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

Journal Article · Sat Oct 01 04:00:00 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), Journal Name: Phys. Rev. A; (United States) Vol. 28:4; ISSN PLRAA
• Country of Publication: United States
• Language: English

Related Subjects

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