Lecture Note for Sept. 11, 2002 (Wed.)

 

I.     Introduction

 

     Exact, analytic wave functions available only for H  (and one-electron ions), relativistic and nonrelativistic, bound and continuum states

 

     Almost exact, nonrelativistic numerical wave functions for He (bound states only)

 

     Various approximate wave functions for all other atoms, molecules, ions (bound states: relativistic and nonrelativistic; continuum states: nonrelativistic)

 

     H is useful for testing theory.

 

     Experiments with H atom are very difficult.  H2 comes in a bottle, but not H.

 

     He is useful for testing theory and experiments.

 

     Chemists are far ahead of physicists in their capability to calculate wave functions for bound states.

 

     Continuum wave functions for atoms are much easier to calculate than those for molecules.

     Atoms:          Central field is valid.

       Molecules:   Central field cannot be used.

 

     In principle, all atomic properties can be calculated if wave functions are known.

 

     In practice, wave functions we can calculate are not good enough for most atoms.

 

     For photon-atom interactions, good wave functions for the target atom are sufficient.

 

     For bound-bound transitions, only bound-state wave functions are needed.

 

     For photoionization, however, bound and continuum wave functions are needed, a more difficult task.

 

     For electron-atom collisions, the required wave functions must include both the incident electron and the target atom.

 

     For a neutral target atom, this means a negative ion wave function with one or two electrons in the continuum.

 

 

     High quality bound-state wave functions can be calculated by applying the variational principle to the total energy (e.g., Hartree-Fock method).

 

     There is no equivalent method for continuum wave functions, because the continuum energy must be kept constant, not subject to the variational principle.

 


     History of Methods to Calculate Wave Functions

 

       Exact nonrelativistic wave function for H, Schrödinger (1925)      

 

       Exact relativistic wave function for H, Dirac (1928)

 

Hartrees self-consistent field (SCF) method, no exchange (1929)

 

       Hylleraas wave function for He (1929)

 

       Hartree-Fock (HF) method (1931)

 

       Multiconfiguration HF (MCHF), Frenkel (1934)

 

       Roothaans analytic basis set method (1951);

 

       Almost exact nonrelativistic wave functions for the bound states of He, Pekeris (1957)

 

Roothaans analytic HF code (~1960)

 

Slater approx. for electron exchange (1951)

 

       Herman-Skilman code/book (1963)

 

       Froese-Fischers numerical HF code, single configuration (~1964)

 

       Cowans Hartree-Slater code (~1965)

 

       Kims analytic relativistic HF code, single configuration (1965)

 

       Desclauxs numerical relativistic HF (= Dirac-Fock) code, single configuration (1969)

 

       Froese-Fischers numerical MCHF code (1972)

 

       Descaluxs numerical MCDF code (1975)

 

       Grants numerical MCDF code (1976)

 

     All these codes required the most advanced mainframes available in the 1970-1990s.

 

     From ~1995, PCs became as powerful as workstations, and now there are PC versions for most of these codes.

 

     The PC-based code for MCDF wave functions that we will practice later has gone through more than 30 years of improvements.

 


     All atomic wave function codes assume a central field model:

 

              ynlm(r,J,j) = Rnl(r)Ylm(J,j)

 

     Since Ylm(J,j) is analytic, only Rnl(r) is calculated by the wave function codes.

 

     The difference between different wave function codes is in the method used to determine the radial function, Rnl(r).

 

     Although the angular parts are analytic, calculating them correctly for the quantum numbers and physical quantities of interest is a major task.

 

     For instance, the angular part for an excited state [Rn]5f36d27s of the neutral uranium (Z=92, ground state=[Rn]5f36d7s2) took 2 weeks of CPU time on an IBM RISC6000 Model 595 workstation.

 

     Therefore, it was difficult to combine the calculation of angular part and radial part because such a long job would have the lowest priority on mainframe computers.

 

     Most quantities that can be calculated with wave functions also have their own angular parts, which are different from the total energy expressions.

 

     Most physical quantities of interest involve upper and lower state wave functions, i.e., off-diagonal matrix elements, while total energies involve only one state.

 

     Previously, the MCDF2002 code consisted of separate parts, MJ92, DF92, PH92, and DW92.

 

       MJ92:    Angular momentum algebra

DF92:    DF wave functions, using the output of MJ92 as input

PH92:    Transition probabilities, using the output of MJ92 for the angular part and the radial functions from DF92; also plane-wave Born excitation cross sections using the output from MJ92 and DF92

DW92:   Distorted-wave Born cross sections for excitation and ionization using the output from MJ92 and DF92

 

     These codes were developed between 1975 and 1987, and frozen around 1990, although minor corrections and improvements were added until 2000.

 

     The development of a new version began in 1995 and is still in progress.  The version presented as part of the present course was frozen in May 2002.  It integrated MJ92, DF92, and PH92, but DW92 has not been integrated yet.

 

     DW92 is the only code that generates continuum wave functions.  This is the reason for the delay in integrating it into MCDF2002.

 

     Continuum wave functions are usually expanded in partial waves:

 

              yelm(r,J,j) = Rel(r)Ylm(J,j)

 

       where e is the continuum electron energy.

 

     More rigorous theories [see Refs. 1—3 of the lecture note] all require the use of both bound and continuum state wave functions, which have been the main source of difficulty in developing such collision theories.

 

     As in the case of bound-state wave function codes, only the radial functions are calculated by different collision cross section codes, while the angular parts are calculated analytically (using computers).

 

     Independent of the development of the MCDF2002 code, Kim and Rudd [Phys. Rev. A 50, 3954 (1994)] developed a collision theory for electron-impact ionization of atoms and molecules, known as the binary-encounter-dipole (BED) model.

 

     A simplified version of the BED model, which is called the binary-encounter-Bethe (BEB) model was also developed at the same time.

 

     The BEB model is very simple to use, and requires a few numbers from the ground-state wave function of the target atom or molecule.

 

     The BEB model does not even use the radial functions, only the binding energy B, kinetic energy U, and electron occupation number N of each bound orbital.

 

     The kinetic energy U of a bound electron is not a measurable quantity, though it is well defined theoretically:

 

              U = < p2/2m>

 

     One cannot obtain U from the binding energy B using the virial theorem, except for one-electron atoms.  The virial theorem is for the ratio of the total potential energy and the total kinetic energy.

 

     Only a wave function code can generate reliable values of U.  The numerical values of U are insensitive to the accuracy of wave functions, unlike the values of B.

 

     The numerical values of B for valence electrons are very sensitive to the quality of wave functions.  (More about this later)

 

     For electron-impact cross sections, first-order perturbation theories such as the Born approximation do not generate accurate results when the incident electron energy is low, about 5 times the threshold energy or less.

 

     This difficulty was partly solved for strong, dipole- and spin-allowed excitations by two scaling methods developed by Kim [Phys. Rev. A 64, 032713 (2001); 65, 022705 (2002)].

 

     Both the BEB model and the scaling of Born cross sections will be covered toward the end of this course.

 

     The BEB model has been successfully tested for both atoms and molecules.

 

     The scaling of Born cross sections has been tested only for atoms, and singly charged atomic ions.

 

     These electron-impact cross sections for atoms can be calculated using the output from the MCDF2002 code, and a PC-based spreadsheet program, such as Excel or Lotus123.

 


     In atomic and molecular computations, atomic units are used, which use the quantities associated with the ground state of H as units.

 

       Length:  Bohr radius, a0 = 0.529 Ǻ

       Mass:     Electron mass, me

       Charge:  Electron charge, e

       Energy:  Hartree = e2/a0 = 2 rydberg = 27.2 eV

       Plancks constant:      = 1     

       Speed of light:     c = 1/a @ 137

Cross section:      a02, but Ǻ2 =10–16 cm2 is popular among experimentalists

       (See Table I)