Explicit Hilbert-space representations of atomic and molecular photoabsorption spectra: Computational studies of Stieltjes-Tchebycheff functions

Journal Article · Sat Oct 01 00:00:00 EDT 1983 · Phys. Rev. A; (United States)

DOI:https://doi.org/10.1103/PhysRevA.28.1957· OSTI ID:5388734

Hermann, M R; Langhoff, P W

Explicit Hilbert-space techniques are reported for construction of the discrete and continuum Schroedinger states required in atomic and molecular photoexcitation and/or photoionization studies. These developments extend and clarify previously described moment-theory methods for determinations of photoabsorption cross sections from discrete basis-set calculations to include explicit construction of underlying wave functions. The appropriate Stieltjes-Tchebycheff excitation and ionization functions of nth order are defined as Radau-type eigenstates of an appropriate operator in an n-term Cauchy-Lanczos basis. The energies of these states are the Radau quadrature points of the photoabsorption cross section, and their (reciprocal) norms provide the corresponding quadrature weights. Although finite-order Stieltjes-Tchebycheff functions are L/sup 2/ integrable, and do not have asymptotic spatial tails in the continuous spectrum, the Radau quadrature weights nevertheless provide information for normalization in the conventional Dirac delta-function sense. Since one Radau point can be placed anywhere in the spectrum, appropriately normalized convergent approximations to any of the discrete or continuum Schroedinger states are obtained from the development. Connections with matrix partitioning methods are established, demonstrating that nth-order Stieltjes-Tchebycheff functions are optical-potential solutions of the matrix Schroedinger equation in the full Cauchy-Lanczos basis.

• Research Organization: Department of Chemistry, Indiana University, Bloomington, Indiana 47405 and Max Planck Institut fuer Astrophysik, Karl-Schwarzschild-Strasse 1, D-8046 Garching bei Muenchen, Federal Republic of Germany
• OSTI ID: 5388734
• Journal Information: Phys. Rev. A; (United States), Vol. 28:4
• Country of Publication: United States
• Language: English

Related Subjects

74 ATOMIC AND MOLECULAR PHYSICS
HYDROGEN
PHOTOIONIZATION
ABSORPTION
ATOMS
CONVERGENCE
EIGENSTATES
HILBERT SPACE
HILBERT TRANSFORMATION
MOLECULES
BANACH SPACE
ELEMENTS
INTEGRAL TRANSFORMATIONS
IONIZATION
MATHEMATICAL SPACE
NONMETALS
SPACE
TRANSFORMATIONS
640302* - Atomic
Molecular & Chemical Physics- Atomic & Molecular Properties & Theory