Electric moment coordinates

The radial deformation of the valence density is accounted for by the expansion-contraction variables (k and k ). The ED parameters P, Pim , k, and k are optimized, along with conventional crystallographic variables (Ra and Ua for each atom), in an LS refinement against a set of measured structure factor amplitudes. The use of individual atomic coordinate systems provides a convenient way to constrain multipole populations according to chemical and local symmetries. Superposition of pseudoatoms (15) yields an efficient and relatively simple analytic representation of the molecular and crystalline ED. Density-related properties, such as electric moments electrostatic potential and energy, can readily be obtained from the pseudoatomic properties [53]. 

The electric moments are not invariant under a shift of the origin of the coordinate system. The first non-vanishing moment is however usually independent of the choice of the coordinates. The origin of coordinate system is thus often chosen as the center of gravity or the center of the charge distribution. 

Wc imagine a co-ordinate system , rj, introduced so that the -axis is in the direction of the angular momentum, and the 77-axis perpendicular to the direction of the field. In this system of coordinates the electric moment may be expressed in the form... 

As an example, let us discuss the case of the electric moment (15) induced by a static external electric field having a uniform gradient. The equation that describes the change of the mixed third-rank electric polarizability tensor (19) with respect to a change to the origin of the coordinate system is... 

The electric moments of a nucleus can be introduced, e.g., in the following way. Let the origin of a Cartesian coordinate system Tn> coincide with the center of the nucleus (O Fig, 2,4a). Then the electric potential Vn(r) of the nucleus is given by... 

For a pair of D and L enantiomers, the values of these two contributions have the same magnitude but opposite signs, and therefore can be envisaged to discriminate between enantiomers in NMR experiments. Similarly, these two, linear and quadratic effects, contribute to the electric dipole induced at any nucleus by the precession of a permanent magnetic dipole. The invariance to a translation of the coordinate system of these induced electric moment and induced magnetic field at the nucleus has been analyzed in details by Lazzeretti et alP ... 

We now consider the combinations because of the sepgurability one can separate the integrals that represent the combination intensities into integrals over dxp and di and those over the coordinates of the vibration. The work of Rademacher and Reiche contains everything that we need. If x, y and z are the components of the electrical moment in the space fixed reference system, then the combinations are determined by the expressions ... 

The rotatory strength is zero by symmetry if the molecule possesses at least one improper axis. Depending on which formulation is employed for the electric moment, R can also be obtained in the length and the velocity form. The length form depends on the coordinate origin, the results employing the velocity form are quite sensitive to details of the wavefunction. 

In the above expressions, s is equal to the smaller of numbers k and 1. Capital letter Mdjin) refers to the m h component of the 2 th electric moment expressed in spherical coordinates and related to the charge distributions associated with the orbital product P J(r) ... 

In addition, there could be a mechanical or electromagnetic interaction of a system with an external entity which may do work on an otherwise isolated system. Such a contact with a work source can be represented by the Hamiltonian U p, q, x) where x is the coordinate (for example, the position of a piston in a box containing a gas, or the magnetic moment if an external magnetic field is present, or the electric dipole moment in the presence of an external electric field) describing the interaction between the system and the external work source. Then the force, canonically conjugate to x, which the system exerts on the outside world is... 

Theoreticians did little to improve their case by proposing yet more complicated and obviously unreUable parameter schemes. For example, it is usual to call the C2 axis of the water molecule the z-axis. The molecule doesn t care, it must have the same energy, electric dipole moment and enthalpy of formation no matter how we label the axes. I have to tell you that some of the more esoteric versions of extended Hiickel theory did not satisfy this simple criterion. It proved possible to calculate different physical properties depending on the arbitrary choice of coordinate system. 

To look ahead a little, there are properties that depend on the choice of coordinate system the electric dipole moment of a charged species is origin-dependent in a well-understood way. But not the charge density or the electronic energy Quantities that have the same value in any coordinate system are sometimes referred to as invariants, a term borrowed from the theory of relativity. 

At the molecular level, electric quadrupoles can lead to useful structural information. Thus, whilst the absence of a permanent electric dipole in CO2 simply means that the molecule is linear, the fact that the electric quadrupole moment is negative shows that our simple chemical intuition of 0 C" 0 is correct. The definition of quadrupole moment is only independent of the coordinate origin when the charges sum to zero and when the electric dipole moment is zero. 

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

The change in the dipole moment with respect to a geometry displacement along a normal coordinate is approximately proportional to the intensity of an IR absorption. In the so-called double harmonic approximation (terminating the expansion at first order in the electric field and geometry), the intensity is (except for some constants)... 

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

In the case of an infrared spectrum, the intensity is related to the square of the infinitesimal change of the electric dipole moment p with respect to the normal coordinates,46 q,... 

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