Size-consistency and orbital-invariance issues revealed by VQE-UCCSD calculations with the FMO scheme.

The fragment molecular orbital (FMO) scheme is one of the popular fragmentation-based methods and has the potential advantage of making the circuit shallow for quantum chemical calculations on quantum computers. In this study, we used a GPU-accelerated quantum simulator (cuQuantum) to perform the electron correlation part of the FMO calculation as unitary coupled-cluster singles and doubles (UCCSD) with the variational quantum eigensolver (VQE) for hydrogen-bonded (FH) 3 $$ {}_3 $$ and (FH) 2 $$ {}_2 $$ -H 2 $$ {}_2 $$ O systems with the STO-3G basis set. VQE-UCCSD calculations were performed using both canonical and localized MO sets, and the results were examined from the point of view of size-consistency and orbital-invariance affected by the Trotter error. It was found that the use of localized MO leads to better results, especially for (FH) 2 $$ {}_2 $$ -H 2 $$ {}_2 $$ O. The GPU acceleration was substantial for the simulations with larger numbers of qubits, and was about a factor of 6.7-7.7 for 18 qubit systems.

Selective Isotope Labeling and LC-Photo-CIDNP Enable NMR Spectroscopy at Low-Nanomolar Concentration.

NMR spectroscopy is a powerful tool to investigate molecular structure and dynamics. The poor sensitivity of this technique, however, limits its ability to tackle questions requiring dilute samples. Low-concentration photochemically induced dynamic nuclear polarization (LC-photo-CIDNP) is an optically enhanced NMR technology capable of addressing the above challenge by increasing the detection limit of aromatic amino acids in solution up to 1000-fold, either in isolation or within proteins. Here, we show that the absence of NMR-active nuclei close to a magnetically active site of interest (e.g., the structurally diagnostic 1Hα-13Cα pair of amino acids) is expected to significantly increase LC-photo-CIDNP hyperpolarization. Then, we exploit the spin-diluted tryptophan isotopolog Trp-α-13C-ß,ß,2,4,5,6,7-d7 and take advantage of the above prediction to experimentally achieve a ca 4-fold enhancement in NMR sensitivity over regular LC-photo-CIDNP. This advance enables the rapid (within seconds) detection of 20 nM concentrations or the molecule of interest, corresponding to a remarkable 3 ng detection limit. Finally, the above Trp isotopolog is amenable to incorporation within proteins and is readily detectable at a 1 µM concentration in complex cell-like media, including Escherichia coli cell-free extracts.

Factors Threatening Central Visual Function of Patients with Advanced Glaucoma: A Prospective Longitudinal Observational Study.

PURPOSE: To identify risk factors for further deterioration of central visual function in advanced glaucoma eyes. DESIGN: Prospective, observational 5-year study. PARTICIPANTS: Advanced glaucoma patients with well-controlled intraocular pressure (IOP), mean deviation (MD) of the Humphrey Field Analyzer (HFA) 24-2 program ≤-20 dB and best-corrected visual acuity (BCVA) of 20/40. METHODS: The HFA 10-2 test and BCVA examination were performed every 6 months, and the HFA 24-2 test was performed every 12 months for 5 years. The Cox proportional hazards model was used to identify risk factors for deterioration of HFA 10-2 and 24-2 results and BCVA. MAIN OUTCOME MEASURES: Deterioration of HFA 10-2 results was defined by the presence of the same ≥3 points with negative total deviation slope ≤-1 dB/year at P < 0.01 on ≥3 consecutive tests, deterioration of HFA 24-2 results by an increase ≥2 in the Advanced Glaucoma Intervention Study score on ≥2 consecutive tests, and deterioration of BCVA by an increase of ≥0.2 logarithm of the minimum angle of resolution (logMAR) on ≥2 consecutive tests. RESULTS: A total of 175 eyes of 175 patients (mean age, 64.1 years; mean baseline IOP, 13.2 mmHg; mean BCVA, 0.02 logMAR; mean HFA 24-2 and 10-2 MD, -25.9 and -22.9 dB, respectively) were included. The probabilities of deterioration in HFA 10-2 and 24-2 results and BCVA were 0.269 ± 0.043 (standard error), 0.173 ± 0.031, and 0.194 ± 0.033, respectively, at 5 years. Lower BCVA at baseline (P = 0.012) was associated significantly with further deterioration of HFA 10-2 results. Better HFA 24-2 MD (P < 0.001) and use of systemic antihypertensive agents (P = 0.009) were associated significantly with further deterioration of HFA 24-2 results, and a greater ß-peripapillary atrophy area-to-disc area ratio (P < 0.001), use of systemic antihypertensive agents (P = 0.025), and lower BCVA (P = 0.042) were associated significantly with further deterioration of BCVA, respectively. CONCLUSIONS: In advanced glaucoma eyes with well-controlled IOP, BCVA, ß-peripapillary atrophy area-to-disc area ratio, and use of systemic antihypertensive agents were significant prognostic factors for further deterioration of central visual function.

Variational quantum eigensolver simulations with the multireference unitary coupled cluster ansatz: a case study of the C2v quasi-reaction pathway of beryllium insertion into a H2 molecule.

Variational quantum eigensolver (VQE)-based quantum chemical calculations have been extensively studied as a computational model using noisy intermediate-scale quantum devices. The VQE uses a parametrized quantum circuit defined through an "ansatz" to generate approximated wave functions, and the appropriate choice of an ansatz is the most important step. Because most chemistry problems focus on the energy difference between two electronic states or structures, calculating the total energies in different molecular structures with the same accuracy is essential to correctly understand chemistry and chemical processes. In this context, the development of ansatzes that are capable of describing electronic structures of strongly correlated systems accurately is an important task. Here we applied a conventional unitary coupled cluster (UCC) and a newly developed multireference unitary coupled cluster with partially generalized singles and doubles (MR-UCCpGSD) ansatzes to the quasi-reaction pathway of Be insertion into H2, LiH molecule under covalent bond dissociation, and a rectangular tetra-hydrogen cluster known as a P4 cluster; these are representative systems in which the static electron correlation effect is prominent. Our numerical simulations revealed that the UCCSD ansatz exhibits extremely slow convergence behaviour around the point where an avoided crossing occurs in the Be + H2 → BeH2 reaction pathway, resulting in a large discrepancy of the simulated VQE energy from the full-configuration interaction (full-CI) value. By contrast, the MR-UCCpGSD ansatz can give more reliable results with respect to total energy and the overlap with the full-CI solution, insisting the importance of multiconfigurational treatments in the calculations of strongly correlated systems. The MR-UCCpGSD ansatz allows us to compute the energy with the same accuracy regardless of the strength of multiconfigurational character, which is an essential property to discuss energy differences of various molecular systems.

Synthesis and Isolation of a Kekulé Hydrocarbon with a Triplet Ground State.

Design, synthesis, and isolation of a Kekulé hydrocarbon with a triplet ground state is described. Its triplet ground state was unambiguously confirmed by ESR experiments, and the structure and fundamental physical properties were also revealed. The key feature of the molecular design is the decrease in the bonding interaction in the singlet state by aromatic stabilization of benzene rings and the increase of the exchange interaction of unpaired electrons which are favorable for the triplet state. These results contribute to the development of hydrocarbon-based organic magnetic materials.

Aurophilic Interactions in Multi-Radical Species: Electronic-Spin and Redox Properties of Bis- and Tris-[(Nitronyl Nitroxide)-Gold(I)] Complexes with Phosphine-Ligand Scaffolds.

Multinuclear AuI complexes with two or three nitronyl nitroxide-2-ide radical anion and phosphine-ligand scaffolds, (NN-Au)2 -1 o, (NN-Au)2 -1 m, and (NN-Au)2 -1 p, have been synthesized to investigate the influence of AuI -AuI (aurophilic) interactions on the properties of multispin molecular systems. The desired complexes were successfully prepared in moderate yields in a one-pot synthesis from the corresponding phosphine ligand, AuI source, parent NN, and sodium hydroxide. Among the prepared complexes, (NN-Au)2 -1 o, in which an aurophilic interaction was clearly observed by crystal structure analysis, showed characteristic spin-spin interactions, electrochemical properties, and solvatochromic behavior. The results from theoretical calculations also suggested that the differences in properties between complex (NN-Au)2 -1 o and the other complexes are due to intramolecular aurophilic interactions.

Tin(II)-Nitrene Radical Complexes Formed by Electron Transfer from Redox-Active Ligand to Organic Azides and Their Reactivity in C(sp3)-H Activation.

A tin(II) complex coordinated by a sterically demanding o-phenylenediamido ligand is synthesized. The ligand is redox-active to reach a tin(II) complex with the diiminobenzosemiquinone radial anion in the oxidation by AgPF6. The tin(II) complex reacts with a series of nosylazides (x-NO2C6H4-SO2-N3; x = o, m, or p) at -30 °C to yield the corresponding nitrene radical bound tin(II) complexes. The nitrene radical complexes exhibit C(sp3)-H activation and amination reactivity.

Bayesian phase difference estimation: a general quantum algorithm for the direct calculation of energy gaps.

Quantum computers can perform full configuration interaction (full-CI) calculations by utilising the quantum phase estimation (QPE) algorithms including Bayesian phase estimation (BPE) and iterative quantum phase estimation (IQPE). In these quantum algorithms, the time evolution of wave functions for atoms and molecules is simulated conditionally with an ancillary qubit as the control, which make implementation to real quantum devices difficult. Also, most of the problems in chemistry discuss energy differences between two electronic states rather than total energies themselves, and thus direct calculations of energy gaps are promising for future applications of quantum computers to real chemistry problems. In the race of finding efficient quantum algorithms to solve quantum chemistry problems, we test a Bayesian phase difference estimation (BPDE) algorithm, which is a general algorithm to calculate the difference of two eigenphases of unitary operators in the several cases of the direct calculations of energy gaps between two electronic states on quantum computers, including vertical ionisation energies, singlet-triplet energy gaps, and vertical excitation energies. In the BPDE algorithm, state preparation is carried out conditionally on the ancillary qubit, and the time evolution of the wave functions in superposition of two electronic states are executed unconditionally. Based on our test, we conclude that BPDE is capable of computing the energy gap with an accuracy similar to BPE without controlled-time evolution simulations and with the smaller number of iterations in Bayesian optimisations.

A probabilistic spin annihilation method for quantum chemical calculations on quantum computers.

A probabilistic spin annihilation method based on the quantum phase estimation algorithm is presented for quantum chemical calculations on quantum computers. This approach can eliminate more than one spin component from the spin contaminated wave functions by single operation. Comparison with the spin annihilation operation on classical computers is given.

Magnetic Properties of π-Conjugated Hybrid Phenoxyl-Nitroxide Radicals with Extended π-Spin Delocalization.

A series of stable and genuinely organic open-shell systems, π-conjugated phenoxyl-nitroxide free radicals (hybrid phenoxyl-nitroxide radicals), have been synthesized and their magnetic properties in the crystalline state investigated, revealing their usefulness as new building blocks for molecular magnetic materials. The salient electronic structure of the hybrid phenoxyl-nitroxide radicals is extended π-spin delocalization from the nitroxide moiety, mediating the localization effect intrinsic to nitroxide radicals. Five representative hybrid radicals containing an aliphatic, aromatic, and heteroaromatic substituent in the side part of the compact hybrid radical centers were synthesized, and their molecular/crystal structures in the crystalline state were determined by X-ray diffraction analyses. CW X-band ESR, 1H-ENDOR spectroscopy, and DFT calculations for the hybrid radicals confirmed that an unpaired spin delocalizes over the whole molecular frame including the nonconjugated fragments, suggesting the possibility of tuning their electronic properties through substituent effects in the crystalline state. Significant influence of the phenoxyl moiety on the electronic structure was analyzed in terms of the g-tensor calculations. The SQUID magnetization measurements revealed that the nitroxides bearing alkyl or aromatic substituents behave as 3D Curie-Weiss paramagnets with weak antiferromagnetic (AFM) (Θ = -1 to -2.6 K) or ferromagnetic (FM) (Θ = +0.33 K) spin-spin exchange interactions. On the other hand, heteroaromatically substituted hybrid phenoxyl-nitroxide showed significant AFM interactions with J/kB = -25.6 K. The analysis of the bulk magnetic properties based on the crystallographic data and DFT calculations revealed competition between the intermolecular AFM and FM interactions which originate from the C-O(phenoxyl)···Me(nitroxide) or (N)O-C(arom) infinite 1D head-to-tail chains and the C(arom)-C(arom) head-over-tail dimers forming 3D networks in their crystal lattices.

Triplet Diradical-Cation Salts Consisting of the Phenothiazine Radical Cation and a Nitronyl Nitroxide.

The spin-spin and magnetic properties of two (nitronyl nitroxide)-(di-p-anisylamine-phenothiazine) diradical cation salts, (DAA-PTZ)+ -NN⋅MBr4 - (M=Ga, Fe), have been investigated. These diradical-cation species were prepared by the cross-coupling of iodophenothiazine DAA-PTZ-I with NN-AuPPh3 followed by oxidation with the thianthrenium radical cation (TA+ ⋅MBr4 - ). These salts were found to be highly stable under aerobic conditions. For the GaBr4 salt, large ferromagnetic intramolecular and small antiferromagnetic intermolecular interactions (J1 /kB =+320 K and J2 /kB =-2 K, respectively) were observed. The magnetic property of the Fe3+ salt was analyzed by using a six-spin model assuming identical intramolecular exchange interaction (J3 /kB =+320 K) and the other exchange interactions (J4 /kB =-7 K and J5 /kB =-4 K). A significant color change was observed in the UV/Vis/NIR absorption spectra upon electrochemical oxidation of the doublet DAA-PTZ-NN to the triplet (DAA-PTZ)+ -NN.

Quantum chemistry on quantum computers: quantum simulations of the time evolution of wave functions under the S2 operator and determination of the spin quantum number S.

Quantum computers have an enormous impact on quantum chemical calculations. Approaches to calculate the energies of atoms and molecules on quantum computers by utilizing quantum phase estimation (QPE) and the variational quantum eigensolver (VQE) have been well documented, and dozens of methodological improvements to decrease computational costs and to mitigate errors have been reported until recently. However, the possible methodological implementation of observables on quantum computers such as calculating the spin quantum numbers of arbitrary wave functions, which is a crucial issue in quantum chemistry, has been discussed less. Here, we propose a quantum circuit to simulate the time evolution of wave functions under an S2 operator, exp(-iS2t)|Ψ, and integrate it into the QPE circuit enabling us to determine the spin quantum number of the arbitrary wave functions. We demonstrate that the spin quantum numbers of up to three spins can be determined by only one qubit measurement in QPE.

Trityl-Aryl-Nitroxide-Based Genuinely g-Engineered Biradicals, As Studied by Dynamic Nuclear Polarization, Multifrequency ESR/ENDOR, Arbitrary Wave Generator Pulse Microwave Waveform Spectroscopy, and Quantum Chemical Calculations.

Trityl and nitroxide radicals are connected by π-topologically controlled aryl linkers, generating genuinely g-engineered biradicals. They serve as a typical model for biradicals in which the exchange (J) and hyperfine interactions compete with the g-difference electronic Zeeman interactions. The magnetic properties underlying the biradical spin Hamiltonian for solution, including J's, have been determined by multifrequency CW-ESR and 1H ENDOR spectroscopy and compared with those obtained by quantum chemical calculations. The experimental J values were in good agreement with the quantum chemical calculations. The g-engineered biradicals have been tested as a prototype for AWG (Arbitrary Wave Generator)-based spin manipulation techniques, which enable GRAPE (GRAdient Pulse Engineering) microwave control of spins in molecular magnetic resonance spectroscopy for use in molecular spin quantum computers, demonstrating efficient signal enhancement of specific weakened hyperfine signals. Dynamic nuclear polarization (DNP) effects of the biradicals for 400 MHz nuclear magnetic resonance signal enhancement have been examined, giving efficiency factors of 30 for 1H and 27.8 for 13C nuclei. The marked DNP results show the feasibility of these biradicals for hyperpolarization.

Reversible Solution π-Dimerization and Long Multicenter Bonding in a Stable Phenoxyl Radical.

Reversible solution π-dimerization is observed in the stable neutral phenoxyl radical 2,6-bis-(8-quinolylamino)-4-(tert-butyl)phenoxyl baqp and is spectroscopically characterized. This behavior, not previously observed for π-extended phenoxyl radicals, is relevant to the formation of long multicenter bonding in the π-dimer at low temperature akin to previously reported phenalenyl radicals. Our experimental data are supported in a quantitative manner by results from density functional theory (DFT) and ab initio molecular orbital theory calculations. Our theoretical results indicate that the solution dimer features strong bonding interactions between the two phenoxyl rings but that the stability of the dimer is also related to dispersion interactions between the flanking nearly parallel quinolyl rings.

Synthesis and Characterization of Dibenzo[a,f]pentalene: Harmonization of the Antiaromatic and Singlet Biradical Character.

Mesityl derivatives of the unknown dibenzopentalene isomer dibenzo[a,f]pentalene were synthesized. The molecular geometry and physical properties of dibenzo[a,f]pentalene were investigated. Dibenzo[a,f]pentalene combines a large antiaromatic and appreciable singlet open-shell character, properties not shared by well-known isomer dibenzo[a,e]pentalene.

Behaviour of DFT-based approaches to the spin-orbit term of zero-field splitting tensors: a case study of metallocomplexes, MIII(acac)3 (M = V, Cr, Mn, Fe and Mo).

Spin-orbit contributions to the zero-field splitting (ZFS) tensor (DSO tensor) of MIII(acac)3 complexes (M = V, Cr, Mn, Fe and Mo; acac = acetylacetonate anion) are evaluated by means of ab initio (a hybrid CASSCF/MRMP2) and DFT (Pederson-Khanna (PK) and natural orbital-based Pederson-Khanna (NOB-PK)) methods, focusing on the behaviour of DFT-based approaches to the DSO tensors against the valence d-electron configurations of the transition metal ions in octahedral coordination. Both the DFT-based approaches reproduce trends in the D tensors. Significantly, the differences between the theoretical and experimental D (D = DZZ - (DXX + DYY)/2) values are smaller in NOB-PK than in PK, emphasising the usefulness of the natural orbital-based approach to the D tensor calculations of transition metal ion complexes. In the case of d2 and d4 electronic configurations, the DSO(NOB-PK) values are considerably underestimated in the absolute magnitude, compared with the experimental ones. The DSO tensor analysis based on the orbital region partitioning technique (ORPT) revealed that the DSO contributions attributed to excitations from the singly occupied region (SOR) to the unoccupied region (UOR) are significantly underestimated in the DFT-based approaches to all the complexes under study. In the case of d3 and d5 configurations, the (SOR → UOR) excitations contribute in a nearly isotropic manner, which causes fortuitous error cancellations in the DFT-based DSO values. These results indicate that more efforts to develop DFT frameworks should be directed towards the reproduction of quantitative DSO tensors of transition metal complexes with various electronic configurations and local symmetries around metal ions.

Analyses of sizable ZFS and magnetic tensors of high spin metallocomplexes.

The fictitious spin-1/2 Hamiltonian approach is the putative method to analyze the fine-structure/hyperfine ESR spectra of high spin metallocomplexes having sizable zerofield splitting (ZFS), thus giving salient principal g-values far from around g = 2 without explicitly providing their ZFS parameters in most cases. Indeed, the significant departure of the g-values from g = 2 is indicative of the occurrence of their high spin states, but naturally they never agree with true g-values acquired by quantum chemical calculations such as sophisticated DFT or ab initio MO calculations. In this work, we propose facile approaches to determine the magnetic tensors of high spin metallocomplexes having sizable ZFS, instead of performing advanced high-field/high-frequency ESR spectroscopy. We have revisited analytical expressions for the relationship between effective g-values and true principal g-values for high spins. The useful analytical formulas for the geff-gtrue relationships are given for S's up to 7/2. The genuine Zeeman perturbation formalism gives the exact solutions for S = 3/2, and for higher S's it is much more accurate than the pseudo-Zeeman perturbation approach documented so far (A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of Transition Metal Ions, 1970; J. R. Pilbrow, J. Magn. Reson., 1978, 31, 479; F. Trandafir et al., Appl. Magn. Reson., 2007, 31, 553; M. Fittipaldi et al., J. Phys. Chem. B, 2008, 112, 3859), in which the E(Sx2 - Sy2) term is putatively treated to the second order. To show the usefulness of the present approach, we exploit FeIII(Cl)OEP (S = 5/2) (OEP: 2,3,7,8,12,13,17,18-octaethylporphyrin) and CoIIOEP (S = 3/2) well magnetically diluted in the diamagnetic host crystal lattice of NiIIOEP. The advantage of single-crystal ESR spectroscopy lies in the fact that the molecular information on the principal axes of the magnetic tensors is crucial in comparing with reliable theoretical results. In high spin states of metallocomplexes with sizable ZFS in pseudo-octahedral symmetry, their fine-structure ESR transitions for the principal z-axis orientation appear in the lower field far from g = 2 at the X-band, disagreeing with the putative intuitive picture obtained using relevant ESR spectroscopy. A ReIII,IV dinuclear complex in a mixed valence state exemplifies the cases, whose fine-structure/hyperfine ESR spectra of the neat crystals have been analyzed in their principal-axis system. The DFT-based/ab initio MO calculations of the magnetic tensors for all the high spin entities in this work were carried out.

Quasi-Restricted Orbital Treatment for the Density Functional Theory Calculations of the Spin-Orbit Term of Zero-Field Splitting Tensors.

A quasi-restricted orbital (QRO) approach for the calculation of the spin-orbit term of zero-field splitting tensors (DSO tensors) by means of density functional theory (DFT) importantly features in the fact that it is free from spin contamination problems because it uses spin eigenfunctions for the zeroth order wave functions. In 2011, however, Schmitt and co-workers pointed out that in the originally proposed QRO working equation some possible excitations were not included in their sum-over-states procedure, which causes spurious DSO contributions from closed-shell subsystems located far from the magnetic molecule under study. We have revisited the derivation of the QRO working equation and modified it, making it include all possible types of excitations in the sum-over-states procedure. We have found that the spurious DSO contribution can be eliminated by taking into account contributions from all possible types of singly excited configuration state functions. We have also found that only the SOMO(α) → SOMO(ß) excited configurations have nonzero contributions to the DSO tensors as long as α and ß spin orbitals have the same spatial distributions and orbital energies. For the DSO tensor calculations, by using a ground state wave function free from spin contamination, we propose a natural orbital-based Pederson-Khanna (NOB-PK) method, which utilizes the single determinant wave function consisting of natural orbitals in conjunction with the Pederson-Khanna (PK) type perturbation treatment. Some relevant calculations revealed that the NOB-PK method can afford more accurate DSO tensors than the conventional PK method as well as the QRO approach in MnII complexes and ReIV-based single molecule magnets.

Quantum Chemistry on Quantum Computers: A Polynomial-Time Quantum Algorithm for Constructing the Wave Functions of Open-Shell Molecules.

Quantum computers are capable to efficiently perform full configuration interaction (FCI) calculations of atoms and molecules by using the quantum phase estimation (QPE) algorithm. Because the success probability of the QPE depends on the overlap between approximate and exact wave functions, efficient methods to prepare accurate initial guess wave functions enough to have sufficiently large overlap with the exact ones are highly desired. Here, we propose a quantum algorithm to construct the wave function consisting of one configuration state function, which is suitable for the initial guess wave function in QPE-based FCI calculations of open-shell molecules, based on the addition theorem of angular momentum. The proposed quantum algorithm enables us to prepare the wave function consisting of an exponential number of Slater determinants only by a polynomial number of quantum operations.