Matrix elements molecular, factorization

The /3-decay matrix element is factorized into an intranuclear matrix element and a molecular matrix element. The former describes the /J transition in the radioactive nucleus, whereas the latter describes the excitation of molecular degrees of freedom induced by the /3 decay. 

In the Condon approximation the molecular matrix element is factorized into electron and vibrational-rotational matrix elements. In this case the... 

The high value of the electron density at the nucleus leads to the enhancement of the electron EDM in heavy atoms. The other possible source of the enhancement is the presence of small energy denominators in the sum over states in the first term of Eq.(29). In particular, this takes place when (Eo — En) is of the order of the molecular rotational constant. (It is imperative that a nonperturbative treatment be invoked when the Stark matrix element e z(v /0 z v / ) is comparable to the energy denominator (Eq En) [33].) Neglecting the second term of the right-hand side of Eq.(29), which does not contain this enhancement factor [8, 27], we get... 

The sum of all these factors for a specific problem is equal to the trace of the matrix of the corresponding characteristic problem4 the sum of their squares equals the sum of the squares of all matrix elements of the corresponding characteristic problem.64 These relations are useful for checking the values of orbital energies. By an equally simple procedure1-4 the values of expansion coefficients yield the values of ir-electron densities (q), bond orders (p), and free valences (F) these quantities are usually presented in the form of a molecular diagram ... 

For the rotovibrational bands, the induced dipole components Bc of Eqs. 6.13 through 6.18 become the vibrational matrix element [281], Eq. 4.20. Furthermore, each XifoAL component now occurs twice, once with molecule 1 vibrating and once with molecule 2 vibrating in the final state. For like molecular pairs, this may be taken into account by removing the factors of 1/2 in Eqs. 6.13, 6.14, 6.16, and 6.17. For dissimilar molecular pairs, the factors 1/2 are absent from the equations quoted. 

Which properties are least well determined by the variational method The basis functions in the LCAO expansion are either Slater orbitals with an exponential factor e r or gaussians, e ar2 r appears explicitly only as a denominator in the SCF equations thus matrix elements are of the form < fc/r 0i> these have the largest values as r->0. Thus the parts of the wavefunction closest to the nuclei are the best determined, and the largest errors are in the outer regions. This corresponds to the physical observation that the inner-shell orbitals contribute most to the molecular energy. It is unfortunate in this respect that the bonding properties depend on the outer shells. 

In this approach k is proportional to the square of the donor-acceptor electronic coupling matrix element (//DA) and a Franck-Condon term that contains the dependence of the ET rate on AGgy, X and factors related to the molecular structure,... 

In order to use the perturbation theory it is necessary that the state vectors in the matrix element Eq. (8) belong to the spectrum of the unperturbed Hamiltonian H0 only. However, this is usually not so, since, in p decay, the initial particles are not the same as the final products of the reaction the initial molecule containing the radioactive atom transforms into a different molecule besides, the ft electron and the neutrino appear. One of the ways to describe the initial and final states using only the H0 Hamiltonian is to use the isotopic spin formalism for both the nucleons and the leptons (/ electron and neutrino). In the appendix (Section V) we present the wave functions of the initial and the final states together with the necessary transformations, which one can use to factorize the initial matrix element Eq. (8) into the intranuclear and the molecular parts. Here we briefly discuss only the approximations necessary for performing such a factorization. 

The weak-interaction Hamiltonian Hp in Eq. (8) acts only on the nucleon and the lepton coordinates, leaving the molecular ones unaffected. Consequently, in order to single out the molecular factor in the matrix element Eq. (8), it is necessary that the wave functions of nucleons and leptons be independent of the molecular variables The wave functions describe the motion of the nucleons inside the /eth nucleus with respect to its center of mass. The position of the latter is described by the radius vector R4, and, generally speaking, the wave functions depend both on Rk and on the internal nucleon coordinates. The wave function of the / electron produced in the final state must also depend on the center-of-mass coordinates of the nuclei. 

Thus, the molecular matrix element Eq. (13) is factorized into electron and vibrational-rotational matrix elements. The electron matrix element is independent of the recoil momentum, and, consequently, the electron degrees of freedom are excited only by the jump in the charge of the radioactive nucleus, whereas the excitation of vibrations and rotations is due to both the jump in the charge and the recoil of the radioactive nucleus. 

Appendix. Modification of the Molecular Hamiltonian and Factorization of the Matrix Element... 

Here d is the z projection of the dipole matrix elements for the spinless S PZ transition of an outer electron. The factor (— )sw in equation (38) characterizes the symmetry in the arrangement of atomic dipoles in mixed S-P atomic states. Spin S and parity w of a molecular state are relevant either to the exchange of electrons (with atomic orbital fixed at nuclei) or to the exchange both of electrons and atomic orbitals. The exchange of orbitals (with atomic electrons attached to corresponding nuclei) is accomplished by the product of these two transformations. This explains the appearance of the spin quantum number in (38). 

Optical absorption in a medium can take place because of the existence of electric or magnetic dipole moments associated with atomic, molecular or crystal entities. Unless otherwise specified, only electric dipoles are considered here. In quantum mechanics, the condition related to the dipole moment for discrete optical absorption appears in terms of transition probability between the initial and final states. It can be formally expressed as the modulus squared of a matrix element involving the wave functions and final states and the electric dipole operator, which reduces, within a proportionality factor to the general displacement coordinate ra ... 

For a linear molecule, the position of the symmetry axis (the molecule-fixed. z-axis) in space is specified by only two Euler angles, / and 7, which are respectively identical to the spherical polar coordinates 6 and (see Fig. 2.4). The third Euler angle, a, which specifies the orientation of the molecule-fixed x- and y-axes, is unaffected by molecular rotation but appears explicitly as an O- dependent phase factor in the rotational basis functions [Eq. (2.3.41)]. Cartesian coordinates in space- and molecule-fixed systems are related by the geometrical transformation represented by the 3x3 direction cosine matrix (Wilson et al., 1980, p. 286). The direction cosine matrix a given by Hougen (1970, p. 18) is obtained by setting a = 7t/2 (notation of Wilson et al, 1980 6 fi,4)=, x = oi 7t/2). The direction cosine matrix is expressed in terms of sines and cosines of 9 and 4>. Matrix elements (J M O la JMQ), evaluated in the JMQ) basis, of the direction cosines, are expressed in terms of the J, M, and quantum numbers. The direction cosine matrix elements of Hougen (1970, p. 31), Townes and Schawlow (1955, p. 96), and Table 2.1 assume the basis set definition derived from Eq. (2.3.40) and the phase choice a = 7t/2 ... 

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). 

Chapters 2, 3, and 5 form the core of this book. Perturbations are defined and simple procedures for evaluating matrix elements of angular momentum operators are presented in Chapter 2. Chapter 3 deals with the troublesome terms in the molecular Hamiltonian that are responsible for perturbations. Particular attention is devoted to the reduction of matrix elements to separately evaluable rotational, vibrational, and electronic factors. Whenever possible the electronic factor is reduced to one- and two-electron orbital matrix elements. The magnitudes and physical interpretations of matrix elements are discussed in Chapter 5. In Chapter 4 the process of reducing spectra to molecular constants and the difficulty of relating empirical-parameters to terms in the exact molecular Hamiltonian are described. Transition intensities, especially quantum mechanical interference effects, are discussed in Chapter 6. Also included in Chapter 6 are examples of experiments that illustrate, sample, or utilize perturbation effects. The phenomena of predissociation and autoionization are forms of perturbation and are discussed in Chapters 7 and 8. 

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