Quasi-harmonic Approximation Research Articles

Crystal structures and mechanical properties of tungsten monocarbide predicted by first-principles investigations

The crystal structure of tungsten monocarbide (WC) is researched from 0 to 650 GPa through first principles calculations. The results verify that the experimental structure (hP2-WC) with the space group P6¯m2 is the most stable phase in a wide range of pressure. Above 231 GPa, a new stable structure (space group P63/mmc, hP4-WC) is found to be the most stable phase, and it will transform to a CsCl-type phase (cF8-WC) around 582 GPa. Phonon calculations reveal that the hP4-WC phase is dynamically stable and may be a metastable structure at ambient conditions. The cF8-WC phase possesses dynamical stability above 20 GPa. The hP4-WC phase is a low compressible material with a large bulk modulus of 377 GPa at zero pressure. The hardness values of hP2-WC and hP4-WC at zero pressure are 32 and 21 GPa, respectively, while the cF8-WC phase possesses a hardness of 21 GPa at 20 GPa, implying that these phases are potential hard materials. The temperature–pressure phase boundary of WC is obtained by means of the quasi-harmonic approximation method. As the temperature increases, the transition pressure from hP2-WC to hP4-WC remained nearly unchanged. The transition pressure between hP4-WC and cF8-WC decreases with the increasing temperature.

Equation of State for Solid Benzene Valid for Temperatures up to 470 K and Pressures up to 1800 MPa

The thermodynamic property data for solid phase I of benzene are reviewed and utilized to develop a new fundamental equation of state (EOS) based on Helmholtz energy, following the methodology used for solid phase I of CO2 by Trusler [J. Phys. Chem. Ref. Data 40, 043105 (2011)]. With temperature and molar volume as independent variables, the EOS is able to calculate all thermodynamic properties of solid benzene at temperatures up to 470 K and at pressures up to 1800 MPa. The model is constructed using the quasi-harmonic approximation, incorporating a Debye oscillator distribution for the vibrons, four discrete modes for the librons, and a further 30 distinct modes for the internal vibrations of the benzene molecule. An anharmonic term is used to account for inevitable deviations from the quasi-harmonic model, which are particularly important near the triple point. The new EOS is able to describe the available experimental data to a level comparable with the likely experimental uncertainties. The estimated relative standard uncertainties of the EOS are 0.2% and 1.5% for molar volume on the sublimation curve and in the compressed solid region, respectively; 8%–1% for isobaric heat capacity on the sublimation curve between 4 K and 278 K; 4% for thermal expansivity; 1% for isentropic bulk modulus; 1% for enthalpy of sublimation and melting; and 3% and 4% for the computed sublimation and melting pressures, respectively. The EOS behaves in a physically reasonable manner at temperatures approaching absolute zero and also at very high pressures.

Thermal expansion of continuous random networks of carbon

Homogeneous and isotropic continuous random networks (CRNs) of carbon were created spanning a large portion of hybridization space, from sp3-rich configurations up to 70% sp2 carbon. For each of these CRNs, the volumetric coefficient of thermal expansion (CTE) was calculated according to the quasi-harmonic approximation using the Brenner potential. The CTE of H-free, sp3-rich CRNs is similar to that of diamond and almost constant up to about 50% sp2-C. The CTE at 300 K varies from about 3.5 ×10−6K−1 to 5.5×10−6K−1, linearly increasing with the sp2-C content for CRNs above ca. 40% sp2-C, in good agreement with previous experimental studies. A small fraction of 10% to 15% of the CRNs, irrespective of composition, exhibit volumetric negative thermal expansion (NTE) over a limited temperature range around 100 K to 200 K.

Thermodynamic and electronic properties of ReN2 polymorphs at high pressure

High pressure synthesis of rhenium nitride pernitride ReN$_2$ with crystal structure unusual for transition metal dinitrides and high values of hardness and bulk modulus attracted significant attention to this system. We investigate the thermodynamic and electronic properties of the P2$_1$/c phase of ReN$_2$ and compare them with two other polytypes, C2/m and P4/mbm phases suggested in the literature. Our calculations of the formation enthalpy at zero temperature show that the former phase is the most stable of the three up to pressure p=170 GPa, followed by the stabilization of the P4/mbm phase at higher pressure. The theoretical prediction is confirmed by diamond anvil cell synthesis of the P4/mbm ReN$_2$ at $\approx$175 GPa. Considering the effects of finite temperature in the quasi-harmonic approximation at p=100 GPa we demonstrate that the P2$_1$/c phase has the lowest free energy of formation at least up to 1000 K. Our analysis of the pressure dependence of the electronic structure of the rhenium nitride pernitride shows a presence of two electronic topological transitions around 18 GPa, when the Fermi surface changes its topology due to an appearance of a new electron pocket at the high-symmetry Y$_2$ point of the Brillouin zone while the disruption of a neck takes place slightly off from the $\Gamma$-A line.

Calculation of Energy for RNA/RNA and DNA/RNA Duplex Formation by Molecular Dynamics Simulation

The development of approaches for predictive calculation of hybridization properties of various nucleic acid (NA) derivatives is the basis for the rational design of the NA-based constructs. Modern advances in computer modeling methods provide the feasibility of these calculations. We have analyzed the possibility of calculating the energy of DNA/RNA and RNA/RNA duplex formation using representative sets of complexes (65 and 75 complexes, respectively). We used the classical molecular dynamics (MD) method, the MMPBSA or MMGBSA approaches to calculate the enthalpy (ΔH°) component, and the quasi-harmonic approximation (Q-Harm) or the normal mode analysis (NMA) methods to calculate the entropy (ΔS°) contribution to the Gibbs energy (Delta G_{{37}}^{^circ } ) of the NA complex formation. We have found that the MMGBSA method in the analysis of the MD trajectory of only the NA duplex and the empirical linear approximation allow calculation of the enthalpy of formation of the DNA, RNA, and hybrid duplexes of various lengths and GC content with an accuracy of 8.6%. Within each type of complex, the combination of rather efficient MMGBSA and Q-Harm approaches being applied to the trajectory of only the bimolecular complex makes it possible to calculate the Delta G_{{37}}^{^circ } of the duplex formation with an error value of 10%. The high accuracy of predictive calculation for different types of natural complexes (DNA/RNA, DNA/RNA, and RNA/RNA) indicates the possibility of extending the considered approach to analogs and derivatives of nucleic acids, which gives a fundamental opportunity in the future to perform rational design of new types of NA-targeted sequence-specific compounds.

First-principles study on the thermodynamic and thermoelastic properties of pyrophyllite at high temperatures

Pyrophyllite [Al2Si4O10(OH)2] is mainly used in the field of refractories and super-hard materials due to its excellent thermal properties such as good heat resistance, stable chemical properties and crystal structure stability during heating. However, its thermodynamic properties have rarely been fully described at the atomic scale. In this study, the thermodynamic and thermoelastic properties of pyrophyllite are reported in detail, based on the first principles. Firstly, the relationship between the Helmholtz free energy and volume (E−V) under different temperatures was fitted using the quasi-harmonic approximation (QHA), which confirmed the thermal expansion effect of pyrophyllite at 0–900 K. Phonon dispersion curves at different temperatures showed that pyrophyllite maintained phase stability within a given temperature range (0–900 K). Furthermore, its elastic behaviors at 0, 300, 450, 600, 750, and 900 K are discussed. As the temperature increased from 0 K to 900 K, the elastic modulus of pyrophyllite decreased to varying degrees, which indicated that high temperature weakened its ability to resist external pressure. This elastic softening phenomenon was caused by the thermal expansion effect. The changes of Paugh's modulus ratio (G/B) clearly indicated that the ductility of pyrophyllite under high temperature (up to 900 K) was greater than that at 0 K. Overall, high temperature increases the volume of pyrophyllite and decreases its strength. Thus, when pyrophyllite is used in the field of refractories, the particle size and amount of pyrophyllite should be reasonably selected to achieve the balance between the volume stability and strength of the material. These findings enrich our theoretical knowledge of pyrophyllite behavior at high temperatures at the atomic scale, which is difficult to determine accurately using physical experiments.

Efficient Gaussian process regression for prediction of molecular crystals harmonic free energies

We present a method to accurately predict the Helmholtz harmonic free energies of molecular crystals in high-throughput settings. This is achieved by devising a computationally efficient framework that employs a Gaussian Process Regression model based on local atomic environments. The cost to train the model with ab initio potentials is reduced by starting the optimization of the framework parameters, as well as the training and validation sets, with an empirical potential. This is then transferred to train the model based on density-functional theory potentials, including dispersion-corrections. We benchmarked our framework on a set of 444 hydrocarbon crystal structures, comprising 38 polymorphs and 406 crystal structures either measured in different conditions or derived from these polymorphs. Superior performance and high prediction accuracy, with mean absolute deviation below 0.04 kJ mol−1 per atom at 300 K is achieved by training on as little as 60 crystal structures. Furthermore, we demonstrate the predictive efficiency and accuracy of the developed framework by successfully calculating the thermal lattice expansion of aromatic hydrocarbon crystals within the quasi-harmonic approximation, and predict how lattice expansion affects the polymorph stability ranking.

Tri-Base Synergy in Sulfuric Acid-Base Clusters

Synergistic effects between different bases can greatly enhance atmospheric sulfuric acid (SA)-base cluster formation. However, only the synergy between two base components has previously been investigated. Here, we extend this concept to three bases by studying large atmospherically relevant (SA)3(base)3 clusters, with the bases ammonia (A), methylamine (MA), dimethylamine (DMA), trimethylamine (TMA) and ethylenediamine (EDA). Using density functional theory—ωB97X-D/6-31++G(d,p)—we calculate the cluster structures and vibrational frequencies. The thermochemical parameters are calculated at 29,815 K and 1 atm, using the quasi-harmonic approximation. The binding energies of the clusters are calculated using high level DLPNO-CCSD(T0)/aug-cc-pVTZ. We find that the cluster stability in general depends on the basicity of the constituent bases, with some noteworthy additional guidelines: DMA enhances the cluster stability, TMA decreases the cluster stability and there is high synergy between DMA and EDA. Based on our calculations, we find it highly likely that three, or potentially more, different bases, are involved in the growth pathways of sulfuric acid-base clusters.

Ab-initio investigation of structural, mechanical, thermodynamic, electronic, magnetic and thermoelectric properties of half-metallic d0 half-Heusler alloys LiXSi (X=Ca, Sr)

Structural, mechanical, thermodynamic, electronic, and magnetic properties of LiXSi (X ​= ​Ca, Sr) d0 half-Heusler alloys have been evaluated by implementing the GGA-PBE approximation in the Density Functional Theory calculations. The Boltzmann transport theory has been implemented for examining the thermoelectric properties of the alloys. The spin-polarized calculations show that LiCaSi and LiSrSi crystallize in the ferromagnetic α-phase equilibrium states. Spin-resolved band structures depict complete spin polarization obtained at the Fermi level and confirm the true half-metallic nature of the alloys. The calculated total magnetic moment is 0.99 ​μB for LiCaSi and 1.00 ​μB for LiSrSi, which very closely follow the Slater-Pauling rule. The observed ferromagnetic characteristics arise mostly from the spin-polarization of the 2p states in Si for both alloys. The LiXSi (X ​= ​Ca, Sr) alloys are found to be mechanically stable according to the Born stability criteria for cubic structured systems. Both the alloys are predicted to be ductile in nature. Important thermodynamic properties such as Debye temperature and CV have been calculated by implementing the quasi-harmonic approximation. The calculations for the transport properties show that the alloys are p-type materials with high Seebeck coefficients and electrical conductivities. The dimensionless thermoelectric figure of merit obtained for both the alloys at 300K is close to unity, which confirms the high thermoelectric efficiency of the alloys.

Quasi-harmonic thermoelasticity of palladium, platinum, copper, and gold from first principles

We calculate the temperature-dependent elastic constants (ECs) of palladium, platinum, copper and gold within the quasi-harmonic approximation using a first-principles approach and evaluating numerically the second derivatives of the Helmholtz free-energy with respect to strain at the minimum of the free-energy itself. We find an overall good agreement with the experimental data although the anomalies of palladium and platinum reported at room temperature are not reproduced. The contribution of electronic excitations is also investigated: we find that it is non-negligible for the C 44 ECs of palladium and platinum while it is irrelevant in the other cases. Its effect is not sufficient to explain the details of the anomalies found by experiments, not even when, in the case of platinum, we take into account the electron–phonon interaction. Lastly, the effect of the exchange and correlation functional is addressed and it is found that it is important at T = 0 K, while all functionals give similar temperature dependencies.

First-principles studies of the structural, elastic, vibrational, and thermodynamic properties of nonlinear chalcogenide LiGaTe2 under pressure

First-principles calculations for the structural, elastic, thermal and vibrational properties of LiGaTe2 under pressure up to 5GPa have been systematically investigated within the framework of the density functional theory. The lattices parameters found a, c and the bulk modulus B under zero pressure and zero temperature are close with the experimental data and the theoretical results. The analysis of the mechanical constants and the frequencies of the phonons found, confirms that the compound LiGaTe2 is dynamically and mechanically stable up to 5GPa. The effect of pressure on elastic constants Cij, bulk modulus B, Young's modulus E, Poisson ratio ν, shear modulus G, and compressibility k of LiGaTe2 are also successfully obtained and discussed. In addition, the thermal properties of LiGaTe2, such the heat capacity CV and Cp, the grüneisen parameter γ and the thermal expansion coefficient α are predicted by the quasi-harmonic approximation (QHA).

High temperature (up to 1200 °C) thermal-mechanical stability of Si and Ni doped CrN framework coatings

High temperature thermal-mechanical stability of tribological thin coatings is extremely important to a large number of applications in modern industries. DC magnetron sputtering of single metallic element (Cr, Si) and alloy (Ni:Cr) targets formed transition metal nitrides film coatings, CrSiN and CrNiN onto M2 steel. High temperature in-situ synchrotron X-ray diffraction, in the range 25 °C–700 °C, obtained experimental data for a range of structural and mechanical properties. Furthermore, experimental room temperature Nanoindentation measurements, made before and after the in-situ heating cycle, provided corresponding hardness and shear modulus results. The structural results identified microstructure and phase transformation changes, while the mechanical results identified microstrain, hardness, elastic modulus and deformation resistance properties of the coatings. Density functional theory (DFT) and quasi-harmonic approximation (QHA) modelled the high temperature thermal and mechanical properties such as: Young's modulus, shear modulus and thermal expansion coefficients (populated up to 1200 °C). Estimates of hardness are made by correlating the bulk phase hardness and shear modulus, of the CrN and Ni phases, as a function of temperature. Results indicate that Si doping enhances the hardness of the CrN framework, increasing from 29 to 36 GPa and improves the coatings elastic modulus, and resistance to deformation. However the addition of Ni reduced these properties. Furthermore, formation of (Cr,Si)N and Ni(Cr) solid solutions is inferred from DFT, Rietveld and lattice constant analysis.

Mechanical and thermodynamics properties of Al2Ca and Mg2Ca under various pressure: A first principle study

The mechanical and thermodynamic properties of Al2Ca and Mg2Ca in the pressure range of 0~100 Gpa were investigated using first-principles calculations. The structural parameters, such as lattice constant ratio, unit cell volume ratio, density, were investigated. The calculated elastic constants satisfy the born’s stability criterion, indicating that they are mechanically stable at normal and high pressure. Mechanical parameters such as bulk modulus, shear modulus, and Young’s modulus of polycrystalline materials have been derived from single-crystal elastic constants. The Poisson’s ratio and anisotropy were investigated. The results show that the B/G value of Mg2Ca is greater than 1.75, indicating it is a ductile phase under various pressures. When the pressure was equal to 40 Gpa, Al2Ca was transferred brittle to toughness, and the bulk modulus, shear modulus, and Young’s modulus of Al2Ca were all larger than those of Mg2Ca, indicating that the comprehensive mechanical properties of Al2Ca are better than those of Mg2Ca. The constant heat capacity obtained by the quasi-harmonic approximation indicates that the ability of Mg2Ca to release or store heat is greater than that of Al2Ca. Moreover, the coefficient of thermal expansion (α) increases exponentially at lower temperatures and linearly at higher temperatures for both alloys.

Ab initio study of the structure and relative stability of MgSiO4H2 polymorphs at high pressures and temperatures

Abstract Using particle swarm optimization with density functional theory, we identify the positions of hydrogen in a hypothetical Mg-end-member of phase egg (MgSiO4H2) and predict the most stable crystal structures with MgSiO4H2 stoichiometry at pressures between 0 and 300 GPa. The particle swarm optimization method consistently and systematically identifies phase H as the energetically most stable structure in the pressure range 10–300 GPa at 0 K. Phase Mg-egg has a slightly higher energy compared to phase H at all relevant pressures, such that the energy difference nearly plateaus at high pressures; however, the combined effects of temperature and chemical substitutions may decrease or even reverse the energy difference between the two structures. We find a new MgSiO4H2 phase with the P43212 space group that has topological similarities to phase Mg-egg and is energetically preferred to phase H at 0–10 GPa and 0 K. We compute the free energies for phase Mg-egg, phase P43212, and phase H at 0–30 GPa within the quasi-harmonic approximation and find that the effect of temperature is relatively small. At 1800 K, the stability field of phase P43212 relative to the other polymorphs increases to 0–14 GPa, while pure phase Mg-egg remains energetically unfavorable at all pressures. Simulated X-ray diffraction patterns and Raman spectra are provided for the three phases. Additionally, the crystallographic information for two metastable polymorphs with the P1 space group is provided. Our results have implications for the deep hydrogen cycle in that we identify two novel potential carrier phases for hydrogen in the mantles of terrestrial planets and assess their stability relative to phase H. We determine that further experimental and computational investigation of an extended compositional space remains necessary to establish the most stable dense hydrated silicate phases.

First-principles calculations to investigate electronic, optical, thermodynamic and thermoelectric properties of new Na6ZnX4 (X=O, S, Se) ternary alloys

We report a first principle investigation on the new type of ternary semiconductor alloys Na6ZnX4 (X = O, S, Se). Structural and elastic properties show that these elements present low formation energy and it's mechanically stable. The obtained electronic energy band gap results show direct electronic transition along the Γ-symmetry direction involving X-s states (X = O, S, Se) in the valence band maximum and mixed contributions from Na-s and Zn-d states. Optical absorption as high as 104 cm−1 is observed in the visible region in the spectrum, with maximum absorbance being as high as 106 cm−1 in the ultraviolet region. Quasi-harmonic approximations are used to investigate the thermodynamic properties, which show the solid characteristics of the alloys even at high temperature range. Our investigation shows Na6ZnSe4 and Na6ZnO4 to be the best materials among the studied compounds for photovoltaic applications and Na6ZnS4 as the best thermoelectric candidate. The electronic band structure and the physical properties reported for the compounds are reveled for the first time.

Computational study of Li3BO3 and Li3BN2 II: Stability analysis of pure phases and of model interfaces with Li anodes

Both ${\mathrm{Li}}_{3}{\mathrm{BO}}_{3}$ and ${\mathrm{Li}}_{3}{\mathrm{BN}}_{2}$ materials have promising properties for use in all solid-state batteries and other technologies dependent on electrolytes with significant ionic conductivity. As the second of a two-part study, the structural properties of ${\mathrm{Li}}_{3}{\mathrm{BO}}_{3}$ and three reported phases of ${\mathrm{Li}}_{3}{\mathrm{BN}}_{2}$ are investigated using first-principles modeling techniques. For $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{Li}}_{3}{\mathrm{BN}}_{2}$, the tetragonal $P{4}_{2}/mnm$ structure reported in the literature is found to be unstable as evidenced by imaginary phonon modes near the M point of its Brillouin zone. Our simulations within the harmonic approximation suggest that the real $\ensuremath{\alpha}$ phase has the orthorhombic space group symmetry $Pmmn$ formed with twice as many formula units and tiny adjustments of the equivalent lattice parameters and fractional coordinates. Extending the analysis of the $Pmmn\phantom{\rule{4pt}{0ex}}\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{Li}}_{3}{\mathrm{BN}}_{2}$ structure to the quasiharmonic approximation improves the agreement between the room-temperature x-ray pattern reported in the literature and the corresponding simulation results. In anticipation of the use of the monoclinic phases of ${\mathrm{Li}}_{3}{\mathrm{BO}}_{3}$ and ${\mathrm{Li}}_{3}{\mathrm{BN}}_{2}$ in Li ion conducting applications, chemical stability is investigated in terms of free-energy differences of possible decomposition and Li reaction processes, finding encouraging results. As further investigations of ${\mathrm{Li}}_{3}{\mathrm{BO}}_{3}$ and $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{Li}}_{3}{\mathrm{BN}}_{2}$ as electrolyte or coating materials, particularly for use with Li metal anodes, idealized electrolyte/Li interfaces were investigated in terms of their geometric, energetic, and electronic properties. The results find the electrolyte/Li interfaces to be quite favorable, perhaps comparable to the pioneering LiPON/Li system.

Vibrational and Thermodynamic Properties of Hydrous Iron-Bearing Lowermost Mantle Minerals

The vibrational and thermodynamic properties of minerals are key to understanding the phase stability and the thermal structure of the Earth’s mantle. In this study, we modeled hydrous iron-bearing bridgmanite (Brg) and post-perovskite (PPv) with different [Fe3+-H] defect configurations using first-principles calculations combined with quasi-harmonic approximations (QHA). Fe3+-H configurations can be vibrationally stable in Brg and PPv; the site occupancy of this defect will strongly affect its thermodynamic properties and particularly its response to pressure. The presence of Fe3+-H introduces distinctive high-frequency vibrations to the crystal. The frequency of these peaks is configuration dependence. Of the two defect configurations, [FeSi′+OH·] makes large effects on the thermodynamic properties of Brg and PPv, whereas [VMg″+FeMg·+OH·] has negligible effects. With an expected lower mantle water concentrations of <1000 wt. ppm the effect of Fe3+-H clusters on properties such as heat capacity and thermal expansion is negligible, but the effect on the Grüneisen parameter γ can be significant (~1.2%). This may imply that even a small amount of water may affect the anharmonicity of Fe3+-bearing MgSiO3 in lower mantle conditions and that when calculating the adiabaticity of the mantle, water concentrations need to be considered.

Thiol Reactivity of N-Aryl α-Methylene-γ-lactams: A Reactive Group for Targeted Covalent Inhibitor Design.

Kinase activity can be modulated reversibly or irreversibly by the reaction of targeted covalent inhibitors with nucleophilic residues in protein active sites. Herein, we present thiol reactivity studies that support α-methylene-γ-lactams as tunable surrogates for the highly reactive α-methylene-γ-lactones. The reactivity of the α-methylene is modulated via the N substituent, and the reaction rates toward glutathione were determined via mass spectrometry. Density functional theory calculations of transition states of thiol additions to α-methylene-γ-lactams revealed that the use of the M06-2X functional with the SMD solvation model and methyl thiolate as a model nucleophile reliably predicts the relative reactivities of the α-methylene-γ-lactams, and quasiharmonic approximations improve the agreement between experiment and computation.

Electron- and hole-doping on ScH2 and YH2: effects on superconductivity without applied pressure

We present the evolution of the structural, electronic, and lattice dynamical properties, as well as the electron–phonon (el–ph) coupling and superconducting critical temperature (T c) of ScH2 and YH2 metal hydrides solid solutions, as a function of the electron- and hole-doping content. The study was performed within the density functional perturbation theory, taking into account the effect of zero-point energy through the quasi-harmonic approximation, and the solid solutions Sc1−x M x H2 (M = Ca, Ti) and Y1−x M x H2 (M = Sr, Zr) were modeled by the virtual crystal approximation. We have found that, under hole-doping (M = Ca, Sr), the ScH2 and YH2 hydrides do not improve their el–ph coupling properties, sensed by λ(x). Instead, by electron-doping (M = Ti, Zr), the systems reach a critical content x ≈ 0.5 where the latent coupling is triggered, increasing λ as high as 70%, in comparison with its λ(x = 0) value. Our results show that T c quickly decreases as a function of x on the hole-doping region, from x = 0.2 to x = 0.9, collapsing at the end. Alternatively, for electron-doping, T c first decreases steadily until x = 0.5, reaching its minimum, but for x > 0.5 it increases rapidly, reaching its maximum value of the entire range at the Sc0.05Ti0.95H2 and Y0.2Zr0.8H2 solid solutions, demonstrating that electron-doping can improve the superconducting properties of pristine metal hydrides, in the absence of applied pressure.

Isotope effect of methane adsorbed on fcc metal (1 1 1) surfaces

By means of the van der Waals density functional and quasi-harmonic approximation, we predict that the adsorption potential energies of CD4 on various fcc metal (111) surfaces are shallower than those of CH4, whereas the equilibrium distances from the surface are larger. This isotope effect originates from the softening of the CH bond pointing toward the surface, leading to the significant lowering of its vibrational frequency and hence the large zero-point energy difference between CH4 and CD4. The CH bond softening is attributed to the hybridization of the lowest unoccupied molecular orbital with the substrate state.