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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 379 records · Page 21
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Neural network potential from bispectrum components: A case study on crystalline silicon

In this article, we present a systematic study on developing machine learning force fields (MLFFs) for crystalline silicon. While the main-stream approach of fitting a MLFF is to use a small and localized training set from molecular dynamics simulations, it is unlikely to cover the global features of the potential energy surface. Additionally, to remedy this issue, we used randomly generated symmetrical crystal structures to train a more general Si-MLFF. Furthermore, we performed substantial benchmarks among different choices of material descriptors and regression techniques on two different sets of silicon data. Our results show that neural network potential fitting with bispectrum coefficients as descriptors is a feasible method for obtaining accurate and transferable MLFFs.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Development of a Geothermal Module in reV: Quantifying the Geothermal Potential While Accounting for the Geospatial Intersection of the Grid Infrastructure and Land Use Characteristics: Preprint

The Renewable Energy Potential (reV) model is a geospatial platform for estimating technical potential and developing renewable energy supply curves, initially developed for wind and solar technologies. The model evaluates deployment constraints, considering land use, environmental, and cultural factors, and estimates the distance to existing grid features to connect future plants (Maclaurin et al., 2021). A pressing deficiency in the reV model, however, is representation of geothermal electricity generation technologies. To address this gap, we developed a novel geothermal generation module for reV that allows for representation and analysis at the same level of detail as other renewable technologies. This paper describes our process for evaluating data sources for the modeling, and presents five preliminary reV geothermal results. More specifically, we present two sets of resource data that represent upper and lower bounds for geothermal potential. We then present several sensitivity runs using the upper bound resource data; the results are encouraging that levelized cost of electricity (LCOE) can be reduced by optimizing the location and estimated capacity of the spatially diverse geothermal resource while considering the distance to existing grid infrastructure. Our preliminary supply curves and levelized cost of electricity (LCOE) results should be considered with care due to the highly uncertainty in geothermal resource potential data. We present median LCOE values for the conterminous U.S. for five scenarios: four hydrothermal (3.5km depth) and one EGS (4.5km depth). The capital and operating costs for each respective technology are modeled. We also compare results using two different resource data sources.

exclusions↗
DOE OSTI2020

Benchmarking Adaptive Variational Quantum Eigensolvers

By design, the variational quantum eigensolver (VQE) strives to recover the lowest-energy eigenvalue of a given Hamiltonian by preparing quantum states guided by the variational principle. In practice, the prepared quantum state is indirectly assessed by the value of the associated energy. Novel adaptive derivative-assembled pseudo-trotter (ADAPT) ansatz approaches and recent formal advances now establish a clear connection between the theory of quantum chemistry and the quantum state ansatz used to solve the electronic structure problem. Here we benchmark the accuracy of VQE and ADAPT-VQE to calculate the electronic ground states and potential energy curves for a few selected diatomic molecules, namely H 2 , NaH, and KH. Using numerical simulation, we find both methods provide good estimates of the energy and ground state, but only ADAPT-VQE proves to be robust to particularities in optimization methods. Another relevant finding is that gradient-based optimization is overall more economical and delivers superior performance than analogous simulations carried out with gradient-free optimizers. The results also identify small errors in the prepared state fidelity which show an increasing trend with molecular size.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Dynamical electron correlation and the chemical bond. II. Recoupled pair bonds in the $a$ 4 Σ - states of CH and CF

We extended our studies of the effect of dynamical electron correlation on the covalent bonds in the AH and AF series (A = B–F) to the recoupled pair bonds in the excited $a$ 4 Σ - states of the CH and CF molecules. Dynamical correlation is energetically less important in the $a$ 4 Σ - states than in the corresponding X 2 Π states for both molecules, which is reflected in smaller changes in bond energies ($D$ e ). Changes in the equilibrium bond distance ($R$ e ) and vibrational frequency ($ω$ e ), on the other hand, are influenced by the changes in the slope and curvature of the dynamical electron correlation energy as a function of the internuclear distance $R$, $E$ DEC ($R$). In the CH($a$ 4 Σ - ) state, these changes are much smaller than in the CH(X 2 Π) state, but in the CF($a$ 4 Σ - ) state, they are larger, reflecting a significant difference in the shapes of $E$ DEC ($R$) curves. At large $R$, the shape of $E$ DEC ($R$) curves for covalent and recoupled pair bonds is similar although different in magnitude. For the CH($a$ 4 Σ - ) state, $E$ DEC ($R$) has a minimum at $R$ = $R$ e + 0.72 Å as the orbitals associated with the formation of the recoupled pair bond switch places. $E$ DEC (R) for the CF($a$ 4 Σ - ) state decreases continuously throughout the bound region of the potential energy curve because the dynamical correlation energy associated with the electrons in the lone pair orbitals is increasing. In conclusion, these results support our earlier conclusion that we still have much to learn about the nature of dynamical electron correlation in molecules.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Hydrocarbon-based membranes cost-effectively manage species transport and increase performance in thermally regenerative batteries

Low-temperature heat (T<130°C) can be utilized by thermally regenerative batteries (TRBs) for power production, allowing the thermal energy to be converted to storable chemical potential energy. However, TRBs suffer from high ohmic losses and ammonia crossover, which has slowed their development. In this study, we examined how the use of six different membranes influenced TRB performance, determined the most influential membrane parameters, and identified promising membrane candidates that cost-effectively increase TRB performance. Of the six membranes examined, an inexpensive, hydrocarbon CEM (Selemion CMVN) had low ammonia crossover without compromising resistance, resulting in good performance across all metrics studied. A thin anion exchange membrane (Sustainion, 50 microns) showed a high peak power density of 82 mW cm –2 due to low resistance, but the average power density and energy density were low due to high ammonia flux. Full discharge curves using Selemion CMVN provided an average power density of 26 ± 7 mW cm –2 with an energy density of 2.9 Wh L –1 , which were large improvements on previous TRBs. A techno-economic analysis showed that Selemion CMVN had the lowest levelized cost of storage ($410 per MWh) at an applied current density of 50 mA cm –2 .

25 ENERGY STORAGE↗
DOE OSTI2023

2023 American Society for Mass Spectrometry (ASMS) 71st Annual Conference on Mass Spectrometry and Allied Topics

Title (20 words): The investigation of the mechanism for the water splitting by holmium oxide nitrate complex in gas-phase Introduction (120 words): The studies for hydrogen from water splitting is important for clean energy economy as molecular hydrogen is a potential energy carrier. Therefore, understanding the fundamental chemistry of water splitting is essential. The water splitting activated by [Ho(NO3)4]- was previously studied. The suggested mechanism was to first eliminate •NO2 from [Ho(NO3)4]- to make [HoO(NO3)3]-. Then, water was added to form [HoO(NO3)3(H2O)]-, followed by eliminating the •OH to form [HoOH(NO3)3]-. However, we proposed a new mechanism with the formation of [Ho(NO3)3]- prior to the water addition. Once water is added to this intermediate to make [Ho(NO3)3(H2O)]-, the •H is eliminated to form [HoOH(NO3)3]-. Here we have used a labeled experiments with 18-O nitrates. The results to support the new mechanism are discussed.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Nonlinear spectroscopy in the condensed phase: The role of Duschinsky rotations and third order cumulant contributions

First-principles modeling of nonlinear optical spectra in the condensed phase is highly challenging because both environment and vibronic interactions can play a large role in determining spectral shapes and excited state dynamics. In this work, we compute two dimensional electronic spectroscopy (2DES) signals based on a cumulant expansion of the energy gap fluctuation operator, with specific focus on analyzing mode mixing effects introduced by the Duschinsky rotation and the role of the third order term in the cumulant expansion for both model and realistic condensed phase systems. We show that for a harmonic model system, the third order cumulant correction captures effects introduced by a mismatch in curvatures of ground and excited state potential energy surfaces, as well as effects of mode mixing. We also demonstrate that 2DES signals can be accurately reconstructed from purely classical correlation functions using quantum correction factors. We then compute nonlinear optical spectra for the Nile red and methylene blue chromophores in solution, assessing the third order cumulant contribution for realistic systems. We show that the third order cumulant correction is strongly dependent on the treatment of the solvent environment, revealing the interplay between environmental polarization and the electronic-vibrational coupling.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Shell effects in quasi-fission for calcium induced reactions forming thorium isotopes

Quantum shell effects induce an asymmetric fission mode in actinides, which disappears in neutron deficient isotopes. Quasi-fission, characterized by a significant mass transfer in heavy ion collisions at low-energies, is expected to be affected by similar shell effects. This is studied in 40−56 Ca+176 reactions with the time-dependent Hartree-Fock approach. All reactions exhibit a mass equilibration process that stops when a heavy fragment with 𝑍 ≃ 54 protons is formed. Unlike the fission of thorium compound nuclei, quasi-fission does not exhibit a transition to symmetric modes in neutron deficient systems. This observation is interpreted in terms of potential energy surfaces that show a persistence of an asymmetric valley with an increasing barrier preventing its population in fission of the most neutron deficient thorium isotopes.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

ELECTRONIC STRUCTURE METHODS AND PROTOCOLS WITH APPLICATION TO DYNAMICS, KINETICS AND THERMOCHEMISTRY

Hydrocarbon combustion involves the reaction dynamics of a tremendous number of species beginning with many-component fuel mixtures and proceeding via a complex system of intermediates to form primary and secondary products. Combustion conditions corresponding to new advanced engines and/or alternative fuels rely increasingly on autoignition and low-temperature-combustion chemistry. In these regimes various transient radical species such as HO2, ROO·, ·QOOH, HCO, NO2, HOCO, and Criegee intermediates play important roles in determining the detailed as well as more general dynamics. A clear understanding and accurate representation of these processes is needed for effective modeling. Given the difficulties associated with making reliable experimental measurements of these systems, computation can play an important role in developing these energy technologies. Accurate calculations have their own challenges since even within the simplest dynamical approximations such as transition state theory, the rates depend exponentially on critical barrier heights and these may be sensitive to the level of quantum chemistry. Moreover, it is well-known that in many cases it is necessary to go beyond statistical theories and consider the dynamics. Quantum tunneling, resonances, radiative transitions, and non-adiabatic effects governed by spin-orbit or derivative coupling can be determining factors in those dynamics. Building upon progress made during a period of prior support through the DOE Early Career Program, this project combines developments in the areas of potential energy surface (PES) fitting and multistate multireference quantum chemistry to allow spectroscopically and dynamically/kinetically accurate investigations of key molecular systems (such as those mentioned above), many of which are radicals with strong multireference character and have the possibility of multiple electronic states contributing to the observed dynamics. An ongoing area of investigation is to develop general strategies for robustly convergent electronic structure theory for global multichannel reactive surfaces including diabatization of energy and other relevant surfaces such as dipole transition. Combining advances in ab initio methods with automated interpolative PES fitting allows the construction of high-quality PESs (incorporating thousands of high-level data) to be done rapidly through parallel processing on high-performance computing (HPC) clusters. In addition, new methods and approaches to electronic structure theory will be developed and tested through applications. This project will explore limitations in traditional multireference calculations (e.g., MRCI) such as those imposed by internal contraction, lack of high-order correlation treatment and poor scaling. Methods such as DMRG-based extended active-space CASSCF and various Quantum Monte Carlo (QMC) methods will be applied (including VMC/DMC and FCIQMC). Insight into the relative significance of different orbital spaces and the robustness of application of these approaches on leadership class computing architectures will be gained. Synergy with other components of this research program such as automated PES fitting and multireference quantum chemistry will be used to address challenges encountered by the standard approaches to computational thermochemistry (those being single-reference quantum chemistry and perturbative treatments of the anharmonic vibrational energy, which break down for some cases of electronic structure or floppy strongly coupled vibrational modes).

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2021

NENCI-2021. I. A large benchmark database of non-equilibrium non-covalent interactions emphasizing close intermolecular contacts

In this work, we present NENCI-2021, a benchmark database of ~8000 Non-Equilibirum Non-Covalent Interaction energies for a large and diverse selection of intermolecular complexes of biological and chemical relevance. To meet the growing demand for large and high-quality quantum mechanical data in the chemical sciences, NENCI-2021 starts with the 101 molecular dimers in the widely used S66 and S101 databases and extends the scope of these works by (i) including 40 cation–π and anion–π complexes, a fundamentally important class of non-covalent interactions that are found throughout nature and pose a substantial challenge to theory, and (ii) systematically sampling all 141 intermolecular potential energy surfaces (PESs) by simultaneously varying the intermolecular distance and intermolecular angle in each dimer. Designed with an emphasis on close contacts, the complexes in NENCI-2021 were generated by sampling seven intermolecular distances along each PES (ranging from 0.7× to 1.1× the equilibrium separation) and nine intermolecular angles per distance (five for each ion–π complex), yielding an extensive database of 7763 benchmark intermolecular interaction energies (E int ) obtained at the coupled-cluster with singles, doubles, and perturbative triples/complete basis set [CCSD(T)/CBS] level of theory. The E int values in NENCI-2021 span a total of 225.3 kcal/mol, ranging from -38.5 to +186.8 kcal/mol, with a mean (median) E int value of -1.06 kcal/mol (-2.39 kcal/mol). In addition, a wide range of intermolecular atom-pair distances are also present in NENCI-2021, where close intermolecular contacts involving atoms that are located within the so-called van der Waals envelope are prevalent—these interactions, in particular, pose an enormous challenge for molecular modeling and are observed in many important chemical and biological systems. A detailed symmetry-adapted perturbation theory (SAPT)- based energy decomposition analysis also confirms the diverse and comprehensive nature of the intermolecular binding motifs present in NENCI-2021, which now includes a significant number of primarily induction-bound dimers (e.g., cation–π complexes). NENCI-2021 thus spans all regions of the SAPT ternary diagram, thereby warranting a new four-category classification scheme that includes complexes primarily bound by electrostatics (3499), induction (700), dispersion (1372), or mixtures thereof (2192). A critical error analysis performed on a representative set of intermolecular complexes in NENCI-2021 demonstrates that the E int values provided herein have an average error of ±0.1 kcal/mol, even for complexes with strongly repulsive E int values, and maximum errors of ±0.2–0.3 kcal/mol (i.e., ~±1.0 kJ/mol) for the most challenging cases. For these reasons, we expect that NENCI-2021 will play an important role in the testing, training, and development of next-generation classical and polarizable force fields, density functional theory approximations, wavefunction theory methods, and machine learning based intra- and inter-molecular potentials.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Experimental and computational investigation of the bond energy of thorium dicarbonyl cation and theoretical elucidation of its isomerization mechanism to the thermodynamically most stable isomer, thorium oxide ketenylidene cation, OTh + CCO

Collision-induced dissociation (CID) of [Th,2C,2O] + with Xe is performed using a guided ion beam tandem mass spectrometer (GIBMS). The only products observed are ThCO + and Th + by sequential loss of CO ligands. The experimental findings and theoretical calculations support that the structure of [Th,2C,2O] + is the bent homoleptic thorium dicarbonyl cation, Th + (CO) 2 , having quartet spin, which is both thermodynamically and kinetically stable enough in the gas phase to be observed in our GIBMS instrument. Analysis of the kinetic energy-dependent cross sections for this CID reaction yields the first experimental determination of the bond dissociation energy (BDE) of (CO)Th + –CO at 0 K as 1.05 ± 0.09 eV. A theoretical BDE calculated at the CCSD(T) level with cc-pVXZ (X = T and Q) basis sets and a complete basis set (CBS) extrapolation is in very good agreement with the experimental result. Although the doublet spin bent thorium oxide ketenylidene cation, OTh + CCO, is calculated to be the most thermodynamically stable structure, it is not observed in our experiment where [Th,2C,2O] + is formed by association of Th + and CO in a direct current discharge flow tube (DC/FT) ion source. Potential energy profiles of both quartet and doublet spin are constructed to elucidate the isomerization mechanism of Th + (CO) 2 to OTh + CCO. The failure to observe OTh + CCO is attributed to a barrier associated with C–C bond formation, which makes OTh + CCO kinetically inaccessible under our experimental conditions. Furthermore, chemical bonding patterns in low-lying states of linear and bent Th + (CO) 2 and OTh + CCO isomers are also investigated.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2022

Ultrafast infrared transient absorption spectroscopy of gas-phase Ni(CO) 4 photodissociation at 261 nm

Here, we employ ultrafast mid-infrared transient absorption spectroscopy to probe the rapid loss of carbonyl ligands from gas-phase nickel tetracarbonyl following ultraviolet photoexcitation at 261 nm. Here, nickel tetracarbonyl undergoes prompt dissociation to produce nickel tricarbonyl in a singlet excited state; this electronically excited tricarbonyl loses another CO group over tens of picoseconds. Our results also suggest the presence of a parallel, concerted dissociation mechanism to produce nickel dicarbonyl in a triplet excited state, which likely dissociates to nickel monocarbonyl. Mechanisms for the formation of these photoproducts in multiple electronic excited states are theoretically predicted with one-dimensional cuts through the potential energy surfaces and computation of spin–orbit coupling constants using equation of motion coupled cluster methods (EOM-CC) and coupled cluster theory with single and double excitations (CCSD). Bond dissociation energies are calculated with CCSD, and anharmonic frequencies of ground and excited state species are computed using density functional theory (DFT) and time-dependent density functional theory (TD-DFT).

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Femtosecond dynamics and coherence of ionic retro-Diels–Alder reactions

Ultrafast tunnel ionization enables femtosecond time-resolved dynamic measurements of the retro-Diels–Alder reactions of positively charged cyclohexene, norbornene, and dicyclopentadiene. Unlike the reaction times of 500–600 ps that are observed following UV excitation of neutral species, on the ionic potential energy surfaces, these reactions occur on a single picosecond timescale and, in some cases, exhibit vibrational coherence. In the case of norbornene, a 270 cm –1 vibrational mode is found to modulate the retro-Diels–Alder reaction.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2026

Long-Range Fit: A Software Package for the Representation and Study of Long-Range Molecular Interactions

Describing intermolecular forces is fundamental to modeling and predicting the behavior of molecular systems. In particular, long-range molecular interactions─with electrostatic, induction, and dispersion as the main components─play a critical role, especially for low-temperature and low-density regimes. Long-range interactions are often described through perturbation theory, representing the electronic charge distribution via a multipolar series of the moments and polarizability tensors corresponding to each molecule. However, while the theory is well established, obtaining the resulting analytical expressions (and their practical implementation) constitutes a highly complex and system-dependent task. To address this challenge, we developed long-range-fit (LRF), an interactive and user-friendly software package designed to automate the generation and fitting of long-range interaction terms for arbitrary molecules in nondegenerate (ground or excited) electronic states. We have derived and implemented all terms up to 15th order, without approximations, via a spherical tensor representation, with symmetry adaptation to all molecular point-group symmetries. The resulting potential energy surface is compatible with most representations of the close interaction region.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Separability of H 2 O molecular potential surfaces in hyperspherical coordinates via adiabatic approximation

An adiabatic scheme for separation of the three-dimensional (3D) nuclear dynamics on the ground electronic Born-Oppenheimer potential energy surface of an H 2 O molecule in hyperspherical coordinates is presented. It is found that the three vibrational modes are weakly coupled and the 3D vibrational wave function can be approximated as a product of three separable functions: one represented by the hyperradius and two by the two hyperangles individually. Additionally, this framework is then used for investigation of the formation and the role of a saddlelike barrier arising in the two hyperspherical angles that is to moderate the OH + H dissociation process. In order to test the validity of the framework, vibrational states with energies up to 19 500 cm -1 are constructed under the assumptions of adiabaticity and separability and compared to full three-dimensional high-precision numerical calculations yielding remarkable correspondence. As a result we present a simple construction scheme for separated molecular vibration states as the first step towards theoretical investigation of laser-driven molecular dynamics of triatomic molecules.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2022

Wind Energy Costs in Puerto Rico Through 2035

In 2019, Puerto Rico codified into law a 100% renewable electricity by 2050 target. Decision makers there need information about the costs and performance of wind energy technologies to incorporate into planning efforts for the future grid. After consulting with key technical stakeholders and accounting for unique conditions in Puerto Rico like increased hurricane risks, we customize NREL's Renewable Energy Potential model to calculate how capital expenditures, operational expenditures, net annual energy generation, and levelized cost of energy (LCOE) vary in time and space for wind energy projects around Puerto Rico. Wind turbines can add value by providing grid services through grid-forming inverters and complement solar energy generation with evening and nighttime electricity production, especially offshore. By 2035, we expect average costs of land-based and offshore wind energy in Puerto Rico to reach $\$$69/MWh (6.9 cents/kWh) and $\$$100/MWh (10 cents/kWh), respectively. While these projections do not include transmission system upgrade costs, they do suggest that low-cost, clean electricity is possible for Puerto Rico.

17 WIND ENERGY↗
DOE OSTI2022

Vibrational levels of a generalized Morse potential

A Generalized Morse Potential (GMP) is an extension of the Morse Potential (MP) with an additional exponential term and an additional parameter that compensate for MP’s erroneous behavior in the long range part of the interaction potential. Because of the additional term and parameter, the vibrational levels of the GMP cannot be solved analytically, unlike the case for the MP. We present several numerical approaches for solving the vibrational problem of the GMP based on Galerkin methods, namely, the Laguerre Polynomial Method (LPM), the Symmetrized LPM, and the Polynomial Expansion Method (PEM), and apply them to the vibrational levels of the homonuclear diatomic molecules B 2 , O 2 , and F 2 , for which high level theoretical near full configuration interaction (CI) electronic ground state potential energy surfaces and experimentally measured vibrational levels have been reported. Overall, the LPM produces vibrational states for the GMP that are converged to within spectroscopic accuracy of 0.01 cm –1 in between 1 and 2 orders of magnitude faster and with much fewer basis functions/grid points than the Colbert–Miller Discrete Variable Representation (CN-DVR) method for the three homonuclear diatomic molecules examined in this study.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Electronic structure theory with molecular point group symmetries on quantum annealers

Quantum computation has the potential to revolutionize quantum chemistry through major speedups in computation times and an exponential reduction in computational resources. Here, we combine the symmetry-adapted Jordan–Wigner encoding based on the full Boolean symmetry group $\mathbb{Z}$$^{k}_{2}$ with our new implementation of the Xia–Bian–Kais (XBK) method for improving the efficiency of electronic structure theory calculations on quantum annealers, particularly by reducing the number of qubits needed to achieve the same accuracy. By providing a more extensive symmetry-adapted encoding (SAE) than previous work, we are able to simulate molecules larger than those previously reported that have been studied using methods developed for quantum annealers and without using an active space. We calculated the potential energy surfaces of H 2 , LiH, He 2 , H 2 O, O 2 , N 2 , Li 2 , F 2 , CO, BH 3 , NH 3 , and CH 4 , with the largest molecule in the STO-6G basis set requiring 16 qubits with our SAE, and compared them with full configuration interaction results. The application of SAE to the XBK method provides an exponential reduction in the size of the Hilbert space and scales well with the size of the problem. It does not introduce significant additional errors for even or large values of a key variational parameter that determines the number of ancilla qubits used in the XBK method’s Hamiltonian embedding, or for certain molecules such as He 2 and H 2 O. Here, we provide an explanation for this behavior and a recommendation on the usage of our method. In addition, we briefly discuss the potential of extracting electronic excited states from our method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗