Natural Orbital Occupation Research Articles

Tuning the Diradical Character of Pentacene Derivatives via Non-Benzenoid Coupling Motifs.

The development of functional organic molecules requires structures of increasing size and complexity, which are typically obtained by the covalent coupling of smaller building blocks. Herein, with the aid of high-resolution scanning tunneling microscopy/spectroscopy and density functional theory, the coupling of a sterically demanded pentacene derivative on Au(111) into fused dimers connected by non-benzenoid rings was studied. The diradical character of the products was tuned according to the coupling section. In particular, the antiaromaticity of cyclobutadiene as the coupling motif and its position within the structure play a decisive role in shifting the natural orbital occupancies toward a stronger diradical electronic character. Understanding these structure-property relations is desirable not only for fundamental reasons but also for designing new complex and functional molecular structures.

An ab initio study on the stability of isolated phosphaalkene synthons.

Using ab initio methods and flexible basis sets, we examined the electronic, geometric, and thermodynamic stabilities of selected phosphaalkene synthons matching the (PCR2)- formula (R = H, CH3, C6H5, C6F5, and Mes). All isolated synthons considered were found to be electronically stable and susceptible to neither fragmentation nor isomerization processes. The structures corresponding to the most stable isomers of the studied phosphaalkene synthons contain a PC double-bond (whose presence was confirmed by natural bond orbital occupancies of σ(P-C) and π(P-C) approaching 2 electrons) and two R substituents connected to the carbon atom in either (PCR2)- (for R = H, CH3, C6H5, and Mes) or (PCF-R-R)- (for R = C6F5) manner. Vertical electron detachment energies (spanning the 0.924-3.118 eV range) characterizing the phosphaalkene synthons were predicted and discussed.

Natural orbitals and two-particle correlators as tools for the analysis of effective exchange couplings in solids.

Using generalizations of spin-averaged natural orbitals and two-particle charge correlators for solids, we investigate the electronic structure of antiferromagnetic transition-metal oxides with a fully self-consistent, imaginary-time GW method. Our findings disagree with the Goodenough-Kanamori (GK) rules that are commonly used for the qualitative interpretation of such solids. First, we found a strong dependence of the natural orbital occupancies on momenta, contradicting GK assumptions. Second, along the momentum path, the character of natural orbitals changes. In particular, the contributions of oxygen 2s orbitals are important, which has not been considered in the GK rules. To analyze the influence of the electronic correlation on the values of effective exchange coupling constants, we use both natural orbitals and two-particle correlators and show that electronic screening modulates the degree of superexchange by stabilizing the charge-transfer contributions, which greatly affects these coupling constants. Finally, we give a set of predictions and recommendations regarding the use of density functional, Green's function, and wave-function methods for evaluating effective magnetic couplings in molecules and solids.

Structural stability and the low-lying singlet and triplet states of BN-n-acenes, n=1-7.

The chemical stability and the low-lying singlet and triplet excited states of BN-n-acenes (n=1-7) were studied using single reference and multireference methodologies. From the calculations, descriptors such as the singlet-triplet splitting, the natural orbital (NO) occupations and aromaticity indexes are used to provide structural and energetic analysis. The boron and nitrogen atoms form an isoelectronic pair of two carbon atoms, which was used for the complete substitution of these units in the acene series. The structural analysis confirms the effects originated from the insertion of a uniform pattern of electronegativity difference within the molecular systems. The covalent bonds tend to be strongly polarized which does not happen in the case of a carbon-only framework. This effect leads to a charge transfer between neighbor atoms resulting in a more strengthened structure, keeping the aromaticity roughly constant along the chain. The singlet-triplet splitting also agrees with this stability trend, maintaining a consistent gap value for all molecules. The BN-n-acenes molecules possess a ground state with monoconfigurational character indicating their electronic stability. The low-lying singlet excited states have charge transfer character, which proceeds from nitrogen to boron.

Explicit-by-Implicit Treatment of Natural Orbital Occupations Using First- and Second-Order Optimization Algorithms: A Comparative Study.

To address the convergence issues in the natural occupation optimization of reduced density matrix functional theory (RDMFT), we recently proposed the explicit-by-implicit (EBI) idea to handle the ensemble N-representability constraint (Yao et al. J. Phys. Chem. Lett. 2021, 12, 6788). This work continues to focus on these issues that can affect the reliability of the electronic structure description in RDMFT; further explores the combination of EBI, as well as the (augmented) Lagrangian methods (both LM and ALM), with both first- and second-order numerical optimization algorithms; and carefully evaluates their performances in natural occupation optimizations of various systems, including strongly correlated systems and large molecules. By comparing both converged energies and elapsed times, it can be seen that the LM and ALM have serious convergence issues for systems of different sizes. In contrast, the optimizations of EBI can converge to better energies with fewer iterations. However, due to the local convergence nature of the Newton's Method (NM) algorithm, EBI@NM still suffers from the local minimum issue for both strongly correlated systems and large molecules. Overall, the combination of EBI with the simple first-order algorithm of gradient descent (GD), namely EBI@GD, consistently provides the lowest converged energies for different types of systems, with the lowest computational scaling. These tests demonstrate the advantages of EBI in the calculations of transition states, strongly correlated systems, and large molecules. Meanwhile, the insights gained from this work are helpful to further develop more efficient algorithms for RDMFT.

An ab initio study on the stability of isolated borata-alkene synthons

• Isolated (H 2 C = BH 2 ) – synthon is a thermodynamically stable system. • (H 2 C = B R 2 ) – synthons ( R = CH 3 , C 6 H 5 , C 6 F 5 , Mes) are also stable as isolated species. • The presence of C = B double bond is confirmed in all borata-alkene systems studied. • Excess electron binding energies of (H 2 C = B R 2 ) – synthons span the 1.70–3.29 eV range. • Na + (H 2 C = B R 2 ) – salts preserve the structure and bonding pattern of embedded synthons. Using ab initio electronic structure methods with flexible basis sets, we examined the electronic, geometric, and thermodynamic stability of selected borata-alkene synthons (H 2 C = B R 2 ) – ( R = H, CH 3 , C 6 H 5 , C 6 F 5 , Mes). The isolated (H 2 C = BH 2 ) – synthon was found to be electronically stable and susceptible to neither fragmentation nor isomerization processes. Geometric and electronic stabilities of the remaining (H 2 C = B R 2 ) – anions were confirmed likewise. The structures of borata-alkene synthons studied contain planar H 2 C = B < fragment with the apparent double-bond character of the C–B connection confirmed by natural bond orbital occupancies of σ(C–B) and π(C–B) bonds approaching 2 e and by the shapes and spatial localizations of the corresponding NBOs. Vertical electron detachment energies (1.70–3.29 eV) of the (H 2 C = B R 2 ) – synthons were predicted and compared to those of the representative alkenyl carbanions. The stability and structures of sodium salts comprising the (H 2 C = B R 2 ) – unit and their ability to preserve the basic properties of the borata-alkene synthons were determined and discussed.

Prethermalization and entanglement dynamics in interacting topological pumps

We investigate the formation of quasisteady states in one-dimensional pumps of interacting fermions at non-integer filling fraction, in the regime where the driving frequency and interaction strength are small compared to the instantaneous single-particle band gap throughout the driving cycle. The system rapidly absorbs energy from the driving field, and approaches a quasisteady state that locally resembles a maximal entropy state subject to the constraint of fixed particle number in each of the system's single-particle Floquet bands. We explore the nature of this quasisteady state through one-body observables including the pumped current and natural orbital occupations, as well as the (many-body) entanglement spectrum and entropy. Potential disorder significantly reduces the amplitude of fluctuations of the quasisteady state current around its universal value, while the lifetime of the quasisteady state remains nearly unaffected for disorder strengths up to the scale of the single-particle band gap. Interestingly, the natural orbital occupations and entanglement entropy display patterns signifying the periodic entangling and disentangling of the system's degrees of freedom over each driving cycle. Moreover, prominent features in the system's time-dependent entanglement spectrum reveal the emergence of new long timescales associated with the equilibration of many-particle correlations.

Resolving correlated states of benzyne with an error-mitigated contracted quantum eigensolver

The simulation of strongly correlated many-electron systems is one of the most promising applications for near-term quantum devices. Here we use a class of eigenvalue solvers (presented in Phys. Rev. Lett. 126, 070504 (2021)) in which a contraction of the Schr\"odinger equation is solved for the two-electron reduced density matrix (2-RDM) to resolve the energy splittings of ortho-, meta-, and para-isomers of benzyne ${\textrm C_6} {\textrm H_4}$. In contrast to the traditional variational quantum eigensolver, the contracted quantum eigensolver solves an integration (or contraction) of the many-electron Schr\"odinger equation onto the two-electron space. The quantum solution of the anti-Hermitian part of the contracted Schr\"odinger equation (qACSE) provides a scalable approach with variational parameters that has its foundations in 2-RDM theory. Experimentally, a variety of error mitigation strategies enable the calculation, including a linear shift in the 2-RDM targeting the iterative nature of the algorithm as well as a projection of the 2-RDM onto the convex set of approximately $N$-representable 2-RDMs defined by the 2-positive (DQG) $N$-representability conditions. The relative energies exhibit single-digit millihartree errors, capturing a large part of the electron correlation energy, and the computed natural orbital occupations reflect the significant differences in the electron correlation of the isomers.

Quantum dynamics of few dipolar bosons in a double-well potential

We study the few-body dynamics of dipolar bosons in one-dimensional double-wells. Increasing the interaction strength, by investigating one-body observables, we study in the considered few-body systems tunneling oscillations, self-trapping and the regime exhibting an equilibrating behaviour. The corresponding two-body correlation dynamics exhibits a strong interplay between the interatomic correlation due to non-local nature of the repulsion and the inter-well coherence. We also study the link between the correlation dynamics and the occupation of natural orbitals of the one-body density matrix.

Self-Consistent-Field Method for Correlated Many-Electron Systems with an Entropic Cumulant Energy

A self-consistent field method is presented within density matrix functional theory. The computational cost for a correlated many-electron calculation is reduced to that of the self-consistent-field Hartree-Fock method, while the accuracy still reaches that of sophisticated configuration interaction based methods. In this method, the two-electron cumulant energy is measured with an information entropy associated with the Fermi-Dirac distribution of the occupation numbers. An eigenvalue equation for the orbitals is obtained, with the eigenvalues (orbital energies) connected to the occupation numbers through the Fermi-Dirac distribution. The occupation numbers for the strongly occupied orbitals are very close to the natural orbital occupation numbers from wave function methods. It covers in a single scheme the nondynamical correlation in weak or breaking bonds as well as the dynamical correlation at all distances. The method is well suited to large-scale potential energy surface calculation and molecular dynamics simulation.

The thermal isomerization of benzvalyne to benzyne.

The isomerization of the highly strained benzvalyne structure to o-benzyne has been investigated using MCSCF and CCSD(T) levels of theory. Two reaction channels were modeled: the disrotatory one which leads directly to the benzyne product, and the conrotatory one which leads to an intermediate which can subsequently lead to the benzyne product. Energies at the MRMP2 level give 22.9 kcal mol-1 for the disrotatory channel and 21.7 and 1.4 kcal mol-1 for the two steps in the conrotatory one. However, the CCSD(T) energies give 19.3 and 14.2 kcal mol-1 for the two conrotatory steps. The first step of the conrotatory channel is significantly higher than the second so is rate determining for this channel. Comparison of the two separate channels shows that the conrotatory has just a slight energetic edge of 1.2 kcal mol-1 at the MRMP2 level. We did not compute the disrotatory channel at the CCSD(T) level due to the significant biradical nature of the wavefunction with natural orbital occupation numbers of 1.2 and 0.8 in the active space.

A Valence-Bond-Based Multiconfigurational Density Functional Theory: The λ-DFVB Method Revisited.

A recently developed valence-bond-based multireference density functional theory, named λ-DFVB, is revisited in this paper. λ-DFVB remedies the double-counting error of electron correlation by decomposing the electron–electron interactions into the wave function term and density functional term with a variable parameter λ. The λ value is defined as a function of the free valence index in our previous scheme, denoted as λ-DFVB(K) in this paper. Here we revisit the λ-DFVB method and present a new scheme based on natural orbital occupation numbers (NOONs) for parameter λ, named λ-DFVB(IS), to simplify the process of λ-DFVB calculation. In λ-DFVB(IS), the parameter λ is defined as a function of NOONs, which are straightforwardly determined from the many-electron wave function of the molecule. Furthermore, λ-DFVB(IS) does not involve further self-consistent field calculation after performing the valence bond self-consistent field (VBSCF) calculation, and thus, the computational effort in λ-DFVB(IS) is approximately the same as the VBSCF method, greatly reduced from λ-DFVB(K). The performance of λ-DFVB(IS) was investigated on a broader range of molecular properties, including equilibrium bond lengths and dissociation energies, atomization energies, atomic excitation energies, and chemical reaction barriers. The computational results show that λ-DFVB(IS) is more robust without losing accuracy and comparable in accuracy to high-level multireference wave function methods, such as CASPT2.

Fragmentation of natural orbital occupation numbers-based diagnostic of differential multireference character in complexes with hydrogen bonds.

We explore the possible route to approximate natural orbital occupation numbers-based diagnostic of differential multireference character of noncovalent energy differences by techniques based on many-body expansion. It turns out that two-body fragmentation of monomers may lead to a reasonable approximation of such a diagnostic in hydrogen-bonded complexes. The results are useful, for example, for assessment of the expected bias cancellation in energy differences of larger systems obtained by single-reference methods.

Effect of adsorption sensitivity of armchair single-walled BN nanotube toward thiocyanate anion: A systematic evaluation of length and diameter effects

The first-principles calculations to investigate the adsorption behavior of thiocyanate anion (SCN−) on the external surface of H-capped (6, 0), (3, 3), and Al and Ga doped (3, 3) single-walled boron nitride nanotubes (SWBNNTs) has been performed. Binding energy and the equilibrium distance corresponding to the most stable state of SCN−/(3, 3) BNNT is found to be −1.747 eV and 1.546 A, respectively, which can represent mainly a polar covalent bond. The calculation results have shown that both doped aluminum and gallium can enhance the adsorption energy of SCN− into the (3, 3) BNNTs. For the SCN− /BNNT complexes, the energy gaps, NBO, dipole moments, natural atomic orbital occupancies and global indices have been computed using DFT theoretical method at B3LYP level and 6–31G** basis set. Finally, a novel type of the adsorbent that can be used to remove of SCN− anion has been reported.

Automatic Selection of Active Orbitals from Generalized Valence Bond Orbitals.

The accurate multireference (MR) calculation of a strongly correlated chemical system usually relies on a correct preselection of a small number of active orbitals from numerous molecular orbitals. Currently, the active orbitals are generally determined by using a trial-and-error method. Such a preselection by chemical intuition and personal experience may be tedious or unreliable, especially for large complicated systems, and accordingly, the construction of active space becomes a bottleneck for large-scale MR calculations. In this work, we propose to automatically select the active orbitals according to the natural orbital occupation numbers by performing black box generalized valence bond calculations. We demonstrate the accuracy of this method through testing calculations of the ground states in various systems, ranging from bond dissociation of diatomic molecules (N2, C2, Cr2) to conjugated molecules (pentacene, hexacene, and heptacene) as well as a binuclear transition-metal complex [Mn2O2(H2O)2(terpy)2]3+ (terpy = 2,2':6,2″-terpyridine) with active spaces up to (30e, 30o) and comparing with the complete active space self-consistent field (CASSCF), density matrix renormalization group (DMRG)-CASSCF references, and other recently proposed inexpensive strategies for constructing active spaces. The results indicate that our method is among the most successful ones within our comparison, providing high-quality initial active orbitals very close to the finally optimized (DMRG-)CASSCF orbitals. The method proposed here is expected to greatly benefit the practical implementation of large active space ground-state MR calculations, for example, large-scale DMRG calculations.

Elucidating the Relationship between Multiradical Character and Predicted Singlet Fission Activity

AbstractThe diradical and multiradical characters of a large and diverse set of 241 predicted singlet fission candidates are determined by multiconfigurational wavefunction methods, specifically natural orbital occupation number (NOON) analysis using the CASSCF method. The ascertained multiradical character strongly supports the current prediction that a good singlet fission candidate, independent of its structure, will tend to be of the weak biradicaloid class as well as having very little tetraradical character. Although the rule remains robust, it is not absolute as a small portion of our potential candidates are predicted to be closed shell. Additionally, a smaller handful of outliers with apparent tetraradical characters are identified, suggesting there may be the possibility, albeit rare, of singlet fission amongst weak tetraradicaloids.

Developing new and understanding old approximations in TDDFT.

When a system has evolved far from a ground-state, the adiabatic approximations commonly used in time-dependent density functional theory calculations completely fail in some applications, while giving qualitatively good predictions in others, and sometimes even quantitative predictions. It is not clearly understood why this is so, and developing practical approximations going beyond the adiabatic approximation remains a challenge. This paper explores three different lines of investigation. First, an expression for the exact time-dependent exchange-correlation potential suggests that the accuracy of an adiabatic approximation is intimately related to the deviation between the natural orbital occupation numbers of the physical system and those of the Kohn-Sham system, and we explore this on some exactly-solvable model systems. The exact expression further suggests a path to go beyond the adiabatic approximations, and in the second part we discuss a newly proposed class of memory-dependent approximations developed in this way. Finally, we derive a new expression for the exact exchange-correlation potential from a coupling-constant path integration.

The ANO-R Basis Set.

In this work, the new ANO-R basis set for all elements of the first six periods is introduced. The ANO-R basis set is an all-electron basis set that was constructed including scalar-relativistic effects of the exact-two component (X2C) Hamiltonian and modeling the atomic nucleus by a Gaussian charge distribution, which makes the basis set suitable for calculations of both light and heavy elements. For high accuracy, it takes advantage of the general contraction scheme and was developed at the CASSCF/CASPT2 level of theory. The distinguishing feature of the ANO-R basis set is its compactness in terms of both primitive and contracted basis functions, thus containing no superfluous functions for a given quality. An optimum number of primitive basis functions was selected based on studying the convergence toward the complete basis set limit for each element individually. The primitive basis sets were then contracted using the density-averaged atomic-natural-orbital (ANO) scheme, and suitable contraction levels were determined solely based on the natural orbital occupation numbers that describe the contribution of each natural orbital to the one-particle density matrix. Rather than following the common "split-valence n-tuple zeta plus polarization functions" structure, the resulting basis sets ANO-R0 to ANO-R3 possess a unique composition for each element, ensuring that no unnecessary functions are included while the basis sets are still balanced across the first six periods (H-Rn).

Sparse non-orthogonal wave function expansions from the extension of the generalized Pauli constraints to the two-electron reduced density matrix.

Generalized Pauli constraints (GPCs) impose constraints in the form of inequalities on the natural orbital occupation numbers of the one electron reduced density matrix (1-RDM), defining the set of pure N-representable 1-RDMs, or 1-RDMs that can be derived from an N-electron wave function. Saturation of these constraints is termed "pinning" and implies a significant simplification of the N-electron wave function as the number of Slater determinants required to fully describe the system is reduced. Recent research has shown pinning to occur for the ground states of atoms and molecules with N = 3 and r = 6, where N is the number of electrons and r is the number of spin orbitals. For N = 4 and r = 8, however, pinning occurs not to the GPCs but rather to inequalities defining the pure N-representable two-electron reduced density matrices (2-RDMs). Using these more general inequalities, we derive a wave function ansatz for a system with four electrons in eight spin orbitals. We apply the ansatz to the isoelectronic series of the carbon atom and the dissociation of linear H4 where the correlation energies are recovered to fractions of a kcal/mol. These results provide a foundation for further developments in wave function and RDM theories based on "pinned" solutions, and elucidate a fundamental physical basis for the emergence of non-orthogonal bases in electronic systems of N ≥ 4.

Curve Effect on Singlet Diradical Contribution in Kekulé-type Diradicals: A Sensitive Probe for Quinoidal Structure in Curved π-Conjugated Molecules.

Curved (non-planar) aromatic compounds have attracted significant research attention in the fields of basic chemistry and materials science. The contribution of the quinoidal structure in the curved π-conjugated structures has been proposed to be the key for materials functions. In this study, the curve effect on the quinoidal contribution was investigated in Kekulé-type singlet diradicals (S-DR1-4) as a sensitive probe for quinoidal structures in curved π-conjugated molecules. The quinoidal contribution in S-DR1-4 was found to increase with increasing the curvature of the curved structure, which was quantitatively analyzed using NBO analysis and the natural orbital occupation numbers computed by the CASSCF method. The curve effect on the singlet-triplet energy gap was examined by the CASPT2 method. The singlet-triplet energy gaps for the highly π-conjugated diradicals were determined for the first time using the CASPT2 method. Substantial quinoidal contribution was found in the curved structures of the delocalized singlet diradicals S-DR1-4, in contrast to its absence in the corresponding triplet states T-DR1-4.