Perturbation matrix elements

Clearly, Bi f embodies the final-level degeneraey faetor gf, the perturbation matrix elements, and the 2n faetor in the earlier expression for Ri f. The spontaneous rate of transition from the exeited to the lower level is found to be independent of photon intensity, beeause it deals with a proeess that does not require eollision with a photon to oeeur, and is usually denoted Ai f. The rate of photon-stimulated upward transitions from state f to state i (gi Rf i = gi Ri f in the present ease) is also proportional to g(cOf,i), so it is written by eonvention as ... 

This paper considers the hyperspherical harmonics of the four dimensional rotation group 0(4) in the same spirit ofprevious investigations [2,11]), where the possibility is considered of exploiting different parametrizations of the 5" hypersphere to build up alternative Sturmian [12] basis sets. Their symmetry and completeness properties make them in fact adapt to solve quantum mechanical problems where the hyperspherical symmetry of the kinetic energy operator is broken by the interaction potential, but the corresponding perturbation matrix elements can be worked out explicitly, as in the case of Coulomb interactions (see Section 3). A final discussion is given in Section 4. 

Have there been examples where the intrachannel dn/dR has been independently determined by the perturbation matrix element, autoionization rate, and V (R) Vn (R) methods ... 

We use the operator on the left-hand side of equation (7.169) as the zeroth-order vibrational Hamiltonian. The remaining terms in the effective electronic Hamiltonian, given for example in equations (7.124) and (7.137), are treated as perturbations. In a similar vein to the electronic problem, we consider only first- and second-order corrections as given in equations (7.68) and (7.69) to produce an effective Hamiltonian 3Q, which is confined to act within a single vibronic state rj, v) only. Once again, the condition for the validity of this approximation is that the perturbation matrix elements should be small compared with the vibrational intervals. It will therefore tend to fail for loosely bound states with low vibrational frequencies. 

Since our i-basis has even inversion symmetry, the matrix elements connected with the perturbation from a given water molecule are independent of whether this molecule is situated on one or on the other side of the central ion. This means that if we want to discuss the perturbation from our six water molecules with octahedrally positioned ligators (point group symmetry Tn), we can as well take into account only three of them, nos. 1, 2, and 3, say, and eventually multiply all perturbation matrix elements by two. One may say that the holohedrized symmetry (9, 21, 22, 23) of the three water molecules around the central ion is T1. ... 

Assume a perturbing electric field of frequency co in the v-direction with perturbation matrix elements (V + W) = 2DV in (16), and consider the perturbation of the magnetic dipole moment s u-component. With the help of the definitions... 

Equations (6.299) and (6.300) show that Onsager s reciprocal rules hold. The Js eq and Jweq have a microscopic definition represented by perturbation matrix elements and a macroscopic definition represented by the equilibrium exchange rate. As long as the criteria of linearization are satisfied, the statistical rate theory may be used to describe systems with temperature differences at an interface besides the driving forces of pressure and concentration differences. 

This set of equations is known as the Heilman-Feynman theorem (see, e.g., Hirschfelder et al. (1954)). It is a set of simple first order differential equations for the energy levels En- But (4.1.53) is not a closed system of differential equations since we do not know the behaviour of the perturbation matrix elements as a function of e. In an attempt to close the system (4.1.53) we compute... 

This result is valid for times t sufficiently short that the ground state i/ o is not significantly depopulated yet sufficiently long so that cut 27t. The dependence on the transition rate on the square of the perturbation matrix element, as assumed in Sect. 4.8, has thus been proven. 

JT active coordinate originates in the 1st order perturbation theory with the Taylor expansion of the perturbation operator being restricted to linear members [13]. For the nuclear coordinate 2/c we demand non-zero value of the 1st order perturbation matrix element... 

Junction parameter within the excitonic model Excitonic perturbation matrix element Transition oscillator strength Two-photon absorption transition strength N,N-Diethyl-4-(2-nitroethenyl)phenylamine Oligomer consisting of N monomers V-shape molecule with N monomers on branches g-generation dendrimer with N monomers on branches fluorene-fluorene junction... 

The perturbation matrix elements < = hy are determined from the eigenfunc-... 

Whenever a perturbation, predissociation, or autoionization is observed, its strength is governed by an off-diagonal matrix element. Although these matrix elements can frequently be calculated from, ab initio wavefunctions or estimated semiempirically from other perturbation-related information, experimentalists typically treat the perturbation matrix element as a purely empirical parameter of no interest other than its capacity to account for the spectrum. This book is based on the premise that perturbation matrix elements have intrinsic molecular structural significance, that their magnitudes are predictable, and that their measured values often provide unexpected clues to a global description of the electronic structure of a molecule. 

Since J+L = J+ 17) the L-uncoupling operator is a one-electron operator, and consequently, in the single-configuration approximation, the configurations describing the two interacting states can differ by no more than one spin-orbital. The electronic part of the perturbation matrix element is then proportional to the same (7r+ l+ spin-electronic perturbation. However, owing to the presence of the J+ operator, the total matrix element of the B(.R)J+L operator between (fi — 1 and ft) is proportional to [J(J + 1) — 0(0 — 1)]1 2-... 

The magnitudes of perturbation matrix elements are seldom tabulated in compilations of molecular constants. If deperturbed diagonal constants are listed, then the off-diagonal perturbation parameters should be listed as well, even though they cannot, without specialized narrative footnotes, be accommodated into the standard tabular format of such compilations. Without specification of at least the electronic part of the interaction parameters, it is impossible to reconstruct spectral line frequencies or intensities thus the deperturbed diagonal constants by themselves have no meaning. 

The purpose of this chapter is to show how to extract useful information from perturbation matrix elements. The uses of this information range from tactical (vibrational and electronic assignments) to insight into the global electronic structure of a molecule or family of molecules. It is tempting to suggest that perturbation matrix elements can contain at least as much structural information and vastly more dynamical information than the usual molecular constants. 

The ultimate test of a deperturbation consists of (1) a demonstration that all observed perturbation matrix elements have the v, J -dependence required by the factorization into electronic and vibrational parts (2) agreement between the observed and ab initio values of Hf2 Rc) (3) verification that the molecular constants for both electronic states are internally consistent [isotope shifts, Dv values calculable from G v) and Bv functions] (Gottscho, et al, 1979 Kotlar, et al, 1980). 

Once a model potential is derived, it is possible to verify and refine this potential, making use of observed perturbation matrix elements and calculated overlap integrals between the vibrational levels of the two interacting electronic states. Although it is usually more convenient to input the analytic form of V(R) into a numerical integration program to calculate overlap integrals (Section 5.1.3), analytic expressions exist for harmonic and Morse (vi vj) factors. 

When the matrix element method fails, two possibilities for establishing the vibrational numbering remain, ab initio He(R) functions and isotope shifts. When Ec appears to he above the highest observed perturbing level, isotope shifts are the method of choice. However, if He(R) is available, then a modified matrix element method may prove successful. Each trial numbering determines hence He (Rj ial). The calculated vibrational overlap should be equal to the observed perturbation matrix element divided by He (R ial). However, if... 

Is the perturbation matrix element J-dependent (heterogeneous perturbation or S-uncoupling in either the perturbed or perturbing state) Figure 5.8 contrasts the level shifts resulting from J-independent perturbation matrix elements with those from matrix elements proportional to J. 

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