Potential matrix elements

The reduced potential matrix elements which couple the internal states n and j are... 

In Section III.D, we shall investigate when this happens. For the moment, imagine that we are at a point of degeneracy. To find out the topology of the adiabatic PES around this point, the diabatic potential matrix elements can be expressed by a hrst order Taylor expansion. 

The potential matrix elements are then obtained by making Taylor expansions around 00, using suitable zero-order diabatic potential energy functions,... 

For diabatic calculations, the equivalent expression uses the diabatic potential matrix elements [218]. When the value of this coupling becomes greater than a pre-defined cutoff, the tiajectory has entered a non-adiabatic region. The propagation is continued from this time, ti, until the trajectoiy moves out of the region at time f2-... 

In this section, diabatization is formed employing the adiabatic-to-diabatic transformation matrix A, which is a solution of Eq. (19). Once A is calculated, the diabatic potential matiix W is obtained from Eq. (22). Thus Eqs. (19) and (22) form the basis for the procedure to obtain the diabatic potential matrix elements. 

Since the nuclear potential is a multiplicative function, the nuclear potential matrix elements are defined by... 

It is evident that this potential leads to a logarithm squared contribution of order a (Za) after substitution in (3.71). One may obtain one more logarithm from the continuous spectrum contribution in (3.71). Due to locality of the potential, matrix elements reduce to the products of the values of the respective wave functions at the origin and the potentials in (3.72). The value of the continuous spectrum Coulomb wave function at the origin is well known (see, e.g., [94]), and... 

The close coupled scheme is described on pp. 306 through 308. Specifically, the intermolecular potential of H2-H2 is given by an expression like Eq. 6.39 [354, 358] the potential matrix elements are computed according to Eq. 6.45ff. The dipole function is given by Eq. 4.18. Vibration, i.e., the dependences on the H2 vibrational quantum numbers vu will be suppressed here so that the formalism describes the rototranslational band only. For like pairs, the angular part of the wavefunction, Eq. 6.42, must be symmetrized, according to Eq. 6.47. 

If the A group is a saturated amine only three parameters remain eaA, a, and p. It should be possible to determine all three from the ligand field potential matrix, or directly from the spectroscopic fit. Our own experience with the ethylenediamine complex, [Cr(en)3]3+, revealed another problem [10] the effects of a and P on the ligand field potential matrix elements and on the calculated band positions, while not identical, are similar enough that they cannot be practically disentangled. It was, however, possible to determine a and P in [Cr(en)3]3+ by using circular dichroism spectra (see Sect. 7). 

The natural choice of the reaction coordinate R, mentioned just before Section 3.1, for describing the channels with the asymptotic arrangement A -(- B is the distance between the centers of mass of A and B. This defines the conceptually simplest set of close-coupling equations. However, the corresponding potential matrix elements Vn,/(R) are difficult to interpret since many channels are coupled to each other strongly in general. Thus, no single potential is expected to predict the physics of the processes under consideration. 

Let us start the discussion with 0=0. These resonances can directly decay to the j = 0 rotational state which automatically has the same helicity 0 = 0. The coupling is mainly provided by the V q ° R) potential matrix element [see Equation (11.9)]. Because of the triangular condition j — j < A < j + j following from the Wigner 3,7-symbol,... 

The matrix elements are composed of the matrix elements of the potential energy 1/rc and the kinetic energy — /d/jp . For the potential matrix elements we obtain... 

Equn. (4.119) for the partial-wave potential matrix element shows why only a finite number of partial-wave T-matrix elements contribute to the scattering. For very large L the centrifugal barrier means that UL kr) is appreciably greater than zero only for values of r greater than r , beyond which V(r) is effectively zero. Note also that there is a range of L for which (fc IIFLllfc") is so small that the Born approximation is valid... 

To compute the potential matrix elements we use the partial-wave expansions of the Coulomb scattering functions in the analogue of (4.118). 

The extension of the matrix solution of section 4.3 for one-electron bound states to the Hartree—Fock problem has many advantages. It results in radial orbitals specified as linear combinations of analytic functions, usually normalised Slater-type orbitals (4.38). This is a very convenient form for the computation of potential matrix elements in reaction theory. The method has been described by Roothaan (1960) for a closed-shell or single-open-shell structure. 

It would be convenient for solving the Lippmann—Schwinger equation (6.73) if we could make the potential matrix elements as small as possible. For example, we could hope to find a transformed equation whose iteration would converge much more quickly. This is achieved by a judicious choice of a local, central potential U, which is called the distorting potential since the problem is reformulated in terms of the distorted-wave eigenstates of U rather than the plane waves of (6.73). An important particular case of U is the Coulomb potential Vc in the case where the target is charged. The Hamiltonian (6.2) is repartitioned as follows... 

The projectile state is represented by a partial-wave expansion (4.188). The quantum numbers of the projectile partial wave and the target state are coupled to total angular momentum P and parity n. The jy-coupling expansion of the potential matrix element is... 

The reduced potential matrix elements are obtained by inverting (7.42). 

This form is used in the LS-coupling representation of the potential matrix element. For jj coupling we use... 

The target states are expressed, according to equn. (7.35) for the full potential matrix elements, in terms of orbitals a) and P). The quantity that relates the orbitals to the target states i ) and i) is the m-scheme density matrix i alap i). Its transformation properties under rotations are important in finding the reduced potential matrix elements. 

The direct reduced potential matrix element in j j coupling is calculated by substituting the full potential matrix elements into (7.37) with the continuum orbitals given by (7.46), the bound orbitals by (7.49), the two-electron potential by (7.60) and the reduced density-matrix element by (7.59). 

The spin—angle integrations are performed by (3.104,107). We use the orthogonality of the spherical harmonics (3.71) and the Clebsch—Gordan coefficients (3.89). Expressing the Clebsch—Gordan coefficients as 3-j symbols by (3.93) the direct reduced potential matrix element becomes... 

The final form for the direct reduced potential matrix element is... 

The direct reduced potential matrix element for LS coupling is given by Bray et al. (1989). In this case the integrations over the spin coordinates <70 and <71 result in the factor (v v)(va vi5), which prohibits spin flip. 

The reduction of the exchange potential matrix element to the form used for computation parallels that for the direct term. We exchange the coordinates Tq, <7q and ri, <7i in the kets of (7.37). 

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