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At least 739 records · Page 41
All sourcesNASA NTRSDOE OSTI
DOE OSTI2016

An alternative derivation of ring-polymer molecular dynamics transition-state theory

In a previous article [T. J. H. Hele and S. C. Althorpe, J. Chem. Phys. 138, 084108 (2013)], we showed that the t → 0{sub +} limit of ring-polymer molecular dynamics (RPMD) rate-theory is also the t → 0{sub +} limit of a new type of quantum flux-side time-correlation function, in which the dividing surfaces are invariant to imaginary-time translation; in other words, that RPMD transition-state theory (RMPD-TST) is a t → 0{sub +} quantum transition-state theory (QTST). Recently, Jang and Voth [J. Chem. Phys. 144, 084110 (2016)] rederived this quantum t → 0{sub +} limit and claimed that it gives instead the centroid-density approximation. Here we show that the t → 0{sub +} limit derived by Jang and Voth is in fact RPMD-TST.

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

Towards a Deeper Fundamental Understanding of (Al,Sc)N Ferroelectric Nitrides

Density functional theory (DFT) calculations, within the virtual crystal alloy approximation, are performed, along with the development of a Landau-type model employing a symmetry-allowed analytical expression of the internal energy and having parameters determined from first principles, to investigate properties and energetics of Al1-xScxN ferroelectric nitrides in their hexagonal forms. These DFT computations and this model predict the existence of two different types of minima, namely, the fourfold-coordinated wurtzite (WZ) polar structure and a five-fold coordinated paraelectric hexagonal phase (denoted as H5), for any Sc composition up to 40%. The H5 minimum progressively becomes the lowest-energy state within hexagonal symmetry as the Sc concentration increases from 0 to 0.4. Furthermore, the model points to several key findings. Examples include the crucial role of the coupling between polarization and strains to create the WZ minimum, in addition to polar and elastic energies, and that the origin of the H5 state overcoming the WZ phase as the global minimum within hexagonal symmetry when increasing the Sc composition mostly lies in the compositional dependency of only two parameters-one linked to the polarization and another one being purely elastic in nature. Other examples are that forcing Al1-xScxN systems to have no or a weak change in lattice parameters when heating them allows us to reproduce their finite-temperature polar properties well and that a value of the axial ratio close to that of the ideal WZ structure implies a large polarization at low temperatures but not necessarily at high temperatures because of the ordered-disordered character of the temperature-induced formation of the WZ state. Such findings should allow for a better fundamental understanding of (Al,Sc)N ferroelectric nitrides, which may be used to design efficient devices having, e.g., low operating voltages.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2016

Structural phase transition and electronic properties in samarium chalcogenides

The electronic structure and high pressure properties of samarium monochalcogenides SmS, SmSe and SmTe have been reported by using tight binding linear muffin-tin-orbital (TB-LMTO) method within the local density approximation (LDA). The total energy as a function of volume is evaluated. It is found that these monochalcogenides are stable in NaCl-type structure under ambient pressure. We predict a structural phase transition from NaCl-type (B{sub 1}-phase) structure to CsCl-type (B{sub 2}-type) structure for these compounds. Phase transition pressures were found to be 1.7, 4.4 and 6.6 GPa, for SmS, SmSe and SmTe respectively. Apart from this, the lattice parameter (a{sub 0}), bulk modulus (B{sub 0}), band structure (BS) and density of states (DOS) are calculated. From energy band diagram we observed that these compounds exhibit metallic character. The calculated values of equilibrium lattice parameter and phase transition pressure are in general good agreement with available data.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS2023

Efficient Calibration of Expensive Computational Models

Accounting for uncertainty when calibrating expensive computational models is a common challenge faced by scientists and engineers. Often Bayesian techniques are adopted to estimate a probability density function over the model parameters given noisy empirical data. The methods used to perform this type of probabilistic calibration are computationally prohibitive in that they require a large number of evaluations of the expensive model. In these cases, surrogate modeling -- that is, using a fast-to-evaluate, lower fidelity stand-in for the original computational model -- may be the only option to alleviate this computational burden. However, the upfront cost of generating training data to build a surrogate model can itself be expensive. As such, it is important to be judicious when selecting training points at which the full-fidelity model is evaluated. Here, an active learning approach is proposed that enables efficient selection of training points using approximate samples of the calibrated parameter probability density function. In this way, the training points can be concentrated in regions where the calibration algorithm requires high model accuracy.

active learning↗
DOE OSTI2024

Tailoring the Electronic Structures and Spectral Properties of ZnO with Irradiation Defects Generated Under Intense Electronic Excitation: A Combined Experimental and DFT Approach

Here, the relationship between the composition of the internal defect states, spectral properties, and correlated electronic structures of wurtzite zinc oxide (ZnO) crystals under 645 MeV Xe 35+ irradiation is systematically investigated, employing experimental characterizations combined with first-principle calculations. Based on the ion irradiation-induced thermal expansion and relaxation processes, the high concentration of vacancy/interstitial defects produced from the transient disordered phase in molten track states trigger photoelectric changes, as follows: i) the generation of internal defect states effectively reduces the intrinsic bandgap (3.25 eV → 2.66 eV); ii) a large number of defective active sites inhibits the recombination between electron–hole pairs, causing dark conductance and photoconductance to increase with increasing damage levels until optimal fluence is achieved. Based on the density functional theory (DFT) with the GGA + U (GGA = generalized gradient approximation) method, the defective models associated with the different electronic structures, density of states, formation energy, and the nature of the chemical bonding are established. The narrowing of the bandgap observed experimentally and the enhancement of carrier concentration originating from the internal electron defect states are qualitatively verified, therefore laying the foundation for designing future nanoscale photoelectronic devices and microelectronics applications.

36 MATERIALS SCIENCE↗
DOE OSTI2015

Two-Yukawa fluid at a hard wall: Field theory treatment

We apply a field-theoretical approach to study the structure and thermodynamics of a two-Yukawa fluid confined by a hard wall. We derive mean field equations allowing for numerical evaluation of the density profile which is compared to analytical estimations. Beyond the mean field approximation, analytical expressions for the free energy, the pressure, and the correlation function are derived. Subsequently, contributions to the density profile and the adsorption coefficient due to Gaussian fluctuations are found. Both the mean field and the fluctuation terms of the density profile are shown to satisfy the contact theorem. We further use the contact theorem to improve the Gaussian approximation for the density profile based on a better approximation for the bulk pressure. The results obtained are compared to computer simulation data.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

Tomography and weak lensing statistics

We provide generic predictions for the lower order cumulants of weak lensing maps, and their correlators for tomographic bins as well as in three dimensions (3D). Using small-angle approximation, we derive the corresponding one- and two-point probability distribution function for the tomographic maps from different bins and for 3D convergence maps. The modelling of weak lensing statistics is obtained by adopting a detailed prescription for the underlying density contrast that involves hierarchal ansatz and lognormal distribution. We study the dependence of our results on cosmological parameters and source distributions corresponding to the realistic surveys such as LSST and DES. We briefly outline how photometric redshift information can be incorporated in our results. We also show how topological properties of convergence maps can be quantified using our results.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2014

Response calculations based on an independent particle system with the exact one-particle density matrix: Polarizabilities

Recently, we have demonstrated that the problems finding a suitable adiabatic approximation in time-dependent one-body reduced density matrix functional theory can be remedied by introducing an additional degree of freedom to describe the system: the phase of the natural orbitals [K. J. H. Giesbertz, O. V. Gritsenko, and E. J. Baerends, Phys. Rev. Lett. 105, 013002 (2010); K. J. H. Giesbertz, O. V. Gritsenko, and E. J. Baerends, J. Chem. Phys. 133, 174119 (2010)]. In this article we will show in detail how the frequency-dependent response equations give the proper static limit (ω → 0), including the perturbation in the chemical potential, which is required in static response theory to ensure the correct number of particles. Additionally we show results for the polarizability for H{sub 2} and compare the performance of two different two-electron functionals: the phase-including Löwdin–Shull functional and the density matrix form of the Löwdin–Shull functional.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

Debye temperature, thermal expansion, and heat capacity of TcC up to 100 GPa

Highlights: • A number of thermodynamic properties of rocksalt TcC are investigated for the first time. • The quasi-harmonic Debye model is applied to take into account the thermal effect. • The pressure and temperature up to about 100 GPa and 3000 K, respectively. - Abstract: Debye temperature, thermal expansion coefficient, and heat capacity of ideal stoichiometric TcC in the rocksalt structure have been studied systematically by using ab initio plane-wave pseudopotential density functional theory method within the generalized gradient approximation. Through the quasi-harmonic Debye model, in which the phononic effects are considered, the dependences of Debye temperature, thermal expansion coefficient, constant-volume heat capacity, and constant-pressure heat capacity on pressure and temperature are successfully predicted. All the thermodynamic properties of TcC with rocksalt phase have been predicted in the entire temperature range from 300 to 3000 K and pressure up to 100 GPa.

36 MATERIALS SCIENCE↗
DOE OSTI2019

Unraveling Excitonic Effects for the First Hyperpolarizabilities of Chromophore Aggregates

Excitonic interactions often significantly affect the optoelectronic properties of molecular materials. However, their role in determining the nonlinear optical response of organic electro-optic materials remains poorly understood. In this paper, we explore the effects of excitonic interactions on the first hyperpolarizability for aggregates of donor–acceptor chromophores. We show that calculations of the first hyperpolarizabilty of chromophore aggregates based on a two-state model agree well with the more rigorous coupled perturbed Hartree–Fock method. We then use both time-dependent density functional theory calculations and the molecular exciton approximation to parametrize the two-state model. Use of the molecular exciton approximation to the two-state model (i) is appropriate for disordered aggregates (unlike band theory), (ii) is computationally efficient enough for calculating the first hyperpolarizability of materials that consist of thousands of interacting chromophores, and (iii) allows the unraveling of the effects of both excitonic interactions and electrostatic polarization of the chromophore electron density by its environment on the first hyperpolarizability of molecular materials. We find that use of the molecular exciton approximation to the two-state model does not introduce significant additional errors compared to those introduced by applying the two-state model alone. We determine that the absolute change to the first hyperpolarizability of chromophore aggregates due to excitonic interactions increases with the size of the aggregate. For all sizes of disordered aggregates of chromophores considered in this paper, the inclusion of excitonic interactions on average decreases the magnitude of the first hyperpolarizability by 12–14% compared to the case of non-interacting chromophores. Lastly, we present a method for analytically calculating the first hyperpolarizability of a one-dimensional periodic array of chromophores within the molecular exciton approximation to the two-state model. This technique can be used to include an approximate correction for excitonic effects when simulating the electro-optic response of disordered and ordered organic materials.

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

Electronic structure of mononuclear Cu-based molecule from density-functional theory with self-interaction correction

In this paper, we investigate the electronic structure of a planar mononuclear Cu-based molecule [Cu(C 6 H 4 S 2 ) 2 ] z in two oxidation states ( z = –2, –1) using density-functional theory (DFT) with Fermi–Löwdin orbital (FLO) self-interaction correction (SIC). The dianionic Cu-based molecule was proposed to be a promising qubit candidate. Self-interaction error within approximate DFT functionals renders severe delocalization of electron and spin densities arising from 3 d orbitals. The FLO-SIC method relies on optimization of Fermi–Löwdin orbital descriptors (FODs) with which localized occupied orbitals are constructed to create SIC potentials. Starting with many initial sets of FODs, we employ a frozen-density loop algorithm within the FLO-SIC method to study the Cu-based molecule. We find that the electronic structure of the molecule remains unchanged despite somewhat different final FOD configurations. In the dianionic state (spin S = 1/2), FLO-SIC spin density originates from the Cu d and S p orbitals with an approximate ratio of 2:1, in quantitative agreement with multireference calculations, while in the case of SIC-free DFT, the orbital ratio is reversed. Overall, FLO-SIC lowers the energies of the occupied orbitals and, in particular, the 3 d orbitals unhybridized with the ligands significantly, which substantially increases the energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) compared to SIC-free DFT results. The FLO-SIC HOMO–LUMO gap of the dianionic state is larger than that of the monoanionic state, which is consistent with experiment. Our results suggest a positive outlook of the FLO-SIC method in the description of magnetic exchange coupling within 3 d -element-based systems.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2014

LyMAS: Predicting large-scale Lyα forest statistics from the dark matter density field

We describe Lyα Mass Association Scheme (LyMAS), a method of predicting clustering statistics in the Lyα forest on large scales from moderate-resolution simulations of the dark matter (DM) distribution, with calibration from high-resolution hydrodynamic simulations of smaller volumes. We use the 'Horizon-MareNostrum' simulation, a 50 h {sup –1} Mpc comoving volume evolved with the adaptive mesh hydrodynamic code RAMSES, to compute the conditional probability distribution P(F{sub s} |δ {sub s}) of the transmitted flux F{sub s} , smoothed (one-dimensionally, 1D) over the spectral resolution scale, on the DM density contrast δ {sub s}, smoothed (three-dimensionally, 3D) over a similar scale. In this study we adopt the spectral resolution of the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS) at z = 2.5, and we find optimal results for a DM smoothing length σ = 0.3 h {sup –1} Mpc (comoving). In its simplest form, LyMAS draws randomly from the hydro-calibrated P(F{sub s} |δ {sub s}) to convert DM skewers into Lyα forest pseudo-spectra, which are then used to compute cross-sightline flux statistics. In extended form, LyMAS exactly reproduces both the 1D power spectrum and one-point flux distribution of the hydro simulation spectra. Applied to the MareNostrum DM field, LyMAS accurately predicts the two-point conditional flux distribution and flux correlation function of the full hydro simulation for transverse sightline separations as small as 1 h {sup –1} Mpc, including redshift-space distortion effects. It is substantially more accurate than a deterministic density-flux mapping ({sup F}luctuating Gunn-Peterson Approximation{sup )}, often used for large-volume simulations of the forest. With the MareNostrum calibration, we apply LyMAS to 1024{sup 3} N-body simulations of a 300 h {sup –1} Mpc and 1.0 h {sup –1} Gpc cube to produce large, publicly available catalogs of mock BOSS spectra that probe a large comoving volume. LyMAS will be a powerful tool for interpreting 3D Lyα forest data, thereby transforming measurements from BOSS and other massive quasar absorption surveys into constraints on dark energy, DM, space geometry, and intergalactic medium physics.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2014

On the room-temperature phase diagram of high pressure hydrogen: An ab initio molecular dynamics perspective and a diffusion Monte Carlo study

The finite-temperature phase diagram of hydrogen in the region of phase IV and its neighborhood was studied using the ab initio molecular dynamics (MD) and the ab initio path-integral molecular dynamics (PIMD). The electronic structures were analyzed using the density-functional theory (DFT), the random-phase approximation, and the diffusion Monte Carlo (DMC) methods. Taking the state-of-the-art DMC results as benchmark, comparisons of the energy differences between structures generated from the MD and PIMD simulations, with molecular and dissociated hydrogens, respectively, in the weak molecular layers of phase IV, indicate that standard functionals in DFT tend to underestimate the dissociation barrier of the weak molecular layers in this mixed phase. Because of this underestimation, inclusion of the quantum nuclear effects (QNEs) in PIMD using electronic structures generated with these functionals leads to artificially dissociated hydrogen layers in phase IV and an error compensation between the neglect of QNEs and the deficiencies of these functionals in standard ab initio MD simulations exists. This analysis partly rationalizes why earlier ab initio MD simulations complement so well the experimental observations. The temperature and pressure dependencies for the stability of phase IV were also studied in the end and compared with earlier results.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2019

Ferromagnetism in (X, Y)-Codoped 4H-GaN Polytype X = Na and Mg, Y = Na and Mg: Ab Initio Study

In this work, the magnetic behavior of 4H-GaN polytype codoped with (X = Na and Mg, Y = Na and Mg) is studied by means of first-principle calculations based on spin-density functional theory within the generalized-gradient approximation. It is found that the most stable configuration reveals the smallest cohesion energy and the X–Y bond length is minimal. (X, Y )-codoped 4H-GaN reveals magnetism behavior, and the total magnetic moment comes mainly from the N host atoms. Interestingly, (Na, Na)-codoped 4H-GaN is found to be half-metallic ferromagnetic with a magnetic moment of 4 μ{sub B}. Results show also that (Mg, Na)- and (Na, Mg)-codoped 4H-GaN are found to be nearly half-metallic ferromagnetic with a magnetic moment of 2.96 and 2.97 μ{sub B}, respectively. Whereas (Mg, Mg)-codoped 4H-GaN have a metallic ferromagnetic behavior with a magnetic moment of 1.87 μ{sub B}. The partial density of states denotes that magnetism is achieved through a p-p-like coupling between the impurity X-2p states of X atom, Y-2p states of Y atom, and N-2p states of N host atoms. These theoretical results make Na–Na-codoped 4H-GaN very promising materials for spintronic technology.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2021

Density-functional theory of material design: fundamentals and applications-I

This article is part-I of a review of density-functional theory (DFT) that is the most widely used method for calculating electronic structure of materials. The accuracy and ease of numerical implementation of DFT methods has resulted in its extensive use for materials design and discovery and has thus ushered in the new field of computational material science. In this article, we start with an introduction to Schrödinger equation and methods of its solutions. After presenting exact results for some well-known systems, difficulties encountered in solving the equation for interacting electrons are described. How these difficulties are handled using the variational principle for the energy to obtain approximate solutions of the Schrödinger equation is discussed. The resulting Hartree and Hartree–Fock theories are presented along with results they give for atomic and solid-state systems. We then describe Thomas–Fermi theory and its extensions which were the initial attempts to formulate many-electron problem in terms of electronic density of a system. Having described these theories, we introduce modern DFT by discussing Hohenberg–Kohn theorems that form its foundations. We then go on to discuss Kohn–Sham (KS) formulation of DFT in its exact form. Next, local density approximation (LDA) is introduced and solutions of KS equation for some representative systems, obtained using the LDA, are presented. We end part-I of the review describing the contents of part-II.

36 MATERIALS SCIENCE↗
DOE OSTI2013

Density of electronic states and dispersion of optical functions of defect chalcopyrite CdGa{sub 2}X{sub 4} (X = S, Se): DFT study

Graphical abstract: - Highlights: • FPLAPW method is used for calculating the electronic and optical properties of CdGa{sub 2}X{sub 4}. • Electronic and optical properties were calculated using LDA, GGA, EVGGA and mBJ. • Band gap conformed that CdGa{sub 2}X{sub 4} are semiconductors fit for UV and visible light. • The ECD shows that change in the bond length and bond nature affect the band gap. • The dielectric tensor components and its derivatives show considerable anisotropy. - Abstract: A density functional theory (DFT) based on full potential linear augmented plane wave (FPLAPW) was used for calculating the electronic structure, charge density and optical properties of CdGa{sub 2}X{sub 4} (X = S, Se) compounds. Local density approximation (LDA), generalized gradient approximation (GGA), Engle Vasko generalized gradient approximation (EVGGA) and recently modified Becke–Johnson (mBJ) were applied to calculate the band structure, total and partial density of states. The investigation of band structures and density of states of CdGa{sub 2}X{sub 4} (X = S, Se) elucidate that mBJ potential show close agreement to the experimental results. The mBJ potential was selected for further explanation of optical properties of CdGa{sub 2}X{sub 4} (X = S, Se). The study of electronic charge density contours shows that change in the bond lengths and bond nature affect the band gap of the compounds. The two non-zero dielectric tensor components and its derivatives show considerable anisotropy between the perpendicular and parallel components. The present work provide accurate information about the combination (hybridization) of orbital, formation of bands and dispersion of non-zero tensor components of CdGa{sub 2}X{sub 4} (X = S, Se)

36 MATERIALS SCIENCE↗
DOE OSTI2015

Components of the Bond Energy in Polar Diatomic Molecules, Radicals, and Ions Formed by Group-1 and Group-2 Metal Atoms

While many transition metal complexes are known to have high multireference character, the multireference character of main-group closed-shell singlet diatomic molecules like BeF, CaO, and MgO has been less studied. Yet, many group-1 and group-2 diatomic molecules do have multireference character, and they provide informative systems for studying multireference character because they are simpler than transition metal compounds. The goal of the present work is to understand these multireference systems better so that, ultimately, we can apply what we learn to more complicated multireference systems and to the design of new exchange-correlation functionals for treating multireference systems more adequately. Fourteen main-group diatomic molecules and one triatomic molecule (including radicals, cations, and anions, as well as neutral closed-shell species) have been studied for this article. Eight of these molecules contain a group-1 element, and six contain a group-2 element. Seven of these molecules are multireference systems, and eight of them are single-reference systems. Fifty-three exchange-correlation functionals of 11 types [local spin-density approximation (LSDA), generalized gradient approximation (GGA), nonseparable gradient approximation (NGA), global-hybrid GGA, meta-GGA, meta-NGA, global-hybrid meta GGA, range-separated hybrid GGA, range-separated hybrid meta-GGA, range-separated hybrid meta-NGA, and DFT augmented with molecular mechanics damped dispersion (DFT-D)] and the Hartree–Fock method have been applied to calculate the bond distance, bond dissociation energy (BDE), and dipole moment of these molecules. All of the calculations are converged to a stable solution by allowing the symmetry of the Slater determinant to be broken. A reliable functional should not only predict an accurate BDE but also predict accurate components of the BDE, so each bond dissociation energy has been decomposed into ionization potential (IP) of the electropositive element, electron affinity of the electronegative bonding partner (EA), atomic excitation energy (EE) to prepare the valence states of the interacting partners, and interaction energy (IE) of the valence-prepared states. Adding Hartree–Fock exchange helps to obtain better results for atomic excitation energy, and this leads to improvements in getting the right answer for the right reason. The following functionals are singled out for reasonably good performance on all three of bond distance, BDE, and dipole moment: B97-1, B97-3, MPW1B95, M05, M06, M06-2X, M08-SO, N12-SX, O3LYP, TPSS, τ-HCTHhyb, and GAM; all but two (TPSS and GAM) of these functionals are hybrid functionals.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2015

Van der Waals coefficients beyond the classical shell model

Van der Waals (vdW) coefficients can be accurately generated and understood by modelling the dynamic multipole polarizability of each interacting object. Accurate static polarizabilities are the key to accurate dynamic polarizabilities and vdW coefficients. In this work, we present and study in detail a hollow-sphere model for the dynamic multipole polarizability proposed recently by two of the present authors (JT and JPP) to simulate the vdW coefficients for inhomogeneous systems that allow for a cavity. The inputs to this model are the accurate static multipole polarizabilities and the electron density. A simplification of the full hollow-sphere model, the single-frequency approximation (SFA), circumvents the need for a detailed electron density and for a double numerical integration over space. We find that the hollow-sphere model in SFA is not only accurate for nanoclusters and cage molecules (e.g., fullerenes) but also yields vdW coefficients among atoms, fullerenes, and small clusters in good agreement with expensive time-dependent density functional calculations. However, the classical shell model (CSM), which inputs the static dipole polarizabilities and estimates the static higher-order multipole polarizabilities therefrom, is accurate for the higher-order vdW coefficients only when the interacting objects are large. For the lowest-order vdW coefficient C{sub 6}, SFA and CSM are exactly the same. The higher-order (C{sub 8} and C{sub 10}) terms of the vdW expansion can be almost as important as the C{sub 6} term in molecular crystals. Application to a variety of clusters shows that there is strong non-additivity of the long-range vdW interactions between nanoclusters.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗