Dipole moments system Hamiltonians

This term (Equation 19.31), linear in F, is zero for a nondegenerate system with no permanent electric dipole moment, whose Hamiltonian is unaffected by the parity operation [94]. In centrosymmetric nondegenerate polymers with no permanent dipole moment, the linear Stark effect ensues from disorder [95]. In a semiclassical approach, the shift in energy caused by a permanent dipole moment my can be expressed as ... 

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... 

Let us suppose that the system of interest does not possess a dipole moment as in the case of a homonuclear diatomic molecule. In this case, the leading term in the electric field-molecule interaction involves the polarizability, a, and the Hamiltonian is of the form ... 

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... 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

An inference of fundamental importance follows from Eqs. (2.3.9) and (2.3.11) When long axes of nonpolar molecules deviate from the surface-normal direction slightly enough, their azimuthal orientational behavior is accounted for by much the same Hamiltonian as that for a two-dimensional dipole system. Indeed, at sin<9 1 the main nonlocal contribution to Eq. (2.3.9) is provided by a term quadratic in which contains the interaction tensor V 2 (r) of much the same structure as dipole-dipole interaction tensor 2B3 > 0, B4 < 0, only differing in values 2B3 and B4. For dipole-dipole interactions, 2B3 = D = flic (p is the dipole moment) and B4 = -3D, whereas, e.g., purely quadrupole-quadrupole interactions are characterized by 2B3 = 3U, B4 = - SU (see Table 2.2). Evidently, it is for this reason that the dipole model applied to the system CO/NaCl(100), with rather small values 0(6 25°), provided an adequate picture for the ground-state orientational structure.81 A contradiction arose only in the estimation of the temperature Tc of the observed orientational phase transition For the experimental value Tc = 25 K to be reproduced, the dipole moment should have been set n = 1.3D, which is ten times as large as the corresponding value n in a gas phase. Section 2.4 will be devoted to a detailed consideration of orientational states and excitation spectra of a model system on a square lattice described by relations (2.3.9)-(2.3.11). 

The dipole moment is the total dipole of the sample, p = Y.i Pi The correlation function describes the response of the system to the weakly coupled radiation field. The effects of the field are modeled by the response of the individual atoms or molecules unaffected by the weak coupling. The Hamiltonian describes the interaction of the field and matter (first-order perturbatiuon theory). The correlation function describes how the perturbed system approaches equilibrium. 

Thus suppose we had included the interaction of the radiation s magnetic field B with the atomic or molecular electrons and nuclei. The Hamiltonian for this interaction is [Equation (1.268)] -B , where p is the magnetic dipole-moment operator for the system. This gives additional terms in cm that are proportional to... 

Most numerical methods for calculating molecular hyperpolarizability use sum over states expressions in either a time-dependent (explicitly including field dependent dispersion terms) or time-independent perturbation theory framework [13,14]. Sum over states methods require an ability to determine the excited states of the system reliably. This can become computationally demanding, especially for high order hyperpolarizabilities [15]. An alternative strategy adds a finite electric field term to the hamiltonian and computes the hyperpolarizability from the derivatives of the field dependent molecular dipole moment. Finite-field calculations use the ground state wave function only and include the influence of the field in a self-consistent manner [16]. 

Here Hq is the molecular Hamiltonian, and fi e(t) is the interaction between the molecule and the laser field in the dipole approximation, where (i is the transition dipole moment of the molecule. Time evolution of the system is determined by the time-dependent Schrodinger equation,... 

The ACF of the dipole moment operator of the fast mode may be written in the presence of Fermi resonances by aid of Eq. (10). Besides, the dipole moment operator at time t appearing in this equation is given by a Heisenberg equation involving the full Hamiltonian (225). The thermal average involved in the ACF must be performed on the Boltzmann operator of the system involving the real... 

In order to demonstrate the efficiency of the present theory for systems with many degrees of freedom, we have applied it to a 4-D model of HCN CNH isomerization (i.e., isomerization in a plane). The system is described in terms of two vectors J n=c for the vector from N to C, and Ru for the vector from the center of mass of the system to H. A fixed spatial Cartesian framework is used, with the a -axis set to be parallel to the initial direction of Rn=c, and the y- and 2 -axes perpendicular to it. The center of mass of NC is assumed to be the same as that of the whole system so that the kinetic part of the Hamiltonian is diagonal. The potential energy surface and the dipole moment are taken from [32]. 

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 ... 

We have shown in Section V.A.2 that a laser field can drive the V-type system into the antisymmetric (trapping) state through the coherent interaction between the symmetric and antisymmetric states. Akram et al. [24] have shown that in the A system there are no trapping states to which the population can be transferred by the laser field. This can be illustrated by calculating the transition dipole moments between the dressed states of the driven A system. The procedure of calculating the dressed states of the A system is the same as for the V system. The only difference is that now the eigenstates of the unperturbed Hamiltonian Ho are 3, N - 1), 1,N), 2,N), and the dressed states are given by... 

The average value of the dipole moment will be calculated by means of Dirac s perturbation theory for nonstationary. states, up to third order the zero order refers to the free molecules in the absence of the field. Let the wave function of the system of the two interacting molecules in- the external field be specified by y, an eigenfunction of the total Hamiltonian H. This wave function y> may be expanded in a complete set of the energy eigenfunctions unperturbed system the index n labels the various unperturbed eigenstates characterized by the energy En. We may then write... 

Let us now formulate an objective, namely, that a control field E(t) (field Ec(t) in Fig. 1) is to be determined which causes a fragmentation. This scenario is sketched in Fig. 2. Intuitively it is clear that, in order to break the bond, energy has to be pumped into the system. In the formalism of LCT, this can be realized if the system s energy (represented by the Hamiltonian F/o,i) increases steadily as a function of time. To specify the interaction with the external field, we have to take the special features of the Nal molecule into consideration Due to the ionic character of state 11) at distances larger than the point where the nonadiabatic coupling is present (marked as Rc in Fig. 1), the dipole moment is linear [106], so that the dipole interaction is W(R,1) = —RE(t). Then, via Eq. (4) and for A = 7/0,l we find... 

Almost all the formalism and the approximation schemes of Sections II and III have a natural extension to systems of polarizable dipolar particles, but the precise details of the extension depend on the way polarizability is introduced into the Hamiltonian. We refer to the two quite distinct Hamiltonian models that have been most thoroughly developed in this context as the constant-polarizability model and the fluctuating-polarizability model. The dielectric behavior of the former was first systematically investigated from a statistical mechanical viewpoint by Kirkwood and by Yvon, who considered the model almost exclusively in the absence of permanent dipole moments. (Kirkwood S subsequently pioneered an exact formulation of the statistical mechanics of polar molecules, but largely as a separate enterprise that did not attempt to treat the polarizability exactly.) The general case of polar-polarizable particles remained only very partially developed ... 

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