Showing papers by "Gustavo E. Scuseria published in 2008"
TL;DR: A revised Perdew-Burke-Ernzerhof generalized gradient approximation is introduced that improves equilibrium properties of densely packed solids and their surfaces.
Abstract: Popular modern generalized gradient approximations are biased toward the description of free-atom energies. Restoration of the first-principles gradient expansion for exchange over a wide range of density gradients eliminates this bias. We introduce a revised Perdew-Burke-Ernzerhof generalized gradient approximation that improves equilibrium properties of densely packed solids and their surfaces.
7,797 citations
TL;DR: By using a calibration curve, it is found that the LR functionals systematically allow an even more consistent description of the low-lying excited-state energies than the conventional hybrids.
Abstract: The π → π* transitions of more than 100 organic dyes from the major classes of chromophores (quinones, diazo, ...) have been investigated using a Time-Dependent Density Functional Theory (TD-DFT) procedure relying on large atomic basis sets and the systematic modeling of solvent effects. These calculations have been performed with pure (PBE) as well as conventional (PBE0) and long-range (LR) corrected hybrid functionals (LC-PBE, LC-ωPBE, and CAM-B3LYP). The computed wavelengths are systematically guided by the percentage of exact exchange included at intermediate interelectronic distance, i.e., the λmax value always follows the PBE > PBE0 > CAM-B3LYP > LC-PBE > LC-ωPBE > HF sequence. The functional giving the best estimates of the experimental transition energies may vary, but PBE0 and CAM-B3LYP tend to outperform all other approaches. The latter functional is shown to be especially adequate to treat molecules with delocalized excited states. The mean absolute error provided by PBE0 is 22 nm (0.14 eV) wit...
766 citations
TL;DR: In this article, a comprehensive first-principles theoretical study of the electronic properties and half-metallic nature of finite rectangular graphene nanoribbons is presented, and the bisanthrene isomer of the C28H14 molecule is identified as the smallest graphene derivative to present a spin-polarized ground state.
Abstract: A comprehensive first-principles theoretical study of the electronic properties and half-metallic nature of finite rectangular graphene nanoribbons is presented. We identify the bisanthrene isomer of the C28H14 molecule to be the smallest graphene derivative to present a spin-polarized ground state. Even at this quantum dot level, the spins are predicted to be aligned antiferromagnetically at the two zigzag edges of the system. As a rule of thumb, we find that zigzag graphene edges that are at least three consecutive units long will present spin polarization if the width of the system is 1 nm or wider. Room temperature detectability of the magnetic ordering is predicted for ribbons with zigzag edges 1 nm and longer. For the longer systems studied, spin wave structures appear in some high spin multiplicity states. Energy gap oscillations with the length of the zigzag edge are observed. The amplitude of these oscillations is found to be smaller than that predicted for infinite ribbons. The half-metallic nature of the ribbons under an external in-plane electric field is found to be preserved even for finite and extremely short systems.
277 citations
TL;DR: In this article, the equivalence between the random phase approximation (RPA) to the ground state correlation energy and a ring-diagram simplification of the coupled cluster doubles (CCD) equations was shown.
Abstract: We present an analytic proof demonstrating the equivalence between the random phase approximation (RPA) to the ground state correlation energy and a ring-diagram simplification of the coupled cluster doubles (CCD) equations. In the CCD framework, the RPA equations can be solved in O(N4) computational effort, where N is proportional to the number of basis functions.
272 citations
TL;DR: Analytic results and preliminary thermochemical tests indicate that this model improves upon the simple, local-density-based exchange hole model of Iikura et al, and enables fully analytic evaluation of range-separated hybrid density functionals.
Abstract: We propose a general model for the spherically averaged exchange hole corresponding to a generalized gradient approximation (GGA) exchange functional. Parameters are reported for several common GGAs. Our model is based upon that of Ernzerhof and Perdew [J. Chem. Phys. 109, 3313 (1998)]. It improves upon the former by precisely reproducing the energy of the parent GGA, and by enabling fully analytic evaluation of range-separated hybrid density functionals. Analytic results and preliminary thermochemical tests indicate that our model also improves upon the simple, local-density-based exchange hole model of Iikura et al. [J. Chem. Phys. 115, 3540 (2001)].
251 citations
TL;DR: In this paper, the exact exchange-correlation hole density around an electron is constructed by interpolating between different approximations suitable for two extreme regions of the electron density, i.e., the normal and abnormal regions.
Abstract: We construct a nonlocal density functional approximation with full exact exchange, while preserving the constraint-satisfaction approach and justified error cancellations of simpler semilocal functionals. This is achieved by interpolating between different approximations suitable for two extreme regions of the electron density. In a ``normal'' region, the exact exchange-correlation hole density around an electron is semilocal because its spatial range is reduced by correlation and because it integrates over a narrow range to $\ensuremath{-}1$. These regions are well described by popular semilocal approximations (many of which have been constructed nonempirically), because of proper accuracy for a slowly varying density or because of error cancellation between exchange and correlation. ``Abnormal'' regions, where nonlocality is unveiled, include those in which exchange can dominate correlation (one-electron, nonuniform high density, and rapidly varying limits), and those open subsystems of fluctuating electron number over which the exact exchange-correlation hole integrates to a value greater than $\ensuremath{-}1$. Regions between these extremes are described by a hybrid functional mixing exact and semilocal exchange energy densities locally, i.e., with a mixing fraction that is a function of position $\mathbf{r}$ and a functional of the density. Because our mixing fraction tends to 1 in the high-density limit, we employ full exact exchange according to the rigorous definition of the exchange component of any exchange-correlation energy functional. Use of full exact exchange permits the satisfaction of many exact constraints, but the nonlocality of exchange also requires balanced nonlocality of correlation. We find that this nonlocality can demand at least five empirical parameters, corresponding roughly to the four kinds of abnormal regions. Our local hybrid functional is perhaps the first accurate fourth-rung density functional or hyper-generalized gradient approximation, with full exact exchange, that is size-consistent in the way that simpler functionals are. It satisfies other known exact constraints, including exactness for all one-electron densities, and provides an excellent fit to the 223 molecular enthalpies of formation of the G3/99 set and the 42 reaction barrier heights of the BH42/03 set, improving both (but especially the latter) over most semilocal functionals and global hybrids. Exact constraints, physical insights, and paradigm examples hopefully suppress ``overfitting.''
209 citations
TL;DR: These results provide the first direct evidence of the room-temperature catalytic hydrodechlorination of a chlorinated solvent, a potentially important pathway for groundwater cleanup, and highlight the exciting prospects of studying catalytic processes in water in situ, like those involved in biomass conversion and proton-exchange membrane fuel cells.
Abstract: Insight into the nature of transient reaction intermediates and mechanistic pathways involved in heterogeneously catalyzed chemical reactions is obtainable from a number of surface spectroscopic techniques. Carrying out these investigations under actual reaction conditions is preferred but remains challenging, especially for catalytic reactions that occur in water. Here, we report the direct spectroscopic study of the catalytic hydrodechlorination of 1,1-dichloroethene in H2O using surface-enhanced Raman spectroscopy (SERS). With Pd islands grown on Au nanoshell films, this reaction can be followed in situ using SERS, exploiting the high enhancements and large active area of Au nanoshell SERS substrates, the transparency of Raman spectroscopy to aqueous solvents, and the catalytic activity enhancement of Pd by the underlying Au metal. The formation and subsequent transformation of several adsorbate species was observed. These results provide the first direct evidence of the room-temperature catalytic hydr...
182 citations
TL;DR: The band energy differences of solids calculated with screened hybrid density functionals, such as the functional of Heyd-Scuseria-Ernzerhof (HSE), reproduce experimental band gaps with a high degree of accuracy.
Abstract: The band energy differences of solids calculated with screened hybrid density functionals, such as the functional of Heyd-Scuseria-Ernzerhof (HSE), reproduce experimental band gaps with a high degree of accuracy. This unexpected result is here rationalized by observing that band energy differences obtained from generalized Kohn-Sham calculations with screened (short-range) Hartree-Fock-type exchange approach the excitation energies obtained via time-dependent density functional calculations with the corresponding unscreened functional. The latter are expected to be the accurate predictions of the experimental optical absorption spectra. While the optimum screening parameter (omega) is system dependent, the HSE standard value of omega=0.11 bohr(-1) represents a reasonable compromise across diverse systems.
146 citations
TL;DR: In this article, the authors extended the performance of the HISS functional for several simple properties and applied it to the dissociation of homonuclear diatomic cations and to the polarizability of linear H2 chains.
Abstract: While hybrid functionals are largely responsible for the utility of modern Kohn-Sham density functional theory, they are not without their weaknesses. In the solid state, the slow decay of their nonlocal Hartree-Fock-type exchange makes hybrids computationally demanding and can introduce unphysical effects. Both problems can be remedied by a screened hybrid which uses exact exchange only at short-range. Many molecular properties, in contrast, benefit from the inclusion of long-range exact exchange. Recently, the authors reconciled these two seemingly contradictory requirements by introducing the HISS functional [ J. Chem. Phys. 2007 , 127 , 221103 ], which uses exact exchange only in the middle range. In this paper, we expand upon our previous work, benchmarking the performance of the HISS functional for several simple properties and applying it to the dissociation of homonuclear diatomic cations and to the polarizability of linear H2 chains to determine the importance of middle-range exact exchange for these systems, which are expected to be sensitive to the asymptotic exchange potential.
145 citations
TL;DR: In this article, a density functional theory study of the electronic structure of copper phthalocyanine (CuPc) using several different (semi)local and hybrid functionals is presented.
Abstract: We present a systematic density functional theory study of the electronic structure of copper phthalocyanine (CuPc) using several different (semi)local and hybrid functionals and compare the results to experimental photoemission data. We show that semilocal functionals fail qualitatively for CuPc primarily because of underbinding of localized orbitals due to self-interaction errors. We discuss an appropriate choice of functional for studies of CuPc/metal interfaces and suggest the Heyd–Scuseria–Ernzerhof screened hybrid functional as a suitable compromise functional.
135 citations
TL;DR: This work proposes a novel method that uses a position-dependent screening function to add substantial flexibility for describing diverse electronic structures and satisfy a high-density scaling constraint better than the fixed screening approximation does.
Abstract: Range-separated (screened) hybrid functionals provide a powerful strategy for incorporating nonlocal exact (Hartree–Fock-type) exchange into density functional theory. Existing implementations of range separation use a fixed system-independent screening parameter. Here, we propose a novel method that uses a position-dependent screening function. These locally range-separated hybrids add substantial flexibility for describing diverse electronic structures and satisfy a high-density scaling constraint better than the fixed screening approximation does.
TL;DR: Semilocal density functional theory predictions for the barrier heights of representative hydrogen transfer, heavy-atom transfer, and nucleophilic substitution reactions are significantly improved in non-self-consistent calculations using Hartree-Fock orbitals.
Abstract: Semilocal density functional theory predictions for the barrier heights of representative hydrogen transfer, heavy-atom transfer, and nucleophilic substitution reactions are significantly improved in non-self-consistent calculations using Hartree–Fock orbitals. Orbitals from hybrid calculations yield related improvements. These results provide insight into compensating for one-electron self-interaction error in semilocal density functional theory.
TL;DR: In this article, the exact-exchange energy density and energy density of a semilocal density-functional approximation are used to model the static correlation, a strongly nonlocal functional of the electron density through a local hybrid functional.
Abstract: The exact-exchange energy density and energy density of a semilocal density-functional approximation are two key ingredients for modeling the static correlation, a strongly nonlocal functional of the electron density, through a local hybrid functional. Because energy densities are not uniquely defined, the conventional (Slater) exact-exchange energy density ${e}_{\text{x}}^{\text{ex}(\text{conv})}$ is not necessarily well suited for local mixing with a given semilocal approximation. We show how to transform ${e}_{\text{x}}^{\text{ex}(\text{conv})}$ in order to make it compatible with an arbitrary semilocal density functional, taking the nonempirical meta-generalized-gradient approximation of Tao, Perdew, Staroverov, and Scuseria as an example. Our additive gauge transformation function integrates to zero, satisfies exact constraints, and is most important where the density is dominated by a single orbital shape. We show that, as expected, the difference between semilocal and exact-exchange energy densities becomes more negative under bond stretching in $\text{He}_{2}{}^{+}$ and related systems. Our construction of ${e}_{\text{x}}^{\text{ex}(\text{conv})}$ by a resolution-of-the-identity method requires uncontracted basis functions.
TL;DR: Using an ab initio TD-DFT approach systematically accounting for the molecular surrounding effects, a large set of functionals has been reviewed for the determination of transition energies and structural parameters as mentioned in this paper.
TL;DR: This article reviews the work on developing novel exchange-correlation functionals that build upon the successes of global hybrids and focuses on more flexible functional forms, including local and range-separated hybrid functionals, constructed to obey known exact constraints and (ideally) to incorporate a minimum of empirical parametrization.
Abstract: Kohn−Sham density functional theory has become a standard method for modeling energetic, spectroscopic, and chemical reactivity properties of large molecules and solids. Density functional theory provides a rigorous theoretical framework for modeling the many-body exchange-correlation effects that dominate the computational cost of traditional wave function approaches. The advent of hybrid exchange-correlation functionals which incorporate a fraction of nonlocal exact exchange has solidified the prominence of density functional theory within computational chemistry. Hybrids provide accurate treatments of properties such as thermochemistry and molecular geometry. But they also exhibit some rather spectacular failures, and often contain multiple empirical parameters. This article reviews our work on developing novel exchange-correlation functionals that build upon the successes of global hybrids. We focus on more flexible functional forms, including local and range-separated hybrid functionals, constructed ...
TL;DR: The results show that B3LYP is the best functional for predicting vibrational frequencies (both fundamental and harmonic); the screened-PBE hybrid (HSE) density functional works best for infrared intensities, and the long-range corrected PBE (LC-omegaPBE), M 06-HF, and M06-L density functionals are almost as good as MP2 for predicting Raman activities.
Abstract: We present an assessment of different density functionals, with emphasis on range-separated hybrids, for the prediction of fundamental and harmonic vibrational frequencies, infrared intensities, and Raman activities. Additionally, we discuss the basis set convergence of vibrational properties of H2O with long-range corrected hybrids. Our results show that B3LYP is the best functional for predicting vibrational frequencies (both fundamental and harmonic); the screened-PBE hybrid (HSE) density functional works best for infrared intensities, and the long-range corrected PBE (LC-ωPBE), M06-HF, and M06-L density functionals are almost as good as MP2 for predicting Raman activities. We show the predicted Raman spectrum of adenine as an example of a medium-size molecule where a DFT/Sadlej pVTZ calculation is affordable and compare our results against the experimental spectrum.
TL;DR: Previously reported "charge-transfer" problems with semilocal TD-DFT excitations in molecules can be deduced from the analysis by taking the limit to infinite lattice constant.
Abstract: We examine the time-dependent density functional theory (TD-DFT) equations for calculating excitation energies in solids with Gaussian orbitals and analytically show that for semilocal functionals, their lowest eigenvalue collapses to the minimum band orbital energy difference. With the introduction of nonlocal Hartree–Fock-type exchange (as in hybrid functionals), this result is no longer valid, and the lowest TD-DFT eigenvalue reflects the appearance of excitonic effects. Previously reported “charge-transfer” problems with semilocal TD-DFT excitations in molecules can be deduced from our analysis by taking the limit to infinite lattice constant.
TL;DR: Spectroscopic data reveal specific interactions between ib uprofen and phospholipid moieties and indicate that the overall hydrophobicity of ibuprofen plays an important role in its intercalation in these membrane mimics.
Abstract: The incorporation of small molecules into lipid bilayers is a process of biological importance and clinical relevance that can change the material properties of cell membranes and cause deleterious side effects for certain drugs. Here we report the direct observation, using surface-enhanced Raman and IR spectroscopies (SERS, SEIRA), of the insertion of ibuprofen molecules into hybrid lipid bilayers. The alkanethiol-phospholipid hybrid bilayers were formed onto gold nanoshells by self-assembly, where the underlying nanoshell substrates provided the necessary enhancements for SERS and SEIRA. The spectroscopic data reveal specific interactions between ibuprofen and phospholipid moieties and indicate that the overall hydrophobicity of ibuprofen plays an important role in its intercalation in these membrane mimics.
TL;DR: The increased accuracy of LC-omegaPBE suggests that the inclusion of 100% Hartree-Fock exchange considered in the definition of this long-range corrected hybrid functional has important consequences for an accurate description of exchange and correlation effects on the electronic structure of open shell systems.
Abstract: The performance of the Heyd-Scuseria-Ernzerhorf (HSE) and single parameter long-range corrected Perdew-Burke-Ernzerhorf (LC-omegaPBE) range-separated hybrids for predicting magnetic coupling constants has been investigated for a broad set of magnetic molecular systems for which accurate experimental data exist. The set includes the H-He-H model system, two organic diradicals with different magnetic behaviors, and a series of Cu dinuclear complexes with a broad range of magnetic coupling values. Both HSE and LC-omegaPBE provide a significant improvement to standard hybrids such as the well-known hybrid Becke-3-parameters exchange with Lee-Yang-Parr correlation (B3LYP) functional. Nevertheless, the performance of these two range-separated hybrid functionals is different: HSE overestimates antiferromagnetic and ferromagnetic interactions in Cu dinuclear complexes (although significantly less than B3LYP), whereas LC-omegaPBE treats ferro- and antiferromagnetic couplings on a much more balanced way. The increased accuracy of LC-omegaPBE suggests that the inclusion of 100% Hartree-Fock exchange considered in the definition of this long-range corrected hybrid functional has important consequences for an accurate description of exchange and correlation effects on the electronic structure of open shell systems. On the other hand, HSE, which was developed with periodic systems in mind, also performs quite well (and better than B3LYP) thus opening the possibility of magnetic coupling studies in metal oxides and other challenging solids.
TL;DR: Some dimetal fullerenes M 2@C 60 (M = Cr, Mo, W) have been studied with computational quantum chemistry methods and it is shown that the transition metal diatomic molecules Cr 2, Mo 2, W 2 form exohedral complexes with C 60.
Abstract: Some dimetal fullerenes M2@C60 (M = Cr, Mo, W) have been studied with computational quantum chemistry methods. The transition metal diatomic molecules Cr2, Mo2, W2 form exohedral complexes with C60, while U2 forms a highly symmetric endohedral compound and it is placed in the center of the C60 cavity. This highly symmetric structure is an artifact due to the small size of the C60 cavity, which constrains U2 at the center. If a larger cavity is used, like C70 or C84, U2 preferentially binds the internal walls of the cavity and the U−U bond no longer exists.
TL;DR: Local hybrid functionals that incorporate a position-dependent admixture of short-range (screened) nonlocal exact [Hartree-Fock-type (HF)] exchange appear to provide a promising extension of existing local and range-separated hybrids.
Abstract: We present local hybrid functionals that incorporate a position-dependent admixture of short-range (screened) nonlocal exact [Hartree-Fock-type (HF)] exchange. We test two limiting cases: screened local hybrids with no long-range HF exchange and long-range-corrected local hybrids with 100% long-range HF exchange. Long-range-corrected local hybrids provide the exact asymptotic exchange-correlation potential in finite systems, while screened local hybrids avoid the problems inherent to long-range HF exchange in metals and small-bandgap systems. We treat these functionals self-consistently using the nonlocal exchange potential constructed from Kohn-Sham orbital derivatives. Generalized Kohn-Sham calculations with screened and long-range-corrected local hybrids can provide accurate molecular thermochemistry and kinetics, comparable to existing local hybrids of full-range exchange. Generalized Kohn-Sham calculations with existing full-range local hybrids provide results consistent with previous non-self-consistent and “localized local hybrid” calculations. These new functionals appear to provide a promising extension of existing local and range-separated hybrids.
TL;DR: An improvement in performance of the atomic orbital Laplace transformed second-order Møller-Plesset (AO-LT-MP2) method for periodic systems is reported using the resolution of identity (RI) technique.
Abstract: An improvement in performance of the atomic orbital Laplace transformed second-order Moller–Plesset (AO-LT-MP2) method for periodic systems is reported using the resolution of identity (RI) technique. Transformation of the two-electron integrals constitutes the main computational bottleneck of the AO-LT-MP2 method. A substitution of regular four-center integrals by their three center counterparts in the RI approximation naturally reduces the computational cost of the integral transformation step. The RI divergence problem in the presence of periodic boundary conditions is solved in our implementation by restricting the fitting domain. Accuracy and computational efficiency of the RI-AO-LT-MP2 approach are assessed on a set of one-dimensional test systems: trans-polyacetylene and anti-transoid polymethineimine.
TL;DR: In this article, the electronic properties and half-metallic nature of finite zigzag carbon nanotubes were investigated. But the results were limited to the case of nanoribbons with spin-polarized ground states.
Abstract: A comprehensive first-principles theoretical study of the electronic properties and half-metallic nature of finite zigzag carbon nanotubes is presented. Unlike previous reports, we find that all nanotubes studied present a spin-polarized ground state, where opposite spins are localized at the two zigzag edges in a long-range antiferromagnetic-type configuration. Relative stability analysis of the different spin states indicates that, for the shorter segments, spin-ordering should be present even at room temperature. The energy gap between the highest occupied and the lowest unoccupied molecular orbitals of the finite systems is found to be inversely proportional to the nanotube’s segment lengths, suggesting a route to control their electronic properties. Similar to the case of zigzag graphene nanoribbons, half-metallic behavior is obtained under the influence of an external axial electric field.
TL;DR: The screened hybrid density functional theory of Heyd, Scuseria, and Enzerhof and high‐resolution photoemission measurement on a single crystal of UO2 shows a slight dispersion in the f‐orbital derived bands in good agreement with the HSE band structure.
Abstract: We present a comparison between the screened hybrid density functional theory of Heyd, Scuseria, and Enzerhof (HSE06) and high-resolution photoemission (PES) measurement on a single crystal of UO(2). Angle-resolved photoemission data show a slight dispersion in the f-orbital derived bands in good agreement with the HSE band structure. The effect of spin-orbit coupling on the HSE band gap has also been calculated and found to be negligible.
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TL;DR: In this paper, the authors show that SIC is fully nonlocal and exact for all one-electron densities, and (more relevantly to the electrical response) tends to maintain an integer number of electrons on each hydrogen chain unit to a greater extent than the semilocal functionals do.
Abstract: Semilocal density functionals such as the local-spin-density and generalized-gradient approximations are known to overestimate the polarizabilities and especially the hyperpolarizabilities of long-chain molecules, the latter by as much as a factor of 10 or more in model hydrogen chains. These quantities are much better predicted by exact-exchange methods such as Hartree-Fock or optimized effective potential. We show here that the semilocal functionals, after full or scaled-down Perdew-Zunger self-interaction correction (SIC), are about as good as the exact-exchange methods for these quantities. As is the case for the exact-exchange methods, SIC is fully nonlocal and exact for all one-electron densities, and (more relevantly to the electrical response) tends to maintain an integer number of electrons on each ${\mathrm{H}}_{2}$ chain unit to a greater extent than the semilocal functionals do. In this study, the SIC energy is minimized directly, without an optimized effective potential.
TL;DR: An approximate similarity metric comparing exact versus generalized gradient approximation (GGA), exchange and parameterized mixing functions using these similarity metrics yields significantly improved thermochemistry, including GGA local hybrids whose thermochemical performance approaches GGA global hybrids.
Abstract: We recently proposed a real-space similarity metric comparing the Kohn–Sham one-particle density matrix to the local spin-density approximation model density matrix [Janesko and Scuseria, J. Chem. Phys. 127, 164117 (2007)]. This metric provides a useful ingredient for constructing local hybrid density functionals that locally mix exact exchange and semilocal density functional theory exchange. Here we present two lines of inquiry: An approximate similarity metric comparing exact versus generalized gradient approximation (GGA), exchange and parameterized mixing functions using these similarity metrics. This approach yields significantly improved thermochemistry, including GGA local hybrids whose thermochemical performance approaches GGA global hybrids.
TL;DR: In this article, the B1 rock-salt-type phase of thorium carbide was studied using Gaussian-type-orbitals and density functional theory using periodic boundary conditions.
TL;DR: In this article, the authors present a comprehensive theoretical study of the polarizability of nanotubes and nanotube bundles modeled as periodic systems using both density functional and Hartree−Fock theories with Gaussian basis set and periodic boundary conditions.
Abstract: We present a comprehensive theoretical study of the polarizability of nanotubes and nanotube bundles modeled as periodic systems. Both static and dynamic fields are considered via our recent implementation of analytical methods for polarizability. The dynamic polarizability results in this article are nanotubes' response to the time-dependent electric fields, in contrast to earlier studies that focused on only static fields. Results are obtained using both density functional and Hartree−Fock theories with a Gaussian basis set and periodic boundary conditions. In addition to elaborating on previous finite-field static polarizability calculations, we investigated how polarizability varies with the field frequency spanning a range of energies up to the band gap values, as well as how polarizabilities are affected by nanotube interactions. These findings have interesting implications for nanoelectronic applications and provide guidance for experimental studies of nanotubes' polarizability.
TL;DR: In this article, the zero-point anharmonic expansion (ZPAE) of Eq. 18, which gives an estimate of the ZPAE that must be subtracted from the experimental low-temperature lattice constant to obtain the static-lattice value, requires the volume per atom.
Abstract: In most of the numbered equations, the symbols V and V0 can refer either to the volume per primitive unit cell or to the volume per atom so long as the energy is defined respectively . But Eq. 18 , which gives an estimate of the zero-point anharmonic expansion ZPAE that must be subtracted from the experimental low-temperature lattice constant to obtain the static-lattice value, requires the volume per atom.1 When calculating the ZPAE of Eq. 18 for a solid whose primitive cell contains two atoms a nonmetal , we inadvertently used the volume per primitive cell. As a result, the ZPAE corrections incorporated into the parenthesized experimental lattice constants of the ten nonmetals in Table III were underestimated by a factor of 2. The revised experimental a0 values adjusted for ZPAE appear in the following table in bold. The static-lattice constants of the other eight solids all metals are not changed, but we include them for completeness and ease of reference along with the experimental Debye temperatures of Refs. 66 and 67.