Matrix Elements between Electronic Wavefunctions

Meg (in atomic units, IDebye = 0.3935 a.u.) is the electronic transition matrix element between the e and g electronic states, assuming the dipole length approximation, (ve is the energy normalized nuclear continuum wavefunction, and fj) is the initial state bound vibrational wavefunction. The overlap integral has units of cm1/2 (see Section 7.5). Note that 10 18 cm2=lMb... 

Such methods have, however, been developed recently. In this volume the basic theory is discussed, as well as the intricate details necessary to arrive at efficient procedures for the evaluation of the energy matrix elements between electronic wavefunctions essential for large scale Cl calculation. 

To consider the nature of this approximation one should notice that the nuclear kinetic energy operator acts both on the electronic and the nuclear parts of the BO wavefunction. Hence, the deviations from the adiabatic approximation will be measured by the matrix elements of the nuclear kinetic energy, T(Q), and of the nuclear momentum. The approximate adiabatic wavefunctions have the following off-diagonal matrix elements between different vibronic states ... 

The mixing coefficients are still unknown, but as will be seen below, in the present context only the a0-coefficient, where a0 < 1, is needed. Hence one can add to the wavefunction in equ. (5.26) the photoelectron s contribution, in order to calculate the matrix element between the initial state (equ. (5.24)) and the selected 2p photoionization channel. Working out the relevant dipole matrix elements, one then has the advantage that the overlap factors are unity if there the electron configurations are the same on both sides, and zero for all other cases. Hence only two ionization channels remain, and they are given by... 

Figure 3.9 Ab initio d/dR and d2/dR2 matrix elements between the E, F and G, K adiabatic states of H2 Bl2 = (-hw 2ad), M2 = ( iad. y 2ad)r where 4>i and 2 are the adiabatic electronic wavefunctions for the E, F and G, K double-minimum states, respectively. Except for the smallest R values, Bi2(R) is Lorentzian. The relationships B12 = —B21 and (Ai2 — A2i) = Bi2 are not satisfied exactly because 4>iad ar d 4>2a

Matrix elements between two n-electron wavefunctions can be factored into spin and spatial (including all information about orbital angular momenta) parts. It has been shown frequently that the Wigner-Eckart theorem can be applied as follows (Langhoff and Kern, 1977 McWeeny, 1965 Cooper and Musher, 1972) ... 

In Eq. (1), FCWD is the Franck-Condon weighted density of states that describes the nuclear contribution to k, and //daW is the electronic coupling matrix element that defines the interaction strength occurring between the reactant and product wavefunctions at the transition state. In the classical limit, the Franck-... 

The interaction matrix element//21 in the integrand must consider the initial and final nuclear states of the energy donor and acceptor in addition to the electronic wavefunctions. However, to the extent that the Bom-Oppenheimer approximation is valid, the nuclei will not move significantly during the instant when the excitation energy jumps between the molecules. //21 thus can be approximated as a product of a purely electronic interaction matrix element (//2i(eo) and two nuclear overlap integrals [cf. Eq. (4.42)] ... 

As previously discussed, according to the Fermi golden rule, the intensity of processes like photoemission and Auger decay is expressed by a transition matrix element between initial and final states of the dipole and, respectively, the Coulomb operator. In both cases the final state belongs to the electronic continuum and we already observed that an representation lacks a number of relevant properties of a continuum wavefunction. Nevertheless, it was also observed that the transition moment, due to the presence of the initial bound wavefunction, implies an integration essentially over the molecular space and then even an l representation of the final state may provide information on the transition process. We consider now a numerical technique that allows us to compute the intensity for a transition to the electronic continuum from the results of I calculations that have the advantage, in comparison with the simple atomic one-center model, to supply a correct multicenter description of the continuum orbital. 

The theorem that states that there is no nonvanishing configuration interaction matrix elements between the ground state determinantal wavefunction and those determinants resulting from the excitation of one electron into an empty orbital of the initial SCF calculation. 

The approximate wavefunctions of the adiabatic approximation are characterized by the following off-diagonal matrix elements between different electronic states [11] ... 

Before returning to the non-BO rate expression, it is important to note that, in this spectroscopy case, the perturbation (i.e., the photon s vector potential) appears explicitly only in the p.i f matrix element because this external field is purely an electronic operator. In contrast, in the non-BO case, the perturbation involves a product of momentum operators, one acting on the electronic wavefimction and the second acting on the vibration/rotation wavefunction because the non-BO perturbation involves an explicit exchange of momentum between the electrons and the nuclei. As a result, one has matrix elements of the form (P/ t)Xf > in the non-BO case where one finds lXf > in the spectroscopy case. A primary difference is that derivatives of the vibration/rotation functions appear in the former case (in (P/(J.)x ) where only X appears in the latter. 

The electron capture processes are driven by non-adiabatic couplings between molecular states. All the non-zero radial and rotational eoupling matrix elements have therefore been evaluated from ab initio wavefunctions. 

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