Dipole velocity

Exciton coupling between the two styryl chromophores in each of the conformers (A)-(C) leads to a negative Cotton effect at around 270 nm, as shown by theoretical analysis using the 71-electron SCF Cl dipole velocity MO method. Therefore, the tram-2,.1-disubstituted spirocyclo-propane-1,9 -fluorene derivative 6, from which the distyryl derivative was obtained, must have accordingly the 2S3S absolute configuration122. 

Fig. 8. Absorption spectra (AB upper curves) and circular dichroism (CD lower curves) of calycanthine (formula in the inset of the figure). The solid lines refer to the experimental spectra and the broken lines to the theoretical curves for the configuration illustrated, calculated by the Pariser-Parr-Pople method using the dipole velocity procedure21,22)...

The absolute configuration of diketone 169 was confirmed by comparison of the experimental CD with that calculated by the SCF-CI-Dipole Velocity MO method361. 

Oscillator strengths can be defined either as dipole velocity quantities or dipole lengths,... 

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 ... 

FIguK 1.23. Spectral characteristics of the alkylidenecyclopropene 14 calculated as a function of the number of configurations used in the Cl procedure a) excitation energies and b) oscillator strengths, using dipole length (—) and dipole velocity ( ) expressions (by permission from Downing et al., 1974). 

In addition to the CD exciton chirality method, we have recently reported that the theoretical calculation of the CD spectra by the jt-electron SCF-CI-dipole velocity MO method8-l4 has become an important tool for determination of the absolute configuration of a variety of twisted and conjugated n-electron systems. In fact, we have recently determined the absolute stereochemistry of (8aS)-(+)-l,8a-dihydro-3,8-dimethylazulene 10, a labile biosynthetic intermediate for 1,4-dimethylazulene 11 isolated from a liverwort, by application of the MO method to the theoretical calculation of the CD spectra of the twisted tetraene system (8a/ )-12.15 In that case, we have also succeeded in the experimental verification of the absolute configuration theoretically determined, by comparison of the CD spectra of the natural product with those of synthetic chiral model compounds (8a5)-(+)-... 

The CD and UV spectra of the compound with a twisted n-electron system can be calculated by the jc-electron Self-Consistent-Field Configuration-Interaction Dipole-Velocity Molecular Orbital method (the Tc-electron SCF-CI-DV MO me-thod).8-10 In the dipole velocity method, the rotational strength I ba and dipole strength Dba which govern the sign and intensity of a CD Cotton effect and the intensity of a UV absorption band, respectively are formulated as follows ... 

Dipole length formula, 23, S6-Slt Dipole moment, 465. See also Electric dipole moment Magnetic dipole moment excipicx, 282 excited-slate. 47-48. 132 induced.130 permanent, 130-31 Dipole strength, IS8 Dipole velocity formula, 2, 56-57 Direct reaction. See Reaction 1,4-Disilabenzene, 105-7 Disproportionation, 228-29, 380, 390, 433 Disrotatory. See Elecirocyclic reaction ... 

It should be noted that in the RPA, the dipole oscillator strengths calculated in dipole velocity, dipole length, or mixed representation and all sum rules would be identical, and the TRK sum rule, Eq. (13), would be fulfilled exactly, that is, be equal to the number of electrons if the computational basis were complete [30,34,35]. Comparison of the oscillator strengths calculated in the different formulations thus gives a measure of the completeness of the computational basis in addition to the fulfillment of the Thomas-Reiche-Kuhn sum rule (vide infra). 

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. 

This form is consistent with the expression for the electric dipole velocity operator... 

Many-body theory starts out from the principle that all wavefunctions (for both ground and excited states) should be calculated in the same atomic field, i.e. from the same Hamiltonian. The perturbative expansion then allows the higher-order corrections to be calculated systematically. It can then be shown [250] that in the pure RPAE, the dipole length and dipole velocity forms of the cross section are precisely equal, by construction. For this reason, the pure RPAE is often referred to as exact, which means simply that it satisfies equation (5.31) exactly, and not that one should necessarily expect it to agree exactly with experiment. 

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