Interaction Hamiltonian electric dipole

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... 

The perturbation theory is the convenient starting point for the determination of the polarizability from the Schrodinger equation, restricted to its electronic part and the electric dipole interaction regime. The Stark Hamiltonian —p. describes the dipolar interaction between the electric field and the molecule represented by its... 

If the system under consideration also possesses an electric dipole moment d and is exposed to an electric field then the interaction Hamiltonian can be written... 

If the transition is of an electric dipole nature, the interaction Hamiltonian can be written as // = p E, where p is the electric dipole moment and E is the electric field of the radiation. The electric dipole moment is given by p =, where r is the... 

Provided that a transition is forbidden by an electric dipole process, it is still possible to observe absorption or emission bands induced by a magnetic dipole transition. In this case, the transition proceeds because of the interaction of the center with the magnetic field of the incident radiation. The interaction Hamiltonian is now written as // = Um B, where is the magnetic dipole moment and B is the magnetic field of the radiation. 

An applied electric field (E) interacts with the electric dipole moment (p,e) of a polar diatomic molecule, which lies along the direction of the intemuclear axis. The applied field defines the space-fixed p = 0 direction, or Z direction, whilst the molecule-fixed q = 0 direction corresponds to the intemuclear axis. Transformation from one axis system to the other is accomplished by means of a first-rank rotation matrix, so that the interaction may be represented by the effective Hamiltonian as follows ... 

Two further aspects need to be considered in order to understand the magnetic resonance spectrum, namely, the effects of an applied magnetic field, and the electric dipole transition probabilities. The effective Hamiltonian describing the interactions with an applied magnetic field, expressed in the molecule-fixed axis system q, is ... 

It is, of course, also necessary to calculate the relative intensities of the hyperfine components of each rotational transition in order to assign the spectrum. As we have seen elsewhere, the perturbation due to the interaction of the microwave electric field E(t) with the molecular electric dipole moment may be represented by the effective Hamiltonian... 

In this section we establish the general conditions under which the electric dipole electromagnetic field interaction may be used to attain selective control over the population of a desired enantiomer. Consider a molecule, described by the total Hamiltonian (including electrons and nuclei) Hmt, which has eigenstates describing the L and D enantiomers, denoted L ) and Dt) (i = 1,2,3,. ..) that satisfy... 

Forced Electric Dipole Transitions. In more recent work, Judd (15) has given further attention to the problem of intensities. According to this work, under certain symmetry restricted circumstances, the Hamiltonian for the interaction of a lanthanide ion with its neighbors can contain spherical harmonics with fc = 1 if the electrons of the rare-earth ion produce an electric field at the nucleus that exactly cancels that... 

I is in general no direct relation between such functions and ionization energies or electron excitation this is because they are not eigenfunctions of a hamiltonian, hence they cannot be associated with an energy. For that reason, we kept the usual designation localized molecular orbitals but with [ the last word in inverted commas orbitals . However, for the interpretation of some other molecular properties, the minimized residual interactions i between quasi-localized molecular orbitals are not very importaint and, so, the direct use of a localized bond description is quite justified. That is the [ Case for properties such as bond energies and electric dipole moments, as well as the features of the total electron density distribution with which those properties are directly associated. 

The Hamiltonian of the crystal with possible ordering of the electric dipole moments [24] additionally to the traditional terms discussed above contains the energy of the polarized crystal, the electron-polarization interaction (similar to the electron-strain interaction), and the interaction with the external electric field. After... 

Here (a)/ are the components of the operator of the quadrupole moment of the molecule in the LSC. Usually in the measurement of the difference of the refraction indices (195) the contribution to the Kerr effect can be separated (Buckingham and Disch, 1968). Therefore in Eq. (196) one can neglect the contribution of the interaction of the dipole moment with the electric field and use the following Hamiltonian instead of H(%) ... 

Because circular dichroism is a difference in absorption for left and right circularly polarized light, its theoretical description includes subtraction of the transition probabilities induced by left and right circularly polarized radiation. The interaction Hamiltonian that determines transition probability includes electric, , and magnetic, B, fields of electromagnetic circularly polarized radiation, and the electric, /i, and magnetic, m, dipole moments of the molecule. 

The total Hamiltonian describing the energies of the systems, electromagnetic field and interactions, in the electric dipole and RWA (rotating-wave approximation) approximations [21], is composed of four terms... 

The explicit form of the interaction Hamiltonian // ,(() consists of a series of multipolar terms, but for most purposes the electric-dipole (El) approximation is sufficient. Although the results are calculated within this approximation for each molecular center detailed analysis of the coupling provides results equivalent to the inclusion of higher-order multipole terms for the pair. The same assumption underlies the well-known coupled-chromophore model of optical rotation (Kuhn 1930 Boys 1934 Kirkwood 1937). The Hamiltonian for the system may thus be written as... 

The role of the geometric factor has frequently been ignored. However, in this special case it takes a simple form. The interaction Hamiltonian between two electric dipoles /xA and separated by a distance R may be written as... 

The full result for the electric dipole-electric dipole resonance interaction arising from the complete Hamiltonian (IV.5) was given by McLone and Power... 

The remainder of this section is devoted to a simplified two-level treatment of the Zeeman and Stark effects in the presence of zero-field Stark effect and field-dependent interactions between basis functions 1M) and 2M). In the presence of a static field directed along the space Z-axis, Mj remains a good quantum number. The Zeeman and Stark Hamiltonians involve the interaction between a magnetic field or electric dipole, /r, in the molecule-fixed axis system and the space-fixed magnetic or electric field, F, parallel to the laboratory direction K. The interaction can be expressed in terms of direction cosines... 

Let us consider how independent /i(i ) 2 effects contribute to the v E) for the hydrogen halides, HX (X = I, Br, and Cl). The curves shown on Fig. 7.6 correspond to relativistic adiabatic potential energy curves (respectively 0 dotted, 0+ dashed, 1 and 2 solid) for HI obtained after diagonalization of the electronic plus spin-orbit Hamiltonians (see Section 3.1.2.2). The strong R-dependence of the electronic transition moment reflects the independence of the relative contributions of the case(a) A-S-Q basis states to each relativistic adiabatic II state. The independent experimental photodissociation cross sections are plotted as solid curves in Fig. 7.7 for HI and HBr. Note that, in addition to the independent variations in the A — S characters of each fl-state caused by All = 0 spin-orbit interactions, all transitions from the X1E+ state to states that dissociate to the X(2P) + H(2S) limit are forbidden in the separated atom limit because they are at best (2Pi/2 <— 2P3/2) parity forbidden electric dipole transitions on the X atom. In the case of the continuum region of an attractive potential, the energy dependence of the dissociation cross section exhibits continuity in the Franck-Condon factor density (see Fig. 7.18 Allison and Dalgarno, 1971 Smith, 1971 Allison and Stwalley, 1973). 

Here R and P are sets of coordinate and momentum operators, respectively. The unperturbed system is described by the Hamiltonian Ho, which is the sum of the kinetic energy (T(P)) and the potential energy (V(R)) operators. The time-dependent perturbation is denoted as W(R,t). In the examples presented below, the latter is an electric dipole interaction with an external field E(t) ... 

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