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Dipole-length form

C2H2 -----, present results (dipole length form) of Reference 17 ... [Pg.110]

This means that the oscillator strengths of an absorption spectrum calculated within the RPA (in a complete orbital basis) will be identical in the dipole length form... [Pg.88]

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],... [Pg.276]

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 ). [Pg.277]

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. [Pg.47]

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... [Pg.323]

Finally, the differential cross section for photoionization, dff/dfl, will be given explicitly for the dipole approximation and the length form of the matrix element by collecting all the individual steps. This cross section is related to the transition rate w by... [Pg.325]

One then obtains the result (see [HJC81], note that the length form of the dipole matrix element is employed here)... [Pg.328]

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))... [Pg.343]

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. [Pg.194]

It has a higher boiling point, so it exists as a liquid at room temperature. Pentane has more sites along its length than methane does where temporary dipoles can form. The dispersion forces add up, so that it takes more energy overall to separate the molecules. This leads to a higher boiling point. [Pg.420]

SAC-CI method was applied to calculate the electronic CD spectrum of uridine [43], Based on theoretical CD and absorption spectra, observed peaks in the experimental spectra were assigned. The rotational strength (R) in the length form [44] was calculated as imaginary part of the inner product of the electric transition dipole moment (ETDM) and magnetic transition dipole moment (MTDM). [Pg.99]

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... [Pg.470]

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. [Pg.183]

In MBPT also, it is a goal to achieve equality of dipole length and velocity forms. This test is often applied to give some idea whether the approximations made in the calculation are sufficiently accurate. [Pg.184]

From these calculations, it is obvious to condude that the first reaction is synchronous with similar bond lengths formed whereas the second one is asynchronous with quite different bond lengths formed. Furthermore, the dipole moments remained the same for GS and TS in the first reaction whereas they are noticeably increased in the TS for the second. All these conclusions strongly support the evidence and interpretation of important spedfic not purely thermal MW effects when asynchronous mechanisms are involved. [Pg.150]

Let us assume that the field is polarized linearly and fhaf Vext(f) is described in the EDA. There is the question of which form to use. It turns out that only in the length form of fhe EDA do fhe expansion coefficients represent directly probability amplitudes, e.g.. Refs. [80, 83, 101]. On the other hand, in the velocity form it is easier to handle numerically the singularity that appears in the free-free dipole mafrix elements [54,105]. Therefore, we choose fhe A(0) p form, in which case, once the solution of the TDSE, 4>(r,f), is obtained, we must multiply it by the phase factor in order for the new coefficients to acquire their correct meaning as probability amplitudes [83, 101]. Furthermore, in order for the initial state at f = 0 to be a state of Ha>j, the "preferential" gauge can be chosen in which A(f) = 0 whenever E(f) becomes zero [[81b], [87]]. [Pg.364]

In order to determine a transition probability, one usually uses a standard amplitude approach. Each of the theoretical approaches to calculation of transition probabilities contains critical factors (configuration interaction or multiconfiguration treatment, spectroscopic coupling schemes and relativistic corrections, exchange-correlation corrections, convergence of results and of the dipole length and velocity forms, accuracy of transition energies, etc.) which need to be adequately taken care of to obtain reliable results (look details in Refs. [2-5]). [Pg.232]

A further simplified form of the dipole-dipole interaction potential is often used for dipole-dipole separation distances that are large compared to 6 = b, the dipole length. Such a situation exists for molecules in the vapor state. Let yac = jrac and expand jac about 6 = 0 in a Taylor series, i.e.. [Pg.16]

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... [Pg.149]


See other pages where Dipole-length form is mentioned: [Pg.13]    [Pg.183]    [Pg.167]    [Pg.13]    [Pg.183]    [Pg.167]    [Pg.79]    [Pg.473]    [Pg.30]    [Pg.13]    [Pg.199]    [Pg.183]    [Pg.13]    [Pg.199]    [Pg.162]    [Pg.95]    [Pg.105]    [Pg.334]    [Pg.183]    [Pg.32]    [Pg.62]    [Pg.62]    [Pg.168]   
See also in sourсe #XX -- [ Pg.167 ]




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Length form

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