Charge Distribution in an External Field

It is often useful to know the energy of interaction U of a charge distribution in the presence of an external electrostatic field E. It is 

Just like the electric quadrupole moment, the electric field gradient matrix can be written in diagonal form for a suitable choice of coordinate axes. 

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Multipoles. One can express [15] the multipole expansion of the energy of a charge distribution in an external field by defining the free energy G of a localized charge distribution p(r), which is placed in an external potential (which has no charge distribution associated with it) ... 

Related to this feature, the two lone pairs of electrons on the oxygen atom render a water molecule polarizable to electric fields from other molecules or from a charged surface. Thus, water molecules can respond to the changing of charge distribution in an external solute (or surface) to lower the energy of an... 

Polarizability Attraction. AU. matter is composed of electrical charges which move in response to (become electrically polarized in) an external field. This field can be created by the distribution and motion of charges in nearby matter. The Hamaket constant for interaction energy, A, is a measure of this polarizability. As a first approximation it may be computed from the dielectric permittivity, S, and the refractive index, n, of the material (15), where is the frequency of the principal electronic absorption... 

In molecular systems the polarization P results from the individual molecular dipoles and has two main contributions. One is associated with the average orientation induced by an external field in the distribution of permanent molecular dipoles. The other results from the dipoles induced in each individual molecule by the local electrostatic field. The characteristic timescale associated with the first effect is that of nuclear orientational relaxation, x , typically 10 " s for small molecule fluids at room temperature. The other effect arises mostly from the distortion of the molecular electronic charge distribution by the external field, and its typical... 

Our discussion of elecfronic effects has concentrated so far on permanent features of the cliarge distribution. Electrostatic interactions also arise from changes in the charge distribution of a molecule or atom caused by an external field, a process called polarisation. The primary effect of the external electric field (which in our case will be caused by neighbouring molecules) is to induce a dipole in the molecule. The magnitude of the induced dipole moment ginj is proportional to the electric field E, with the constant of proportionahty being the polarisability a ... 

The presence of an electrostatic field subjects a material to a stress. Such fields can arise from the presence of free charges in the material or from the action of an external field. A number of authors (Hogan, H12, H13 Graf, G8 Doyle, Moffett, and Vonnegut, D6 and others) give the outward pressure on a liquid sphere with a uniformly distributed surface charge as... 

The asymmetry of distributions of positive and negative charges in a molecule can have important effects on its physical properties as well as on its chemical reactivity. This asymmetry can have two quite different origins, illustrated by the permanent and induced dipole moments. A molecule such as HF has a permanent dipole moment which results from the electronegativities of the H and F atoms but even a symmetrical molecule such as H2C=CH2 can acquire an induced dipole moment in an external electric field. 

The potential outside the charge distribution and due to it is simply related to the moments, as is the interaction energy when an external field is applied.14 The multipole moments are thus very useful quantities and have been extensively applied in the theory of intermolecular forces, particularly at long range where the electrostatic contribution to the interaction may be expanded in moments. Their values are related to the symmetry of the system thus, for instance, a plane of symmetry indicates that the component of n perpendicular to it must be zero. Such multipoles are worth calculating in their own right. 

Let us describe the charge distribution induced in a molecule by an external field, in terms of a set of mutually interacting dipoles centred at the nuclear positions. The dipole induced at the 7th site by the reaction field can then be expressed as ... 

Nuclei with a spin number 7 of one or higher have a nonspherical charge distribution. This asymmetry is described by an electrical quadrupole moment, which, as we shall see later, affects the relaxation time and, consequently, the linewidth of the signal and coupling with neighboring nuclei. In quantum mechanical terms, the spin number 7 determines the number of orientations a nucleus may assume in an external uniform magnetic field in accordance with the formula 27+1. We are concerned with the proton whose spin number 7 is 1/2. 

On the other hand, in some materials which have a symmetry less than cubic, an electric field gradient (EFG) due to the concrete charge distribution in the material can be present [137-140], If a localized charge distribution is immersed in an external electric field with potential electrostatic energy of the system is [144]... 

It is required to calculate the electrostatic interaction between molecules, winch possess a localized charge distribution, mathematically described by p(r") (where 7 = x,/ + x2j + x3 k in rectangular coordinates, and = x, x2 = y, and x3 = z), and are immersed in an external electric field with a potential, V( r ), given by a solid adsorbent. Subsequently, the electrostatic energy of the adsorbate-adsorbent system can be determined with the help of the following expression [46] ... 

There are N/V electrons, each of charge —e, per unit volume in the metal. These will have a certain distribution of velocities (the Maxwell distribution or the Fermi distribution), of which we need only the property that the mean velocity is zero, in the absence of an external field. In the field E the force on each charge will be — eE, so that, if p is the component of momentum of an electron in the direction of the field (which we take to be along the x axis), we shall have... 

The dipole polarizability of DBT has been measured experimentally by refractometry techniques, and evaluated theoretically with ab initio and DFT methods in the A electronic ground state. The molecular dipole polarizability a is the linear response of a molecular electronic distribution to the action of an external electric field Such an external field causes charge rearrangements in the molecular structure that are reflected in changes in the permanent molecular dipole moment <2001JP0709>. 

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