Reference density

The reference most commonly used for solids tuid liquids is water at iiiaxiniuin density, which occurs at 4°C tliis reference density is 1.000 g/cni 1000 kg/iii or 62.43 Ib/ft . 

Morrison, R. C., Zhao, Q., 1995, Solution to the Kohn-Sham Equations Using Reference Densities from Accurate, Correlated Wave Functions for the Neutral Atoms Hehum Through Argon , Phys. Rev. A, 51, 1980. 

The value of the mis deviation from the reference density can be deceptively low, due to the fact that in the intermolecular regions the model density is virtually the same as the one made of spherical-valence shells, which was used as a NUP. The agreement between the MaxEnt map and the reference model is very close in those regions. 

The core and valence monopole populations used for the MaxEnt calculation were the ones of the reference density (electrons in the asymmetric unit iw = 12.44 and nvalence = 35.56). The phases and amplitudes for this spherical-atom structure, union of the core fragment and the NUP, are already very close to those of the full multipolar model density to estimate the initial phase error, we computed the phase statistics recently described in a multipolar charge density study on 0.5 A noise-free data [56],... 

Figure 6(b) shows the difference between the MaxEnt valence density and the reference density, in the COO- plane. The error peaks in the bonding and lone-pair regions, where the deformation features are systematically lower than the reference map (negative contours). The deviation from the reference is largest in the region around the Cl atom valence shell, and reaches -0.406 e A 3. 

Sections of the density from one of these fits, which we will refer to as calculation B, are shown in Figure 7 the MaxEnt deformation density in the COO- plane is shown in Figure 7(a) Figure 7(b) is the difference between the MaxEnt valence density and the reference density in the same plane. The lower noise content of the data is clearly visible, when the map is compared with the one for calculation A in particular, the lone pairs on the oxygen atoms are better defined. The rms deviation from the reference is as low as 0.023 e A 3. 

This finding is more evident in the density profiles in Figure 8 both calculations A and B produce too low a density in the Cl-01 bond. Close to the carbon atom, the profiles depart from the reference density to yield a more atom-like picture of the bond. This bias is milder for low-noise data, because of a tighter constraint from the data. 

In DFT the total energy is expressed as a functional of the electron density p of the molecular system of interest. The derivation of the SCC-DFTB method starts by choosing a reference density po as a superposition of densities p of the neutral atoms a constituting the molecular system,... 

The remaining four energy contributions depend on the reference density only. This is an important observation, which allows to combine these contribution into the so called repulsive energy term Erep pu, which is treated in a simplified way by approximating it by a sum of two-body potentials [44], Erep[po] = Hap Uap Rap)... 

An early attempt to obtain insight from the molecular electron density was to subtract a reference density from it. The resulting difference density, Ap(r), introduced by Daudel and others is then simply ... 

Clearly the form of a deformation density depends crucially on the definition of the reference state used in its calculation. A deformation density is therefore meaningful only in terms of its reference state, which must be taken into account in its interpretation. As we will see shortly, the theory of AIM provides information on bonding directly from the total molecular electron density, thereby avoiding a reference density and its associated problems. But first we discuss experimentally obtained electron densities. 

If we are interested only in the determination of a molecular structure, as most chemists have been, it suffices to approximate the true molecular electron density by the sum of the spherically averaged densities of the atoms, as discussed in Section 6.4. A least-squares procedure fits the model reference density preKr)t0 the observed density pobs(r) by minimizing the residual density Ap(r), defined as follows ... 

The model reference density pref is a good approximation to the dominant part of p appearing very close to the nuclei, and so Ap(r) will be very small everywhere and is assumed to be experimental noise. If the peaks in p are located, then the nuclear positions are known and the structure is resolved. Because they have no core, hydrogen atoms produce only very small maxima, and thus their positions are difficult to locate with any accuracy. If it is important to locate their positions accurately, this can be done by neutron diffraction. Neutrons are scattered by nuclei rather than electrons, and so the positions of the nuclei are obtained directly. Neutron diffraction is particularly important for the accurate determination of the positions of hydrogen atoms. 

Morrison, R. C., Q. Zhao, R. C. Morrison, and R. G. Parr. 1995. Solution of the Kohn-Sham equations using reference densities from accurate, correlated wave functions for the neutral atoms helium through argon. Phys. Rev. A51, 1980. 

The final ingredient that enters the calculation is the density factor pw. This is the actual density of water appropriate to the thermodynamic state intended in the calculation. For the usual case of 1 atm. pressure and 298K, this is I gem 3. The reference density in the electronic structure calculations is p° = 1 atm//entropic cost of sequestering water in the metal-water complexes, the free energies should be adjusted by —mRT In (pi 2o/p ) = —mRTIn (1354). With these inputs the excess chemical potential is readily composed as per (9.50), provided the optimal value of m is known. This is found by composing the excess chemical potential for different assumed m values and identifying the most stable case. For the dication transition metals studied, this is found to be six, consistent with experiment [12]. 

Pref reference density for specific gravity (mass/volume)... 

The governing equations are nondimensionalized using the fuel slot width, D, the difference between the inlet fuel and oxidizer velocities, AU, and Tq-The jet Reynolds number. Re = DAUpo/po, where po and po are the reference density and viscosity, respectively, is chosen as 5000. The nondimensional Damkohler (Da = DpoAf/(AUMWq)), Zel dovich (Ze = Ea/ RT)o), and heat... 

The difference Ap(r) between the total electron density p(r) and a reference density pref(r), is a measure for the adequacy of the reference density in representing the system. Difference densities Ap(r) are obtained by Fourier summation in which the coefficients AF are equal to the difference between the observed and calculated structure factors. If k is the scale factor, as defined in chapter 4, the difference structure factor AF is given by... 

A common reference density, first used by Roux and Daudel (1955), is the superposition of spherical ground-state atoms, centered at the nuclear positions. It is referred to as the promolecule density, or simply the promolecule, as it represents the ensemble of randomly oriented, independent atoms prior to interatomic bonding. It is a hypothetical entity that violates the Pauli exclusion principle. Nevertheless, the promolecule is electrostatically binding if only the electrostatic interactions would exist, the promolecule would be stable (Hirshfeld and Rzotkiewicz 1974). The difference density calculated with the promolecule reference state is commonly called the deformation density, or the standard deformation density. It is the difference between the total density and the density corresponding to the sum of the spherical ground-state atoms located at the positions R... 

Topological analysis of the total density has a considerable advantage over the use of the deformation densities in that it is reference-density independent. There is no need to define hybridized atoms to analyze the nature of covalent bonding, and the ambiguity when using the standard deformation density, noted above in the discussion on propellanes, does not occur. 

Dg is the geometric diameter, pp is the density of the particle, neglecting the buoyancy effects of air, p is the reference density (1 g cm 3), and k is a shape factor, which is 1.0 in the case of a sphere. Because of the effect of particle density on the aerodynamic diameter, a spherical particle of high density will have a larger aerodynamic diameter than its geometric diameter. However, for most substances, pp 10 so that the difference is less than a factor of 3 (Lawrence Berkeley Laboratory, 1979). Particle densities are often lower than bulk densities of pure substances due to voids, pores, and cracks in the particles. 

I = complex impedance, B = conductivity bridge, C method unknown (not provided in the reference) graphical data provided in the reference. Density... 

If the most obvious reference state is problematic, as is the case if the ground state has orbital degeneracy, an alternative choice of reference can be useful provided a TDDFT calculation starting from the alternative reference can access all of the states of interest (including the problematic ground state). Given a reference density... 

One such model that has promise is density functional tight-binding (DFTB) theory. In DFTB, we begin by expressing the energy associated with a reference density po(r) as... 

The type of dimensionless representation of the material function affects the (extended) pi set within which the process relationship is formulated (for more information see Ref. 5). When the standard representation is used, the relevance list must include the reference density po instead of p and incorporate two additional parameters po. Tq. This leads to two additional dimensionless numbers in the process characteristics. With regard to the heat transfer characteristics of a mixing vessel or a smooth straight pipe, Eq. (27), it now follows that... 

Assume that both the velocity and density disturbances are infinitesimally small, and that they represent the density as a reference density p0 plus a perturbation ... 

Beilstein reference Density Refractive index Melting point Boiling point Flash point Solubility in 100 parts solvent... 

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