Hamiltonian electron electric dipole moment

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, H is the field-free semiempirical ir-electronic Hamiltonian and ft is the electric dipole moment operator, ji is expanded in terms of n as... 

The action of outside fields in the orbital space is also described very simply. The contribution to the one-electron Hamiltonian Hi that is due to an outside electric field E is = , where m = er is the electric dipole moment operator. In simple models such as Hiickel or PPP, the electron position operator f is diagonal in the A,B basis. Using the centroid of electron charge rQ = ( +... 

Effective electric dipole moment, 14 Effective electronic Hamiltonian, 1 Effective molecular response functions, 35 Electric dipole moment, 14... 

Hamiltonian = matrix element of the Hamiltonian H I = nuclear spin I = nuclear spin operator /r( ), /m( ) = energy distributions of Mossbauer y-rays = Boltzmann constant k = wave vector L(E) = Lorentzian line M = mass of nucleus Ml = magnetic dipole transition m = spin projection onto the quantization axes = 1 — a — i/3 = the complex index of refraction p = vector of electric dipole moment P = probability of a nuclear transition = tensor of the electric quadrupole q = eZ = nuclear charge R = reflectivity = radius-vector of the pth proton = mean-square radi-S = electronic spin T = temperature v =... 

The first term of (3.289) represents a translational Stark effect. A molecule with a permanent dipole moment experiences a moving magnetic field as an electric field and hence shows an interaction the term could equally well be interpreted as a Zeeman effect. The second term represents the nuclear rotation and vibration Zeeman interactions we shall deal with this more fully below. The fourth term gives the interaction of the field with the orbital motion of the electrons and its small polarisation correction. The other terms are probably not important but are retained to preserve the gauge invariance of the Hamiltonian. For an ionic species (q 0) we have the additional translational term... 

The inversion operation i which leads to the g/u classification of the electronic states is not a true symmetry operation because it does not commute with the Fermi contact hyperfine Hamiltonian. The operator i acts within the molecule-fixed axis system on electron orbital and vibrational coordinates only. It does not affect electron or nuclear spin coordinates and therefore cannot be used to classify the total wave function of the molecule. Since g and u are not exact labels, it was realised by Bunker and Moss [265] that electric dipole pure rotational transitions were possible in ll], the g/u symmetry breaking (and simultaneous ortho-para mixing) being relatively large for levels very close to the dissociation asymptote. The electric dipole transition moment for the 19,1 19,0 rotational transition in the ground electronic state was calculated... 

Finite-field methods were first used to calculate dipole polarizabilities by Cohen and Roothaan [66]. For a fixed field strength V, the Hamiltonian potential energy term for the interaction between the electric field and ith electron is just The induced dipole moment with the applied field can be calculated from the Hartree-Fock wavefunction by integrating the dipole moment operator with the one-electron density since this satisfies the Hellmann-Feyman theorem. With the usual dipole moment expansion. 

A fundamental difference exists with respect to a priori knowledge of electric versus magnetic dipole moments. Electric moments depend on the detailed shape of electronic wavefunctions and must always be measured experimentally or computed from accurate ab initio wavefunctions. In contrast, magnetic moments are directly proportional to angular momenta and may be computed from the eigenvectors of an effective Hamiltonian without knowledge of the electronic wavefunctions. 

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). 

The finite field procedure is the most often used procedure because of two main advantages (1) it is very easy to implement, and (2) it can be applied to a wide range of quantum mechanical methods. To calculate the energy in the presence of a uniform electric field of strength F, an F-r term needs to be added to the one-electron Hamiltonian. This interaction term can be constructed from just the dipole moment integrals over the basis set. Any ab initio or semiem-pirical method can then be used to solve the problem, with or without electron correlation. It is the ability to obtain properties from highly correlated methods that makes finite field calculations usually the most accurate available. 

All the calculations reported in this work were done on a DEC 20-60 using a modified version of the GAUSSIAN 80 series of program (6). Standard ST0-3G minimal basis set (7) was considered. Polarizabilities were calculated by the finite-field SCF method of Cohen and Roothaan (8) which is virtually equivalent to the analytic Coupled Hartree-Fock scheme. A term yf, describing the interaction between the electric field, E, and the molecule is added to the unperturbed molecular Hamiltonian, H y is the total dipole moment of the molecule. At the Hartree-Fock level, the electric field appears explicitly in the one-electron part of the modified Fock operator, F( ),... 

In the last three sections we have considered the effect of a time-dependent external electric field r,t) and a magnetic induction B r,t) on the motion of an electron and denoted the corresponding potentials with 4> r,t) and A r,t). In the present section we want to collect all the terms and derive our final expression for the molecular electronic Hamiltonian. However, we will not restrict ourselves to the case of external fields because in the following chapters we want to study also interactions with other sources of electromagnetic fields such as magnetic dipole moments and electric quadrupole moments of the nuclei, the rotation of the molecule as well as interactions with field gradients. Therefore, we do not include the superscripts B and on the vector and scalar potential in this section. On the other hand, we will assume that the perturbations are time independent. The time-dependent case is considered in Section 3.9. 

The wave is assumed to interact with a stationary, one-electron atom whose instantaneous dipole moment is given by = -er. For transitions involving the absorption or emission of a single electric dipole photon it is sufficient to take the classical interaction energy -p.E(t) as the perturbation operator. The small additional contribution to the Hamiltonian of the system is thus given by... 

A scheme for the treatment of the solvent effects on the electronic absorption spectra in solution had been proposed in the framework of the electrostatic SCRF model and quantum chemical configuration interaction (Cl) method. Within this approach, the absorption of the light by chromophoric molecules was considered as an instantaneous process. Tliere-fore, during the photon absorption no change in the solvent orientational polarization was expected. Only the electronic polarization of solvent would respond to the changed electron density of the solute molecule in its excited (Franck-Condon) state. Consequently, the solvent orientation for the excited state remains the same as it was for the ground state, the solvent electronic polarization, however, must reflect the excited state dipole and other electric moments of the molecule. Considering the SCRF Hamiltonian... 

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