Velocity dipole operator

Variation principle 18, 154, 222 VB (valence bond) model 94 Vector 4 Vector docking 57 Vector potential 294 Vector space 220 Vector, cross product 6 Vector, dot product 5 Vectors, orthogonal 6 Velocity dipole operator 193 Velocity relaxation 253... 

The second-rank tensor P,y (m) depends on the velocity dipole operator, while Mif(co) depends on the velocity dipole operator and on the magnetic dipole operator and finally T (co) on the velocity dipole operator and the velocity form of the electric quadrupole operator, respectively. Their mathematical expressions are reported and described in detail in Chapter 2. Once more, like we did for TPA, invoking the BO approximation and integrating over the electronic coordinates, the TPCD intensity between vibronic states can be written in terms of elements of electronic transition tensors Pej/it, rj, co), Me,f x, rj, co), and T yr(x, rf, co) between the vibrational states and Z5(/)) associated with the initial and final electronic states 0,) and 0/), respectively. 

A similar calculation using the velocity form of the dipole operator leads to... 

The origin with respect to which the electric quadrupole and magnetic dipole operators are defined is indicated by the superscript. jiPp is the /3 component of the velocity operator. The connection between the quadrupole moment referred to or - for example the centre of nuclear masses - and the EQC is... 

In order to calculate R from equation 3, we have to evaluate the integrals iin n and mj,. n, with the MO jr and jr given in equations 5a and 5b. We assume only nearest-neighbour interactions and equal -integrals in all the integrals for the pairs (1,2) and (3,4), as justified by symmetry. The electric dipole will be described by the velocity operator V, in order to ensure origin-independent results, and equation 6 follows ... 

In Tables -A, we report oscillator strengths for some fine structure transitions in neutral fluorine, chlorine, bromine and iodine, respectively. Two sets of RQDO/-values are shown, those computed with the standard dipole length operator g(r) = r, and those where core-valence correlation has been explicitly introduced, Eq. (10). As comparative data, we have included in the tables /-values taken from critical compilations [15,18], results of length and velocity /-values by Ojha and Hibbert [17], who used large configuration expansions in the atomic structure code CIVS, and absolute transition probabilities measured through a gas-driven shock tube by Bengtson et al. converted... 

Relativistic corrections of order v2/c2 to the non-relativistic transition operators may be found either by expanding the relativistic expression of the electron multipole radiation probability in powers of v/c, or semiclas-sically, by replacing p in the Dirac-Breit Hamiltonian by p — (l/c)A (here A is the vector-potential of the radiation field) and retaining the terms linear in A. Calculations show that in the general case the corresponding corrections have very complicated expressions, therefore we shall restrict ourselves to the particular case of electric dipole radiation and to the main corrections to the length and velocity forms of this operator. 

Let us consider the intercombination transitions. Then, we shall retain only the corrections containing the spin operator in the expansion. To find the form of the operator describing the electric multipole intercombination transitions and absorbing the main relativistic corrections, one has to retain in the corresponding expansion the terms containing spin operator S = a and to take into account, for the quantities of order v/c, the first retardation corrections, whereas, for the quantities of order v2/c2 one must neglect the retardation effects. Then the velocity form of the electric dipole transition probability may be written as follows ... 

Calculations show that cross-sections obtained in the Hartree-Fock approximation utilizing length and velocity forms of the appropriate operator, may essentially differ from each other for transitions between neighbouring outer shells, particularly with the same n. However, they are usually close to each other in the case of photoionization or excitation from an inner shell whose wave function is almost orthogonal with the relevant function of the outer open shell. In dipole approximation an electron from a shell lN may be excited to V = l + 1, but the channel /— / + prevails. For configurations ni/f1 n2l 2 an important role is... 

Within the dipole approximation, one can have different forms for the dipole matrix element (see [BSa57]). The form presented so far is called the momentum form (or the velocity form) because the relevant operator contains the momentum p ... 

For the evaluation of probabilities for spin-forbidden electric dipole transitions, the length form is appropriate. The velocity form can be made equivalent by adding spin-dependent terms to the momentum operator. A sum-over-states expansion is slowly convergent and ought to be avoided, if possible. Variational perturbation theory and the use of spin-orbit Cl expansions are conventional alternatives to elegant and more recent response theory approaches. 

The use of the dipole velocity formulation requires the matrix elements of the linear momentum operator p. From the commutation relation of H with r, Linderberg (l%7) showed that the following expression is obtained in the ZDO approximation ... 

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