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In this paper, we study the correspondence between the three-dimensional and one-dimensional physical quantity in the spherically symmetric gravitational fields and we obtain a conjecture that the total energy enclosed by the two-dimensional sphere is equal to twice the change in vacuum zero-point energy of a one-dimensional circle that is the boundary of a hemispherical sphere, we can naturally obtain Schwarzschild metric without relying on Einstein's gravitational field equation under this conjecture, the result leads us to conclude that gravity originates from the change of zero-point energy in the one-dimensional vacuum.

We give a vacuum description with zero-point density for virtual fluctuations. One of the goals is to explain the origin of the vacuum permittivity and permeability and to calculate their values. In particular, we improve on existing calculations by avoiding assumptions on the volume occupied by virtual fluctuations. We propose testing of the models that assume a finite lifetime of virtual fluctuation. If during its propagation, the photon is stochastically trapped and released by virtual pairs, the propagation velocity may fluctuate. The propagation time fluctuation is estimated for several existing models. The obtained values are measurable with available technologies involving ultra-short laser pulses, and some of the models are already in conflict with the existing astronomical observations. The phase velocity is not affected significantly, which is consistent with the interferometric measurements.

Fourier-transform microwave spectroscopy has been applied for the 13C/18O-substituted tropolone to observe tunneling-rotation transitions as well as pure rotational transitions. The tunneling-rotation transitions were observed between the 13C-4 and -6 forms as well as between 13C-3 and -7, between 13C-1 and -2, and between 18O-8 and -9 (we denote these tunneling pairs as 13C-46, etc., below) although they have an asymmetric tunneling potential due to the difference in the zero point energy (ZPE). From the observed tunneling splittings ΔEij (0.9800–1.6824 cm−1), the differences in ZPE Δij for the 13C-46, -37, -12, and 18O-89 species are derived to be −0.1104, 0.5652, −1.3682, and 1.3897 cm−1 to agree well with the DFT calculation. The state mixing ratio of the tunneling states decreases drastically from (44%:56%) to (8.7%:91.3%) for 13C-46 and 18O-89 with an increase in the asymmetry Δij of the tunneling potential function. The observed tunneling-rotation interaction constants Fij decrease from 16.001 to 9.224 cm−1 as the differences in ZPE Δij increase, while the diagonal tunneling-rotation interaction constants Fu increase from 1.767 to 13.70 cm−1, explained well by the mixing ratio of the tunneling states. [ABSTRACT FROM AUTHOR]

To study the effect of hydrogen isotope substitution on the uptake of water during formation of clathrate hydrates, the harmonic intermolecular librational modes of selected water clusters (X2O)n with n = 2–6 and hydrogen isotopes X = H, D, and T are studied. The effects of the quantum mechanical zero-point energy (ZPE) in each cluster on the binding energies of the H2O, D2O, and T2O clusters are determined, with ZPE leading to the smallest binding energies in the H2O clusters and the largest binding energies in the T2O clusters. Corrections for anharmonicity of the librational modes are considered, and these bring the frequency ranges of the calculated intermolecular librational modes in the clusters to the experimental ranges of the librational modes in the infrared spectra of H2O and D2O solid ice and clathrate hydrate phases, and liquid H2O water. These calculations show the expected ranges of the binding energy of tritiated water onto a solid ice and clathrate hydrate surface and can help quantify the isotopic enrichment on a growing clathrate hydrate phase from the solution.

The two-phase thermodynamic (2PT) method is used to compute the zero point energy (ZPE) of several liquids and their mixtures. The 2PT method uses the density of states (DoS), which is computed from the velocity autocorrelation (VAC) function obtained from a short classical molecular dynamics trajectory. By partitioning the VAC and the DoS of a fluid into solid and gaslike components, quantum mechanical corrections to thermodynamical properties can be computed. The ZPE is obtained by combining the partition function of the quantum harmonic oscillator with the vibrational part of the solidlike DoS. The resulting ZPE is found to be in excellent agreement with both experimental and ab initio results. Solvent effects such as hydrogen bonding and polarization can be included by the utilization of ab initio density functional theory based molecular dynamics simulations. It is found that these effects significantly influence the DoS of water molecules. The obtained results demonstrate that the 2PT model is a powerful method for efficient ZPE calculations, in particular, to account for solvent effects and polarization. [ABSTRACT FROM AUTHOR]

An electronic zero-point energy (ZPE) adjustment protocol is presented within the context of the symmetrical quasiclassical (SQC) quantization of the electronic oscillator degrees of freedom (DOF) in classical Meyer-Miller (MM) vibronic dynamics for the molecular dynamics treatment of electronically nonadiabatic processes. The "adjustment" procedure maintains the same initial and final distributions of coordinates and momenta in the electronic oscillator DOF as previously given by the SQC windowing protocol but modifies the ZPE parameter in the MM Hamiltonian, on a per trajectory basis, so that the initial nuclear forces are precisely those corresponding to the initial electronic quantum state. Examples demonstrate that this slight modification to the standard SQC/MM approach significantly improves treatment of the multistate nonadiabatic dynamics following a Franck-Condon type vertical excitation onto a highly repulsive potential energy surface as is typical in the photodissociation context. [ABSTRACT FROM AUTHOR]

Mixed quantum classical (MQC)-initial value representation (IVR) is a recently introduced semiclassical framework that allows for selective quantization of the modes of a complex system. In the quantum limit, MQC reproduces the semiclassical Double Herman–Kluk IVR results, accurately capturing nuclear quantum coherences and conserving zero-point energy. However, in the classical limit, although MQC mimics the Husimi-IVR for real-time correlation functions with linear operators, it is significantly less accurate for non-linear correlation functions with errors even at time zero. Here, we identify the origin of this discrepancy in the MQC formulation and propose a modification. We analytically show that the modified MQC approach is exact for all correlation functions at time zero, and in a study of zero-point energy (ZPE) flow, we numerically demonstrate that it correctly obtains the quantum and classical limits as a function of time. Interestingly, although classical-limit MQC simulations show the expected, unphysical ZPE leakage, we find that it is possible to predict and even modify the direction of ZPE flow through selective quantization of the system, with the quantum-limit modes accepting energy but preserving the minimum quantum mechanically required energy. [ABSTRACT FROM AUTHOR]

In this work, we characterize the temperature dependence of kinetic properties in heavy atom tunneling reactions by means of molecular dynamics simulations, including nuclear quantum effects (NQEs) via Path Integral theory. To this end, we consider the prototypical Cope rearrangement of semibullvalene. The reaction was studied in the 25–300 K temperature range observing that the inclusion of NQEs modifies the temperature behavior of both free energy barriers and dynamical recrossing factors with respect to classical dynamics. Notably, while in classical simulations the activation free energy shows a very weak temperature dependence, it becomes strongly dependent on temperature when NQEs are included. This temperature behavior shows a transition from a regime where the quantum effects are limited and can mainly be traced back to zero point energy, to a low temperature regime where tunneling plays a dominant role. In this regime, the free energy curve tunnels below the potential energy barrier along the reaction coordinate, allowing much faster reaction rates. Finally, the temperature dependence of the rate constants obtained from molecular dynamics simulations was compared with available experimental data and with semi‐classical transition state theory calculations, showing comparable behaviors and similar transition temperatures from thermal to (deep) tunneling regime. [ABSTRACT FROM AUTHOR]

To study the effect of hydrogen isotope substitution on the uptake of water during formation of clathrate hydrates, the harmonic intermolecular librational modes of selected water clusters (X2O)n with n = 2–6 and hydrogen isotopes X = H, D, and T are studied. The effects of the quantum mechanical zero-point energy (ZPE) in each cluster on the binding energies of the H2O, D2O, and T2O clusters are determined, with ZPE leading to the smallest binding energies in the H2O clusters and the largest binding energies in the T2O clusters. Corrections for anharmonicity of the librational modes are considered, and these bring the frequency ranges of the calculated intermolecular librational modes in the clusters to the experimental ranges of the librational modes in the infrared spectra of H2O and D2O solid ice and clathrate hydrate phases, and liquid H2O water. These calculations show the expected ranges of the binding energy of tritiated water onto a solid ice and clathrate hydrate surface and can help quantify the isotopic enrichment on a growing clathrate hydrate phase from the solution. [ABSTRACT FROM AUTHOR]

We investigate the Casimir effect in the systems that consist of parallel but misaligned finite-size plates from the point of view of zero-point energy. We elaborate that the zero-point energies of the radiation field in the perfect conductor systems would generate a tangential Casimir force, and explore the properties and consequences of this tangential force in various conductor systems. Thereafter, we generalize our discussion to dielectrics. After calculating the total zero-point energies of the surface modes in the multilayered systems, we show that the tangential force also exists in dielectrics. We obtain the finite-conductivity corrections to the tangential force for imperfectly conducting plates, and calculate the finite-temperature corrections to the force. The typical strength of the tangential force suggests it might be observable.

In results from thousands of trials and dozens of variations, tests for measurement artifacts, and replications, metal/insulator/metal/Casimir cavity devices produce electric power, apparently by tapping ambient zero-point energy (ZPE). A simple calculation shows that the power potentially available from the ZPE quantum vacuum is an immense 5 gigawatts per square meter. The devices tap a tiny fraction of that, but still deliver a practical power density of 70 watts per square meter. The devices are designed to circumvent the apparent impediments to ZPE harvesting, i.e., that ZPE is the universal ground state, and that ZPE fluctuations are extremely short lived and virtual. If the source ultimately proves to be ZPE, what is the operating principle behind the energy harvesting, and how can the results be reconciled with known physical law? A notional operating principle can be understood as a direct analog to the optical phenomenon of frustrated total internal reflection. Tapping ZPE does not violate the second law of thermodynamics based on the conventional quantum interpretation of ZPE, but ambiguities regarding the source of ZPE leave the issue unresolved.

The feature vector mapping used to represent chemical systems is a key factor governing the superior data efficiency of kernel based quantum machine learning (QML) models applicable throughout chemical compound space. Unfortunately, the most accurate representations require a high dimensional feature mapping, thereby imposing a considerable computational burden on model training and use. We introduce compact yet accurate, linear scaling QML representations based on atomic Gaussian many-body distribution functionals (MBDF) and their derivatives. Weighted density functions of MBDF values are used as global representations that are constant in size, i.e., invariant with respect to the number of atoms. We report predictive performance and training data efficiency that is competitive with state-of-the-art for two diverse datasets of organic molecules, QM9 and QMugs. Generalization capability has been investigated for atomization energies, highest occupied molecular orbital–lowest unoccupied molecular orbital eigenvalues and gap, internal energies at 0 K, zero point vibrational energies, dipole moment norm, static isotropic polarizability, and heat capacity as encoded in QM9. MBDF based QM9 performance lowers the optimal Pareto front spanned between sampling and training cost to compute node minutes, effectively sampling chemical compound space with chemical accuracy at a sampling rate of ∼ 48 molecules per core second. [ABSTRACT FROM AUTHOR]

We give a vacuum description with zero-point density for virtual fluctuations. One of the goals is to explain the origin of the vacuum permittivity and permeability and to calculate their values. In particular, we improve on existing calculations by avoiding assumptions on the volume occupied by virtual fluctuations. We propose testing of the models that assume a finite lifetime of virtual fluctuation. If during its propagation, the photon is stochastically trapped and released by virtual pairs, the propagation velocity may fluctuate. The propagation time fluctuation is estimated for several existing models. The obtained values are measurable with available technologies involving ultra-short laser pulses, and some of the models are already in conflict with the existing astronomical observations. The phase velocity is not affected significantly, which is consistent with the interferometric measurements. [ABSTRACT FROM AUTHOR]

Abstract Following on our earlier work in this area, here we examine in some detail the physical mechanism involved in the Bose–Einstein condensation process. In particular we emphasise the significance of the zero value of the chemical potential at and below the critical temperature. The molar zero-point energy (ZPE) for an ideal gas of He4 atoms in our new analysis is estimated and found to be very close to that calculated for an ideal Fermi gas of He3 atoms under the same conditions. This gives numerical support to our theory. We also show how the theory is consistent with the presence of a density maximum in liquid He4. Highlights • Study of the role of the chemical potential in Bose–Einstein condensation. • Formation of a condensate in the single particle energy state at the transition point. • Calculation of the value of the zero point energy of a mole of He4 atoms. • Similarity of this value to that of the zero point energy of a mole of He3 atoms. • Consistency of the new theory with the two-fluid model for liquid He-II. [ABSTRACT FROM AUTHOR]

In research articles and patents several methods have been proposed for the extraction of zero-point energy from the vacuum. None of the proposals have been reliably demonstrated, yet they remain largely unchallenged. In this paper the underlying thermodynamics principles of equilibrium, detailed balance, and conservation laws are presented for zero-point energy extraction. The proposed methods are separated into three classes: nonlinear processing of the zero-point field, mechanical extraction using Casimir cavities, and the pumping of atoms through Casimir cavities. The first two approaches are shown to violate thermodynamics principles, and therefore appear not to be feasible, no matter how innovative their execution. The third approach, based upon stochastic electrodynamics, does not appear to violate these principles, but may face other obstacles. Initial experimental results are tantalizing but, given the lower than expected power output, inconclusive.

A valence coordinate H2NOH ground state potential energy surface accurate for all levels up to 6000 cm−1 relative to trans zero point energy has been generated at the coupled-cluster single double triple-F12/aug-cc-pVTZ level encompassing the trans and cis as well as the N–H2 permutational conformers. All cis and trans fundamentals and a complete set of eigenfunctions up to about 3100 cm−1 have been calculated and assigned using the improved relaxation method of the Heidelberg multi-configuration time-dependent Hartree package and an exact expression for the kinetic energy in valence coordinates generated by the TANA program. The average and maximal error to all observed transitions is about 6.3 and 14.6 cm−1, respectively. Local cis eigenfunctions exist with up to two quanta in the isomerization mode ν9. Although no significant inversion splittings have been found up to the considered 3100 cm−1, they are expected within the fundamental energy range in view of the calculated 4261 cm−1 H2 permutation/inversion barrier height. The cis-NH2 symmetric stretch fundamental shows a Fermi resonance with a splitting of about 10 cm−1. [ABSTRACT FROM AUTHOR]

Abstract Traditional quantum field theory can lead to enormous zero-point energy, which markedly disagrees with experiment. Unfortunately, this situation is built into conventional canonical quantization procedures. For identical classical theories, an alternative quantization procedure, called affine field quantization, leads to the desirable feature of having a vanishing zero-point energy. This procedure has been applied to renormalizable and nonrenormalizable covariant scalar fields, fermion fields, as well as general relativity. Simpler models are offered as an introduction to affine field quantization.

A full-dimensional, permutationally invariant potential energy surface (PES) for the glycine amino acid is reported. A precise fit to energies and gradients calculated at the density functional theory (DFT)/B3LYP level of electronic-structure theory with Dunning's aug-cc-pVDZ basis set is performed involving 20 000 low-energy points and associated Cartesian gradients plus about 50 000 additional higher-energy points. The fact that newly calculated DFT/B3LYP energies for the main stationary points are close to the coupled-cluster single-double-triple [CCSD(T)] values, recently reported in the literature, provides reassurance about the accuracy of the constructed PES. Eight conformers and numerous saddle points are identified and characterized by describing geometries, relative stability, and harmonic frequencies. Stochastic and dynamical approaches are employed to study the vibrational ground state. Specifically, diffusion Monte Carlo simulations and approximate quantum dynamics, performed by means of the adiabatic switching semiclassical initial value representation technique, provide zero-point energies in excellent agreement with each other. The PES we report is sufficiently complete to permit spectroscopic and dynamical studies on glycine, which may be of interest to the biochemical and astrochemistry communities. [ABSTRACT FROM AUTHOR]

Density functional theory (DFT) including van der Waals (vdW) interactions and accounting for zero-point energy (ZPE) is believed to provide a good description of crystalline ice phases [B. Pamuk et al., Phys. Rev. Lett. 108, 193003 (2012)]. Given the computational cost of DFT, it is not surprising that extensive phonon calculations, which yield the ZPE, have only been done for a limited amount of ice structures. Computationally convenient force fields on the other hand are the method of choice for large systems and/or dynamical simulations, e.g., of supercooled water. Here, we present a systematic comparison for seven hydrogen-ordered crystalline ice phases (Ih, IX, II, XIII, XIV, XV, and VIII) between many commonly used nonpolarizable force fields and density functionals, including some recently developed meta-GGA functionals and accounting for vdW interactions. Starting from the experimentally determined crystal structures, we perform space-group-constrained structural relaxations. These provide the starting point for highly accurate phonon calculations that yield effectively volume-dependent ZPEs within the quasiharmonic approximation. In particular, when including ZPE, the force fields show a remarkably good performance for equilibrium volumes and cohesive energies superior to many density functionals. A decomposition of the cohesive energies into intramolecular deformation, electrostatic, and vdW contributions quantifies the differences between force fields and DFT. Results for the equilibrium volumes and phase transition pressures for all studied force fields are much more strongly affected by ZPE than all studied density functionals. We track this down to significantly smaller shifts of the O–H-stretch modes and compare with experimental data from Raman spectroscopy. [ABSTRACT FROM AUTHOR]

A model of an extended manifold for the Dirac spinor field is considered. Two Lagrangians related by charge-parity-time-mass symmetry are constructed for a pair of the Dirac spinor fields with each spinor field defined in a separate manifold. An interaction between the matter fields in the manifolds is introduced through gravity. A fermionic effective action of the general system is constructed and a tadpole one-loop spinor diagram and part of the one-loop vacuum diagrams with two external gravitational off-shell fields which contribute to the effective action are calculated. It is demonstrated that among different versions of the second spinor Lagrangian there is a special one for which a cancellation of the mentioned diagrams in the total effective action takes place. As a result, the diagrams do not contribute to the cosmological constant, as well there is a zero contribution of the zero point energies of the spinor fields to the action. The non-zero leading order value of the cosmological constant for each manifold in the framework is proportional to the trace of an momentum–energy tensor of each separated manifold or difference of the tensors of the related manifolds. The result is depending on the chosen model of interaction of gravitational fields with fermions, the different possibilities are discussed. An appearance of the dark matter in the model is shortly discussed as well as further applications of the approach and it is interconnections with other cosmology models. [ABSTRACT FROM AUTHOR]

In this article, we present a machine learning model to obtain fast and accurate estimates of the molecular Hessian matrix. In this model, based on a random forest, the second derivatives of the energy with respect to redundant internal coordinates are learned individually. The internal coordinates together with their specific representation guarantee rotational and translational invariance. The model is trained on a subset of the QM7 dataset but is shown to be applicable to larger molecules picked from the QM9 dataset. From the predicted Hessian, it is also possible to obtain reasonable estimates of the vibrational frequencies, normal modes, and zero point energies of the molecules. [ABSTRACT FROM AUTHOR]

A new approach for preventing zero-point energy (ZPE) violation in quasi-classical trajectory (QCT) simulations is presented and applied to H2CO “roaming” reactions. Zero-point energy may be problematic in roaming reactions because they occur at or near bond dissociation thresholds and these channels may be incorrectly open or closed depending on if, or how, ZPE has been treated. Here we run QCT simulations on a “ZPE-corrected” potential energy surface defined as the sum of the molecular potential energy surface (PES) and the global harmonic ZPE surface. Five different harmonic ZPE estimates are examined with four, on average, giving values within 4 kJ/mol—chemical accuracy—for H2CO. The local harmonic ZPE, at arbitrary molecular configurations, is subsequently defined in terms of “projected” Cartesian coordinates and a global ZPE “surface” is constructed using Shepard interpolation. This, combined with a second-order modified Shepard interpolated PES, V, allows us to construct a proof-of-concept ZPE-corrected PES for H2CO, Veff, at no additional computational cost to the PES itself. Both V and Veff are used to model product state distributions from the H + HCO → H2 + CO abstraction reaction, which are shown to reproduce the literature roaming product state distributions. Our ZPE-corrected PES allows all trajectories to be analysed, whereas, in previous simulations, a significant proportion was discarded because of ZPE violation. We find ZPE has little effect on product rotational distributions, validating previous QCT simulations. Running trajectories on V, however, shifts the product kinetic energy release to higher energy than on Veff and classical simulations of kinetic energy release should therefore be viewed with caution. [ABSTRACT FROM AUTHOR]

Ambimodal reactions involve a single transition state leading to multiple products. To assess the ratio of the bifurcation products, quasiclassical trajectories (QCTs) are initiated from transition state geometries that are randomly sampled using the zero‐point energy (ZPE) plus thermal energy of a molecule's vibrational modes. However, in the QCTs, the influence of ZPE percentage in the resulting bifurcation ratio and time gap between formation of bonds is not well understood. To benchmark the effect of varying the percentage of the molecules' inherent ZPE used in the simulation, three organic reactions with distinct mechanisms that all display post‐transition state bifurcation were selected. The three reactions studied include an intramolecular [6 + 4]/[4 + 2] cycloaddition, an asynchronous nitrene insertion, and an SN2/addition to a carbonyl on an α‐bromoketone. These reactions encompass various modes of bond formation and cleavage, offering a broad view of the influence of ZPE scaling in evaluating the product ratio resulted from post‐transition state bifurcation. Exploring means of tuning selectivity in simulation enables development of more accurate organic reaction modeling for synthetic prediction. This work will establish a basis for future QCT studies of organic reactions. [ABSTRACT FROM AUTHOR]

It was long believed that there is a zero-point energy in the form of hx=2 for massive particles, which is obtained from Schrodinger equation for the harmonic oscillator model. In this paper, it is shown, by the Dirac oscillator, that there is no such a zero-point energy. It is argued that when a particles wave function can spread in the whole space, it can be static. This does neither violate wave-particle duality nor uncertainty relationship. Dirac equation correctly describes physical reality, while Schrodinger equation does not when it is not the nonrelativistic approximation of Dirac equation with a certain model. The conclusion that there is no zero-point energy in the form of hx=2 is applied to solve the famous cosmological constant problem for massive particles. [ABSTRACT FROM AUTHOR]

Density functional theory electronic structure calculations were employed to study a proposed novel type of interaction between NH3 and the metallic ring of cyclic trinuclear Ag(I) clusters of the general formula cyclo-Ag3(μ2-L)3 bearing 15 different bridging organic ligands L. Scan single point energy calculations revealed that the most stable orientation of NH3 is that with a N atom facing the center of the silver metallic ring and bridging the three metal centers. Starting with this orientation we performed geometry optimizations of the cyclo-Ag3(μ2-L)3(μ3-NH3) complexes. The optimized geometries retain the initial orientation of NH3. The zero point energy corrected interaction energies, D0, of NH3 with the metallic ring are found in the range 4.2–12.4 kcal mol−1 while the distance of NH3 from the center of the ring is in the range 2.317–2.519 Å, both indicative of a relatively strong interaction. A multitude of theoretical methods employed revealed that the interaction of NH3 with the silver metallic ring is mainly of ionic nature with a contribution of van der Waals forces. Molecular structure–property relationships indicate that the magnitude of D0 increases upon increasing the charge of the silver metal centers and decreasing the LUMO energy of the cyclo-Ag3(μ2-L)3 clusters. [ABSTRACT FROM AUTHOR]