Dipole length operator

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... 

In Eq. (14), /max is the maximum of the orbital angular momentum quantum numbers of the active electron in either the initial or final states, I nl, n l ) is the radial transition integral, that contains only the radial part of both initial and final wavefunctions of the jumping electron and a transition operator. Two different forms for this have been employed, the standard dipole-length operator, P(r) = r, and another derived from the former in such a way that it accounts explicitly for the polarization induced in the atomic core by the active electron [9],... 

Clearly, the n electron cloud of pyridine looks much more like that of benzene than is usually postulated in the Htickel theory. As a matter of fact, the dipole moment of 3.11 D (exp 2.20 D), calculated from the SCF wave function of ref. 107> using the dipole length operator, includes a very small it component (0.325 D) oriented in the same direction as a large a component (2.785 D) 115>s). 

The oscillator strengths are calculated in the INDO/S model by utilizing the dipole length operator including all one-center terms. 

Fourth, oscillator strengths using the dipole-length operator are generally overestimated by a factor of two or three for the more intense bands. This is a well-known feature of CIS calculations and is vastly improved with the inclusion of doubles, but the model has not been parameterized at the CISD level, and there are reasons for not doing so. Rather, work is underway to parameterize the model at the RPA level,which is a more consistent scheme for calculating excitation energies than is the CISD technique. 

As in the case of LS coupling, when there are N equivalent electrons in the outer shell, both the line strength and the oscillator strength should be multiplied by N as well as by the corresponding CFP [ 10,12]. As in the LS scheme the two forms of the electric dipole length transition operator have been employed here in the calculation of the radial transition integral, I nl, n l ). 

Dipole length, 65 Dipole moments, excited states, 103 operator, 131 oscillating, 49... 

Previous work has not investigated if commutation relations are conserved upon transformation to effective operators. Many important consequences emerge from particular commutation relations, for example, the equivalence between the dipole length and dipole velocity forms for transition moments follows from the commutation relation between the position and Hamiltonian operators. Hence, it is of interest to determine if these consequences also apply to effective operators. In particular, commutation relations involving constants of the motion are of central importance since these operators are associated with fundamental symmetries of the system. Effective operator definitions are especially useful... 

Another important application of Theorem V is that (Corollary V.2) the dipole length and dipole velocity transition moments are equivalent when computed with state-independent effective operators obtained with norm-preserving mappings. According to definition A (see Table I), these computations evaluate o( p /8)o, and (a r )3)oWith... 

Table III summarizes Theorems V-VII and their corollaries along with similar results for the other state-independent effective operator definitions. Appendix E demonstrates the analogs of Theorems V-VII, except the conservation by definitions A" and A " of [H, C] for C a constant of the motion which commutes separately with and V. This last point is proven in paper II. The analogs of Corollaries V.l and V.2 are obtained similarly to, respectively. Corollaries V.l and V.2. Just as with Corollary V.2, none of the equivalences between the dipole length and dipole velocity transition moments for definitions A", A , or A , / = I-IV, produces a sum rule for transition moments (see Appendix D).

Section IV proves that the conservation of the commutation relation (4.12) between H and the position operator f leads to the equivalence of the dipole length and dipole velocity transition moments computed with certain effective operator definitions. Contrary to the similar equivalence for transition moments computed with true operators, however, this does not yield a sum rule. Many other sum rules follow from commutation relations between true operators. In view of the many useful applications of sum rules [141, 142] the existence of sum rules for quantities computed using effective operators is of interest and will be studied elsewhere [79]. A potential application lies in determining the amount, or proportion, of transition strengths carried by a particular state or group of states [142, 143]. 

This appendix demonstrates that the equivalence between the dipole length and dipole velocity transition moments, computed using effective operators, does not produce a sum rule for these moments. The proof is first provided for effective operator definition A, and then modifications required for definitions A", A", A", A, and A, i = I-IV are described. 

While O Eq. 5.11 is called the length gauge or dipole-length gauge expression, O Eq. 5.45 is often called the mixed gauge form since it involves both the electric dipole operator and the momentum operator. The length and mixed gauge polarizabilities are equivalent due to the equation of motion... 

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. 

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... 

The fine-structure constant a indicates that first-order perturbation theory has been applied the linear dependence on the photon energy Eph is due to the length form of the dipole operator used in equ. (2.1), and the wavenumber k compensates the 1 /k which appears if the absolute squared value of the continuum wavefunction is used (see equ. (7.29)). The summations over the magnetic quantum numbers M, of the photoion and ms of the photoelectron s spin are necessary because no observation is made with respect to these substates. Due to the closed-shell structure of the initial state with f — 0 and M = 0, the averaging over the magnetic quantum numbers M simply yields unity and is omitted. 

The dipole matrix element on the right-hand side of equ. (8.19b) is called the length form of the matrix element, because the vector r acts as the photon operator (see the discussion of equ. (1.28a) in which the name dipole approximation is also explained). Equ. (8.16) can then be replaced by... 

In the present context the two transition operators of relevance are that for photoionization which is given in the dipole approximation, and within the length form, by (see equ. (1.28a))... 

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