Hamiltonian partial wave

Consider first the case where the scattering projection of the Hamiltonian, PHMP, can induce only elastic scattering (referred to as an uncoupled or elastic continuum). In this situation the partial wave in Eq. (37) reduces to... 

Table 8 Second-order many-body perturbation theory corrections to beryllium-like ions using non-relativistic (E ), Dirac-Coulomb (E ) and Dirac-Coulomb-Breit (E ) hamiltonians, obtained using the atomic precursor to BERTHA, known as SWIRLES. Basis sets are even-tempered S-spinors of dimension N= 17, with exponent sets, Xi generated by Xi = abi-i, with a = 0.413, and p = 1.376. Angular momenta in the range 0 < / < 6 have been included in the partial wave expansion of each second-order energy, and the total relativistic correction toE has been collected as Ef. All energies in hartree.

The Hamiltonian of relative motion is given by Eq. 5.30. The states i), I/), of this Hamiltonian, Jf, which we will call the initial and final states of a given spectroscopic transition, are associated with eigenenergies Ef. The wavefunctions of relative motion are obtained by introducing spherical coordinates R, 8, (p and the partial wave expansion, according... 

Figure 4.19 The partial-wave singlet (full curves) and triplet (broken curves) absorption cross sections in e+ + H(1s) collisions, plotted versus the incident positron energy measured from the threshold energy for positronium formation. Results of hyperspherical closecoupling calculations including the absorption potential —iVabs in the Hamiltonian. Note that the thresholds Etu for the full and broken curves are different by 0.841 meV, the hyperfme splitting. Figure from Ref. [16].

The B-spline K-matrix method follows the close-coupling prescription a complete set of stationary eigenfunctions of the Hamiltonian in the continuum is approximated with a linear combination of partial wave channels (PWCs) [Pg.286]

This work introduced the concept of a vibronic R-matrix, defined on a hypersurface in the joint coordinate space of electrons and intemuclear coordinates. In considering the vibronic problem, it is assumed that a matrix representation of the Schrodinger equation for N+1 electrons has been partitioned to produce an equivalent set of multichannel one-electron equations coupled by a matrix array of nonlocal optical potential operators [270], In the body-fixed reference frame, partial wave functions in the separate channels have the form p(q xN)YL(0, radial channel orbital function i/(q r) and antisymmetrized in the electronic coordinates. Here 0 is a fixed-nuclei A-electron target state or pseudostate and Y] is a spherical harmonic function. Both and i r are parametric functions of the intemuclear coordinate q. It is assumed that the target states 0 for each value of q diagonalize the A-electron Hamiltonian matrix and are orthonormal. 

We have not yet implemented the fully adiabatic theory represented by Equation 5. That theory bears some resemblance to the bending-corrected rotating linear model (BCRLM)(16-18). In this model a partial wave hamiltonian is given by... 

To obtain the scattering path hamiltonian we would begin with the general body-fixed hamiltonian in the variables (R.v.r)(13) and simply re-express it in terms of the variables (t.n.r) using the above transformation. For simplicity we consider the zero partial wave, J 0, and we obtain... 

Even for moderate size matrices M, the partial-wave methods therefore require orders of magnitude more computer time than the solution of the eigenvalue problem (1.19). Furthermore, the formalisms of these methods are complicated, and perturbations are difficult to include because (1.21) are not derived from the variational principle for the one-electron Hamiltonian. 

The Hamiltonian matrix for the ionization run includes coupling between either diabatic neutral state and all ion partial wavefunctions The number of quadrature points for the photoelectron kinetic energy Nk was 50 for a maximum kinetic energy of 5.0 eV. Partial waves up to / = 9 were included in these calculations. An analysis of the resulting spectra shows that waves with I < 5 accounted for over 99 % of the ion population. 

For each partial wave J and parity e, the Hamiltonian and wave packet are discretized in the BF frame in mixed representation [21, 64, 80, 89,160] discrete variable representation (DVR) is employed for the two radial degrees of freedom and finite basis representation (FBR) of normalized associated Legendre function i jK(O) for the angular degree of freedom. Thus the wave packet in the BF frame is written as... 

Where h is Planck s constant over 2n h = 1.05 x 10 J), D, is the partial derivative, here with respect to time, iFis the wave function of the system, and H is the Hamiltonian, an energy operator which in the molecular case may be written... 

Differential equations of pure mechanical systems generate transformation groups for which the Lebesgue measure is invariant this statement is called the Liouville theorem. Major results of the modern theory of dynamic systems are connected with physical sciences, mostly with mechanics. Differential equations of physics may refer to particle or planetary motions described by ordinary differential equations, or to wave motion described by partial differential equations. Dissipative effects are neglected in all these systems, and so the emphasis is on conservative or Hamiltonian systems. 

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