Moment operators defined

In (1.4), jj, is the dipole moment operator defined as Serj, with e representing the electron charge and rj representing the vector distance from the kth electron to the centre of positive charge of the molecule. 

It follows that the spherical components of the electric dipole operator will transform like spherical harmonics of order unity under any arbitrary rotation of the coordinate system. A quantity which transforms under rotations like the spherical harmonic Y (0,( i) is said to be an irreducible tensor operator T of rank k and projection q where the projection quantum number can take any integer value from -k to +k. Since any arbitrary function of 0 and < can generally be expanded as a sum of spherical harmonics, it is usually possible to express any physical operator in terms of irreducible tensor operators. For instance, the electric quad-rupole moment operator defined by equation (4.45) can be shown to be a tensor of rank 2 (Problem 5.6). 

The first term in the brackets is the expectation value of the square of the dipole moment operator (i.e. the second moment) and the second term is the square of the expectation value of the dipole moment operator. This expression defines the sum over states model. A subjective choice of the average excitation energy As has to be made. 

For infrared absorption the operator 0 is the dipole moment M defined by... 

In a different form, the traceless moment operators can be written as the Cartesian spherical harmonics c,mp multiplied by r, which defines the spherical harmonic electrostatic moments ... 

The first- and second-order Zeeman effect coefficients in the expansion of equation (62) are defined by the quantum numbers which specify the atomic energy level. They are in general a function of the direction of the magnetic field with respect to the axis of quantization of the wave functions. They are obtained by the use of the magnetic moment operator for the appropriate direction, q = x,y ox z ... 

The dipole operator d is a vector defined in the body-fixed frame of the molecule. Consequently, the transition dipole moment /a defined in (2.35) is a vector field with three components each depending — like the potential — on R, r, and 7. For a parallel transition the transition dipole lies in the plane defined by the three atoms and for a perpendicular transition it is perpendicular to this plane. Following Balint-Kurti and Shapiro, the projection of /z, which is normally calculated in the body-fixed coordinate system, on the space-fixed z-axis, which is assumed to be parallel to the polarization of the electric field, can be written as... 

All terms involving the electronic charge moment operators are defined as ... 

In Eq. (18), we recognize the first quantization fcth-order electronic multipole moment operator (r - Ro). An analogous expansion of r - Rm 1 would yield the nuclear multipole moment operator (Rm — Rq). The higher-order multipole moments generally depend on the choice of origin however, to simplify the notation, we omit any explicit reference to this dependence. The partial derivatives in Eq. (18) are elements of the so-called interaction tensors defined as... 

We now collect together terms in Po, Pr and pt. We shall need the electric dipole moment operator which we define by... 

The difference between the definitions of the shift operators J and the spherical tensor components T, (./) should be noted because it often causes confusion. Because J is a vector and because all vector operators transform in the same way under rotations, that is, according to equation (5.104) with k = 1, it follows that any cartesian vector V has spherical tensor components defined in the same way (see table 5.2). There is a one-to-one correspondence between the cartesian vector and the first-rank spherical tensor. Common examples of such quantities in molecular quantum mechanics are the position vector r and the electric dipole moment operator pe. 

Important connections between optical and thermal ET may be established if the matrix elements of the dipole moment operator p) as well as the Hamiltonian are used [12, 30], In particular, we note two expressions defined within the above TSA the first of these is given by Eq. 74 ... 

Fluorescence is defined simply as the electric dipole tranation from an excited electronic state to a lower state, usually the ground state, of the same multiplicity. Mathematically, the probability of an electric-dipole induced electronic transition between specific vibronic levels is proportional to R f where Rjf, the transition moment integral between initial state i and final state f is given by Eq. (1), where represents the electronic wavefunction, the vibrational wavefunctions, M is the electronic dipole moment operator, and where the Born-Oppenheimer principle of parability of electronic and vibrational wavefunctions has been invoked. The first integral involves only the electronic wavefunctions of the stem, and the second term, when squared, is the familiar Franck-Condon factor. 

As mentioned in the Introduction, we do not have know the left-hand eigenstates of Hn if we are not interested in properties other than the energy. However, we cannot avoid left-hand eigenstates in various property calculations. For example, the EOMCC transition moments involving the operator 0 (0 is, for example, the dipole moment operator) are defined as [34]... 

Another way to define this dispersion is to consider the expectation value of the quadratic moment operator with its origin at the centroid of charge ... 

However, there are two common operationally defined types of structure that are determined from effective moments of inertia. The more common, the so-called effective or r0 structure, is somewhat loosely defined. In practice, any structural parameter that requires for its determination fitting one or more of the second moment relations is designated as r0. r0 structures are not uniquely defined since, for any over-determined system, the value of structural parameters obtained depends somewhat on the manner in which the data are treated and the values are isotopically dependent. This problem is examined in more detail by Schwendeman (this volume). 

The second common type of operationally defined structure is the so-called substitution or rt structure.10 The structural parameter is said to be an rs parameter whenever it has been obtained from Cartesian coordinates calculated from changes in moments of inertia that occur on isotopic substitution at the atoms involved by using Kraitchman s equations.9 In contrast to r0 structures, rs structures are very nearly isotopically consistent. Nonetheless, isotope effects can cause difficulties as discussed by Schwendeman. Watson12 has recently shown that to first-order in perturbation theory a moment of inertia calculated entirely from substitution coordinates is approximately the average of the effective and equilibrium moments of inertia. However, this relation does not extend to the structural parameters themselves, except for a diatomic molecule or a very few special cases of polyatomics. In fact, one drawback of rs structures is their lack of a well-defined relation to other types of structural parameters in spite of the well-defined way in which they are determined. It is occasionally stated in the literature that r, parameters approximate re parameters, but this cannot be true in general. For example, for a linear molecule Watson12 has shown that to first order ... 

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