Showing papers in "Molecular Physics in 1984"
TL;DR: In this paper, a molecular dynamics simulation method which can generate configurations belonging to the canonical (T, V, N) ensemble or the constant temperature constant pressure ensemble was proposed, which is tested for an atomic fluid (Ar) and works well.
Abstract: A molecular dynamics simulation method which can generate configurations belonging to the canonical (T, V, N) ensemble or the constant temperature constant pressure (T, P, N) ensemble, is proposed. The physical system of interest consists of N particles (f degrees of freedom), to which an external, macroscopic variable and its conjugate momentum are added. This device allows the total energy of the physical system to fluctuate. The equilibrium distribution of the energy coincides with the canonical distribution both in momentum and in coordinate space. The method is tested for an atomic fluid (Ar) and works well.
8,110 citations
TL;DR: In this paper, a detailed Monte Carlo study of the phase diagram of infinitely thin hard platelets is presented, and a weak first-order isotropicnematic transition is observed.
Abstract: We present the results of a detailed Monte Carlo study of the phase diagram of infinitely thin hard platelets. A weak first order isotropicnematic transition is observed. The equation of state in the isotropic regime is compared with several current theories, none of which is found to be fully satisfactory. The density dependence of the nematic order parameter is found to be compatible with a ‘critical’ exponent β=0·25. A study of the fluctuations of the order parameter in the isotropic phase casts doubt on the applicability of the Landau-de Gennes expression for the free energy. We observe that the relation between the nematic order parameters and is compatible with the predictions of mean-field theory. Practical aspects of the computation are discussed. A novel method to compute the pressure in a constant-volume Monte Carlo run is presented.
601 citations
TL;DR: In this paper, the LJ fluid at a density and temperature close to the triple point is determined and compared for molecular dynamics computer simulations using (N, V, E), (N V, T, T), ( N, P, H) and (N P, T) ensembles.
Abstract: Dynamic and static properties of the LJ fluid at a density and temperature close to the triple point are determined and compared for molecular dynamics computer simulations using (N, V, E), (N, V, T), (N, P, H) and (N, P, T) ensembles. As expected, the mean values of thermodynamic properties for all the different ensembles show good agreement. For the velocity autocorrelation function and the time dependence of the mean square displacement it is shown that the constant temperature and/or constant pressure algorithms used produce results which are identical, within statistical accuracy, to those obtained using the constant energy ensemble. The equations of motion are presented in a readily implementable form.
325 citations
TL;DR: In this paper, a density functional model based on the coarse-grained density distribution is used to analyse the melting of hard spheres and discs, and the role of the attractive interaction is modelled in a mean field approximation, to get the full solid-liquid-vapour phase diagram of a system with Lennard-Jones interactions.
Abstract: A density functional model, based on the coarse grained density distribution is used to analyse the melting of hard spheres and discs. The role of the attractive interaction is modelled in a mean field approximation, to get the full solid-liquid-vapour phase diagram of a system with Lennard-Jones interactions.
306 citations
TL;DR: In this article, a perturbational method is described for calculating the interaction energy of two molecules in the region where the overlap between their wave-functions is significant, without orthogonalization.
Abstract: A perturbational method is described for calculating the interaction energy of two molecules in the region where the overlap between their wave-functions is significant. By working directly with a basis of determinants constructed from the SCF orbitals of the separated molecules, without orthogonalization, it is possible to avoid many of the disadvantages of other methods.
248 citations
TL;DR: In this paper, a simple free energy functional, which incorporates both local thermodynamics and short ranged correlations, is formulated and applied to the calculation of the density profile of fluids near hard walls.
Abstract: A simple free energy functional, which incorporates both ‘local’ thermodynamics and short ranged correlations, is formulated and applied to the calculation of the density profile of fluids near hard walls. For hard sphere fluids the calculated profiles are in reasonable agreement with Monte Carlo results. For a Lennard-Jones liquid the profiles exhibit the phenomenon of wetting by gas; the oscillations in the density profiles become much less pronounced and a layer of gas develops near the wall as the bulk density approaches its value at coexistence. Such behaviour was found earlier in Monte Carlo simulations but is not accounted for by existing integral equation theories based on closures of the wall-particle Ornstein-Zernike equation.
240 citations
TL;DR: The statistical mechanics of inhomogeneous systems are usefully discussed from at least two complementary approaches: (i) the virial route, in which calculations of the structure and the thermodynamic properties are based on knowledge of the intermolecular forces in conjunction with the distribution function hierarchy, and (ii) the fluctuation theory route, based on formal results linking the equilibrium structure and properties of a system to its response to a change in external field.
Abstract: The statistical mechanics of inhomogeneous systems is usefully discussed from at least two complementary approaches: (i) the virial route, in which calculations of the structure and the thermodynamic properties are based on knowledge of the intermolecular forces in conjunction with the distribution function hierarchy, and (ii) the fluctuation theory route, based on formal results linking the equilibrium structure and properties of a system to its response to a change in external field. This paper makes use of both of these approaches to discuss the properties of a fluid in the presence of a strong localized external field. Particular attention is paid to the limiting case when the external field acts as a hard wall. Fluctuation theory yields a unified approach to interfaces stabilized by one body external fields of arbitrary strength, the strength determining the stability of the interfacial boundary with respect to wave-like fluctuations; with the weak field limit identified as a free fluid-fluid interfa...
238 citations
TL;DR: In this article, the parity-violating weak neutral current perturbation of the groundstate electronic energy has been calculated, by ab initio methods, for the (S)-isomer of hydrogen peroxide, over a range of standard Gaussian basis sets.
Abstract: The parity-violating weak neutral current perturbation of the groundstate electronic energy has been calculated, by ab initio methods, for the (S)-isomer of hydrogen peroxide, over a range of standard Gaussian basis sets. The STO-N-31G, but not the STO-NG, sets give energy shifts E pv which converge with increasing N towards the corresponding E pv value given by an extended basis. Calculations for α-amino acids, peptides, model helical systems, with the STO-6-31G basis, or with an analogous set for corre-sponding sulphur analogues, show that the E pv shift is sensitive in sign and magnitude to the molecular conformation generally. It is found that L-alanine and the L-peptides in the α-helix and the β-sheet conformation have a lower ground state energy than the corresponding D-enantiomers due to the WNC interaction. The energy shift E pv only approximates to an expected Z 5 proportionality, and it does not correlate consistently in sign with the handedness of a helical system.
228 citations
TL;DR: In this article, the authors derived explicit expressions for the difference in refractive index and absorption coefficient of a chiral molecule in a magnetic field parallel and antiparallel to the light beam, and magneto-chiral analogues of the Faraday A-, B- and C-terms introduced.
Abstract: It was pointed out recently that the absorption coefficient of a chiral molecule should be shifted slightly in a magnetic field parallel to a light beam in any polarization state. This suggestion is developed further by considering an analogous refractive index shift and by discussing these magneto-chiral phenomena in the unified context of effects generated by the ‘magnetic’ (time-odd) parts of the complex optical activity tensors. Explicit expressions, in terms of molecular property tensors, are derived for the difference in refractive index and absorption coefficient of a chiral molecule in a magnetic field parallel and antiparallel to the light beam, and magneto-chiral analogues of the Faraday A-, B- and C-terms introduced. A rough estimate of the magneto-chiral birefringence indicates that it should be observable using a modified Rayleigh refractometer. The feasibility of observing magneto-chiral dichroism A-, B- and C-terms in different types of chiral molecules is also considered. The magneto-chira...
228 citations
TL;DR: In this paper, for a system of 512 Stockmayer particles with μ*2 = 3·0 and I* = 0·025 at ρ*= 0·822 and T* = 1·15, in a series of extensive simulations, in which the boundary conditions (spherical reaction field vs. lattice summation technique (LS)) have been varied in a systematic way.
Abstract: Recently a computer adapted theory suitable for evaluating the dielectric properties of polar systems has been proposed. It rests on the assumption that the system behaves like a macroscopic dielectric and that the modifications of dipolar interactions do not affect its dielectric constant. In the present paper these assumptions are tested, for a system of 512 Stockmayer particles with μ*2 = 3·0 and I* = 0·025 at ρ* = 0·822 and T* = 1·15, in a series of extensive simulations, in which the boundary conditions (spherical reaction field (RF) vs. lattice summation technique (LS)) have been varied in a systematic way. Within the limits of statistical accuracy all simulations give identical results for the static as well as for the frequency dependent dielectric constant, namely e(0) = 66 and a relaxation time of τD* ≃ 1·0 for the almost Debye-like behaviour of e(ω). For particle numbers of 256 and upward the system studied behaves like a macroscopic dielectric; the bulk dielectric constant is independent of N ...
214 citations
TL;DR: In this article, the dielectric permittivity of a model fluid representing methyl cyanide has been studied with a view to testing a recently developed theory of the frequency dependent dielectrics.
Abstract: The dielectric permittivity of a model fluid representing methyl cyanide has been studied with a view to testing a recently developed theory of the frequency dependent dielectric permittivity. Although there are large correlations between the orientations of dipoles the Kirkwood g factor is close to the trivial value of unity; this contrasts with the behaviour of Stockmayer and similar model fluids and is a consequence of the molecular shape. The time dependence of both the longitudinal and transverse components of the dipole density is obtained and related to the relaxation of the single molecule dipole correlation function. The applicability of the ‘dipolaron’ description to the oscillation observed in the relaxation of the longitudinal component is examined.
TL;DR: In this paper, the authors derived explicit expressions for the terms up to and including R -5 in the R -1 expansion of the electrostatic energy of two molecules of arbitrary symmetry, in a simple form which is suitable for use in model potentials.
Abstract: The spherical multipole expansion has been used to derive explicit expressions for the terms up to and including R -5 in the R -1 expansion of the electrostatic energy of two molecules of arbitrary symmetry, in a simple form which is suitable for use in model potentials. We show also how the corresponding forces and torques can be readily derived from these energy expressions, and give explicitly the forces and torques which would be required for a molecular dynamics simulation of a fluid of linear molecules.
TL;DR: In this article, some simple theories are described for calculating rate constants for neutral chemical reactions in the gas phase which are dominated by long-range forces and have no barriers in the potential energy surface.
Abstract: Some simple theories are described for calculating rate constants for neutral chemical reactions in the gas phase which are dominated by long-range forces and have no barriers in the potential energy surface. The methods involve combining an adiabatic capture approach with the quantum mechanical sudden approximation of non-reactive energy transfer. A version of the technique is shown to work well in calculations on the O + OH reaction. The methods are applied to the diatom-diatom reactions SO + OH, NO + BrO, NO + ClO, CN + O2 and CH + O2. The results provide new information on the origin of the negative temperature dependence of the rate constants for reactions such as these.
TL;DR: In this article, a model of spherical n/6 molecules and of linear two-centre Lennard-Jones molecules is considered and the orthobaric densities and the vapour pressures are calculated by using perturbation theory at liquid densities.
Abstract: Model fluids consisting of spherical n/6 molecules and of linear two-centre Lennard-Jones molecules are considered. The orthobaric densities and the vapour pressures are calculated by using perturbation theory at liquid densities and second virial coefficients at gas densities. In order to classify the results the deviations from corresponding states principle are investigated. For spherical molecules, a steeper repulsive potential lowers the vapour pressure considerably but hardly effects the liquid saturation density. For linear molecules, an increasing elongation lowers the vapour pressure and increases the liquid saturation density. Finally, comparison with real Ar, CH4, N2, F2, C2H6, and CO2 is made.
TL;DR: In this article, the authors introduce a model for the association of spherically symmetric atoms into diatomic molecules, which is based on the Percus-Yevick approximation.
Abstract: We introduce models for the association of spherically symmetric atoms into diatomic molecules. In the Percus-Yevick approximation, we solve analytically for the equilibrium properties of a one-parameter family of such models corresponding to .4 + B ~ A B association. We focus in detail on the behaviour of the association constant, comparing alternative definitions of that quantity. We find that our results are insensitive to the details of the repulsive core of the .4B potential as long as it is several times larger than kBT (kB = Boltzmann's constant, T= temperature). As a result, we conclude that a limiting-case model in which the repulsive core of the JIB potential is a hard core is of particular interest and utility. The mathematics of the Percus-Yevick approximation simplifies considerably in this limiting-case analysis.
TL;DR: The variational method for the determination of ro-vibrational energy levels has been extended to the rovibronic energy levels of Renner-Teller triatomic molecules as discussed by the authors.
Abstract: The variational method for the determination of ro-vibrational energy levels has been extended to the ro-vibronic energy levels of Renner-Teller triatomic molecules. A previously derived hamiltonia...
TL;DR: The vibrational distribution of the methyl fragment resulting from the photodissociation of CH3I at 248 nm were obtained by TOF photofragment spectroscopy in this article.
Abstract: The vibrational distributions of the methyl fragment resulting from the photodissociation of CH3I at 248 nm are obtained by TOF photofragment spectroscopy. Only excitation of the v 2 (umbrella) mode is observed and the distribution peaks strongly at v = 2 for the electronically adiabatic pathway (I 2 P 1/2) but extends up to ν = 8 for the minor (∼ 30 per cent) non-adiabatic pathway (I 2 P 3/2). Angular measurements show that both pathways result from an initial parallel (3 Q 0 ←N) transition followed by a curve crossing to produce the ground state I fragment. The coupling of the curves is vibronic in nature and probably involves the v 6 methyl rocking mode. The lifetime of the photodissociation is found to be ∼ 1 × 10-13 s.
TL;DR: In this article, the non-additive contribution to the crystal binding energies, as defined by the difference ΔE(0) between the experimental results and those evaluated from the two-body potentials, is relatively large.
Abstract: A fundamental problem exists regarding the rigorous representation of the non-additive part of the many-body potential. This is illustrated by considering the contributions to the crystal binding energies at 0 K, E(0), of the rare gases (Ne, Ar, Kr and Xe) arising from reliable two-body potentials and a variety of non-additive terms including dispersion and first-order exchange energies. The non-additive contribution to the crystal binding energies, as defined by the difference ΔE(0) between the experimental results and those evaluated from the two-body potentials, is relatively large. It is shown that while this difference corresponds (in magnitude) to the contribution arising from the non-additive dispersion energies, there can be a significant cancellation between this contribution to ΔE(0) and that arising from the non-additive first-order three-body exchange energies with a resulting disagreement with experiment. Various possible non-additive interactions are considered in an attempt to resolve this ...
TL;DR: In this paper, the authors used the mean square displacement as well as the velocity-correlation function of the center of mass of particles of kind 1 to determine the mutual diffusion coefficient D 12 in a binary model mixture.
Abstract: Molecular dynamics calculations have been performed to determine the mutual diffusion coefficient D 12 i.e. the dynamic cross correlations in a binary model mixture. The mean square displacement as well as the velocity-correlation function of the centre of mass of particles of kind 1 have been used for the determination. From various pilot runs it turned out that using runs of a length of 105 time-steps the cross correlations can be very accurately evaluated in terms of D 12/D 12 0, where D 12 0 is the pure mutual diffusion coefficient available through the two self diffusion coefficients of the mixture. Whilst the overall error for this quotient D 12/D 12 0 is found to be 3 per cent, the inaccuracy in D 12 itself is slightly larger because of the additional evaluation of the thermodynamic factor for non-ideal mixtures. Systematic errors have been discovered in earlier molecular dynamics calculations of the D 12, which are primarily due to the use of too short runs. Our calculations resulted in positive c...
TL;DR: In this article, a statistical-mechanical theory for the interfacial properties of molecular fluids characterized by strongly anisotropic pairwise potentials was developed, which is a generalization to molecular fluids of the mean-field (or van der Waals) theory of non-uniform fluids of spherical molecules.
Abstract: We have developed a statistical-mechanical theory for the interfacial properties of molecular fluids characterized by strongly anisotropic pairwise potentials. The theory, which is a generalization to molecular fluids of the mean-field (or van der Waals) theory for the Helmholtz free energy of non-uniform fluids of spherical molecules, was applied to the nematic-isotropic and nematic-vapour interfaces of a simple model of a nematic liquid crystal. Our model exhibits three fluid phases—a nematic, an isotropic liquid and a vapour—which can co-exist in pairs, and which also simultaneously co-exist at a low temperature triple point, T tr. We present explicit results for the global phase diagram of the model which are in good qualitative agreement with data on real liquid crystals. We have studied in detail the various fluid interfaces occurring in different regions of the phase diagram. We present results of numerical calculations for the density and orientational order parameter profiles of a nematic-isotrop...
TL;DR: In this paper, the exact rovibrational levels for the H3 + and H2D+ molecules were calculated using a recently published accurate potential, and the implications for astrophysics, the interpretation of the infrared spectrum of H3+ near its dissociation limit and the unassigned spectrum of h2D+, were discussed.
Abstract: Variationally exact rovibrational levels for the H3 + and H2D+ molecules are calculated using a recently published accurate potential. Vibrational fundamentals are v A 1 = 3191 cm-1 and vE = 2494 (2521·6) cm-1 for H3 + and v 1 = 3000 cm-1, v 2 = 2184 cm-1 and v 3 = 2310 cm-1 for H2D+. For H3 + calculated ground state rotational constants are B 0 = 43·51 (43·57) cm-1, C 0 = 20·59 (20·71) cm-1, DJ 0 = 0·04 (0·05) cm-1, DJ K 0 = -0·07 (-0·10) cm-1 and DK = 0·04 (0·04) cm-1 (where experimental results are given in parenthesis). An attempt is made to stabilize many vibrational states. We thus reassess the results of Carney and Porter. The implications for astrophysics, the interpretation of the infrared spectrum of H3 + near its dissociation limit and the unassigned spectrum of H2D+ are discussed.
TL;DR: In this article, the molecular rotation-vibration hamiltonian is written in local coordinates and a variational technique for calculating accurate eigenfunctions and eigenvalues is described.
Abstract: The molecular rotation-vibration hamiltonian is written in local coordinates and a variational technique for calculating accurate eigenfunctions and eigenvalues is described. It is shown that the local mode approach is rapidly convergent and no more difficult to apply than the normal mode method. A simple four-parameter Morse potential function is fitted to 56 vibrational levels of H2O, HDO and D2O, and used to predict all ro-vibrational levels of up to 5 vibrational quanta with J = 0, 1 and 2. The agreement between calculated and observed frequencies is remarkably good for a model containing so few empirical parameters.
TL;DR: In this paper, the intermolecular pair potentials used in these calculations were derived with a new model which was introduced recently, and calculated data on thermodynamics, static structure, self-diffusion coefficient and reorientational correlation times are compared with experimental and other molecular dynamics results.
Abstract: Results of molecular dynamics simulations are reported for CH3F, CHF3, CH3Cl, CH3CN, CO2 and CS2. The intermolecular pair potentials used in these calculations were derived with a new model which was introduced recently. The calculated data on thermodynamics, static structure, self-diffusion coefficient and reorientational correlation times are compared with experimental and other molecular dynamics results.
TL;DR: In this paper, the adsorption of N2 molecules on a structured graphite surface has been studied using the molecular dynamics technique using two Lennard-Jones centers and partial charges to represent the quadrupole moment.
Abstract: The adsorption of N2 molecules on a structured graphite surface has been studied using the molecular dynamics technique. N2 is modelled using two Lennard-Jones centres and partial charges to represent the quadrupole moment. The surface, which is represented by a rigid external field, has longitudinal structure and the adsorbate molecules form a √3 × √3 commensurate solid with herringbone orientational ordering in two sublattices. The model exhibits a rotational transition to plastic crystal at 33 K (experiment 27 K). The simulated transition temperature does not vary markedly with coverage. The adsorbate shows little residual herringbone ordering above the phase transition, but a six-fold ordering due to the crystal field of the adsorbate appears.
TL;DR: In this paper, an integral equation for the full particle-particle correlation functions in molecular fluids is obtained by expressing the relevant quantities in terms of the graphical language of the interaction site formalism.
Abstract: An integral equation for the full particle-particle correlation functions in molecular fluids is obtained by expressing the relevant quantities in terms of the graphical language of the interaction site formalism. The equation is of the form of the Ornstein-Zernike equation for an atomic mixture, but is mathematically equivalent to the ‘proper’ integral equation of Chandler, Silbey, and Ladanyi. The simple form results in part from the introduction of a convenient method which yields the topologically correct interaction site graph chain sums. Further, by direct analogy with the usual graphical analysis of atomic fluids, we obtain a closure relation which is formally exact in the same sense as that in the usual atomic theory. Numerical results obtained using closures which are the graphical analogues of the usual atomic Percus-Yevick and hypernetted chain approximations are discussed for model homonuclear Lennard-Jones diatomic fluids. These preliminary results suggest that such direct generalizations are...
TL;DR: In this article, a simple, intermolecular potential function has been derived empirically to yield good thermodynamic and structural results for liquid hydrogen fluoride, and the function was tested in Monte Carlo statistical mechanics simulations for the liquid at temperatures of 0°C and -70°C at 1 atm.
Abstract: A simple, intermolecular potential function has been derived empirically to yield good thermodynamic and structural results for liquid hydrogen fluoride. The function was tested in Monte Carlo statistical mechanics simulations for the liquid at temperatures of 0°C and -70°C at 1 atm. The average errors in the computed densities and energies are 1 and 5 per cent, respectively. The temperature dependence of the structural results is also analysed by means of radial distribution functions and hydrogen bond distributions. As expected, hydrogen bonded chains dominate the liquid's structure. Enhanced structure and hydrogen bonding are evident as the temperature is lowered. In view of the simplicity of the potential function and the quality of the results, the potential is well suited for simulations of dilute solutions including studies of the solvation of carbonium ions in a model superacid solvent.
TL;DR: The orientational order of α,ω-bis(4,4′-cyanobiphenyloxy)alkanes has been studied using deuterium N.R. spectroscopy.
Abstract: The orientational order of the α,ω-bis(4,4′-cyanobiphenyloxy)alkanes, new nematogens with a flexible core linking two rigid groups has been studied using deuterium N.M.R. spectroscopy. The order parameter for the para-axis of the cyanobiphenyl group, determined at the nematic-isotropic transition, is found to be considerably greater for members of the homologous series with an even number of methylene groups than for those with an odd number.
TL;DR: In this article, the energy, ground-state properties, polarizability and dipole shielding factor of one-electron systems confined in boxes are studied as functions of box size, R. The systems are the unbound electron, hydrogenic atom, harmonic oscillator and z 2p oscillators in various 1D and 3D boxes (cartesian and spherical).
Abstract: The energy, ground-state properties, polarizability and dipole shielding factor of one-electron systems confined in boxes are studied as functions of box size, R. The systems are the unbound electron, hydrogenic atom, harmonic oscillator and z 2p oscillators in various 1D and 3D boxes (cartesian and spherical). Exact polarizabilities α are compared with the variational Kirkwood, Buckingham and Unsold bounds. For all boxes with an attractive internal potential the curve α(R) has a characteristic s shape. Some discussion of α(R) for ions in crystals in given in terms of these models.
TL;DR: In this article, a general procedure for constructing the potential energy surfaces of small polyatomic molecules and van der Waals molecules has been used to obtain explicit functions for HeH2 and HeLi2 potentials.
Abstract: A general procedure recently proposed by one of the authors for constructing the potential energy surfaces of small polyatomic molecules and van der Waals molecules has been used to obtain explicit functions for HeH2 ([Xtilde], 1 A′), HeLi2 ([Xtilde], 1 A′) and HO2 ([Xtilde], 2 A″) The HeH2 and HeLi2 potentials compare well the best available correlated ab initio energies, and should be suitable for studying both vibration-rotation inelasticity in atom-diatom collisions and atom (H, Li) recombination in the presence of He atoms For HO2, the surface conforms with the ab initio data of Melius and Blint for the 3-body energy, and shows good agreement with available spectroscopic data for the energy and geometry of the minima referring to the equilibrium triatomic Two secondary minima which have chemical interest are predicted One refers to a T-shaped (C 2v ) H-O2 weakly bound (℞ 25 kJ mol-1 with respect to the H + O2 dissociation limit) species, while the other is related to a linear (C ∞v ) weak hydroge
TL;DR: In this article, an intermolecular pair potential energy function has been developed for nitrogen which is able to reconcile theoretical results for the system with experimental data for various bulk properties, including low temperature lattice properties and second virial coefficients.
Abstract: An intermolecular pair potential energy function has been developed for nitrogen which is able to reconcile theoretical results for the system with experimental data for various bulk properties. The long and short range forms of the potential are known from quantum mechanical calculations. Modifications to the quadrupole-quadrupole energy due to electron charge overlap effects have been included. In the well region, many different representations have been studied and have been assessed by fitting to experimental data, chiefly for the low temperature lattice properties and second virial coefficients. A number of functions have been derived which are able to reproduce the available information within experimental error. Corrections for non-additive three body terms have been included in calculating the lattice properties, and the first order quantum corrections to the virial coefficients were also included. The dilute gas shear viscosity (based on the Monchick-Mason approximation) was also calculated and u...