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. H_{2} 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)

Hartree¡¯s self-consistent
field (SCF) method, no exchange (1929)

Hylleraas
wave function for He (1929)

Hartree-Fock
(HF) method (1931)

Multiconfiguration
HF (MCHF), Frenkel (1934)

Roothaan¡¯s
analytic basis set method (1951);

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

Roothaan¡¯s analytic HF code
(~1960)

Slater approx. for electron
exchange (1951)

Herman-Skilman
code/book (1963)

Froese-Fischer¡¯s
numerical HF code, single configuration (~1964)

Cowan¡¯s
Hartree-Slater code (~1965)

Kim¡¯s
analytic relativistic HF code, single configuration (1965)

Desclaux¡¯s
numerical relativistic HF (= Dirac-Fock) code, single configuration (1969)

Froese-Fischer¡¯s
numerical MCHF code (1972)

Descalux¡¯s
numerical MCDF code (1975)

Grant¡¯s
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:

*y*_{nlm}*(r,**J**,**j**) = R _{nl}(r)Y_{lm}(*

¡¤ Since *Y _{lm}(*

¡¤ The difference between different wave
function codes is in the method used to determine the radial function, *R _{nl}(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]5f^{3}6d^{2}7s of the neutral uranium (Z=92,
ground state=[Rn]5f^{3}6d7s^{2}) 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:

*y*_{e}_{lm}*(r,**J**,**j**) = R*_{e}_{l}*(r)Y _{lm}(*

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*
= <* p^{2}/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, *a*_{0} = 0.529
Ǻ

Mass: Electron mass, *m _{e}*

Charge: Electron charge, *e*

Energy: Hartree = *e*^{2}/*a*_{0
}= 2 rydberg = 27.2 eV

Planck¡¯s
constant: ©¤ =
1

Speed of
light: *c* =
1/*a* @ 137

Cross section: *a*_{0}^{2},
but Ǻ^{2 }=10^{–16} cm^{2} is popular among
experimentalists

(See Table
I)