Electrical moments quantum-mechanical

Once an approximation to the wavefunction of a molecule has been found, it can be used to calculate the probable result of many physical measurements and hence to predict properties such as a molecular hexadecapole moment or the electric field gradient at a quadrupolar nucleus. For many workers in the field, this is the primary objective for performing quantum-mechanical calculations. But from... 

The quantum mechanical argument used in deriving the original electronegativity scale involved the amount of ionic character of a normal covalent bond A—B, and it was evident that the amount of ionic character and accordingly the value of the electric dipole moment of the bond would be closely correlated with the difference Ax = xA — xB of the two atoms A and B. In the first edition of The Nature of the Chemical Bond (1939) the following equation was advanced ... 

In his pioneering work Baetzold used the Hartree-Fock (HF) method for quantum mechanical calculations for the cluster structure (the details are summarized in Reference 33). The value of the HF procedure is that it yields the best possible single-determinant wave function, which in turn should give correct values for expectation values of single-particle operators such as electric moments and... 

In the ideal case of free Eu + ions, we first must observe that the components of the electric dipole moment, e x, y, z), belong to the irreducible representation in the full rotation group. This can be seen, for instance, from the character table of group 0 (Table 7.4), where the dipole moment operator transforms as the T representation, which corresponds to in the full rotation group (Table 7.5). Since Z)° x Z) = Z) only the Dq -> Fi transition would be allowed at electric dipole order. This is, of course, the well known selection rule A.I = 0, 1 (except for / = 0 / = 0) from quantum mechanics. Thus, the emission spectrum of free Eu + ions would consist of a single Dq Ei transition, as indicated by an arrow in Figure 7.7 and sketched in Figure 7.8. 

Although the relation between the vibrational g factor and the derivative of electric dipolar moment, equation (10), is formally equivalent to the relation between the rotational g factor and this dipolar moment, equation (9), there arises an important distinction. The derivative of the electrical dipolar moment involves the linear response of the ground-state wave function and thus a non-adiabatic expression for a sum over excited states similar to electronic contributions to the g factors. The vibrational g factor can hence not be partitioned in the same as was the rotational g factor into a contribution that depends only on the ground-state wave function and irreducible non-adiabatic contribution. Nevertheless g "(R) is treated as such. A detailed expression for ( ) in terms of quantum-mechanical operators and a sum over excited states, similar to equations (11) and (12), is not yet reported. 

Electric conductivity of ion-radical salts arises from the mobility of their unpaired electrons. At the same time, each of the unpaired electrons possesses a magnetic moment. This small magnetic moment is associated with the electron quantum-mechanical spin. Spin-originated magnetism as a phenomenon is described in many sources (see, e.g., monographs by Khan 1993, Bauld 1997, Itokh and Kinoshita 2001 and reviews by Miller 2000, Miller and Epstein 1994, 1995, Wudl and Thompson 1992). This section is, naturally, devoted to the organic magnets based on ion-radicals. 

Dipolar ions like CN and OH can be incorporated into solids like NaCl and KCl. Several small dopant ions like Cu and Li ions get stabilized in off-centre positions (slightly away from the lattice positions) in host lattices like KCl, giving rise to dipoles. These dipoles, which are present in the field of the crystal potential, are both polarizable and orientable in an external field, hence the name paraelectric impurities. Molecular ions like SJ, SeJ, Nf and O J can also be incorporated into alkali halides. Their optical spectra and relaxation behaviour are of diagnostic value in studying the host lattices. These impurities are characterized by an electric dipole vector and an elastic dipole tensor. The dipole moments and the orientation direction of a variety of paraelectric impurities have been studied in recent years. The reorientation movements may be classical or involve quantum-mechanical tunnelling. 

Molecular electronic dipole moments, pi, and dipole polarizabilities, a, are important in determining the energy, geometry, and intermolecular forces of molecules, and are often related to biological activity. Classically, the pKa electric dipole moment pic can be expressed as a sum of discrete charges multiplied by the position vector r from the origin to the ith charge. Quantum mechanically, the permanent electric dipole moment of a molecule in electronic state Wei is defined simply as an expectation value ... 

One of the more profound manifestations of quantum mechanics is that this curve does not accurately describe reality. Instead, because the motions of electrons are correlated (more properly, the electronic wave functions are correlated), the two atoms simultaneously develop electrical moments that are oriented so as to be mutually attractive. The force associated with tills interaction is referred to variously as dispersion , the London force, or the attractive van der Waals force. In the absence of a permanent charge, the strongest such interaction is a dipole-dipole interaction, usually referred to as an induced dipole-induced dipole interaction, since the moments in question are not permanent. Such an interaction has an inverse sixtli power dependence on the distance between the two atoms. Thus, the potential energy becomes increasingly negative as the two noble gas atoms approach one another from infinity. 

Not only must the difference E2 - E1 be correct for absorption but also there must always be a change in the dipole moment of the molecule in going from one energy level to another. Only when this is true can the electric field of the light wave interact with the molecule. A further limitation comes from the symmetry properties of the wave functions associated with each energy level. Quantum mechanical considerations... 

The intermolecular forces of adhesion and cohesion can be loosely classified into three categories quantum mechanical forces, pure electrostatic forces, and polarization forces. Quantum mechanical forces give rise both to covalent bonding and to the exchange interactions that balance tile attractive forces when matter is compressed to the point where outer electron orbits interpenetrate. Pure electrostatic interactions include Coulomb forces between charged ions, permanent dipoles, and quadrupoles. Polarization forces arise from the dipole moments induced in atoms and molecules by the electric fields of nearby charges and other permanent and induced dipoles. 

The first term in the square bracket in this equation is the electric monopole moment, which is equal to the nuclear charge, Ze. The second term in the square bracket is the electric dipole moment while the third term in the square bracket is the electric quadmpole moment. For a quantum mechanical system in a well-defined quantum state, the charge density p is an even function, and because the dipole moment involves the product of an even and an odd function, the corresponding integral is identically zero. Therefore, there should be no electric dipole moment or any other odd electric moment for nuclei. For spherical nuclei, the charge density p does not depend on 0, and thus the quadmpole moment Q is given by... 

We have performed a series of semiempirical quantum-mechanical calculations of the molecular hyperpolarzabilities using two different schemes the finite-field (FF), and the sum-over-state (SOS) methods. Under the FF method, the molecular ground state dipole moment fJ.g is calculated in the presence of a static electric field E. The tensor components of the molecular polarizability a and hyperpolarizability / are subsequently calculated by taking the appropriate first and second (finite-difference) derivatives of the ground state dipole moment with respect to the static field and using... 

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