Electron Density Matrix Research Articles

Second-Order Mass-Weighting Scheme for Atom-Centered Density Matrix Propagation Molecular Dynamics.

The atom-centered density matrix propagation (ADMP) method is an extended Lagrangian approach to ab initio molecular dynamics, which includes the density matrix in an orthonormalized atom-centered Gaussian basis as additional, fictitious, electronic degrees of freedom, classically propagated along with the nuclear ones. A high adiabaticity between the nuclear and electronic subsystems is mandatory in order to keep the trajectory close to the Born-Oppenheimer (BO) surface. In this regard, the fictitious electronic mass μ, being a symmetric, nondiagonal matrix in its most general form, represents a free parameter, exploitable to optimize the propagation of the electronic density. Although mass-weighting schemes in ADMP exist, a systematic procedure to define an optimal value of the fictitious masses is not available yet. In this work, in order to rationally evaluate the electronic mass, fictitious electronic normal modes are defined through the diagonalization of the Hessian of the electronic density matrix. If the same frequency is imposed on all such modes (compatible with the chosen integration time step), then the corresponding μ matrix can be calculated and then employed for the following propagation. Analysis of several ADMP test simulations reveals that such Hessian-based mass-weighting approach is able to ensure, together with a 0.1/0.2 fs time steps, a high separation between the (real) nuclear and the (fictitious) electronic frequencies, which determines a high adiabaticity. This high, unprecedented, accuracy in the propagation leads, in turn, to low errors in the estimated nuclear vibrational frequencies, making the ADMP method totally comparable to a fully converged BO molecular dynamics simulation but more computationally efficient. This work, therefore, contributes to a further development of the ADMP ab initio molecular dynamics method, aimed at improving its accuracy through a more rational evaluation of the fictitious electronic mass parameter.

Time Resolved Quantum Tomography in Molecular Spectroscopy by the Maximal Entropy Approach.

Attosecond science offers unprecedented precision in probing the initial moments of chemical reactions, revealing the dynamics of molecular electrons that shape reaction pathways. A fundamental question emerges: what role, if any, do quantum coherences between molecular electron states play in photochemical reactions? Answering this question necessitates quantum tomography─the determination of the electronic density matrix from experimental data, where the off-diagonal elements represent these coherences. The Maximal Entropy (MaxEnt) based Quantum State Tomography (QST) approach offers unique advantages in studying molecular dynamics, particularly with partial tomographic data. Here, we explore the application of MaxEnt-based QST on photoexcited ammonia, necessitating the operator form of observables specific to the performed measurements. We present two methodologies for constructing these operators: one leveraging Molecular Angular Distribution Moments (MADMs) which accurately capture the orientation-dependent vibronic dynamics of molecules and another utilizing Angular Momentum Coherence Operators to construct measurement operators for the full rovibronic density matrix in the symmetric top basis. A key revelation of our study is the direct link between Lagrange multipliers in the MaxEnt formalism and the unique set of MADMs. Additionally, we visualize the electron density within the molecular frame, demonstrating charge migration across the molecule. Furthermore, we achieve a groundbreaking milestone by constructing, for the first time, the entanglement entropy of the electronic subsystem─a metric that was previously inaccessible. The entropy vividly reveals and quantifies the effects of coupling between the excited electron and nuclear degrees of freedom. Consequently, our findings open new avenues for research in ultrafast molecular spectroscopy within the broader domain of quantum information science, offering profound implications for the study of molecular systems under excitation using quantum tomographic schemes.

Ultrafast X-ray Diffraction Probe of Coherent Spin-State Dynamics in Molecules.

We propose an approach to probe coherent spin-state dynamics of molecules using circularly polarized hard X-ray pulses. For the dynamically aligned nitric oxide molecules in a coherent superposition spin-orbit coupled electronic state that can be prepared through stimulated Raman scattering, we demonstrate the capability of ultrafast X-ray diffraction to not only reveal the quantum beating of the coherent spin-state wave packet but also image the spatial spin density of the molecule. With a circularly polarized ultrafast X-ray diffraction signal, we show that the electronic density matrix can be retrieved. The spatiotemporal resolving power of ultrafast X-ray diffraction paves the way for tracking transient spatial wave function in molecular dynamics involving the spin degree of freedom.

Spin-Vibronic Dynamics in Open-Shell Systems beyond the Spin Hamiltonian Formalism.

Vibronic coupling has a dramatic influence over a large number of molecular processes, ranging from photochemistry to spin relaxation and electronic transport. The simulation of vibronic coupling with multireference wave function methods has been largely applied to organic compounds, and only early efforts are available for open-shell systems such as transition metal and lanthanide complexes. In this work, we derive a numerical strategy to differentiate the molecular electronic Hamiltonian in the context of multireference ab initio methods and inclusive of spin-orbit coupling effects. We then provide a formulation of open quantum system dynamics able to predict the time evolution of the electron density matrix under the influence of a Markovian phonon bath up to fourth-order perturbation theory. We apply our method to Co(II) and Dy(III) molecular complexes exhibiting long spin relaxation times and successfully validate our strategy against the use of an effective spin Hamiltonian. Our study sheds light on the nature of vibronic coupling, the importance of electronic excited states in spin relaxation, and the need for high-level computational chemistry to quantify it.

An Updating of the IONORT Tool to Perform a High-Frequency Ionospheric Ray Tracing

This paper describes the main updates characterizing the new version of IONORT (IONOsperic Ray Tracing), a software tool developed at Istituto Nazionale di Geofisica e Vulcanologia to determine both the path of a high frequency (HF) radio wave propagating in the ionospheric medium, and the group time delay of the wave itself along the path. One of the main changes concerns the replacement of a regional three-dimensional electron density matrix, which was previously taken as input to represent the ionosphere, with a global one. Therefore, it is now possible to carry out different ray tracings from whatever point of the Earth’s surface, simply by selecting suitable loop cycles thanks to the new ray tracing graphical user interface (GUI). At the same time, thanks to a homing GUI, it is also possible to generate synthetic oblique ionograms for whatever radio link chosen by the user. Both ray tracing and homing GUIs will be described in detail providing at the same time some practical examples of their use for different regions. IONORT software finds practical application in the planning of HF radio links, exploiting the sky wave, through an accurate and thorough knowledge of the ionospheric medium. HF radio waves users, including broadcasting and civil aviation, would benefit from the use of the IONORT software (version 2023.10).

Density-Corrected Density Functional Theory for Molecular Properties.

Density-corrected (DC) density functional theory (DFT) has been proposed to overcome difficulties related to the self-interaction error. The procedure uses the Hartree-Fock electron density (matrix) non-self-consistently in conjunction with an approximate functional. DC-DFT has so far mainly been tested for total energy differences, whereas other types of molecular properties have not been evaluated systematically. This work focuses on the performance of DC-DFT for molecular properties, namely, dipole moments, static polarizabilities, and electric field gradients (EFGs) at atomic nuclei. Accurate reference data were generated from coupled-cluster theory to assess the performance of DC and self-consistent DFT calculations for twelve molecules, including diatomics with transition metals. DC-DFT does no harm in dipole moment calculations, but it negatively impacts the polarizability in at least one case. DC-DFT performs well for EFGs, even for the difficult case of CuCl.

Real-Space and Real-Time Propagation for Correlated Electron-Nuclear Dynamics Based on Exact Factorization.

We present coupled equations of motion for correlated electron-nuclear dynamics for real-space and real-time propagation with a proper electron-nuclear correlation (ENC) from the exact factorization. Since the original ENC term from the exact factorization is non-Hermitian, the numerical instability arises as we propagate an electronic wave function. In this paper, we propose a Hermitian-type ENC term which depends on the electron density matrix and the nuclear quantum momentum. Moreover, we show that the Hermitian property of the electron-nuclear correlation term can capture quantum (de)coherence with a stable numerical real-space and real-time propagation. As an application, we demonstrate a real-space and real-time propagation of an electronic wave function coupled to trajectory-based nuclear motion for a one-dimensional model Hamiltonian. Our approach can capture nonadiabatic phenomena as well as quantum decoherence in excited state molecular dynamics. In addition, we propose a scheme to extend the current approach to many-body electronic states based on real-time time-dependent density functional theory, testing the nonadiabatic dynamics of a simple molecular system.

Tensor-Train Thermo-Field Memory Kernels for Generalized Quantum Master Equations.

The generalized quantum master equation (GQME) approach provides a rigorous framework for deriving the exact equation of motion for any subset of electronic reduced density matrix elements (e.g., the diagonal elements). In the context of electronic dynamics, the memory kernel and inhomogeneous term of the GQME introduce the implicit coupling to nuclear motion and dynamics of electronic density matrix elements that are projected out (e.g., the off-diagonal elements), allowing for efficient quantum dynamics simulations. Here, we focus on benchmark quantum simulations of electronic dynamics in a spin-boson model system described by various types of GQMEs. Exact memory kernels and inhomogeneous terms are obtained from short-time quantum-mechanically exact tensor-train thermo-field dynamics (TT-TFD) simulations and are compared with those obtained from an approximate linearized semiclassical method, allowing for assessment of the accuracy of these approximate memory kernels and inhomogeneous terms. Moreover, we have analyzed the computational cost of the full and reduced-dimensionality GQMEs. The scaling of the computational cost is dependent on several factors, sometimes with opposite scaling trends. The TT-TFD memory kernels can provide insights on the main sources of inaccuracies of GQME approaches when combined with approximate input methods and pave the road for the development of quantum circuits that implement GQMEs on digital quantum computers.

Density matrix of electrons coupled to Einstein phonons and the electron-phonon entanglement content of excited states

We derive the exact reduced density matrix for the electrons in an analytically solvable electron-phonon model. Here, the electrons are described as a Luttinger liquid that is coupled to Einstein phonons. We further derive analytical expressions for the electron-phonon entanglement, its spectrum and mutual information at finite and zero temperature as well as for excited states. The entanglement entropy is additive in momentum for the quasi-particle excited states, but not in the electron-phonon coupled eigenmodes of the system.

Scalable Ehrenfest Molecular Dynamics Exploiting the Locality of Density-Functional Tight-Binding Hamiltonian.

To explore the science behind excited-state dynamics in high-complexity chemical systems, a scalable nonadiabatic molecular dynamics (MD) technique is indispensable. In this study, by treating the electronic degrees of freedom at the density-functional tight-binding level, we developed and implemented a reduced scaling and multinode-parallelizable Ehrenfest MD method. To achieve this goal, we introduced a concept called patchwork approximation (PA), where the effective Hamiltonian for real-time propagation of the electronic density matrix is partitioned into a set of local parts. Numerical results for giant icosahedral fullerenes, which comprise up to 6000 atoms, suggest that the scaling of the present PA-based method is less than quadratic, which yields a significant advantage over the conventional cubic scaling method in terms of computational time. The acceleration by the parallelization on multiple nodes was also assessed. Furthermore, the electronic and structural dynamics resulting from the perturbation by the external electric field were accurately reproduced with the PA, even when the electronic excitation was spatially delocalized.

On simulating the dynamics of electronic populations and coherences via quantum master equations based on treating off-diagonal electronic coupling terms as a small perturbation.

Quantum master equations provide a general framework for describing the dynamics of electronic observables within a complex molecular system. One particular family of such equations is based on treating the off-diagonal coupling terms between electronic states as a small perturbation within the framework of second-order perturbation theory. In this paper, we show how different choices of projection operators, as well as whether one starts out with the time-convolution or the time-convolutionless forms of the generalized quantum master equation, give rise to four different types of such off-diagonal quantum master equations (OD-QMEs), namely, time-convolution and time-convolutionless versions of a Pauli-type OD-QME for only the electronic populations and an OD-QME for the full electronic density matrix (including both electronic populations and coherences). The fact that those OD-QMEs are given in terms of the interaction picture makes it non-trivial to obtain Schrödinger picture electronic coherences from them. To address this, we also extend a procedure for extracting Schrödinger picture electronic coherences from interaction picture populations recently introduced by Trushechkin in the context of time-convolutionless Pauli-type OD-QME to the other three types of OD-QMEs. The performance of the aforementioned four types of OD-QMEs is explored in the context of the Garg-Onuchic-Ambegaokar benchmark model for charge transfer in the condensed phase across a relatively wide parameter range. The results show that time-convolution OD-QMEs can be significantly more accurate than their time-convolutionless counterparts, particularly in the case of Pauli-type OD-QMEs, and that rather accurate Schrödinger picture coherences can be obtained from interaction picture electronic inputs.

Local perturbations of periodic systems. Chemisorption and point defects by GoGreenGo.

We present a software package GoGreenGo-an overlay aimed to model local perturbations of periodic systems due to either chemisorption or point defects. The electronic structure of an ideal crystal is obtained by worldwide-distributed standard quantum physics/chemistry codes, and then processed by various tools performing projection to atomic orbital basis sets. Starting from this, the perturbation is addressed by GoGreenGo with use of the Green's functions formalism, which allows evaluating its effect on the electronic structure, density matrix, and energy of the system. In the present contribution, the main accent is made on processes of chemical nature, such as chemisorption or doping. We address a general theory and its computational implementation supported by a series of test calculations of the electronic structure perturbations for benchmark model solids: simple, face-centered, and body-centered cubium systems. In addition, more realistic problems of local perturbations in graphene lattice, such as lattice substitution, vacancy, and "on-top" chemisorption, are considered.

Deciphering the role of end-capped acceptor units for amplifying the photovoltaic properties of donor materials for high-performance organic solar cell applications

Non-fullerene organic solar cells (OSCs) deliver the highest efficiency gains overall in reported literature. Efforts are being made to refine efficacies and stabilities of organic solar cells through the designing of acceptor molecules that contain powerful electron-withdrawing groups. Here, we computed four acceptors (Y1-Y4) by end-capped alterations on reference R and optimize photophysical, optoelectronic and photovoltaic properties. Therefore, certain properties such as orientation of FMO’s, excitation and binding energy, open-circuit voltage (Voc), transition density matrix and reorganizational energy of hole and electron are observed in comparison with reference. The calculated molecular structures of Y1 and Y4 show a high red-shift, while that of Y2 and Y3 display slightly blue-shift, along with very fine excitation energies and high charge mobilities. All these molecules (Y1-Y4) and the reference R presents band-gaps as small as 1.5–2.5 eV, and considerable charge transfer potential. This theoretical system shows that end-capped acceptors alteration is quite remarkable to establish desired optoelectronic properties. Thus, Y1-Y4 are suggested to researchers aiming at the potential commercialization of highly efficient solar cells systems.

Correlation-driven phenomena in periodic molecular systems from variational two-electron reduced density matrix theory.

Correlation-driven phenomena in molecular periodic systems are challenging to predict computationally not only because such systems are periodically infinite but also because they are typically strongly correlated. Here, we generalize the variational two-electron reduced density matrix (2-RDM) theory to compute the energies and properties of strongly correlated periodic systems. The 2-RDM of the unit cell is directly computed subject to necessary N-representability conditions such that the unit-cell 2-RDM represents at least one N-electron density matrix. Two canonical but non-trivial systems, periodic metallic hydrogen chains and periodic acenes, are treated to demonstrate the methodology. We show that while single-reference correlation theories do not capture the strong (static) correlation effects in either of these molecular systems, the periodic variational 2-RDM theory predicts the Mott metal-to-insulator transition in the hydrogen chains and the length-dependent polyradical formation in acenes. For both hydrogen chains and acenes, the periodic calculations are compared with previous non-periodic calculations with the results showing a significant change in energies and increase in the electron correlation from the periodic boundary conditions. The 2-RDM theory, which allows for much larger active spaces than are traditionally possible, is applicable to studying correlation-driven phenomena in general periodic molecular solids and materials.

Simulating energy transfer dynamics in the Fenna-Matthews-Olson complex via the modified generalized quantum master equation.

The generalized quantum master equation (GQME) provides a general and formally exact framework for simulating the reduced dynamics of open quantum systems. The recently introduced modified approach to the GQME (M-GQME) corresponds to a specific implementation of the GQME that is geared toward simulating the dynamics of the electronic reduced density matrix in systems governed by an excitonic Hamiltonian. Such a Hamiltonian, which is often used for describing energy and charge transfer dynamics in complex molecular systems, is given in terms of diabatic electronic states that are coupled to each other and correspond to different nuclear Hamiltonians. Within the M-GQME approach, the effect of the nuclear degrees of freedom on the time evolution of the electronic density matrix is fully captured by a memory kernel superoperator, which can be obtained from short-lived (compared to the time scale of energy/charge transfer) projection-free inputs. In this paper, we test the ability of the M-GQME to predict the energy transfer dynamics within a seven-state benchmark model of the Fenna-Matthews-Olson (FMO) complex, with the short-lived projection-free inputs obtained via the Ehrenfest method. The M-GQME with Ehrenfest-based inputs is shown to yield accurate results across a wide parameter range. It is also found to dramatically outperform the direct application of the Ehrenfest method and to provide better-behaved convergence with respect to memory time in comparison to an alternative implementation of the GQME approach previously applied to the same FMO model.

Designing of 5,10-Dihydroindolo [3,2-b] Indole (DINI) Based Donor Materials for Small Molecule Organic Solar Cells

In this paper, four small molecules B1, B2, B3 and B4 based on donor–acceptor–donor–acceptor–donor (D-A-D-A-D) combination were designed by making structural modifications in R. The designed molecules contain 5,10-dihydro-indolo [3,2-b] indole central donor core and different benzo-thiadiazole and fluorine substituted benzothiadiazole (FBT) acceptor units. These molecules have different subunits introduced on 5,10-dihydroindolo [3,2-b] indole central core like benzo [1,2,5] thiadiazole in (B1), 5-Fluoro-benzo [1,2,5] thiadiazole in (B2), 5-Methyl-benzo [1,2,5] thiadiazole in (B3), 2-Fluoro-2-methyl-2-H-benzotriazole unit in (B4), flanked with [2,2’,5’,2”] terthiophene as spacer (S) and triphenyl amine as a common end-capped donor in all the molecules (B1– B4). The optoelectronic properties of these molecules were studied by performing density functional theory (DFT) at CAM-B3LYP. Among all the designed structures, B2 showed maximum absorption (457[Formula: see text]nm) due to its strong electron withdrawing 5-Fluoro-benzo [1,2,5] thiadiazole acceptor unit. Other opto-electronic properties were analyzed through reorganization energies, density of electronic states and transition density matrix (TDM) to estimate the photovoltaic potential of these newly designed molecules. Low exciton binding energies and comparable values of open circuit voltage than R indicate the worth of these candidates to be used in future solar energy driven devices.

Nonadiabatic Exciton and Charge Separation Dynamics at Interfaces of Zinc Phthalocyanine and Fullerene: Orientation Does Matter.

Interface orientation between zinc phthalocyanine (ZnPc) and fullerene (C60) affects their interfacial charge separation dynamics; however, the underlying physical origin is still elusive. In this work, we have employed the time-dependent density functional theory (TDDFT) method to explore excited-state properties of ZnPc and C60 heterojunctions with both face-on and edge-on configurations. Spectroscopically bright absorption is from locally excited (LE) singlet excitons within ZnPc. In the face-on configuration, LE excitons are much higher in energy than charge-transfer (CT) excitons, thereby making charge separation process favorable. However, in the edge-on configuration, LE excitons are the lowest ones and CT ones are higher in energy; thus, charge separation is not efficient. Subsequently, we have carried out TDDFT-based nonadiabatic dynamics method to simulate photoinduced exciton and charge separation dynamics of ZnPc and C60 heterojunctions with both edge-on and face-on configurations. In the former, there are no exciton transfer and charge separation processes observed within 300 fs simulation time; while, in the latter, fragment-based electronic transition density matrix analysis reveals that only LE excitons |C60ZnPC*⟩ and CT excitons |C60-ZnPC+⟩ are involved. The exciton transfer from |C60ZnPC*⟩ to |C60-ZnPC+⟩ is completed within about 100 fs in which charge separation takes place with electron-hole distances increasing from 1.0 to 4.5 Å. This exciton transfer process is essentially in company with electron transfer from ZnPc to C60 but almost not involving hole transfer. These gained insights not only rationalize experiments but also enrich our knowledge to design donor-acceptor orientations to optimize organic photovoltaic performance.

The spectrum of the atomic orbital overlap matrix and the locality of the virtual electronic density matrix

It is well known that near-linear dependencies in the atomic orbital (AO) basis will impede electronic-structure calculations since the inverse AO overlap matrix will be ill-defined. However, small eigenvalues will also impact the locality of the virtual density matrix. The virtual density matrix is relevant for developing efficient local approaches in electronic-structure theory, and recent literature shows that orthogonal molecular orbitals (MOs) are not optimal for exploiting locality. As size of molecules treated is increasing and high-quality basis sets are used, the problem is becoming more pronounced and challenges similar to those for extended systems appear. Here it is shown that the spectrum of the AO overlap matrix puts severe restrictions on the locality of the virtual density matrix, and that locality cannot be recovered by excluding components corresponding to near-singular eigenvalues. The effect is seen even when eigenvalues are orders of magnitude from being near-singular, and occurs also for small basis sets such as cc-pVDZ depending on the molecular system. Non-orthogonal orbitals do not constitute a solution to the problem, but they may be more convenient since lack of locality in the density matrix can be shifted from MO coefficients to the inverse MO overlap matrix.

Implementation of Coherent Switching with Decay of Mixing into the SHARC Program.

Simulation of electronically nonadiabatic dynamics is an important tool for understanding the mechanisms of photochemical and photophysical processes. Two contrasting methods in which the electrons are treated quantum mechanically while the nuclei are treated classically are semiclassical Ehrenfest dynamics and trajectory surface hopping; neither method in its original form includes decoherence. Decoherence in the context of electronically nonadiabatic dynamics refers to the gradual collapse of a coherent quantum mechanical electronic state under the scrutiny of nuclear motion into a mixture of stable pointer states. This is modeled in the coherent switches with decay of mixing (CSDM) method by the decay of the off-diagonal elements of the electronic density matrix. Here, we present an implementation of CSDM in the SHARC program; a key element of the new implementation is the use of a different propagator than that used previously in the ANT program.

Efficient Evaluation of Molecular Electrostatic Potential in Large Systems

An algorithm for the efficient computation of molecular electrostatic potential is reported. It is based on the partition/expansion of density into (pseudo) atomic fragments with the method of Deformed Atoms in Molecules, which allows to compute the potential as a sum of atomic contributions. These contributions are expressed as a series of irregular spherical harmonics times effective multipole moments and inverse multipole moments, including short-range terms. The problem is split into two steps. The first one consists of the partition/expansion of density accompanied by the computation of multipole moments, and its cost depends on the size of the basis set used in the computation of electron density within the Linear Combination of Atomic Orbitals framework. The second one is the actual computation of the electrostatic potential from the quantities calculated in the first step, and its cost depends on the number of computation points. For a precision in the electrostatic potential of six decimal figures, the algorithm leads to a dramatic reduction of the computation time with respect to the calculation from electron density matrix and integrals involving basis set functions.