The Interaction Hamiltonian

In order to include all types of discrimination we first give a complete hamil-tonian (II. 1), expressed as the sum of the isolated molecule hamiltonians Hs, and H i, their electrostatic interaction H-e, the coupling by radiation, and the hamiltonian for the radiation field. 

The sums in expression (II.2) run over the electrons i and the nuclei p (charge Zp) of molecule a, and over j and j in b. pf is the particle momentum operator. 

The expansion gives the interaction operator as a series of multipole-multipole operators, which can be taken term by term. The quantities Q nt) are the components of multipole moments referred to axes fixed in the molecules, primes distinguishing quantities for centre b. The moments transform as spherical harmonics, and are defined in (II.5), 

When the expectation value of (II.4) is taken over the product of the ground state wave functions for molecules a and b, the result is the sum of coupling energies for the permanent electric moments in the two molecules. If the moments are known, or taken as parameters, the electrostatic interaction energy is known for a chosen orientation, and can be compared for d and I species. By forming averages over angles we can make the calculation for molecules in relative rotational motion. First it is useful to examine the symmetry restrictions imposed by chiral character, and to see how the moments in one chiral enantiomer are related to those in the other. 

---

In order to evaluate equation B1.2.6, we will consider the electric field to be in the z-direction, and express the interaction Hamiltonian as... 

In diatomic VER, the frequency Q is often much greater than so VER requires a high-order multiphonon process (see example C3.5.6.1). Because polyatomic molecules have several vibrations ranging from higher to lower frequencies, only lower-order phonon processes are ordinarily needed [34]- The usual practice is to expand the interaction Hamiltonian > in equation (03.5.2) in powers of nonnal coordinates [34, 631,... 

The non-bonded interaction energy, the van-der-Waals and electrostatic part of the interaction Hamiltonian are best determined by parametrizing a molecular liquid that contains the same chemical groups as the polymers against the experimentally measured thermodynamical and dynamical data, e.g., enthalpy of vaporization, diffusion coefficient, or viscosity. The parameters can then be transferred to polymers, as was done in our case, for instance in polystyrene (from benzene) [19] or poly (vinyl alcohol) (from ethanol) [20,21]. 

In other words that a negaton initially in a state of momentum p, energy Vp2 + m2 helicity s, would remain forever in that state (since it does not interact with anything). Let us, however, compute the left-hand side of Eq. (11-123) with the -matrix given in terms of the interaction hamiltonian (11-121). To lowest order the diagrams indicated in Fig. 11-6 contribute and give rise to the following contribution to the matrix element of S between one-particle states... 

Similar considerations lead to the transformation properties of the one-photon states and of the photon in -operators which create photons of definite momentum and helicity. We shall, however, omit them here. Suffice it to remark that the above transformation properties imply that the interaction hamiltonian density Jf mAz) = transforms like a scalar under restricted inhomogeneous Lorentz transformation... 

The interaction hamiltonian density jin A x) transforms like a scalar ... 

The basic physics governing the measurement of the EDM in all types of electrically neutral systems is almost the same as discussed in this section. If the system under consideration has a magnetic moment p and is exposed to a magnetic field B, then the interaction Hamiltonian can be written... 

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... 

The relativistic treatment of electron EDM begins by replacing the nonrelativistic Hamiltonian H and the interaction Hamiltonian Hi by their relativistic counterparts... 

Consequently, we introduce the second approximation which is to use an approximate electrostatic potential in Eq.(4-21) to determine inter-fragment electronic interaction energies. Thus, the electronic integrals in Eq. (4-21) are expressed as a multipole expansion on molecule J, whose formalisms are not detailed here. If we only use the monopole term, i.e., partial atomic charges, the interaction Hamiltonian is simply given as follows ... 

If the interaction Hamiltonian in the Coulomb term is expanded in a series about the separation vector, the first term of the expansion is a dipole-dipole interaction, the second a dipole-quadrupole interaction, etc.<4> Again reverting to a classical analog (dipole oscillators), the energy of interaction between the two dipoles is inversely proportional to the third power of the... 

The interaction Hamiltonian with the specified short-range interactions between the monomers is given by the following expression [124,125] ... 

The interaction between both subsystems is cast into a form where the physical charge density of the surrounding medium rm(i X) = appears explicitly, and the interaction Hamiltonian describes now the coupling of the solute... 

Empirically corrected DFT theories almost invariably go back to second-order perturbation theory with expansion of the interaction Hamiltonian in inverse powers of the intermolecular distance, leading to R 6, R x, and R 10 corrections to the energy in an isotropic treatment (odd powers appear if anisotropy is taken into account [86]). 

The interaction Hamiltonian is decomposed into a sum of terms characterized by their commutation relations with Sz ... 

We know from Chapter 1 that the probability P,f of indncing an optical transition from a state i to a state / is proportional to (1 //1), where in the matrix element Ip, and P f denote the eigenfnnctions of the ground and excited states, respectively, and H is the interaction Hamiltonian between the incoming light and the system (i.e., the valence electrons of the center). In general, we can assnme that // is a sinnsoidal... 

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. 

The interaction Hamiltonian that appears in Equation (5.37) can involve different types of interactions namely, multipolar (electric and/or magnetic) interactions and/or a quantum mechanical exchange interaction. The dominant interaction is strongly dependent on the separation between the donor and acceptor ions and on the nature of their wavefunctions. 

In order to obtain a formula for the energy difference between the complete ground state energy E and the noninteracting energy Eks, and thus for E c, the interaction Hamiltonian H = H- is supplemented by a dimensionless coupling strength parameter g in such a way,... 

The interaction Hamiltonian contains the operator A, corresponding to the vector potential A of the electromagnetic field.2 Excluding magnetic scattering, the interaction Hamiltonian is given by... 

In the vicinity of the atomic absorption edges, the participation of free and bound excited states in the scattering process can no longer be ignored. The first term in the interaction Hamiltonian of Eq. (1.11) leads, in second-order perturbation theory, to a resonance scattering contribution (in units of classical electron scattering) equal to (Gerward et al. 1979, Blume 1994)4... 

We assume that initially, the system is brought in contact with its environment (rather than being in equilibrium with it), which corresponds to factorizing initial conditions Ptot(O) = P(0) Ob- Th environment is in a steady state Og, [pg, Hg] = 0, so it is more adequate to speak of a bath. Tracing over the bath in Eq. (4.154) then gives the change of system state Ap = o(t) - p(0) over time t, which we must insert into Eq. (4.150). We further assume a vanishing bath expectation value of the interaction Hamiltonian,... 

---