Permittivity of the vacuum

N is the number of point charges within the molecule and Sq is the dielectric permittivity of the vacuum. This form is used especially in force fields like AMBER and CHARMM for proteins. As already mentioned, Coulombic 1,4-non-bonded interactions interfere with 1,4-torsional potentials and are therefore scaled (e.g., by 1 1.2 in AMBER). Please be aware that Coulombic interactions, unlike the bonded contributions to the PEF presented above, are not limited to a single molecule. If the system under consideration contains more than one molecule (like a peptide in a box of water), non-bonded interactions have to be calculated between the molecules, too. This principle also holds for the non-bonded van der Waals interactions, which are discussed in Section 7.2.3.6. 

Table 8-2 lists several physical properties pertinent to our concern with the effects of solvents on rates for 40 common solvents. The dielectric constant e is a measure of the ability of the solvent to separate charges it is defined as the ratio of the electric permittivity of the solvent to the permittivity of the vacuum. (Because physicists use the symbol e for permittivity, some authors use D for dielectric constant.) Evidently e is dimensionless. The dielectric constant is the property most often associated with the polarity of a solvent in Table 8-2 the solvents are listed in order of increasing dielectric constant, and it is evident that, with a few exceptions, this ranking accords fairly well with chemical intuition. The dielectric constant is a bulk property. 

All equations given in this text appear in a very compact form, without any fundamental physical constants. We achieve this by employing the so-called system of atomic units, which is particularly adapted for working with atoms and molecules. In this system, physical quantities are expressed as multiples of fundamental constants and, if necessary, as combinations of such constants. The mass of an electron, me, the modulus of its charge, lei, Planck s constant h divided by lit, h, and 4jt 0, the permittivity of the vacuum, are all set to unity. Mass, charge, action etc. are then expressed as multiples of these constants, which can therefore be dropped from all equations. The definitions of atomic units used in this book and their relations to the corresponding SI units are summarized in Table 1-1. 

In the equation s is the measured dielectric constant and e0 the permittivity of the vacuum, M is the molar mass and p the molecular density, while Aa and A (po2) are the isotope effects on the polarizability and the square of the permanent dipole moment respectively. Unfortunately, because the isotope effects under discussion are small, and high precision in measurements of bulk phase polarization is difficult to achieve, this approach has fallen into disfavor and now is only rarely used. Polarizability isotope effects, Aa, are better determined by measuring the frequency dependence of the refractive index (see below), and isotope effects on permanent dipole moments with spectroscopic experiments. 

In Equation 12.13, N is the number density of molecules in the beam of radiation (and is thus inversely proportional to the molar volume, Vm), and o is the permittivity of the vacuum. A useful and widely employed method to evaluate the sum in Equation 12.13 leads via the closure approximation to a one-term equation commonly known as the dispersion relation,... 

The above-mentioned formula applies when there is a vacuum between the plates 0 is the permittivity of the vacuum ... 

Formally the capacitance C of a plane capacitor having plates of equal area S in parallel configuration, separated by a distance d, is given by equation (17.2), where Eq and Er are the permittivity of the vacuum and the relative permittivity of the dielectric material, respectively ... 

Here, a is the charge of the wire per unit length and eo is the electric permittivity of the vacuum (eo = 8.85 x 10 C /Nm ). Superposing the fields of the two oppositely charged wires, we obtain the following expression for the electric field on the x axis... 

To convert the potential energy from cgs units to SI units (joules), the expression shown must be multiplied by 1/4ttc0, where e0 (the permittivity of the vacuum) is 8.854 x KT12 C2/J m. 

In the above formulae the permeability of the vacuum can be replaced by the permittivity of the vacuum since qIIqC2 = 1. 

In fact we shall always work in a system of atomic units which is described in Appendix l.Ato this chapter. Roughly speaking, choice of Planck s constant divided by (ft), the charge on the proton (to avoid sign absurdities), and the mass of the electron as units fixes all mechanical quantities. More controversially, perhaps, choosing 47r o (the permittivity of the vacuum) as unity simplifies our Hamiltonian since we are always concerned with electrostatic interactions. 

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