Ground State Potential Energy Curves Research Articles

Computational Analysis of Chemical Reactions Using a Variational Quantum Eigensolver Algorithm without Specifying Spin Multiplicity.

The analysis of a chemical reaction along the ground-state potential energy surface in conjunction with an unknown spin state is challenging because electronic states must be separately computed several times using different spin multiplicities to find the lowest energy state. However, in principle, the ground state could be obtained with just a single calculation using a quantum computer without specifying the spin multiplicity in advance. In the present work, ground-state potential energy curves for PtCO were calculated as a proof-of-concept using a variational quantum eigensolver (VQE) algorithm. This system exhibits a singlet-triplet crossover as a consequence of the interaction between Pt and CO. VQE calculations using a statevector simulator were found to converge to a singlet state in the bonding region, while a triplet state was obtained at the dissociation limit. Calculations performed using an actual quantum device provided potential energies within ±2 kcal/mol of the simulated energies after error mitigation techniques were adopted. The spin multiplicities in the bonding and dissociation regions could be clearly distinguished even in the case of a small number of shots. The results of this study suggest that quantum computing can be a powerful tool for the analysis of the chemical reactions of systems for which the spin multiplicity of the ground state and variations in this parameter are not known in advance.

Efficient Construction of Involutory Linear Combinations of Anticommuting Pauli Generators for Large-Scale Iterative Qubit Coupled Cluster Calculations.

We present an efficient method for construction of a fully anticommutative set of Pauli generators (elements of the Pauli group) from a commutative set of operators that are composed exclusively from Pauli x̂ operators (purely X generators) and sorted by an associated numerical measure, such as absolute energy gradients. Our approach uses the Gauss-Jordan elimination applied to a binary matrix that encodes the set of X generators to bring it to the reduced row-echelon form, followed by the construction of an anticommutative system in a standard basis by means of a modified Jordan-Wigner transformation and returning to the original basis. The algorithm complexity is linear in the size of the X set and quadratic in the number of qubits. The resulting anticommutative sets are used to construct the qubit coupled cluster Ansatz with involutory linear combinations of anticommuting Paulis (QCC-ILCAP) proposed in J. Chem. Theory Comput. 2021, 17 (1), 66-78. We applied the iterative qubit coupled cluster method with the QCC-ILCAP Ansatz to calculations of ground-state potential energy curves for symmetric stretching of the water molecule (36 qubits) and dissociation of N2 (56 qubits).

A Study of NbMo and NbMo- by Anion Photoelectron Spectroscopy.

Photoelectron spectra of the niobium-molybdenum diatomic anion, obtained at 488 and 514 nm, display vibrationally resolved transitions from the ground state and one excited electronic state of the anion to the ground state and one excited electronic state of the neutral molecule. The electron affinity of NbMo is measured to be 1.130 ± 0.005 eV. Its 2Δ3/2 spin-orbit component is observed to lie 870 ± 20 cm-1 above its previously identified 2Δ5/2 ground state. For 93Nb98Mo, vibrational energies measured for levels up to v = 4 for the 2Δ5/2 and 2Δ3/2 states give harmonic frequency and anharmonicity constant values of ωe = 492 ± 12 cm-1 and ωexe = 8.0 ± 3.2 cm-1, the former value corresponding to a force constant of 6.80 ± 0.35 mdyn/Å. These two vibrational parameters suggest a bond dissociation energy that is too low by at least a factor of 3, indicating that the ground state potential energy curve of NbMo deviates markedly from a Morse potential at higher energies. An excited electronic state of NbMo, assigned as a 2Σ+ state, is observed at 2900 ± 25 cm-1 (T0). Vibrational energies up to v = 8 in this excited state give values of ωe = 544 ± 8 cm-1 and ωexe = 1.9 ± 1.2 cm-1 for 93Nb98Mo. The former value corresponds to a high vibrational force constant of 8.30 ± 0.25 mdyn/Å. Both doublet states of the neutral molecule are accessed from the anion ground state, which is assigned as 1Σ+. For the 93Nb98Mo- anion, the fundamental vibrational frequency (ΔG1/2) is 484 ± 15 cm-1. Electron affinity data indicate that the bond dissociation energy of NbMo- is 0.213 ± 0.005 eV greater than that of neutral NbMo, whose previously reported value then gives D0 = 4.85 ± 0.27 eV for the anion. An excited state of the anion lying 3050 ± 25 cm-1 (T0) above its ground state is assigned as 3Δ, and the energies of its spin-orbit components above the 3Δ3 lowest energy level are measured to be 450 ± 20 cm-1 (3Δ2) and 1100 ± 20 cm-1 (3Δ1). Their uneven spacing suggests that the energy of the 3Δ2 level is lowered by interaction with a higher energy Ω = 2 anion state. The vibrational frequency (ΔG1/2) for the 3Δ1 and 3Δ2 states is measured to be 433 ± 20 cm-1. Bond length differences among the observed states are estimated from Franck-Condon fits to vibrational band intensity profiles. When combined with the previously reported NbMo bond length, these provide bond length estimates for the ground state of the anion (1.940 ± 0.025 Å) and for the observed excited states. These species offer extreme examples of multiple metal-metal bonding, with formal bond orders of 51/2 for the 2Δ ground and 2Σ+ excited doublet states of NbMo, 6 for the singlet ground state of the anion, and 5 for its low-lying triplet state. The relationships among their bonding properties and those of related multiply bonded transition metal dimers are discussed.

Spontaneous electron emission vs dissociation in internally hot silver dimer anions.

Referring to a recent experiment, we theoretically study the process of a two-channel decay of the diatomic silver anion (Ag2 -), namely, the spontaneous electron ejection giving Ag2 + e- and the dissociation leading to Ag- + Ag. The ground state potential energy curves of the silver molecules of diatomic neutral and negative ions were calculated using proper pseudo-potentials and atomic basis sets. We also estimated the non-adiabatic electronic coupling between the ground state of Ag2 - and the ground state of Ag2 + e-, which, in turn, allowed us to estimate the minimal and mean values of the electron autodetachment lifetimes. The relative energies of the rovibrational levels allow the description of the spontaneous electron emission process, while the description of the rotational dissociation is treated with the quantum dynamics method as well as time-independent methods. The results of our calculations are verified by comparison with the experimental data.

Hybrid Quantum-Classical Neural Network for Calculating Ground State Energies of Molecules.

We present a hybrid quantum-classical neural network that can be trained to perform electronic structure calculation and generate potential energy curves of simple molecules. The method is based on the combination of parameterized quantum circuits and measurements. With unsupervised training, the neural network can generate electronic potential energy curves based on training at certain bond lengths. To demonstrate the power of the proposed new method, we present the results of using the quantum-classical hybrid neural network to calculate ground state potential energy curves of simple molecules such as H, LiH, and BeH. The results are very accurate and the approach could potentially be used to generate complex molecular potential energy surfaces.

Parametrisation of the optimised effective potential method based on the Coulson–Fischer wave function for excited states

ABSTRACT This article further develops the Coulson–Fischer formalism in conjunction with the optimised effective potential method [V.N. Glushkov and S. Wilson, Molec. Phys. 110, 149 (2012)] to incorporate both exchange and static correlation effects for excited states having the same symmetry as the ground state. By using this formalism, we compute complete potential energy curves for the ground and first excited states of the molecule and the isoelectronic heteronuclear ion. Effective potentials are generated in according to a novel density functional theory [A.K. Theophilou, J. Chem. Phys. 149, 074104 (2018)] in which the Kohn–Sham (KS) like potential has spherical symmetry around each nucleus. In this way, the inconsistencies of standard density functional theory concerning the asymmetry of the KS potential are remedied. In our implementation, an effective potential which satisfies this requirement is expressed as a direct mapping of the external potential. Different approximations for the effective potentials are investigated. It is shown that applications of the proposed formalism support good agreement with the exact Coulson–Fischer theory for both the ground and first excited state potential energy curves and thus a qualitatively correct description of dissociation.

Is excited state intramolecular proton transfer frustrated in 10-hydroxy-11H-benzo[b]fluoren-11-one?

Recently, Piechowska and coworker found that hydroxybenzofluorenone 10-hydroxy11H-benzo[b]fluoren-11-one (10-HHBF) does not show dual fluorescence, which is in contrast to its well-known analogue 1-hydroxy-11H-benzo[b]fluoren-11-one (1-HHBF) [Dyes Pigm. 2019, 165, 346-353.]. Based on the increased donor-acceptor distance and the lower stability of the excited state tautomer of former, they believe that different from 1-HHBF, ESIPT is not occurring in 10-HHBF. In the preset work, in order to clarify whether ESIPT would take place in 10-HHBF, we have optimized the four-state geometrical structures (ground state S0, first singlet excited state S1, transition state S1-TS and after proton transfer S1-PT), carried out the Natural Population Analysis and scanned the ground-state and excited-state potential energy curves of 1-HHBF and 10-HHBF at TD-CAM-B3LYP/6-311+g(2d,2p)/IEFPCM (cyclohexane) theory level. It is found that ESIPT should take place in both 1-HHBF and 10-HHBF and the Gibbs free energy diagram further indicates that the ESIPT process is more favorable in 10-HHBF than in 1-HHBF.

All-electron calculation of ground state vibration-rotation energy levels of <sup>7</sup>Li<sub>2</sub><sup>(0, ±1)</sup> molecular systems

The investigation of spectroscopic information is important for understanding the mechanisms of molecular photochemical and photophysical reactions. As a prototype to study the electronic structures and spectra of diatomic molecular systems, the vibration-rotational spectra of alkali dimer and its ions have aroused considerable research interest in the last two decades. Single-reference and multi-reference coupled cluster theory in combination with correlation consistent Gaussian basis set are adopted to study the ground-state potential energy curves of <sup>7</sup>Li<sub>2</sub><sup>(0,± 1)</sup> molecular systems. The correlation effect and relativistic effect of all the electrons are taken into account in the calculation. And the spectroscopic constants, including the equilibrium internuclear distance <i>R</i><sub>e</sub>, the harmonic vibrational frequency <i>ω</i><sub>e</sub>, the anharmonic constant <i>ω</i><sub>e</sub><i>x</i><sub>e</sub>, the equilibrium rotational constant <i>B</i><sub>e</sub>, and the dissociation energy <i>D</i><sub>e</sub> of the molecular system and vibration-rotational energy level information of the ground states are obtained by solving the radial Schrödinger equations. The calculated spectroscopic constants of the neutral and positive ion system accord well with the experimental values; however for the negative ion system, the calculation of equilibrium internuclear distance needs further improving, and other spectroscopic constants are consistent well with the experimental values. The present computational results indicate that the ground state wave functions of neutral and positive ion systems have obvious single reference configuration characteristics, while the ground state of negative ion molecule system should be described with multireference configuration wave functions. The vibration-rotational energy levels of ground state with different theoretical methods are in good agreement with the experimental values. The vibrational-rotational energy levels and spectroscopic constants of neutral and positive ion systems are well reproduced, and some experimental information about spectrum is still lacking. Although the difference among the equilibrium internuclear distances for the ground state of the negative ion, obtained from different theoretical methods are still existent, the results of the vibrational level interval accord well with each other. This study provides useful information about spectrum for accurately investigating the electronic structures and spectra of the ground state of Li<sub>2</sub> molecular system and its two isotopic molecules, especially for the negative ion system with little information about spectrum.

Qubit Coupled Cluster Method: A Systematic Approach to Quantum Chemistry on a Quantum Computer.

A unitary coupled cluster (UCC) form for the wave function in the variational quantum eigensolver has been suggested as a systematic way to go beyond the mean-field approximation and include electron correlation in solving quantum chemistry problems on a quantum computer. Although being exact in the limit of including all possible coupled cluster excitations, practically, the accuracy of this approach depends on the number and type of terms are included in the wave function parametrization. Another difficulty of UCC is a growth of the number of simultaneously entangled qubits even at the fixed Fermionic excitation rank. Not all quantum computing architectures can cope with this growth. To address both problems, we introduce a qubit coupled cluster (QCC) method that starts directly in the qubit space and uses energy response estimates for ranking the importance of individual entanglers for the variational energy minimization. Also, we provide an exact factorization of a unitary rotation of more than two qubits to a product of two-qubit unitary rotations. Thus, the QCC method with the factorization technique can be limited to only two-qubit entanglement gates and allows for very efficient use of quantum resources in terms of the number of coupled cluster operators. The method performance is illustrated by calculating ground-state potential energy curves of H2 and LiH molecules with chemical accuracy, ≤1 kcal/mol, and a symmetric water dissociation curve.

A Jeziorski-Monkhorst fully uncontracted multi-reference perturbative treatment. I. Principles, second-order versions, and tests on ground state potential energy curves.

The present paper introduces a new multi-reference perturbation approach developed at second order, based on a Jeziorski-Mokhorst expansion using individual Slater determinants as perturbers. Thanks to this choice of perturbers, an effective Hamiltonian may be built, allowing for the dressing of the Hamiltonian matrix within the reference space, assumed here to be a CAS-CI. Such a formulation accounts then for the coupling between the static and dynamic correlation effects. With our new definition of zeroth-order energies, these two approaches are strictly size-extensive provided that local orbitals are used, as numerically illustrated here and formally demonstrated in the Appendix. Also, the present formalism allows for the factorization of all double excitation operators, just as in internally contracted approaches, strongly reducing the computational cost of these two approaches with respect to other determinant-based perturbation theories. The accuracy of these methods has been investigated on ground-state potential curves up to full dissociation limits for a set of six molecules involving single, double, and triple bond breaking together with an excited state calculation. The spectroscopic constants obtained with the present methods are found to be in very good agreement with the full configuration interaction results. As the present formalism does not use any parameter or numerically unstable operation, the curves obtained with the two methods are smooth all along the dissociation path.

Alternative definition of excitation amplitudes in multi-reference state-specific coupled cluster.

A central difficulty of state-specific Multi-Reference Coupled Cluster (MR-CC) in the multi-exponential Jeziorski-Monkhorst formalism concerns the definition of the amplitudes of the single and double excitation operators appearing in the exponential wave operators. If the reference space is a complete active space (CAS), the number of these amplitudes is larger than the number of singly and doubly excited determinants on which one may project the eigenequation, and one must impose additional conditions. The present work first defines a state-specific reference-independent operator T∼^m which acting on the CAS component of the wave function |Ψ0m⟩ maximizes the overlap between (1+T∼^m)|Ψ0m⟩ and the eigenvector of the CAS-SD (Singles and Doubles) Configuration Interaction (CI) matrix |ΨCAS-SDm⟩. This operator may be used to generate approximate coefficients of the triples and quadruples, and a dressing of the CAS-SD CI matrix, according to the intermediate Hamiltonian formalism. The process may be iterated to convergence. As a refinement towards a strict coupled cluster formalism, one may exploit reference-independent amplitudes provided by (1+T∼^m)|Ψ0m⟩ to define a reference-dependent operator T^m by fitting the eigenvector of the (dressed) CAS-SD CI matrix. The two variants, which are internally uncontracted, give rather similar results. The new MR-CC version has been tested on the ground state potential energy curves of 6 molecules (up to triple-bond breaking) and two excited states. The non-parallelism error with respect to the full-CI curves is of the order of 1 mEh.

Theoretical insight into the excited-state intramolecular proton transfer mechanisms of three amino-type hydrogen-bonding molecules

Excited-state intramolecular proton transfer (ESIPT) dynamics of the amino-type hydrogen-bonding compound 2-(2′-aminophenyl)benzothiazole (PBT-NH2) as well as its two derivatives 2-(5′-cyano-2′-aminophenyl)benzothiazole (CN-PBT-NH2) and 2-(5′-cyano-2′-tosylaminophenyl)benzothiazole (CN-PBT-NHTs) were studied by the time-dependent density functional theory (TD-DFT) approach with the B3LYP density functional, and their absorption and emission spectra were also explored at the same level of theory. A good agreement is observed between the theoretical simulations and experimental spectra, indicating that the present calculations are reasonably reliable. In addition, it is also found that the energy barriers of the first excited singlet state of the three targeted molecules along the ESIPT reaction are computed to be 0.38, 0.34 and 0.12eV, respectively, showing the trend of gradual decrease, which implies that the introduction of the electron-withdrawing cyano or tosyl group can facilitate the occurrence of the ESIPT reaction of these amino-type H-bonding systems. Following the ESIPT, both CN-PBT-NH2 and CN-PBT-NHTs dye molecules can undergo the cis-trans isomerization reactions in the ground-state and excited-state potential energy curves along the C2-C3 bond between benzothiazole and phenyl moieties, where the energy barriers of the trans-tautomer→cis-tautomer isomerizations in the ground states are calculated to be 0.83 and 0.34eV, respectively. According to our calculations, it is plausible that there may exist the long-lived trans-tautomer species in the ground states of CN-PBT-NH2 and CN-PBT-NHTs.

Metastable homonuclear diatomic trications [formula omitted] for elements X from the first three rows of the periodic table

Motivated by a systematic stability scan of the homonuclear diatomic systems with elements from hydrogen to argon, we investigated in detail the ensuing five candidates for metastable trications, viz., O23+, P23+, S23+, Cl23+, and Ar23+ by multireference configuration interaction computations. Each of these tricationic systems exhibits a barrier in its ground state potential energy curve leading to quasi-bound nuclear motion. The resulting lifetimes vary drastically; while S23+ and Cl23+ in their lowest vibrational states can be regarded as practically stable, the predicted lifetimes of O23+ and Ar23+ are too short to allow a mass spectroscopic detection of these systems by current experimental capabilities. In the case of Ar23+, the theoretically predicted existence of quasibound vibrational levels also depends on whether or not relativistic corrections are taken into account.

Second-Order Perturbation Theory for Generalized Active Space Self-Consistent-Field Wave Functions.

A multireference second-order perturbation theory approach based on the generalized active space self-consistent-field (GASSCF) wave function is presented. Compared with the complete active space (CAS) and restricted active space (RAS) wave functions, GAS wave functions are more flexible and can employ larger active spaces and/or different truncations of the configuration interaction expansion. With GASSCF, one can explore chemical systems that are not affordable with either CASSCF or RASSCF. Perturbation theory to second order on top of GAS wave functions (GASPT2) has been implemented to recover the remaining electron correlation. The method has been benchmarked by computing the chromium dimer ground-state potential energy curve. These calculations show that GASPT2 gives results similar to CASPT2 even with a configuration interaction expansion much smaller than the corresponding CAS expansion.

Algorithms for calculating mass-velocity and Darwin relativistic corrections with n-electron explicitly correlated Gaussians with shifted centers.

Algorithms for calculating the leading mass-velocity (MV) and Darwin (D) relativistic corrections are derived for electronic wave functions expanded in terms of n-electron explicitly correlated Gaussian functions with shifted centers and without pre-exponential angular factors. The algorithms are implemented and tested in calculations of MV and D corrections for several points on the ground-state potential energy curves of the H2 and LiH molecules. The algorithms are general and can be applied in calculations of systems with an arbitrary number of electrons.

The ground-state potential energy curve of the radium dimer from relativistic coupled cluster calculations.

The potential energy curve for the ground-state of radium dimer (Ra2) is provided by means of atomic and molecular relativistic coupled cluster calculations. The short-range part of this curve is defined by an equilibrium bond length of 5.324 Å, a dissociation energy of 897 cm(-1), and a harmonic vibrational frequency of 20.5 cm(-1). The asymptotic behavior at large interatomic distances is characterized by the van der Waals coefficients C6 = 5.090 × 10(3), C8 = 6.978 × 10(5), and C10 = 8.786 × 10(7) atomic units. The two regions are matched in an analytical potential to provide a convenient representation for use in further calculations, for instance, to model cold collisions between radium atoms. This might become relevant in future experiments on ultracold, optically trapped, radioactive radium atoms that are used to search for a permanent electric dipole moment.

Relativistic state-specific multireference perturbation theory incorporating improved virtual orbitals: Application to the ground state single-bond dissociation.

Using four-component (4c) relativistic spinors, we present a computationally economical relativistic ab initio method for molecular systems employing our recently proposed second-order state-specific multireference perturbation theory (SSMRPT) incorporating the improved virtual orbital-complete active space configuration interaction (IVO-CASCI) reference wavefunction. The resulting method, 4c-IVO-SSMRPT [calculate one state at a time] is tested in pilot calculations on the homonuclear dimers including Li(2), Na(2), K(2), Rb(2), F(2), Cl(2), and Br(2) through the computations of the ground state potential energy curves (PECs). As SSMRPT curbs intruder effects, 4c-IVO-SSMRPT is numerically stable. To our knowledge, the SSMRPT in the 4c relativistic framework has not been explored in the past. Selective spectroscopic constants that are closely related to the correct shape and accuracy of the energy surfaces have been extracted from the computed PECs. For the halogen molecules, a relativistic destabilization of the bond has been found. Relativistic and electron correlation effects need to be incorporated to get reliable estimates. Our results are in good accordance with reference theoretical and experimental data which manifests the computational accuracy and efficiency of the new 4c-IVO-SSMRPT method. The method opens for an improved description of MR systems containing heavy elements. The inexpensiveness of IVO-CASCI makes 4c-IVO-SSMRPT method promising for studies on large systems of heavy elements.

EOMCC over excited state Hartree-Fock solutions (ESHF-EOMCC): An efficient approach for the entire ground state potential energy curves of higher-order bonds

The equation-of-motion coupled cluster (EOMCC) method based on the excited state Hartree-Fock (ESHF) solutions is shown to be appropriate for computing the entire ground state potential energy curves of strongly correlated higher-order bonds. The new approach is best illustrated for the homolytic dissociation of higher-order bonds in molecules. The required multireference character of the true ground state wavefunction is introduced through the linear excitation operator of the EOMCC method. Even at the singles and doubles level of cluster excitation truncation, the nonparallelity error of the ground state potential energy curve from the ESHF based EOMCC method is small.

The van der Waals potentials of MgCa, MgSr, MgBa, CaSr, CaBa, and SrBa

Based on the facts that the potential energy curves of the homo-nuclear group 2 dimers (group IIA metal), except Be2, are conformal, and they can be described by the Tang–Toennies potential model, a set of simple combining rules are proposed for the parameters of the reduced potentials of the hetero-nuclear dimers. Together with the well-established combining rules of the range parameters of the exponential repulsion and the known dispersion coefficients, these rules enable us to determine the ground state potential energy curves of MgCa, MgSr, MgBa, CaSr, CaBa, and SrBa from those of Mg2, Ca2, Sr2, and Ba2. The determined potentials are comparable to some ab initio calculations and in excellent agreement with the experiment.

Does DFT-SAPT method provide spectroscopic accuracy?

Ground state potential energy curves for homonuclear and heteronuclear dimers consisting of noble gas atoms from He to Kr were calculated within the symmetry adapted perturbation theory based on the density functional theory (DFT-SAPT). These potentials together with spectroscopic data derived from them were compared to previous high-precision coupled cluster with singles and doubles including the connected triples theory calculations (or better if available) as well as to experimental data used as the benchmark. The impact of midbond functions on DFT-SAPT results was tested to study the convergence of the interaction energies. It was shown that, for most of the complexes, DFT-SAPT potential calculated at the complete basis set (CBS) limit is lower than the corresponding benchmark potential in the region near its minimum and hence, spectroscopic accuracy cannot be achieved. The influence of the residual term δ(HF) on the interaction energy was also studied. As a result, we have found that this term improves the agreement with the benchmark in the repulsive region for the dimers considered, but leads to even larger overestimation of potential depth De. Although the standard hybrid exchange-correlation (xc) functionals with asymptotic correction within the second order DFT-SAPT do not provide the spectroscopic accuracy at the CBS limit, it is possible to adjust empirically basis sets yielding highly accurate results.