Active Space Model Research Articles

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).

Role of Spin Polarization and Dynamic Correlation in Singlet-Triplet Gap Inversion of Heptazine Derivatives.

The new generation of proposed light-emitting molecules for organic light-emitting diodes (OLEDs) has raised considerable research interest due to its exceptional feature─a negative singlet-triplet (ST) gap violating Hund's multiplicity rule in the excited S1 and T1 states. We investigate the role of spin polarization in the mechanism of ST gap inversion. Spin polarization is associated with doubly excited determinants of certain types, whose presence in the wave function expansion favors the energy of the singlet state more than that of the triplet. Using a perturbation theory-based model for spin polarization, we propose a simple descriptor for prescreening of candidate molecules with negative ST gaps and prove its usefulness for heptazine-type molecules. Numerical results show that the quantitative effect of spin polarization decreases linearly with the increasing highest occupied molecular orbital-lowest unoccupied molecular orbital (HOMO-LUMO) exchange integral. Comparison of single- and multireference coupled-cluster predictions of ST gaps shows that the former methods provide good accuracy by correctly balancing the effects of doubly excited determinants and dynamic correlation. We also show that accurate ST gaps may be obtained using a complete active space model supplemented with dynamic correlation from multireference adiabatic connection theory.

Downfolding of many-body Hamiltonians using active-space models: Extension of the sub-system embedding sub-algebras approach to unitary coupled cluster formalisms.

In this paper, we discuss the extension of the recently introduced subsystem embedding subalgebra coupled cluster (SES-CC) formalism to unitary CC formalisms. In analogy to the standard single-reference SES-CC formalism, its unitary CC extension allows one to include the dynamical (outside the active space) correlation effects in an SES induced complete active space (CAS) effective Hamiltonian. In contrast to the standard single-reference SES-CC theory, the unitary CC approach results in a Hermitian form of the effective Hamiltonian. Additionally, for the double unitary CC (DUCC) formalism, the corresponding CAS eigenvalue problem provides a rigorous separation of external cluster amplitudes that describe dynamical correlation effects-used to define the effective Hamiltonian-from those corresponding to the internal (inside the active space) excitations that define the components of eigenvectors associated with the energy of the entire system. The proposed formalism can be viewed as an efficient way of downfolding many-electron Hamiltonian to the low-energy model represented by a particular choice of CAS. In principle, this technique can be extended to any type of CAS representing an arbitrary energy window of a quantum system. The Hermitian character of low-dimensional effective Hamiltonians makes them an ideal target for several types of full configuration interaction type eigensolvers. As an example, we also discuss the algebraic form of the perturbative expansions of the effective DUCC Hamiltonians corresponding to composite unitary CC theories and discuss possible algorithms for hybrid classical and quantum computing. Given growing interest in quantum computing, we provide energies for H2 and Be systems obtained with the quantum phase estimator algorithm available in the Quantum Development Kit for the approximate DUCC Hamiltonians.

The electronic complexity of the ground-state of the FeMo cofactor of nitrogenase as relevant to quantum simulations.

We report that a recent active space model of the nitrogenase FeMo cofactor, proposed in the context of simulations on quantum computers, is not representative of the electronic structure of the FeMo cofactor ground-state. A more representative model does not affect much certain resource estimates for a quantum computer such as the cost of a Trotter step, while strongly affecting others such as the cost of adiabatic state preparation. Thus, conclusions should not be drawn from the complexity of quantum or classical simulations of the electronic structure of this system in this active space. We provide a different model active space for the FeMo cofactor that contains the basic open-shell qualitative character, which may be useful as a benchmark system for making resource estimates for classical and quantum computers.

Ab initio MCDHF calculations of electron–nucleus interactions

We present recent advances in the development of atomic ab initio multiconfiguration Dirac–Hartree–Fock theory, implemented in the GRASP relativistic atomic structure code. For neutral atoms, the deviations of properties calculated within the Dirac–Hartree–Fock (DHF) method (based on independent particle model of an atomic cloud) are usually dominated by electron correlation effects, i.e. the non-central interactions of individual electrons. We present the recent advances in accurate calculations of electron correlation effects in small, medium, and heavy neutral atoms. We describe methods of systematic development of multiconfiguration expansions leading to systematic, controlled improvement of the accuracy of the ab initio calculations. These methods originate from the concept of the complete active space (CAS) model within the DHF theory, which, at least in principle, permits fully relativistic calculations with full account of electron correlation effects. The calculations within the CAS model on currently available computer systems are feasible only for very light systems. For heavier atoms or ions with more than a few electrons, restrictions have to be imposed on the multiconfiguration expansions. We present methods and tools, which are designed to extend the numerical calculations in a controlled manner, where multiconfiguration expansions account for all leading electron correlation effects. We show examples of applications of the GRASP code to calculations of hyperfine structure constants, but the code may be used for calculations of arbitrary bound-state atomic properties. In recent years it has been applied to calculations of atomic and ionic spectra (transition energies and rates), to determinations of nuclear electromagnetic moments, as well as to calculations related to interactions of bound electrons with nuclear electromagnetic moments leading to violations of discrete symmetries.

Time-dependent multiconfiguration self-consistent-field method based on the occupation-restricted multiple-active-space model for multielectron dynamics in intense laser fields

The time-dependent multiconfiguration self-consistent-field method based on the occupation-restricted multiple active space model is proposed (TD-ORMAS) for multielectron dynamics in intense laser fields. Extending the previously proposed time-dependent complete-active-space self-consistent-field method [TD-CASSCF; Phys. Rev. A, {\bf 88}, 023402 (2013)], which divides the occupied orbitals into core and active orbitals, the TD-ORMAS method {\it further} subdivides the active orbitals into an arbitrary number of subgroups, and poses the {\it occupation restriction} by giving the minimum and maximum number of electrons distributed in each subgroup. This enables highly flexible construction of the configuration interaction (CI) space, allowing a large-active-space simulation of dynamics, e.g., the core excitation or ionization. The equations of motion both for CI coefficients and spatial orbitals are derived based on the time-dependent variational principle, and an efficient algorithm is proposed to solve for the orbital time derivatives. In-depth descriptions of the computational implementation are given in a readily programmable manner. The numerical application to the one-dimensional lithium hydride cluster models demonstrates that the high flexibility of the TD-ORMAS framework allows for the cost-effective simulations of multielectron dynamics, by exploiting systematic series of approximations to the TD-CASSCF method.

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.

Spin-adapted density matrix renormalization group algorithms for quantum chemistry

We extend the spin-adapted density matrix renormalization group (DMRG) algorithm of McCulloch and Gulacsi [Europhys. Lett. 57, 852 (2002)] to quantum chemical Hamiltonians. This involves using a quasi-density matrix, to ensure that the renormalized DMRG states are eigenfunctions of Ŝ(2), and the Wigner-Eckart theorem, to reduce overall storage and computational costs. We argue that the spin-adapted DMRG algorithm is most advantageous for low spin states. Consequently, we also implement a singlet-embedding strategy due to Tatsuaki [Phys. Rev. E 61, 3199 (2000)] where we target high spin states as a component of a larger fictitious singlet system. Finally, we present an efficient algorithm to calculate one- and two-body reduced density matrices from the spin-adapted wavefunctions. We evaluate our developments with benchmark calculations on transition metal system active space models. These include the Fe(2)S(2), [Fe(2)S(2)(SCH(3))(4)](2-), and Cr(2) systems. In the case of Fe(2)S(2), the spin-ladder spacing is on the microHartree scale, and here we show that we can target such very closely spaced states. In [Fe(2)S(2)(SCH(3))(4)](2-), we calculate particle and spin correlation functions, to examine the role of sulfur bridging orbitals in the electronic structure. In Cr(2) we demonstrate that spin-adaptation with the Wigner-Eckart theorem and using singlet embedding can yield up to an order of magnitude increase in computational efficiency. Overall, these calculations demonstrate the potential of using spin-adaptation to extend the range of DMRG calculations in complex transition metal problems.

Four-Electron, Three-Orbital Model for the Low-Energy Electronic Structure of Cationic Diarylmethanes: Notes on a “Pauling Point”

We examine a four-electron, three-orbital complete active space self-consistent field (SA-CASSCF) and multistate multireference perturbation theory (MS-MRPT2) model of the electronic structure associated with the two lowest-lying electronic excitations of a series of cationic diarylmethanes related to Michler's Hydrol Blue. These dyes are of interest because of the sensitivity of their excited-state dynamics to environmental influence in biological and other condensed phases. We show that the model corresponds to an easily understandable physical approximation where the dye electronic structure is mapped onto the π-electron system of an allyl anion. We show that reported trends in solution-state absorbance bands and transition dipole moments associated with the first two electronic excitations can be described within reasonable accuracy by the model. We also show, for Michler's Hydrol Blue, that the four-electron, three-orbital model provides a more balanced description of the electronic difference densities associated with electronic excitation calculated with the full π-electron space than can be achieved with active space models intermediate between these limits. The valence excitation energies predicted by the model are not sensitive to the underlying basis set, so that considerable computational savings may be possible by using split-valence basis sets with a limited number of polarization functions. We conclude that the model meets the criteria for a "Pauling Point": a point where the cancellation of large errors leads to physically balanced model, and where further elaboration degrades, rather than improves, the quality of description. We advocate that this Pauling Point be exploited in condensed-phase dynamical models where the computational overhead associated with the electronic structure must kept to a minimum (for example, nonadiabatic dynamics simulations coupled to QM/MM environmental models).

Correction to constrained coupled cluster doubles models based on the second coupled cluster central moment

We develop a correction for the coupled cluster version of the perfect pairing (PP) model. The correction is based on finding modified values of the PP amplitudes such that the second coupled cluster central moment defined in the space of all valence single and double substitutions vanishes and, subject to this constraint, minimizing the deviation between the modified and unmodified PP amplitudes with respect to a chosen metric. We discuss how this correction can be generalized to other constrained doubles models, such as local correlation and active-space models. While the correction is not strictly size consistent and retains some of the deficiencies of the PP model, numerical results indicate that much of the missing active-space coupled cluster singles and doubles correlation energy is recovered.

Orbital-optimized coupled-cluster theory does not reproduce the full configuration-interaction limit

It is shown that due to the mixing of the usual projection approach of coupled cluster with variational orbital optimization, orbital-optimized coupled cluster (OCC) fails to reproduce the full configuration-interaction (full CI) limit when the cluster operator becomes complete. It is pointed out that the fulfillment of the projected singles equations, which define the orbital gradient in Brueckner coupled cluster (BCC), is mandatory for a correct behavior. As numerical examples we present general OCC and BCC calculations up to the full CI limit on CH(2) and an active-space model of ozone. The observed deviations of OCC from full CI are of the order of the correlation error obtained in calculations with up to quadruples excitations. Thus the failure of OCC may be considered tolerable in more approximate calculations but clearly prohibitive for any benchmark application. For applications to active-space models a hybrid approach for OCC is suggested in which for active particle-hole rotations the Brueckner orbital gradient is employed, whereas for the remaining orbital rotations the variational orbital gradient is retained.

Can coupled cluster singles and doubles be approximated by a valence active space model?

A new, efficient approximation for coupled cluster singles and doubles (CCSD) is proposed in which a CCSD calculation is performed in a valence active space followed by a second-order perturbative correction to account for the inactive singles and doubles cluster amplitudes. This method, denoted VCCSD(SD), satisfactorily reproduces CCSD results in a variety of test cases, including spectroscopic constants of diatomic molecules, reaction energies, the Cope rearrangement, and other relative energies. Use of VCCSD alone is significantly less satisfactory. Formally, the O2V4 scaling of CCSD is reduced to o2v2V2, where o is the number of active occupied orbitals, v is the number of active virtual orbitals, and V is the total number of virtual orbitals. We also investigate the role of orbital optimizations and the appropriate choice of an active space in such methods.

Spin−Orbit Coupling in Biradicals. 1. The 2-Electrons-in-2-Orbitals Model Revisited

The two-electrons-in-two-orbitals active space model of electronic structure of biradicals and biradicaloids is extended to a Hamiltonian that incorporates the usual kinetic and electrostatic energy terms, and also outside electric and magnetic fields, spin−spin dipolar terms, and one- and two-electron spin−orbit coupling terms. It leads to a more rigorous version of the Salem−Rowland rules for the dependence of T1−S0 spin−orbit coupling (SOC) in biradicals and biradicaloids on molecular structure and conformation. For large T1−S0 SOC in a bitopic biradical, (i) the localized orbitals A and B that are singly occupied in the T1 state either interact covalently or one of them is sufficiently lower in energy to have nearly double occupancy in the S0 state, (ii) on at least one atom of reasonably high atomic number one p orbital contributes strongly to A and another to B, and (iii) the atomic contributions add constructively rather than destructively. The nature of this addition is such that inverse heavy ato...

A wave operator description of molecular photodissociation processes using the Floquet formalism

During the last few years, the Floquet matrix approach to the theory of intense-field multiphoton dissociation processes has undergone a revival of interest which is due to the emergence of new powerful algorithms within the complex quasivibrational energy formalism. The method presented here shows that the Bloch wave operator theory is, in the framework of this theory, an efficient tool to select the active space which participates principally in the dissociation process. An illustrative numerical example reveals that the one dimensional active space model, used mainly to reproduce cw laser experimental results, fails when the laser is tuned to bound state–bound state transitions.

Application of the multiconfiguration time-dependent Hartree–Fock method to CH+: The auxiliary active space

The multiconfigurational time-dependent Hartree–Fock (MCTDHF) method is used to study the excitation energies and transition moments of the CH+ ion. A consistent approach to the selection of configurations based on the complete active space (CAS) model is successfully applied. In this approach a large set of orbitals is included in the active space without a substantial increase in the number of configurations, since only for a smaller subset of orbitals is the configuration expansion complete and the remaining orbitals active are considered an auxiliary set. For CH+ the active space includes the valence and additional orbitals, with the valence orbitals constituting the complete subset. Extra configurations involving single and double excitations from the valence to the additional (auxiliary) orbitals are included both for the MCSCF reference state and in the MCTDHF. It is demonstrated that this active space accurately mimics MCSCF and MCTDHF calculations with an enlarged complete active space which includes both the original CAS and the auxiliary orbitals. The calculations for the 1 Π and 1 Σ+ excited states at the equilibrium geometry show that accurate excitation energies and transition moments are obtained in this way. For most of these states for which there are other good ab initio results available, we obtain comparable excitation energies and transition moments. In addition to 1 Π and 1 Σ+, low lying states of other symmetries are obtained and discussed.

Biradicals and biradicaloids: a unified view

A generalized version of the two-electron two-orbital active space model of biradicals and biradicaloids is described in simple non-mathematical terms. The distinctions between homosymmetric, weakly heterosymmetric, critically heterosymmetric, strongly heterosymmetric, and nonsymmetric biradicaloids are pointed out and the consequences for their different behavior are elaborated. Particular attention is paid to structural effects on the singlet - triplet, and lowest singlet - first excited singlet energy gaps, and on the consequences of the latter in photochemistry. The material is illustrated on examples of photochemical reactions involving sigma bond dissociation, pericyclic processes, and geometrical isomerization.

Male Catches of Spodoptera litura (F.) in Pheromone Traps under Fluctuating Wind Velocity : Validity of the Active Space Model

Male Catches of Spodoptera litura (F.) in Pheromone Traps under Fluctuating Wind Velocity : Validity of the Active Space Model

Core hole induced charge‐transfer reactions in disubstituted benzenes

AbstractThe application of the charge‐transfer concept on core electron ionization in donor–acceptor molecules is analyzed. A single excitation picture, involving the highest occupied and lowest unoccupied molecular orbitals is compared with a full π‐valence orbital active space model. The connections between the notion of charge transfer in donor–acceptor species with the shake‐up picture used for core photoelectron spectra of small molecules and with the dynamical screening concept applied on surface–adsorbate spectra are discussed.