Conceptual density functional theory under pressure: Part I. XP-PCM method applied to atoms.

High pressure chemistry offers the chemical community a range of possibilities to control chemical reactivity, develop new materials and fine-tune chemical properties. Despite the large changes that extreme pressure brings to the table, the field has mainly been restricted to the effects of volume changes and thermodynamics with less attention devoted to electronic effects at the molecular scale. This paper combines the conceptual DFT framework for analyzing chemical reactivity with the XP-PCM method for simulating pressures in the GPa range. Starting from the new derivatives of the energy with respect to external pressure, an electronic atomic volume and an atomic compressibility are found, comparable to their enthalpy analogues, respectively. The corresponding radii correlate well with major known sets of this quantity. The ionization potential and electron affinity are both found to decrease with pressure using two different methods. For the electronegativity and chemical hardness, a decreasing and increasing trend is obtained, respectively, and an electronic volume-based argument is proposed to rationalize the observed periodic trends. The cube of the softness is found to correlate well with the polarizability, both decreasing under pressure, while the interpretation of the electrophilicity becomes ambiguous at extreme pressures. Regarding the electron density, the radial distribution function shows a clear concentration of the electron density towards the inner region of the atom and periodic trends can be found in the density using the Carbó quantum similarity index and the Kullback-Leibler information deficiency. Overall, the extension of the CDFT framework with pressure yields clear periodic patterns.

Protein Intake Falls below 0.6 g•kg-1•d-1 in Healthy, Older Patients Admitted for Elective Hip or Knee Arthroplasty.

OBJECTIVE: Hospitalization is generally accompanied by changes in food intake. Patients typically receive hospital meals upon personal preference within the framework of the food administration services of the hospital. In the present study, we assessed food provision and actual food and snack consumption in older patients admitted for elective hip or knee arthroplasty. DESIGN: A prospective observational study. SETTING: Orthopedic nursing ward of the Maastricht University Medical Centre+. PARTICIPANTS: In the present study, n=101 patients (age: 67±10 y; hospital stay: 6.1±1.8 d) were monitored during hospitalization following elective hip or knee arthroplasty. MEASUREMENTS: Energy and protein provided by self-selected hospital meals and snacks, and actual energy and protein (amount, distribution, and source) consumed by patients was weighed and recorded throughout 1-6 days. RESULTS: Self-selected meals provided 6.5±1.5 MJ•d-1, with 16, 48, and 34 En% provided as protein, carbohydrate, and fat, respectively. Self-selected hospital meals provided 0.75±0.16 and 0.79±0.21 g•kg-1•d-1 protein in males and females, respectively. Actual protein consumption averaged merely 0.59±0.18 and 0.50±0.21 g•kg-1•d-1, respectively. Protein consumption at breakfast, lunch, and dinner averaged 16±8, 18±9, and 20±6 g per meal, respectively. CONCLUSIONS: Though self-selected hospital meals provide patients with ~0.8 g•kg-1•d-1 protein during short-term hospitalization, actual protein consumption falls well below 0.6 g•kg-1•d-1 with a large proportion (~32%) of the provided food being discarded. Alternative strategies are required to ensure maintenance of habitual protein intake in older patients admitted for elective orthopedic surgery.

Molecular dynamics simulations of the structure and the morphology of graphene/polymer nanocomposites.

The structure and morphology of three polymer/graphene nanocomposites have been studied using classical molecular dynamics (MD) simulations. The simulations use 10-monomer oligomeric chains of three polymers: polyethylene (PE), polystyrene (PS) and polyvinylidene fluoride (PVDF). The structure of the polymer chains at the graphene surface has been investigated and characterized by pair correlation functions (PCF), g(r), g(θ) and g(r,θ). In addition, the influence of the temperature on the graphene/polymer interactions has been analysed for each of the three polymer/graphene nanocomposite systems. The results indicate that graphene induces order in both the PE and PVDF systems by providing a nucleation site for crystallisation, steering the growth of oligomer crystals according to the orientation of the graphene sheet, whereas the PS system remains disordered in the presence of graphene. The overall results are in line with the findings in a recent quantumchemical study by some of the present authors.

A computational study on the role of noncovalent interactions in the stability of polymer/graphene nanocomposites.

Understanding the interaction between graphene and polymers is of essential interest when designing novel nanocomposites with reinforced mechanical and electrical properties. In this computational study, the interaction of pristine graphene (PG) and graphene oxide (GO) with a series of functional groups, representative of the functionalised buildings blocks occurring in different polymers, and attached to aliphatic and aromatic chains, is analyzed using dispersion-corrected semi-empirical methods (PM6-D3H4X) and density functional theory calculations with empirical dispersion corrections. Functional groups include alkyl, hydroxyl, aldehyde, carboxyl, amino and nitro groups, and the binding energies of these groups with graphene derivatives (PG and GO) are determined. Nitro- and carbonyl groups display stronger interactions in both aliphatic and aromatic chains. The importance of dispersion-type and non-covalent interactions (NCI) in general, which typically, double the interaction energies, is revealed. The results are interpreted in an extensive NCI analysis in order to characterize the different types of NCI, providing a better understanding of the nature of the interaction (π-π stacking, CH-π bonding, H-bonding and lone pair-π interaction) at stake. In order to highlight the influence of polymer structure/conformation on top of that of their functional groups, the binding of three polymers, polyethylene (PE), polystyrene (PS) and polyvinylidene fluoride (PVDF), on pristine graphene is also investigated. Our calculations indicate that, although all polymers exhibit evident attractive interactions with the graphene sheet, the overall interaction is strongly influenced by the specific polymer structure. Thus, three main conformations of PVDF (the so-called α, ß and γ, ε conformations) are analyzed and we find that, although the α-conformer with a trans-gauche-trans-gauche (TGTG') conformation is the lowest energy conformer, the ß-conformation of PVDF with the hydrogen atoms facing the graphene ("F-up") has the strongest interaction with the graphene surface among the polymers under consideration. Taken together, our computational approach sheds light on the character and importance of non-covalent graphene-polymer functional group interactions combined with the structural/conformational properties of the polymer, which are at stake in the design of novel nanocomposites with reinforced mechanical and electrical properties.

Understanding the molecular switching properties of octaphyrins.

Several expanded porphyrins switch between Hückel, Möbius and twisted-Hückel topologies, encoding different aromaticity and NLO properties. Since the topological switch can be induced by different external stimuli, expanded porphyrins represent a promising platform to develop molecular switches for molecular electronic devices. In order to determine the optimum conditions for efficient molecular switches from octaphyrins, we have carried out a comprehensive quantum chemical study focusing on the conformational preferences and aromaticity of [36]octaphyrins. Different external stimuli for triggering the topological switch have been considered in our work, such as protonation and redox reactions. Importantly, the structure-property relationships between the molecular conformation, the number of π-electrons and aromaticity in octaphyrins have been established by using energetic, magnetic, structural and reactivity descriptors. Remarkably, we found that the aromaticity of octaphyrins is highly dependent on the π-conjugation topology and the number of π-electrons and it can be modulated by protonation and redox reactions. A non-aromatic figure-eight conformation is strongly preferred by neutral [36]octaphyrins that switches to a Möbius aromatic conformation upon protonation. Such a change of topology involves an aromaticity switch in a single molecule and is accompanied by a drastic change in the NLO properties. In contrast, the twisted-Hückel topology remains the most stable one in the oxidized and reduced species, but the aromaticity is totally reversed upon redox reactions. Aromaticity is shown to be a key concept in expanded porphyrins, determining the electronic, magnetic and NLO properties of these macrocycles.

Conduction of molecular electronic devices: qualitative insights through atom-atom polarizabilities.

The atom-atom polarizability and the transmission probability at the Fermi level, as obtained through the source-and-sink-potential method for every possible configuration of contacts simultaneously, are compared for polycyclic aromatic compounds. This comparison leads to the conjecture that a positive atom-atom polarizability is a necessary condition for transmission to take place in alternant hydrocarbons without non-bonding orbitals and that the relative transmission probability for different configurations of the contacts can be predicted by analyzing the corresponding atom-atom polarizability. A theoretical link between the two considered properties is derived, leading to a mathematical explanation for the observed trends for transmission based on the atom-atom polarizability.

Revealing the thermodynamic driving force for ligand-based reductions in quinoids; conceptual rules for designing redox active and non-innocent ligands.

Metal and ligand-based reductions have been modeled in octahedral ruthenium complexes revealing metal-ligand interactions as the profound driving force for the redox-active behaviour of orthoquinoid-type ligands. Through an extensive investigation of redox-active ligands we revealed the most critical factors that facilitate or suppress redox-activity of ligands in metal complexes, from which basic rules for designing non-innocent/redox-active ligands can be put forward. These rules also allow rational redox-leveling, i.e. the moderation of redox potentials of ligand-centred electron transfer processes, potentially leading to catalysts with low overpotential in multielectron activation processes.

Evaluating interaction energies of weakly bonded systems using the Buckingham-Hirshfeld method.

We present the finalized Buckingham-Hirshfeld method (BHD-DFT) for the evaluation of interaction energies of non-bonded dimers with Density Functional Theory (DFT). In the method, dispersion energies are evaluated from static multipole polarizabilities, obtained on-the-fly from Coupled Perturbed Kohn-Sham calculations and partitioned into diatomic contributions using the iterative Hirshfeld partitioning method. The dispersion energy expression is distributed over four atoms and has therefore a higher delocalized character compared to the standard pairwise expressions. Additionally, full multipolar polarizability tensors are used as opposed to effective polarizabilities, allowing to retain the anisotropic character at no additional computational cost. A density dependent damping function for the BLYP, PBE, BP86, B3LYP, and PBE0 functionals has been implemented, containing two global parameters which were fitted to interaction energies and geometries of a selected number of dimers using a bi-variate RMS fit. The method is benchmarked against the S22 and S66 data sets for equilibrium geometries and the S22x5 and S66x8 data sets for interaction energies around the equilibrium geometry. Best results are achieved using the B3LYP functional with mean average deviation values of 0.30 and 0.24 kcal/mol for the S22 and S66 data sets, respectively. This situates the BHD-DFT method among the best performing dispersion inclusive DFT methods. Effect of counterpoise correction on DFT energies is discussed.

Exploring the structure-aromaticity relationship in Hückel and Möbius N-fused pentaphyrins using DFT.

N-fused pentaphyrins (NFP) are the stable forms of fully meso-aryl pentaphyrins(1.1.1.1.1). In order to determine the optimum conditions for viable Möbius topologies of these porphyrinoids, the conformational preferences, Hückel-Möbius interconversion pathways and aromaticity of [22] and [24]NFP have been investigated using density functional theory calculations. The conformation of the macrocycle is shown to be strongly dependent on the oxidation state and the macrocyclic aromaticity. [22]NFP prefers a highly aromatic and relatively strain-free Hückel conformation. However, antiaromatic Hückel and weakly aromatic Möbius conformers coexist in dynamic equilibrium in [24]NFP. The Hückel-Möbius aromaticity switch requires very low activation energy barriers (Ea = 3-4 kcal mol(-1)). Interestingly, the balance between Möbius and Hückel conformations in [24]NFP can be controlled by meso-substituents. The structure-property relationship between the molecular conformation, number of π electrons and aromaticity has been established in our study using energetic, magnetic, structural, and reactivity descriptors of aromaticity. Although the Möbius topology is indeed accessible for [24]NFP, it does not exhibit a distinct macrocyclic aromaticity mainly due to the large dihedral angles around the molecular twist. Regarding the computational methodology, B3LYP and M06 show the best overall performance for describing the experimental geometries of NFP and, importantly, our computational results support the experimental evidence available for N-fused pentaphyrins.

Atomic electron affinities and the role of symmetry between electron addition and subtraction in a corrected Koopmans approach.

The essential aspects of zero-temperature grand-canonical ensemble density-functional theory are reviewed in the context of spin-density-functional theory and are used to highlight the assumption of symmetry between electron addition and subtraction that underlies the corrected Koopmans approach of Tozer and De Proft (TDP) for computing electron affinities. The issue of symmetry is then investigated in a systematic study of atomic electron affinities, comparing TDP affinities with those from a conventional Koopmans evaluation and electronic energy differences. Although it cannot compete with affinities determined from energy differences, the TDP expression yields results that are a significant improvement over those from the conventional Koopmans expression. Key insight into the results from both expressions is provided by an analysis of plots of the electronic energy as a function of the number of electrons, which highlight the extent of symmetry between addition and subtraction. The accuracy of the TDP affinities is closely related to the nature of the orbitals involved in the electron addition and subtraction, being particularly poor in cases where there is a change in principal quantum number, but relatively accurate within a single manifold of orbitals. The analysis is then extended to a consideration of the ground state Mulliken electronegativity and chemical hardness. The findings further emphasize the key role of symmetry in determining the quality of the results.

Evaluating London Dispersion Interactions in DFT: A Nonlocal Anisotropic Buckingham-Hirshfeld Model.

In this work, we present a novel model, referred to as BH-DFT-D, for the evaluation of London dispersion, with the purpose to correct the performance of local DFT exchange-correlation functionals for the description of van der Waals interactions. The new BH-DFT-D model combines the equations originally derived by Buckingham [Buckingham, A. D. Adv. Chem. Phys1967, 12, 107] with the definition of distributed multipole polarizability tensors within the Hirshfeld method [Hirshfeld, F.L. Theor. Chim. Acta1977, 44, 129], resulting in nonlocal, fully anisotropic expressions. Since no damping function has been introduced yet into the model, it is suitable in its present form for the evaluation of dispersion interactions in van der Waals dimers with no or negligible overlap. The new method is tested for an extended collection of van der Waals dimers against high-level data, where it is found to reproduce interaction energies at the BH-B3LYP-D/aug-cc-pVTZ level with a mean average error (MAE) of 0.20 kcal/mol. Next, development steps of the model will consist of adding a damping function, analytical gradients, and generalization to a supramolecular system.

Partitioning of higher multipole polarizabilities: numerical evaluation of transferability.

In this work, the partitioning of higher multipole polarizabilities, such as dipole-quadrupole, quadrupole-dipole, and quadrupole-quadrupole polarizabilities, into atomic contributions is studied. Partitioning of higher multipole polarizabilities is necessary in the study of accurate interaction energies where dispersion interactions are of importance. The fractional occupation Hirhsfeld-I (FOHI) method is used to calculate the atomic polarizabilities and is briefly explained together with the methodology for partitioning of the polarizabilities. The atomic multipole polarizabilities are calculated for different sets of molecules, linear alkanes, water clusters, and small organic molecules with different functional groups. It is found that the atomic and group contributions of the dipole and quadrupole polarizabilities are transferable as a function of the functional groups.

A new approach to local hardness.

The applicability of the local hardness as defined by the derivative of the chemical potential with respect to the electron density is undermined by an essential ambiguity arising from this definition. Further, the local quantity defined in this way does not integrate to the (global) hardness-in contrast with the local softness, which integrates to the softness. It has also been shown recently that with the conventional formulae, the largest values of local hardness do not necessarily correspond to the hardest regions of a molecule. Here, in an attempt to fix these drawbacks, we propose a new approach to define and evaluate the local hardness. We define a local chemical potential, utilizing the fact that the chemical potential emerges as the additive constant term in the number-conserving functional derivative of the energy density functional. Then, differentiation of this local chemical potential with respect to the number of electrons leads to a local hardness that integrates to the hardness, and possesses a favourable property; namely, within any given electron system, it is in a local inverse relation with the Fukui function, which is known to be a proper indicator of local softness in the case of soft systems. Numerical tests for a few selected molecules and a detailed analysis, comparing the new definition of local hardness with the previous ones, show promising results.

Energy surface, chemical potentials, Kohn-Sham energies in spin-polarized density functional theory.

On the basis of the zero-temperature grand canonical ensemble generalization of the energy E[N,N(s),v,B] for fractional particle N and spin N(s) numbers, the energy surface over the (N,N(s)) plane is displayed and analyzed in the case of homogeneous external magnetic fields B(r). The (negative of the) left-/right-side derivatives of the energy with respect to N, N(↑), and N(↓) give the fixed-N(s), spin-up, and spin-down ionization potentials/electron affinities, respectively, while the derivative of E[N,N(s),v,B] with respect to N(s) gives the (signed) half excitation energy to the lowest-lying state with N(s) increased (or decreased) by 2. The highest occupied and lowest unoccupied Kohn-Sham spin-orbital energies are identified as the corresponding spin-up and spin-down ionization potentials and electron affinities. The excitation energies to the lowest-lying states with N(s)±2 can be obtained as the differences between the lowest unoccupied and the opposite-spin highest occupied spin-orbital energies, if the (N,N(s)) representation of the Kohn-Sham spin-potentials is used. The cases where the convexity condition on the energy does not hold are also discussed. Finally, the discontinuities of the energy derivatives and the Kohn-Sham potential are analyzed and related.

On the applicability of local softness and hardness.

Global hardness and softness and the associated hard/soft acid/base (HSAB) principle have been used to explain many experimental observed reactivity patterns and these concepts can be found in textbooks of general, inorganic, and organic chemistry. In addition, local versions of these reactivity indices and principles have been defined to describe the regioselectivity of systems. In a very recent article (Chem.-Eur. J. 2008, 14, 8652), the present authors have shown that the picture of these well-known descriptors is incomplete and that the understanding of these reactivity indices must be "reinterpreted". In fact, the local softness and hardness contain the same "potential information" and they should be interpreted as the "local abundance" or "concentration" of their corresponding global properties. In this contribution, we analyze the implications of this new point of view for the applicability of these well-known descriptors when comparing two sites in three situations: two sites within one molecule, two sites in two different, but noninteracting molecules, and two sites in two different, but interacting, molecules. The implications on the HSAB principle are highlighted, leading to the discussion of the role of the electrostatic interaction.

A benchmark theoretical study of the electronic ground state and of the singlet-triplet split of benzene and linear acenes.

A benchmark theoretical study of the electronic ground state and of the vertical and adiabatic singlet-triplet (ST) excitation energies of benzene (n=1) and n-acenes (C(4n+2)H(2n+4)) ranging from naphthalene (n=2) to heptacene (n=7) is presented, on the ground of single- and multireference calculations based on restricted or unrestricted zero-order wave functions. High-level and large scale treatments of electronic correlation in the ground state are found to be necessary for compensating giant but unphysical symmetry-breaking effects in unrestricted single-reference treatments. The composition of multiconfigurational wave functions, the topologies of natural orbitals in symmetry-unrestricted CASSCF calculations, the T1 diagnostics of coupled cluster theory, and further energy-based criteria demonstrate that all investigated systems exhibit a (1)A(g) singlet closed-shell electronic ground state. Singlet-triplet (S(0)-T(1)) energy gaps can therefore be very accurately determined by applying the principles of a focal point analysis onto the results of a series of single-point and symmetry-restricted calculations employing correlation consistent cc-pVXZ basis sets (X=D, T, Q, 5) and single-reference methods [HF, MP2, MP3, MP4SDQ, CCSD, CCSD(T)] of improving quality. According to our best estimates, which amount to a dual extrapolation of energy differences to the level of coupled cluster theory including single, double, and perturbative estimates of connected triple excitations [CCSD(T)] in the limit of an asymptotically complete basis set (cc-pVinfinityZ), the S(0)-T(1) vertical excitation energies of benzene (n=1) and n-acenes (n=2-7) amount to 100.79, 76.28, 56.97, 40.69, 31.51, 22.96, and 18.16 kcal/mol, respectively. Values of 87.02, 62.87, 46.22, 32.23, 24.19, 16.79, and 12.56 kcal/mol are correspondingly obtained at the CCSD(T)/cc-pVinfinityZ level for the S(0)-T(1) adiabatic excitation energies, upon including B3LYP/cc-PVTZ corrections for zero-point vibrational energies. In line with the absence of Peierls distortions, extrapolations of results indicate a vanishingly small S(0)-T(1) energy gap of 0 to approximately 4 kcal/mol (approximately 0.17 eV) in the limit of an infinitely large polyacene.

Nonuniqueness of magnetic fields and energy derivatives in spin-polarized density functional theory.

The effect of the recently uncovered nonuniqueness of the external magnetic field B(r) corresponding to a given pair of density n(r) and spin density n(s)(r) on the derivative of the energy functional of spin-polarized density functional theory, and its implications for the definition of chemical reactivity descriptors, is examined. For ground states, the nonuniqueness of B(r) implies the nondifferentiability of the energy functional E(v,B)[n,n(s)] with respect to n(s)(r). It is shown, on the other hand, that this nonuniqueness allows the existence of the one-sided derivatives of E(v,B)[n,n(s)] with respect to n(s)(r). Although the N-electron ground state can always be obtained from the minimization of E(v,B)[n,n(s)] without any constraint on the spin number N(s)=integraln(s)(r)dr, the Lagrange multiplier mu(s) associated with the fixation of N(s) does not vanish even for ground states. Mu(s) is identified as the left- or right-side derivative of the total energy with respect to N(s), which justifies the interpretation of mu(s) as a (spin) chemical potential. This is relevant not only for the spin-polarized generalization of conceptual density functional theory, the spin chemical potential being one of the elementary reactivity descriptors, but also for the extension of the thermodynamical analogy of density functional theory for the spin-polarized case. For higher-order reactivity indices, B(r)'s nonuniqueness has similar implications as for mu(s), leading to a split of the indices with respect to N(s) into one-sided reactivity descriptors.

Conceptual DFT: the chemical relevance of higher response functions.

In recent years conceptual density functional theory offered a perspective for the interpretation/prediction of experimental/theoretical reactivity data on the basis of a series of response functions to perturbations in the number of electrons and/or external potential. This approach has enabled the sharp definition and computation, from first principles, of a series of well-known but sometimes vaguely defined chemical concepts such as electronegativity and hardness. In this contribution, a short overview of the shortcomings of the simplest, first order response functions is illustrated leading to a description of chemical bonding in a covalent interaction in terms of interacting atoms or groups, governed by electrostatics with the tendency to polarize bonds on the basis of electronegativity differences. The second order approach, well known until now, introduces the hardness/softness and Fukui function concepts related to polarizability and frontier MO theory, respectively. The introduction of polarizability/softness is also considered in a historical perspective in which polarizability was, with some exceptions, mainly put forward in non covalent interactions. A particular series of response functions, arising when the changes in the external potential are solely provoked by changes in nuclear configurations (the "R-analogues") are also systematically considered. The main part of the contribution is devoted to third order response functions which, at first sight, may be expected not to yield chemically significant information, as turns out to be for the hyperhardness. A counterexample is the dual descriptor and its R analogue, the initial hardness response, which turns out to yield a firm basis to regain the Woodward-Hoffmann rules for pericyclic reactions based on a density-only basis, i.e. without involving the phase, sign, symmetry of the wavefunction. Even the second order nonlinear response functions are shown possibly to bear interesting information, e.g. on the local and global polarizability. Its derivatives may govern the influence of charge on the polarizability, the R-analogues being the nuclear Fukui function and the quadratic and cubic force constants. Although some of the higher order derivatives may be difficult to evaluate a comparison with the energy expansion used in spectroscopy in terms of nuclear displacements, nuclear magnetic moments, electric and magnetic fields leads to the conjecture that, certainly cross terms may contain new, intricate information for understanding chemical reactivity.

Dual descriptors within the framework of spin-polarized density functional theory.

Spin-polarized density functional theory (SP-DFT) allows both the analysis of charge-transfer (e.g., electrophilic and nucleophilic reactivity) and of spin-polarization processes (e.g., photophysical changes arising from electron transitions). In analogy with the dual descriptor introduced by Morell et al. [J. Phys. Chem. A 109, 205 (2005)], we introduce new dual descriptors intended to simultaneously give information of the molecular regions where the spin-polarization process linking states of different multiplicity will drive electron density and spin density changes. The electronic charge and spin rearrangement in the spin forbidden radiative transitions S(0)-->T(n,pi(*)) and S(0)-->T(pi,pi(*)) in formaldehyde and ethylene, respectively, have been used as benchmark examples illustrating the usefulness of the new spin-polarization dual descriptors. These quantities indicate those regions where spin-orbit coupling effects are at work in such processes. Additionally, the qualitative relationship between the topology of the spin-polarization dual descriptors and the vertical singlet triplet energy gap in simple substituted carbene series has been also discussed. It is shown that the electron density and spin density rearrangements arise in agreement with spectroscopic experimental evidence and other theoretical results on the selected target systems.

Influence of confinement on atomic and molecular reactivity indicators in DFT.

Spatial confinement of atoms and molecules influences electronic structures, energy spectra, and chemical reactivity. A simple potential barrier approach involving a single parameter is used to study confinement in both atoms and molecules, focusing on the reactivity of the systems through the HOMO-LUMO gap, which is linked to the global chemical hardness. Both atoms and molecules are shown to respond with an increase in hardness when confined. The results suggest that previous observations of a HOMO-LUMO gap decrease for guest molecules in zeolites cannot be assigned exclusively to electron confinement.