Hamiltonian radial

From a comparison of the operator (9.55) with the non-relativistic one-body Hamiltonian operator (see Eq. A5 of [21]), one observes that the angular coefficients of the radial functions Ip n l,nl) r) are identical to those of the one-elecdon Hamiltonian radial integrals In i,ni anticipated from McWeeny analysis [10]. These angular coefficients can be derived by working out the matrix elements of a one-particle scalar operator Fp between configuration state functions with u open shells, as explicitly derived by Gaigalas et al. [22]. 

Choosing appropriate units for the charge, the Hamiltonian for radial motion can be written... 

The principal differences from Eq. (68) lie in the form of the potential W (r) and in the presence of the term j cos 0, of which the latter arises from the dependence of the geometric phase on the radius of the encircling path. The eigenvalues of hr are no longer doubly degenerate, but a precisely equivalent Kramer s twin radial Hamiltonian may be derived from the complex conjugate of Eq. (71). 

A note of caution should be given here regarding the Hamiltonian matrix in Eq. [13]. It is not difficult to see that singularities can arise when the radial coordinates approach zero, which in turn could result in serious convergence... 

Dunham achieved inter-relations between term coefficients l i through use of an intermediate radial function V(J ) in an effective hamiltonian for motion of atomic nuclei of this form. 

That effective hamiltonian according to formula 29, with neglect of W"(R), appears to be the most comprehensive and practical currently available for spectral reduction when one seeks to take into account all three principal extramechanical terms, namely radial functions for rotational and vibrational g factors and adiabatic corrections. The form of this effective hamiltonian differs slightly from that used by van Vleck [9], who failed to recognise a connection between the electronic contribution to the rotational g factor and rotational nonadiabatic terms [150,56]. There exists nevertheless a clear evolution from the advance in van Vleck s [9] elaboration of Dunham s [5] innovative derivation of vibration-rotational energies into the present effective hamiltonian in formula 29 through the work of Herman [60,66]. The notation g for two radial functions pertaining to extra-mechanical effects in formula 29 alludes to that connection between... 

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [9], The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component Dirac-Fock-Breit (DFB) functions. The spherical symmetry of atoms leads to the separation of the one-electron equation into radial and spin-angular parts [10], The radial four-spinor has the so-called large component the upper two places and the small component Q, in the lower two. The quantum number k (with k =j+ 1/2) comes from the spin-angular equation, and n is the principal quantum number, which counts the solutions of the radial equation with the same k. Defining... 

The radial functions Pmi r) and Qn ir) may be obtained by numerical integration [16,17] or by expansion in a basis (for recent reviews see [18,19]). Since the Dirac Hamiltonian is not bound from below, failure to observe correct boundary conditions leads to variational collapse [20,21], where admixture of negative-energy solutions may yield energies much below experimental. To avoid this failure, the basis sets used for expanding the large and small components must maintain kinetic balance [22,23]. 

The general philosophy is in fact identical to that used for the radial variables. The reader is referred to Ref. 87 for further details of the manner in which the operation of the Hamiltonian operator on the wavepacket is accomplished. 

Radial equation. In the laboratory frame, the two-atom Hamiltonian is given by Eq. 5.28. Transformation of Jtf n to center of mass and intermolecular separation coordinates, Eq. 5.29, allows the separation of these variables as was seen above, p. 207. 

The VR term of the Hamiltonian represents the kinetic energy of the relative motion of the two electrons. It can be broken into radial and angular parts using... 

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