambiguously describing this evolution in peridinin, and light-harvesting carotenoids in general, has proven elusive experimentally and computationally. To offer a balanced description of the bright- and dark-state electronic structures, we here apply ab initio multireference perturbation theory quantum chemistry-the density matrix renormalization group self-consistent field and complete-active-space self-consistent field with second-order N-electron valence perturbation theory. At traditional bright- (S2) and dark-state (S1) optimized geometries, we find that an additional correlated triplet pair state and ICT state are derived from the canonical polyene Bu (S3) and 3Ag (S4) dark singlet excited states, respectively. Whereas the S3 state's physical properties are insensitive to peridinin's allene-tail donor and lactone ring acceptor functionalization, the S4 state exhibits a markedly enhanced oscillator strength and highest occupied molecular orbital-lowest unoccupied molecular orbital transition density. These changes suggest that the ICT character stems from mixing between the bright S2 and putatively dark S4.

Matrix product operators, matrix product states, and ab initio density matrix renormalization group algorithms.

Current descriptions of the ab initio density matrix renormalization group (DMRG) algorithm use two superficially different languages: an older language of the renormalization group and renormalized operators, and a more recent language of matrix product states and matrix product operators. The same algorithm can appear dramatically different when written in the two different vocabularies. In this work, we carefully describe the translation between the two languages in several contexts. First, we describe how to efficiently implement the ab initio DMRG sweep using a matrix product operator based code, and the equivalence to the original renormalized operator implementation. Next we describe how to implement the general matrix product operator/matrix product state algebra within a pure renormalized operator-based DMRG code. Finally, we discuss two improvements of the ab initio DMRG sweep algorithm motivated by matrix product operator language: Hamiltonian compression, and a sum over operators representation that allows for perfect computational parallelism. The connections and correspondences described here serve to link the future developments with the past and are important in the efficient implementation of continuing advances in ab initio DMRG and related algorithms.

On the calculation of complete dissociation curves of closed-shell pseudo-onedimensional systems via the complete active space method of increments.

The method of increments (MoI) has been employed using the complete active space formalism in order to calculate the dissociation curve of beryllium ring-shaped clusters Be(n) of different sizes. Benchmarks obtained through different quantum chemical methods including the ab initio density matrix renormalization group were used to verify the validity of the MoI truncation which showed a reliable behavior for the whole dissociation curve. Moreover we investigated the size dependence of the correlation energy at different interatomic distances in order to extrapolate the values for the periodic chain and to discuss the transition from a metal-like to an insulator-like behavior of the wave function through quantum chemical considerations.

Spin orbit coupling for molecular ab initio density matrix renormalization group calculations: Application to g-tensors.

Spin Orbit Coupling (SOC) is introduced to molecular ab initio density matrix renormalization group (DMRG) calculations. In the presented scheme, one first approximates the electronic ground state and a number of excited states of the Born-Oppenheimer (BO) Hamiltonian with the aid of the DMRG algorithm. Owing to the spin-adaptation of the algorithm, the total spin S is a good quantum number for these states. After the non-relativistic DMRG calculation is finished, all magnetic sublevels of the calculated states are constructed explicitly, and the SOC operator is expanded in the resulting basis. To this end, spin orbit coupled energies and wavefunctions are obtained as eigenvalues and eigenfunctions of the full Hamiltonian matrix which is composed of the SOC operator matrix and the BO Hamiltonian matrix. This treatment corresponds to a quasi-degenerate perturbation theory approach and can be regarded as the molecular equivalent to atomic Russell-Saunders coupling. For the evaluation of SOC matrix elements, the full Breit-Pauli SOC Hamiltonian is approximated by the widely used spin-orbit mean field operator. This operator allows for an efficient use of the second quantized triplet replacement operators that are readily generated during the non-relativistic DMRG algorithm, together with the Wigner-Eckart theorem. With a set of spin-orbit coupled wavefunctions at hand, the molecular g-tensors are calculated following the scheme proposed by Gerloch and McMeeking. It interprets the effective molecular g-values as the slope of the energy difference between the lowest Kramers pair with respect to the strength of the applied magnetic field. Test calculations on a chemically relevant Mo complex demonstrate the capabilities of the presented method.

The ab-initio density matrix renormalization group in practice.

The ab-initio density matrix renormalization group (DMRG) is a tool that can be applied to a wide variety of interesting problems in quantum chemistry. Here, we examine the density matrix renormalization group from the vantage point of the quantum chemistry user. What kinds of problems is the DMRG well-suited to? What are the largest systems that can be treated at practical cost? What sort of accuracies can be obtained, and how do we reason about the computational difficulty in different molecules? By examining a diverse benchmark set of molecules: π-electron systems, benchmark main-group and transition metal dimers, and the Mn-oxo-salen and Fe-porphine organometallic compounds, we provide some answers to these questions, and show how the density matrix renormalization group is used in practice.

Scalar Relativistic Calculations of Hyperfine Coupling Constants Using Ab Initio Density Matrix Renormalization Group Method in Combination with Third-Order Douglas–Kroll–Hess Transformation: Case Studies on 4d Transition Metals

We have developed a new computational scheme for high-accuracy prediction of the isotropic hyperfine coupling constant (HFCC) of heavy molecules, accounting for the high-level electron correlation effects, as well as the scalar-relativistic effects. For electron correlation, we employed the ab initio density matrix renormalization group (DMRG) method in conjunction with a complete active space model. The orbital-optimization procedure was employed to obtain the optimized orbitals required for accurately determining the isotropic HFCC. For the scalar-relativistic effects, we initially derived and implemented the Douglas-Kroll-Hess (DKH) hyperfine coupling operators up to the third order (DKH3) by using the direct transformation scheme. A set of 4d transition-metal radicals consisting of Ag atom, PdH, and RhH2 were chosen as test cases. Good agreement between the isotropic HFCC values obtained from DMRG/DKH3 and experiment was archived. Because there are no available gas-phase values for PdH and RhH2 radicals in the literature, the results from the present high-level theory may serve as benchmark data.

Investigation of metal–insulator-like transition through theab initiodensity matrix renormalization group approach

We have studied the metal--insulator-like transition in pseudo-one-dimensional systems, i.e., lithium and beryllium rings, through the ab initio density matrix renormalization group (DMRG) method. Performing accurate calculations for different interatomic distances and using quantum information theory, we investigated the changes occurring in the wave function between a metallic-like state and an insulating state built from free atoms. We also discuss entanglement and relevant excitations among the molecular orbitals in the Li and Be rings and show that the transition bond length can be detected using orbital entropy functions. Also, the effect of different orbital bases on the effectiveness of the DMRG procedure is analyzed comparing the convergence behavior.

Ab initio density matrix renormalization group study of magnetic coupling in dinuclear iron and chromium complexes

The applicability of ab initio multireference wavefunction-based methods to the study of magnetic complexes has been restricted by the quickly rising active-space requirements of oligonuclear systems and dinuclear complexes with S > 1 spin centers. Ab initio density matrix renormalization group (DMRG) methods built upon an efficient parameterization of the correlation network enable the use of much larger active spaces, and therefore may offer a way forward. Here, we apply DMRG-CASSCF to the dinuclear complexes [Fe2OCl6](2-) and [Cr2O(NH3)10](4+). After developing the methodology through systematic basis set and DMRG M testing, we explore the effects of extended active spaces that are beyond the limit of conventional methods. We find that DMRG-CASSCF with active spaces including the metal d orbitals, occupied bridging-ligand orbitals, and their virtual double shells already capture a major portion of the dynamic correlation effects, accurately reproducing the experimental magnetic coupling constant (J) of [Fe2OCl6](2-) with (16e,26o), and considerably improving the smaller active space results for [Cr2O(NH3)10](4+) with (12e,32o). For comparison, we perform conventional MRCI+Q calculations and find the J values to be consistent with those from DMRG-CASSCF. In contrast to previous studies, the higher spin states of the two systems show similar deviations from the Heisenberg spectrum, regardless of the computational method.

Linear and nonlinear optical properties of indeno[2,1-b]fluorene and its structural isomers.

Linear and non-linear optical properties of indeno[2,1-b]fluorene (1) and its structural isomers with 20 π-electrons have been studied using many body methods. As compared to other π electron systems of similar conjugation length, the experimentally measured optical gap (vertical excitation energy) and the singlet-triplet gap of 1 are quite small. The diradical character calculated using the ab initio density matrix renormalization group (DMRG) of 1 is the largest among its isomers, which explains its lowest singlet-triplet gap. The optical gaps calculated by the DMRG method, employing the long-range interacting Pariser-Parr-Pople model, show good agreement with the experimental values as compared to those calculated by DFT, MP2, and CASPT2 methods. Using correlated Hamiltonian and the DMRG approach, the third harmonic generation coefficient is found to correlate well with the diradical character.