PROGRAM RCG MOD11
CALCULATION OF ATOMIC ENERGY LEVELS AND SPECTRA
Robert D. Cowan
Los Alamos National Laboratory
August 1993; revised March 2001
I. Introduction 2
II. Source Programs 4
III. Input/Output Units 8
IV. Input 9
(1) optional control cards 9
(2) cfp decks 10
(3) rescaling card 12
(4) calculational deck 12
(A) control card 13
(B-C) configuration cards 18
(D) energy parameter cards 19
radial multipole-integral cards 21
review 22
V. Output 23
VI. Array Dimensions 25
VII. Memory Requirements 28
VIII. Execution Times 28
IX. Modifications for Other Computers 28
X. Rescaling of Input Data 29
XI. Photoionization Cross Sections Q 30
XII. Autoionization Transition Probabilities 34
(a) Kinetic energies of autoionized electrons 36
(b) Branching ratios and dielectronic recombination 36
(c) Autoionization contributions to collisional ionization 38
(d) Autoionization contributions to collisional excitation 39
XIII. Plane-wave-Born Collision Strengths 41
XIV. List of Principal Variables 43
XV. Program Usage and Example
(A) Program Location at Los Alamos
(B) Execution
(C) Sample Input and Output
XVI. Sample Monitor Screen Output
Appendix--Notes on level designations
I. Introduction
RCG is a FORTRAN77 program initially coded (1964-65) on the
IBM 7030 ("Stretch"), and subsequently modified (with addition of
many new features and options) to run on other IBM, CDC, and CRAY
mainframes, various VAXs and MicroVAXs, SUN and IBM RISC
workstations, and finally Power Macintosh computers using the
Language Systems or Absoft FORTRAN compilers and PCs using the free
GNU g77 compiler. It has also been successfully used on Apollo,
Hewlett-Packard, and SGI computers. On 32-bit-word machines, it
should be run in double-precision mode--preferably using, if
available, a large-exponent-range option (10±350 rather than
10±38) such as the VAX G_FLOATING compiler option. To make the
program easily useable on both 64-bit and 32-bit machines, the
following program modifications have been incorporated:
(1) PROGRAM cards have been included both with and without
file-definition names included, and there have also been included
file-name definitions via OPEN statements. On CRAY and CYBER
machines, the simple PROGRAM card can be commented out and iii in
the main program set to 2, which causes the OPEN statements to be
bypassed. On VAXs and similar computers, the file-definition
PROGRAM card is the one to be commented out, and iii set to 1 to
define all file names via OPEN statements. (iii may alternatively
be set to zero, in which case some file names are set by typing
them in interactively.)
(2) In all subroutines there are included IMPLICIT
REAL*8(A-H,O-Z) statements, which may be commented out if
necessary for 64-bit-word machines.
(3) Generic library subroutine names (e.g., MAX in place
of AMAX1 or DMAX1) have been used so that the compiler will
automatically use the single-precision or double-precision
version, depending on the type of the argument variables. In all
argument lists, constants have been replaced by variable names,
the variable being given a value in a replacement statement such
as TWO=2.0, so that in 32-bit-word machines it will automatically
be converted to double precision.
(4) In subroutines PLEV and CALCFC, there were 25-fold-or-
so nested DO loops, which exceeded the maximum size allowed in the
VAX compiler (20-fold). This problem has been eliminated by
moving the inner-most several DO loops to subroutines PLEVDOLP and
CAFCDOLP, respectively.
(5) In subroutine SECONDS, there are included a single
statement T=SECOND(T) appropriate to the CRAY, and also sets of
timing-routine statements appropriate to VAXs or Macintoshes, PCs,
SUNs, and IBM RISCs. Inappropriate sections of the routine are
to be commented out (or a new one added as needed for the computer
in question, or one can simply set T=0.0 if the computer has no
internal timer clock routine).
(6) All ENCODE and DECODE statements used in older
versions of the program have been removed. This has been
facilitated by defining a number of variables as CHARACTER type.
The basic purpose of RCG is to compute the angular factor
of various matrix elements in the theory of atomic structure and
spectra. The program employs Racah-algebra techniques, and input
decks containing coefficients of fractional parentage (cfp) for
each subshell lw involved in the electron configurations
(n1l1)w1 (n2l2)w2 ... (nqlq)wq (1)
present in the calculation. Any occupied subshell nlw may be (and
in practice always is) deleted from the set (1) if it is filled
(w=4l+2) in every configuration involved in a given calculation.
A short description of the basic theory behind the program
(except for configuration-interaction effects) may be found in
R. D. Cowan, J. Opt. Soc. Am. 58, 808 and 924 (1968). Full
details are given in R. D. Cowan, The Theory of Atomic Structure
and Spectra (University of California Press, Berkeley, 1981)--
especially Chapters 16 and 18--hereinafter referred to simply as
"TASS".
The angular factors in question are:
(a) the trivial (unit-matrix) coefficient of Eav, the
center-of-gravity energy of each configuration;
(b) the coefficients fk, gk, and d of the single-
configuration direct and exchange Coulomb-interaction (Fk and Gk)
and spin-orbit-interaction (z) radial integrals, and the
coefficients rdk and rek of the direct and exchange configuration-
interaction Coulomb radial integrals Rk, which are involved in the
calculation of the Hamiltonian (energy-level) matrix elements;
(c) the magnetic-dipole matrix elements, and the angular
coefficients of the electric-dipole (t=1) and electric-quadrupole
(t=2) reduced radial matrix elements
Pll'(t) = < l || r(t) C(t) || l' > . (2)
Also possible are angular coefficients of certain effective-
Coulomb-interaction operators a, b, g, T1, and T2, and "illegal-k"
operators Fk and Gk used in representing weak configuration-
interaction effects (TASS, Sec. 16-7), and also coefficients of
spherical-Bessel-function radial integrals
< l || jt(Kr) C(t) || l' > , (3)
(TASS, Secs. 18-12 and 18-13).
These angular coefficients may be used as input to programs
(such as RCE Mod 20) for least-squares fitting of experimental
energy levels.
If numerical values of the radial integrals Eav, Fk, Gk, z,
Rk are provided (either by considering them as adjustable parameters
determined by least-squares fitting of experimental levels, or
using ab initio theoretical values computed from atomic radial
wavefunctions), then energy levels and intermediate-coupling
eigenvectors are computed. If numerical values of the electric-
multipole integrals are supplied, then the energy levels and
eigenvectors are used for computation of spectrum-line wavelengths
and the associated oscillator strengths and radiative transition
probabilities. In practice, values of these radial integrals (and
indeed the entire RCG input file) are obtained via a calculation
with the atomic-wavefunction programs RCN/RCN2.
Options are available in RCG for the calculation of
photoionization cross sections, autoionization transition
probabilities, and plane-wave-Born electron-impact collision
strengths.
II. Source Programs
The following brief discussion of the function of each
subroutine or function program provides a rough outline of the
basic calculational procedure. For simplicity, the following
discussion uses the default values of the various disk-file names,
but these can be easily changed. In places, terminology will be
used that dates from the time when computer input consisted of
punched cards: The word "card" may be used to refer to a line of
an input file or FORTRAN source file, and the word "deck" to refer
to an input file--set of cards--or portion thereof. Characters
"punched" in specific columns of these cards are, of course, to be
typed into the corresponding columns of the input line.
Statements that various information is "printed" means that
information is written to the output print file IW=9, but in many
cases only if certain print options are in effect.
MAIN Reads various control cards from the input file (disk unit
IR=10, external name ing11), and calls the various major subroutines
according to the information thereon.
CUVFD (Calculate U,V,f,d) Defines three disk file numbers ID2,
ID3, ID4 (normally 72, 73, 74). Reads coefficient-of-fractional-
parentage (cfp) decks (including term quantum numbers aiLiSi and
_ _ _
parent quantum numbers aiLiSi) for each subshell (li)wi that may
be involved in any of the configurations (1) for which
calculations are to be made later, computes coefficients of
fractional grandparentage (cfgp) if pertinent, and writes all this
on binary disk 72; calculates matrix elements of U(r) and V(r1)
and writes them on disk 73; and calculates angular coefficients
for Fk(ii), for a, b, g (for dw subshells, also T, T1, T2) if so
requested, and for zi, and writes them on disk 74.
CIJKF Calls S3J0SQ to calculate matrix elements
< li || C(k) || lj > . (4)
LOCDSK Locates the first record of information for lw on disk 72,
73, or 74.
LNCUV (ln,C,U,V) Reads input cards specifying the subshells liwi
involved in each configuration, and calculates (via CIJKF) and
prints a table of values of the matrix elements (4). This table
is not used in further calculations, but is for information only.
PLEV (preliminary levels) Using the liwi values read by LNCUV,
reads the terms of liwi from disk 72, and vectorially adds quantum
numbers LiSi to set up tables of all possible quantum numbers LiSi
and Ji (i.LE.q).
PFGD Sets up preliminary tables of the coefficients fk, gk, d, etc.
of Fk(ii), a, b, g, T, T1, T2, and zetai (obtained from disk 74) and
of Fk(ij) and Gk(ij) (computed with the aid of subroutines CIJKF,
RDIJ, and REIJ, using matrix elements of U and V obtained from
disk 73), and writes these tables on disk 20.
PRK Computes preliminary tables of coefficients rk of the
configuration-interaction parameters Rk(ij,i'j') with the aid of
cfp and cfgp from disk 72, U and V matrix elements from disk 73,
and subroutines CLASS1 to CLAS11, RDIJ, and REIJ. Writes these
tables on disk 20.
RDIJ Used in computing coefficients fk and rdk of direct Coulomb-
interaction parameters Fk(ij) and Rk(ij,i'j').
REIJ Used in computing coefficients gk and rek of exchange
parameters Gk(ij) and Rk(ij,j'i').
CLASS1-CLAS11 Used in computing coefficients of Rk for the eleven
possible classes of configuration interaction.
CALCFC (calculate final coefficients) For each possible value of
the total-angular-momentum quantum number J, selects those sets of
quantum numbers aiLiSiLiSi (1.LE.i.LE.q) found by PLEV that can
give this value of J, computes the LS-JJ transformation matrix, and
writes all this on disk unit IL=31 (first parity) or IL=32 (second
parity) and on IC=41 (both parities). Reads the preliminary tables
of coefficients from disk 20, and sets up final coefficient
matrices for all parameters (except Eav) and writes them on disk
unit IC=41. Calls SPRIN to print matrices if desired, and also
calls CPL37. If so requested, writes non-zero coefficient matrix
elements on disk 19 for use by the Argonne National Laboratory
least-squares level fitting program.
CPL37 (coupling 3 to 7) If desired, calculates quantum numbers
for, and transformation matrices to, coupling representations
number 3 to 7 (LS=1, JJ=2--see JOSA article mentioned in Sec. I or
page 15 below for definitions. Writes this information on disk
IC=41.
SPRIN Multipurpose matrix-print routine, to print angular-
coefficient matrices for Fk (KPAR=-1), Gk (0), z (+1), Rk (-2), for
multipole transitions (+2), and for energy (+3) or eigenvector
(+4) matrices, with more-or-less adequately labeled rows and
columns. Also transforms energy-coefficient and mupole matrices
when calculation in the JJ representation is desired, and computes
and prints Land g-values when called for LS eigenvectors. (Also,
writes mupole matrices on disk IC=41.)
SPRN37 Called by SPRIN to read transformation matrices from disk
IC=41, and transform and print eigenvectors in representations 3
to 7.
MUPOLE Reads quantum numbers from disks IL=31 and/or 32, cfp from
disk 72, and U(2) from disk 73 (for electric quadrupole), and
computes angular coefficients for line-strength calculations (or
for plane-wave Born calculations). Calls SPRIN to print matrices
and write on disk IC=41.
ENERGY Reads parameter values (Eav, and Coulomb and spin-orbit
radial integrals) from data input cards on disk IR=10; for each J,
reads quantum numbers and the LS-JJ transformation matrix from disk
IC=41, reads coefficient matrices from IC and computes and
diagonalizes the energy matrix, and writes eigenvalues (sorted in
numerically increasing order) and eigenvectors on disk IE=31
(first parity) or 32 (second parity). Computes autoionization
transition probabilities if appropriate, and writes them on disk
IE. Calls SPRIN to print the energy and eigenvector matrices.
CALCV If so instructed, called by ENERGY (for each J) to calculate
the diagonal elements of the coefficient matrices in the
intermediate-coupling representation (the representation in which
the energy matrix is diagonal). These elements represent the
derivatives of the various eigenvalues with respect to the various
parameters, and provide information needed for a trial-and-error
adjustment of the parameters to produce desired changes in the
eigenvalues. (These are the elements that are calculated and used
in least-squares programs such as RCE for the systematic iterative
fitting of theoretical eigenvalues to experimental energy levels.
In RCG, these elements are simply printed out for use in rough
eye-ball parameter adjustments.)
LVDIST Called by ENERGY, if desired, to calculate the statistical
distribution of energy-level statistical weight, and plot on film
via PLOJB (deleted in the present version of RCG). Also
calculates (and plots) that skewed-Gaussian curve that best fits the
distribution (see TASS, Sec. 21-3 for definitions and examples).
SPECTR Called by ENERGY. Reads values of reduced mupole radial
matrix integrals (2), calculated by RCN2, from the input file on
disk 10. For each possible pair of values of (J,J'), reads the
angular-coefficient mupole matrix from disk 41 and eigenvalues and
vectors from disk IE=31 and/or 32, differences eigenvalues to
compute wavelengths of spectrum lines, and multiplies the mupole
matrix from either side by the appropriate eigenvector matrix (and
multiplies the resulting matrix elements by the appropriate radial
integral) to compute intermediate-coupling line strengths,
oscillator strengths, and radiative transition probabilities.
Spectrum-line information is printed after being sorted by (a)
levels in the first set of configurations (first parity), (b)
levels in the second set of configurations (second parity), and/or
(c) wavelength. In the first two cases, a total transition
probability and lifetime are computed for each level of the given
set with resect to all possible transitions to lower-energy levels
included in the opposite set.
WNDIST (wavenumber distribution) If desired, called from SPECTR
to calculate [and plot] oscillator-strength distributions (see
TASS, Sec. 21-4); writes this on disk unit 11 to provide input
information for program RADRATE.
BORN Can be called from ENERGY to calculate plane-wave-Born
collision strengths (using interpolation routine AKNINT), and
excitation rate coefficients (using routines RCOEFF, E1, and
CSEVL).
UNCPLA and UNCPLB Compute the uncoupling coefficients Ua and Ub
for reduced matrix elements, as defined in TASS, Eqs. (12.26) and
(12.27).
RECPSH, RECPJP, RECPEX Compute the shift, jump, and exchange
recoupling coefficients defined in TASS, Eqs. (13.64)-(13.66).
S3J0SQ, S9J, S6J, DELSQ, CALCFCT Compute the square of the 3-j
symbol with magnetic quantum numbers all zero, the 9-j symbol, and
the 6-j symbol, using a table of factorials computed by CALCFCT.
SORT, ORDER Sort an array of numbers into numerically increasing
order, and correspondingly rearrange up to 12 additional arrays.
MLEW Dummy program (called by ENERGY) to call matrix-
diagonalization routines TRED2 and TQL2.
RCEINP (RCE input) Called by ENERGY to write transformation and
coefficient matrices on binary disk unit 2, and an input formatted
file on disk 11, for the least-squares energy-level fitting
program RCE.
ELECDEN Assumes the availability of the file tape2n computed by
RCN, which contains one-electron radial wavefunctions for each
orbital of each configuration in the present RCG calculation.
Uses these wavefunctions to compute spherically averaged radial
electron-density distributions for each configuration. Then uses
each computed eigenvector to calculate a spherically averaged
radial electron-density distribution for each configuration-
interaction, intermediate-coupling eigenlevel.
III. Input/Output Units
Disk-storage unit numbers used, with external and internal
names, are as follows:
Ext. Int. Default val. Usage
------ --- ------------ ------------------------------------------
ing11 ir 10 input
outg11 iw 9 printed output
tape2n 3 radial-wavefunction input (optional)
elecden 4 electron-density output
tape2e 2 coeffs. for use in least-squares prog. RCE
outgine 11 formatted input for use in program RCE
tape72 id2 72 single-subshell quantum nos., cfp, and cfgp
tape73 id3 id2+1 U(r), V(r1)
tape74 id4 id2+2 single-subshell Coulomb and spin-orbit
coeffs.
20 preliminary Coulomb and spin-orbit
coefficients
il 31,32 quantum nos., transformation matrices
ic 41 quantum nos., transf. and final coef.
matrices
ie 31,32 energy levels, eigenvectors,
autoionization transition probs.
19 coeffs. for use in Argonne Lab
least-squares level-fitting program
11 special-purpose level and osc.-strength
output
3,13 special-purpose dielectronic-recomb. output
Normally, ID2-ID4 are small files; 20, IL, and IE are of
intermediate length; IC is the largest. Actual sizes depend on
the complexity of the set of configurations being run. Units 31
(ILA) and ID4 may share the same I/O buffer, as ILA is not used
until ID4 is no longer needed. Default values may be readily
changed by changing the six statements starting with statement 60
of the main program, by corresponding choice of the value of ID2T
read from the input file at statement 85, and appropriate changes
in the file numbers on the PROGRAM card or in OPEN statments.
IV. Input
We here discuss input-data setup for simple bound-state level
and spectrum calculations. Modifications for special-purpose
calculations involving continuum states will be discussed in Secs.
XI to XIII. Sample input decks and output listings are provided
in Sec. XV.
Briefly, input data consist of the following:
(1) Two types of optional control cards.
(2) If requested by one of the above control cards, a
set of cfp decks, followed by a card with a negative integer in
columns 9-12 (signaling the end of the set of cfp decks).
(3) An optional rescale card (see Sec. X).
(4) One or more calculational decks (usually provided
ready-to-run from output of an RCN/RCN2 calculation); each deck
starts with a control card, and ends with a card containing
"-99999999." in columns 21-30.
(5) A card with a negative integer in columns 1-5,
causing an exit from RCG. (This card is automatically provided by
RCN2).
(6) Any unused input data cards may be stored here if
desired.
*****************************************************************
NOTE: Before actual calculational runs can be made, a
first RCG run must be made in which the input deck includes the
cfp decks indicated in item (2) above; this results in the
calculation of the files 72, 73, 74 discussed earlier, which are
required in all subsequent calculations. An input file called
ing11k is supplied that includes the cfp decks for this first
calculation (with its name changed to ing11). This is discussed
in greater detail in section (2) below, but it is suggested that
on a first reading one skip the discussion of optional control
cards, and at this point jump directly to the discussion of
standard calculational decks in section (4) below.
*****************************************************************
Details of the input are as follows.
(1a) There may be one or two optional control cards of the
following form:
cols. variable format
------ ---------------- --------
1-4 NCLSKP(K) > 0 I4
5 K I1
6-10 NOTKP(K) I5
11-80 MULS1(I),LHS1(I) 14(I4,A1)
K must be 1 for configurations of the first parity, or 2 for
configurations of the second parity. If NOTKP(K) is greater than
zero then only those basis states (for configurations of parity K
and serial number greater than or equal to NCLSKP(K) will be
retained that have one of the NOTKP(K) ("no. of LS terms to be
kept") values of multiplicity and total orbital angular momentum L
specified in columns 11 to 10+5*NOTKP; for example (using carats
to denote blank columns),
^^^12^^^^3^^^4S^^^2D^^^2S
means keep only 4S, 2D, and 2S basis states for all configurations
of the second parity. These control cards need be included only
if LS-term truncation of this type is desired. [Notes: (a) If more
than one truncation control card with given K is included, only
the final one will be effective; (b) Truncation cards may appear
either before control card (1b), or following (1b) and the
associated cfp decks and end card (if any).]
(1b) There may be an optional control card of the form:
cols. variable format comments
----- ------------ ------ -------------------------------
1-5 integer (=3) I5 defines control card of type 1b
6-7 ILNCUV I2 extra output print if > 0 (and
ID2 > 0)
8-10 ID2 I3 file number (default=ID2=72)
11-15 FLBDN E5.1 default=0.001
16-20 FLBDX E5.1 default=500000
21-30 DELEKEV F10.5 default=0.0005 keV
31-40 EIONRY F10.5 default=0.0 Ry
41-45 NPTKEV I5 default=0
46-70 TKEV(I),I=2,6 5F5.3 default=0.0 keV [NTKEV (.LE.5)
determined by the number of
non-zero TKEV]
71-80 EMINA F10.5 default=0.0 (same units as energy
levels)
The variables FLBDN and FLBDX specify the minimum and maximum
wavelengths (lambda) of spectrum lines to be retained in
subroutine SPECTR; the variable DELEKEV is used in WNDIST in
defining the histogram bin-width for calculating the oscillator-
strength distribution; and the variables EIONRY, NPTKEV, TKEV, and
EMINA concern special-purpose dielectronic-recombination
calculations in SPECTR (see Sec. XII , pages 37 and 41). This
control card need be included only if cfp decks are included
and/or if non-default values of the other variables are desired;
there must be such a card (with ID2 non-zero) before a set of cfp
decks, and may be another (with ID2=0) following these decks.
(2) If ID2 is non-zero (normally=72), then this value will
determine values for the file numbers ID3(=ID2+1) and ID4(=ID2+2)
as well (see Sec. III). In addition, the control card (1b) must
then (and only then) be followed by a set of cfp decks, and by an
end card with negative integer in columns 9 to 12; the subroutine
CUVFD is called to process these cfp decks and write information
on disks ID2-ID4, and a non-zero ILNCUV produces printed output of
some of the computed information. Except as noted below, there
must be a cfp deck for each lw involved in any configuration
specified in the calculational decks (4); if subshells lw and lw-2
will both be involved in a set of interacting configurations, then
a cfp deck for lw-1 must also be included (so that cfgp for lw can
be computed). All decks of given l should be grouped together,
and must be arranged in order of increasing w. Decks for l0 and
l1 are needed only if cfgp for l2 are required, so that decks for
g, h, i, ... , electrons are normally never needed. The input
file ING11K provided with the program RCG contains cfp decks for
all sw, pw, dw, f0 to f4 and f11 to f14, and for g0, g1, g2, h0,
h1, and h2. The file cfp contains also f5 to f10 decks, but these
require larger dimensions of several variables than those used in
the code provided (see the comment cards near the beginning of the
main program). Normally, all the cfp decks included in ING11K can
be used in calculation of disk files ID2-ID4. (The presence of
unneeded cfp decks causes no harm except for extra computer time
in CUVFD--see TASS, Tables 16-1 and 16-2--which needs to be
executed only once, and a small amount of extra disk search time
in reading data from disks ID2-ID4 if subshells of large l are
needed in a given calculation.)
The detailed format of each cfp deck need not be discussed
here. We need comment only on the form of the first card of each
deck, which contains:
column 4: the spectrosopic letter code for l
(s,p,d,f,g,h,i,k,..., according as the one-
electron angular momentum is
0,1,2,3,4,5,6,7,..., respectively.
columns 5-8: w (format I4)
columns 9-12: number of LS terms of lw (format I4)
columns 13-16: Number of parents (terms of lw-1) (format I4)
The size of a calculation (particularly for configurations
involving an fw subshell) can be reduced by including only a
limited number of terms of the subshell lw in setting up quantum
states of the complete configuration. To invoke such a
truncation, the number of terms of lw to be included is placed in
columns 21-24, and the terms themselves are placed in columns 29-
32, 33-36, ... 73-76 (and if necessary in columns 1-4, 5-8, ... of
succeeding cards). The first column for each term contains the
value of the multiplicity 2S+1, the second column contains the
letter symbol for L (S,P,D,F,G,H,I,K, ...), and the last two
columns contain any necessary (left-adjusted) serial number to
distinguish different terms of the same LS; this serial number
must match that used in the body of the cfp deck, which follows
the convention used by C. W. Nielson and G. F. Koster,
"Spectroscopic Coefficients for the pn, dn, and fn Configurations"
(The M.I.T. Press, Cambridge, Mass., 1963). (Some examples are
included in the cfp decks provided in ING11K for f2, f3, etc.,
though with the number of terms equal zero so that there is no
truncation.)
The control card (1b) with ID2 > 0, together with cfp decks
and end card, must be included on a first run of RCG--using, for
example, ING11K for the input deck ING11--in order to produce the
files ID2, ID3, and ID4 (normally TAPE72, TAPE73, and TAPE74). If
these files, produced on such a run, are saved and made available
to subsequent runs, then items (1b) and (2) may be deleted from
all further runs [except, of course, that a (1b) card with ID2=0
must be included if non-default values are needed for variables
other than ID2]. It will of course be necessary to recompute
files ID2-ID4 if new subshells need to be added, or if truncation
of LS terms is to be added or changed.
The action of the variable ILNCUV controls the amount of
information from subroutine CUVFD written to the output file outg11:
ILNCUV=0, name of subshell and computing time only
1, tables of cfp and cfgp
2, coefficient names and times
3, coefficient names and times, and coefficient values
4, all the above
(3) For details of the optional rescaling card, see Sec. X).
(4) Calculational desks
NOTE: In the most common usage of RCG, the calculational deck is
automatically prepared by execution of programs RCN and RCN2, so
that all the details below can be ignored except if one wishes to
make changes for special purposes, or for some reason wishes to
set up an input deck by hand. [The output deck from RCN2 is named
"out2ing," and the name needs to be changed to "ing11" for use as
the RCG input deck.]
Each calculational deck consists of the following:
(A) A control card, specifying among other things the
number of configurations of each parity that are involved in the
calculation.
(B) A set of configuration-definition cards, one for
each configuration of the first parity.
(C) A set of configuration-definition cards, one for
each configuration of the second parity. This will be an empty
set if column 16 of the control card contains a zero.
(D) Zero, or one, or several, sets of parameter-value
cards; a set may have any one of three forms:
(i) For a diagonalization, a set consists of:
(a) Parameter values for each configuration of
the first parity.
(b) Configuration-interaction parameter values
for each pair of interacting configurations
of the first parity (if any).
(c) If IQUAD (col. 50 of the control card A) is
1 or > 2, one or more sets of electric
quadrupole reduced-matrix-element cards for
configurations of the first parity.
(d) Same as (a), for configs. of the second
parity (if any).
(e) Same as (b), for configs. of the second
parity (if any).
(f) If IQUAD >1, same as (c) except for the
second parity.
(g) If both parities are present, zero or more
sets of electric-dipole reduced-matrix-
element cards, for all pairs of
configurations of opposite parity.
(ii) To write coefficient matrix elements for each
parity on file 2, a single pseudo-parameter
card containing "-55555555." in columns 21-30.
If this option is used, then columns 9-10 of
the control card (A) must contain a negative
integer; the absolute value of this integer is
the quantity NOCSET used in least-squares
energy-level-fitting program RCE. This
pseudo-parameter card may have one or more
sets of genuine parameter cards (i) preceding
it and/or following it.
(iii) A single pseudo-parameter card containing
"-99999999." in columns 21-30 signals the end
of the calculational deck (4). Any number of
similar decks may follow.
A. Calculational deck control card
The control card (A) is of the following form:
(i) If it contains a negative integer in columns 1-5, it
is the data card (5) above, causing an exit from RCG. Any cards
that follow are not read, but form part of the bone-pile (6).
(ii) If it contains a zero in columns 1-5, this is a
rescaling card (see Sec. X below); it should be followed by a
genuine card (iv) below.
(iii) If it contains a positive integer NOCSET in columns
8-10 and a 1 or 2 in column 5, this is a signal to search through
the file on unit 2 until this CSET (set of coefficient matrices)
is found. This pseudo control card is not followed by cards (B)-
(D), but is immediately followed by a genuine control card (iv)--
normally containing a negative number in columns 9-10 to specify
the serial number of a new CSET that is going to be added on to
unit 2, to provide input for least-squares energy-level-fitting
program RCE. [In practice, this option is never used, CSETs being
computed as needed and written onto a new file on unit 2, rather
than being added onto an old one.]
(iv) A genuine control card is of the following form:
cols. format variable name normal value
----- ------ ------------------- ------------------------
1-5 I5 KCPL 1
6-7 I1,I2 NCK(K), K=1,2 blank
9-10 I2 NOCSET blank; see (iii) above
11-15 I1,2I2 NSCONF(I,1) )
) I=1,3 see below
16-20 I1,2I2 NSCONF(I,2) )
21 I1 IABG blank
22 I1 IV blank
23-25 I3 NLSMAX blank
26-27 I2 NLT11 blank
28-30 I3 NEVMAX blank
31-39 9I1 KCPLD(I), I=1,9 blank (except for Born calc.)
40 I1 IELPUN blank
41-44 2F2.1 SJNK(1), SJXK(1) blank
45-48 2F2.1 SJNK(2), SJXK(2) blank
49 I1 IMAG blank (for no M1 trans.)
50 I1 IQUAD blank (for no E2 trans.)
51-60 F10.5 UENRGY 1000.0
61-65 F5.5 DMIN 0.0
66 I1 ILNCUV blank
67 I1 IPLEV blank
68 I1 ICPC blank
69 I1 ICFC blank
70 I1 IDIP blank
71 I1 IENGYD 0
72 I1 ISPECC 7
73-74 I2 IW6 -6 (blank for batch runs)
75-76 I2 IPCT blank
77-78 I2 ICTC blank
79-80 I2 ICTBCD blank
The significance of these quantities is as follows.
KCPL: < 0, an exit card, (i) above
= 0, a rescaling card, (ii) above
= 1, calculation to be done in LS (or SL) represent.
= 2, calculation to be done in JJ representation
NCK(K): If non-zero, then in all spectrum-line lists and
plane-wave-Born calculations, only those transitions
are included that involve levels belonging to the
first NCK(K) configurations. For parity-changing
(electric-dipole) transitions, K=1 and 2 represent
the first and second parities; for non-parity-
changing transitions, K=1 and 2 represent lower and
upper level, respectively. [Default is 50, except
NCK(1)=1 for plane-wave-Born calculations.]
NOCSET: Must be blank or a negative integer; see under (iii)
above.
NSCONF(1,K): number of subshells for configurations of parity K
NSCONF(2,K): number of configurations of parity K
NSCONF(3,K): number of successive configurations for which
interactions will be included [normally=NSCONF(2,K)]
If NSCONF(3,K)=0, then following the configuration-
definition cards there must be a card containing
INTEST(I), I=1,80 (format 80I1); interaction will be
included between configurations with serial numbers
J1 and J2 if INTEST(I) > 0, where I is computed as
follows:
J2X=NSCONF(2,K)
J1X=J2X-1
I=0
DO 799 J1=1,J1X
J2N=J1+1
DO 799 J2=J2N,J2X
I=I+1
799 CONTINUE
If -7.LE.NSCONF(3,K).LE.-1, interactions will be
included if the first configuration has serial number
J1.LE.MAX(-NSCONF(3,K),IPCT). If NSCONF(3,K)=-8, all
interactions will be included. [Use this value with
K=2 to calculate photoionization cross-sections.]
If NSCONF(3,K)=-9, interactions will be included if
the first configuration has serial number J1=1, or
if the two configurations have successive serial
numbers (J2=J1+1).
IABG: > 0, include effective-operator parameters a (for pw)
or a,b,T,T1,T2 (for dw), or a,b,g (for fw); see
TASS, Sec.16-7.
=2 or 4, include illegal-k effective-operator
parameters Fk(ij) and Gk(ij); TASS, Sec. 16-7.
> 2, use SL instead of LS coupling (for straight LS
coupling only; continue to use LS for compound
couplings such as LSLK, LSJK, LSJLKS, etc.).
IV: If non-zero, call CALCV to calculate and print V
matrices (derivatives of eigenvalues with respect to
parameter values).
NLSMAX: Calculate eigenvectors in representations 3 to 7 if
matrix size NLS is equal to or less than NLSMAX.
The representations are [J. Opt. Sos. Am. 58, 808 (1968)]:
(1) LS: {[((a1L1S1)L1S1, a2L2S2)L2S2,... aqLqSq]LqSq}Jq
(2) JJ: {[[(a1L1S1J1)J1, (a2L2S2J2)]J2,... (aqLqSqJq)]}Jq
(3) JJJK: {[( ... Jq-1)Jq-1, Lq]K,Sq}Jq
(4) LSLK: {[(( ... Lq-1)Lq-1, Lq)Lq, ( ... Sq-1)Sq-1]K,Sq}Jq
(5) LSJK: {[((... Lq-1)Lq-1, ( ... Sq-1)Sq-1)Jq-1,Lq]K, Sq}Jq
(6) LSJLKJ: {[((... Lq-2)Lq-2, ( ... Sq-2)Sq-2)Jq-2,
(Lq-1Lq)L]K, (Sq-1Sq)S}Jq
(7) LSJLSJ: {[(... Lq-2)Lq-2, ( ... Sq-2)Sq-2]Jq-2,
[(Lq-1Lq)L, (Sq-1Sq)S]J }Jq
[Note: In the above expressions, letters L, S, and J following a ),
], or } are generally to be interpreted as script letters, which
cannot be represented in an ASCII file.]
NLT11: For all except the last configuration of each
parity, include no more than the first NLT11
LS-terms of l1w1 (default=119).
NEVMAX: For each J, print at most the eigenvectors having
the NEVMAX smallest eigenvalues (default=500).
KCPLD(I): If > 0, do not print eigenvectors in the
representation I (I=1 to 7) defined under NLSMAX.
If KCPLD(3) > 4, then a plane-wave-Born calculation
is to be made, and KCPLD(3) to KCPLD(7) are
interpreted differently; see Sec.XIII.
If KCPLD(3) = 3 or 4, intermediate-coupling, CI
radial electron-density distributions will be
calculated provided tape2n from an RCN calculation
is present; if = 3, cross-term contributions are
not included.
KCPLD(9)=IPRINT: If > 6, delete energy-matrix print.
If > 7, delete all eigenvector and purity prints.
If > 8, delete eigenvalue and autoionization
probability prints.
IELPUN: If = 1, write eigenvalues on unit 11 (special-
purpose option).
If > 1, write multiplet level strengths on unit 19
(ditto).
SJNK(K): For parity K, exclude matrices with J < SJNK(K)
(default=0.0).
SJXK(K): For parity K, exclude matrices with J > SJXK(K)
(default=99.0).
For example, to include only J=0 for first parity and
J=1 for second parity, put .0.51.1. in columns 41-48.
IMAG: If > 0, calculate magnetic-dipole transitions for
the first, second, or both parities, according as
IMAG=1, 2, or 3.
IQUAD: Similar to IMAG, except for electric quadrupole
transitions (or parity-conserving plane-wave-Born
excitations if KCPLD(3) ³ 5). Note that the
corresponding value of IQUAD must be used in column
50 of the G5INP card in program RCN2 in order to
calculate the required radial integrals.
UENRGY: Unit (in cm-1) of all energy-parameter values on the
parameter cards in this calculational deck
(1000.0 if parameter values are in kilokaysers,
8065.48 if in eV, 109737 if in Ry).
DMIN: Delete spectrum lines for which S/X < DMIN, where
S is the line strength of the transition, and X is
the largest < nl || r || n'l' >**2 for all of the
transition arrays included in the calculation; S/X is
the quantity printed in the spectrum line list in the
column following the wavelength. Typical values of
S/X for strong lines are 2 to 5; an appropriate
value of DMIN to delete weak lines is 0.005 to 0.05.
For modifications of these remarks in certain cases,
see Sec. VI, pages 25-26).
If DMIN < 0.0, in line lists sorted by energy, the
only lines retained are those that involve the lowest-
energy level of the parity in question.
ILNCUV: If > 0, print C(k) matrix elements (4).
IPLEV: If > 0, print preliminary quantum numbers in
subroutine PLEV.
ICPC: If > 0, print prelim. ang. coefs. in PFGD and PRK;
if=1, print only parameter name and configuration(s),
if=2, same as 1 plus single-configuration coeffs.,
if=3, same as 1 plus config-interaction coeffs.,
if=4, same as 1 plus all coeffs.
If > 4, write angular coefficients on unit 11 in a
form suitable for input to program RCN, for making
LS-term HF calculations.
If = 9, skip matrix diagonalization part of program.
(No parameter, multipole, nor "-99999999." cards are
to be included in the input deck).
ICFC: If > 0, print coef. matrices in subroutine CALCFC;
if=1, print only LS- and JJ-representation quantum
numbers,
if=2, print also the LS-JJ transformation matrix,
if=3, same as 2 plus coef. matrices for single-conf.
parameters,
if=4, same as 2 plus coef. matrices for c-i params,
if ³ 5, same as 2 plus coef. matrices for all params.
IDIP: If > 0, print J values of each multipole matrix;
if > 1, print angular multipole matrix, and matrix
of the squares of the elements.
IENGYD: If = 0, print full energy matrix;
if = 1, do not print matrix;
if = 2, print only first NEVMAX rows and columns;
if > 2, print first 11*IENGYD rows and columns.
ISPECC: = 1, 3, 5, or 7, print spectrum lines sorted by
levels of first parity
= 2, 3, 6, or 7, print spectrum lines sorted by
levels of second parity
= 4 to 8, print spectrum lines sorted by wavelength
> 7, call LVDIST and WNDIST; wavelength sort printed
only if 8. Default value is 7.
Must be > 5 to obtain values of BRNCH, etc.,
see Sec. XII.
IW6: If < 0, information on the progress of the
calculation is sent to unit 6 (the monitor screen).
If > 0, normal output is sent to unit 6 instead of
to unit 9; not a practical option because of the
large volume of output.
IPCT: Used only in connection with NSCONF(3,K), see above.
ICTC: If.NE.0, use previously computed file on disk unit 41,
skipping calls of PLEV, PFGD, PRK, CALCFC, and MUPOLE;
TAPE72 is still required, but not TAPE73 nor TAPE74.
ICTBCD: If > 0, write coefficient matrix elements on disk
unit 19 for input to Argonne National Laboratory
least-squares program.
But if NOCSET.NE.0, set ICTBCD = 0 and write only
disk 2, for input to least-squares program RCE.
B-C. Configuration Cards
Each configuration is of the form
l1w1 l2w2 ... lqwq
and each of the NSCONF(2,1) + NSCONF(2,2) configuration cards is
constructed accordingly in the form
l1, w1, l2, w2, ... lq, wq
with format 8(A1,I2,2X), and with li written as the appropriate
letter symbol s,p,d,f,g,h,i,k,... , and wi right-adjusted, even
if less than 10. Any subshell that is filled in all configurations
of this deck may be omitted. The maximum number of subshells
required for configuration specification is limited to 8 in this
program (not only by dimensions used, but also by format restrictions.
Certain restrictions must be observed in setting up the
configurations. If only one parity is involved, then
NSCONF(I,2)=0, all I; if both parities are present, then
NSCONF(1,1)=NSCONF(1,2). For given i, li must be the same in all
configurations, as must also be the principal quantum number,
though it is not explicitly listed; there are three exceptions to
this restriction, all pertaining to the case of a singly occupied
subshell lj [wj=1] when all subsequent subshells in that
configuration are empty:
(1) For given parity, lj may be the same in several
configurations, with only the (unspecified) principal quantum
number differing.
(2) lj may have different values for opposite parities.
Thus, for example, the set of configurations 3s2 3p2, 3s2 3p 4p,
3s2 3p 5p, 3s2 3p 4f, 3s2 3p 5f, 3s 3p3, 3s2 3p 4s, 3s2 3p 5s,
3s2 3p 3d, 3s2 3p 4d, 3s2 3p 5d in Si I could be set up in the form
s 2 p 2 p 0 f 0
s 2 p 1 p 1 f 0
s 2 p 1 p 1 f 0
s 2 p 1 p 0 f 1
s 2 p 1 p 0 f 1
s 1 p 3 s 0 d 0
s 2 p 1 s 1 d 0
s 2 p 1 s 1 d 0
s 2 p 1 s 0 d 1
s 2 p 1 s 0 d 1
s 2 p 1 s 0 d 1
and columns 11-20 of the control card would contain 4^5^54^6^6,
where the carats represent blanks (or zeroes). [If one wished to
include configuration interaction only between 3p2 and 3p 4p, 3p 4p
and 3p 5p, 3p 5p and 3 p4f, and 3p 4f and 3p 5f, the control numbers
would be changed to 4^5^14^6^6, etc. For this and other options,
see the discussion under NSCONF(3,K) above.]
(3) If the final subshell j=q is never more than singly
occupied, then lj (as well, perhaps, as the principal quantum
number) may be different in different configurations, even of the
same parity. (To all intents and purposes, the dimensional
limitation q.LE.8 is then seldom any restriction whatever. In an
RCN2 calculation, use of the minimum possible value of q may be
forced by setting ICON = 2 on the G5INP control card.) In the
example above, the outer p, f, s, and d electron may be placed in
the third subshell in all configurations, with columns 11-20 of
the control card then being 3^5^53^6^6 (or 3^5^13^6^6, etc.).
Dimensions of cfp, and of U and V matrix elements are such
that any multiply occupied fw subshell should be l1w1 (even then,
dimensions are too small for f5 to f10), and any multiply occupied
dw subshell should come next.
D. Energy Parameter Cards
The first parameter card for each configuration contains
any desired BCD identification (e.g., element and configuration)
in columns 1-18. Columns 21-70 contain values of the first five
parameters [format F10.5,4(F9.4,1X)], the value of Eav occupying
columns 21-30; the total number of parameters may be placed in
columns 19-20. Any additional parameter values are put on
additional cards in columns 1-70 [format 7(F9.4,1X)]. Units for
all parameters are defined by the number in columns 51-60 (in
cm-1) of the control card; for cards obtained from an RCN2
calculation, this number is 1000.0, and the energy unit for
parameter values is kK (1000 cm-1). Energy levels (eigenvalues)
are printed in the same units as those used for the parameter
values. The units need to be specified only for purposes of
calculating wavelengths, oscillator strengths, and transition
probabilities. [Note: The units can be changed by means of the
optional rescale control card; see Sec. X.]
Parameters for each configuration are arranged in the
following order:
Eav
Fk(l1,l1)
Fk(l2,l2)
.
.
.
zeta1
zeta2
.
.
.
Fk(l1,l2)
Fk(l1,l3)
.
.
.
Fk(l2,l3)
.
.
.
Fk(lq-1,lq)
Gk(l1,l2)
Gk(l1,l3)
.
.
.
Gk(l2,l3)
.
.
.
Gk(lq-1,lq)
In this list, "Fk" represents F2, F4, ... Fm [m=min(2li,2lj)], and
"Gk" represents G|li-lj|, ... Gli+lj, with index k incremented by 2.
There are no Fk(li,li) unless 2.LE.wi.LE.4li; there are no Fk(li,lj)
nor Gk(li,lj), i < j, unless 1.LE.w.LE.4l+1 for both wi and wj; and
there are no Fk of either type unless both l are greater than zero.
There is no zetai unless 1.LE.wi.LE.4li+1 and li > 0. If IABG > 0,
any parameters a,b,g,T,T1,T2 for subshell i follow the corresponding
Fk(li,li). If IABG = 2 or 4, then "Fk" represents F1, F2, F3, ... Fm,
and k likewise increases in unit steps for the Gk.
If desired, the first parameter card for each configuration
may contain, in columns 71-72, BCD identification of the source of
parameter values--for example, "LS" for least-squares, "HF" for
Hartree-Fock, "HR" for HFR, "HX" for Hartree-plus-statistical-
exchange, etc. In columns 73-74, 75-76, 77-78, and 79-80 may be
included two-digit scale factors for the Fk(li,li), zetai, Fk(li,lj),
and Gk(li,lj), respectively; "50" means 0.50, "85" means 0.85,
"99" means 1.00, and "01" means 0.001. These scale factors are
factors that have already been applied to obtain the parameter
values punched on the card, and are for identification only,
except as discussed in Sec. X.
Configuration-interaction parameter cards are like single-
configuration cards except that the parameter values are
Rk(ij,i'j') (all possible k) and Rk(ij,j'i') (all possible k). If
more than one set of values (ij,i'j') is possible, the order is an
odometer order similar to that for the ij in Fk(ij) and Gk(ij).
The only pertinent scale factor is that in columns 79-80.
For parameter-value cards output by RCN2, the tenth column of
each parameter field (columns 30, 40, ... 70 on the first card;
columns 10, 20, ... 70 on continuation cards) contains a single-
digit-integer code that provides correlation with the various
scale factors in columns 73-80; see the discussion of parameter
rescaling in Sec. X.
D. (cont) Radial-Multipole-Integal Cards
These cards provide values of Racah's reduced matrix element
P = < l || r(t) || l' >
( l t l')
= (-1)**l [(2l+1)(2l'+1)]1/2 ( ) Int.(r**t PlPl' dr) (5)
( 0 0 0 )
in units of (e*a0)**t, e the electronic charge and ao the Bohr unit
of length; t = 1 or 2 for electric dipole or quadrupole radiation,
respectively). No radial integral cards are needed for
magnetic-dipole calculations.
Columns 1-18 and 21-38 contain BCD information (like that
on the single-configuration parameter card) for the two
configurations involved in the transition. In the case of
quadrupole radiation, the two configurations may be the same or
may be two different configurations of the same parity; if they
are the same configuration, the code (incorrectly in some cases)
uses a quadrupole integral only for electrons li in the last
(greatest-i) non-s-electron, non-filled subshell. For electric
radiation there is always one and only one card for each pair of
configurations, even when the value of P is necessarily zero because
of selection rules, as for example in the case of s p3 - s2 p p'
or s2 p s - s2 p f. The value of P is punched in columns 41-50
(format F10.7); columns 51-64 contain "(nl//rt//n'l')" where n and
n' are two-digit integers and l and l' are BCD symbols (e.g., ^3p
and ^3d, for identification only), and t is 1 or 2 as appropriate;
the "(" and the value of t must be punched because the code uses
these punches to distinguish between parameter and multipole
cards.
Additional optional information is
/
FRAC = Int.(r**t PP' dr) / Int. (|r**t PP'| dr) (6)
/
in columns 65-70 (format (F6.4), and identification similar to
that in columns 71-72 of the parameter cards.
All of the above detail is automatically included on all
input cards when the RCG input file is prepared by running RCN and
RCN2.
D. (review): Detailed Arrangement of Parameter and Multipole Cards
(1) Energy parameters for configurations of the first
parity, one card (or set of cards if there are more than 5
parameters) for each configuration 1,2,3,4... in succession.
(2) Configuration-interaction parameter cards, one card
(or set of cards) for each pair of configurations for which
interactions are not identically zero (because of selection rules)
and are not excluded through use of special values of NSCONF(3,K)
(see A. above), in the order (1,2), (1,3), (1,4) ... (2,3), (2,4)
...(3,4) ... .
(3) If IQUAD =1 or 3, a set of quadrupole cards in the order
(1,1), (1,2), (1,3), (1,4) ... (2,1), (2,2), (2,3), (2,4) ...
(3,1), (3,2) ... .
(3a) If desired, additional set(s) of quadrupole cards
containing different values of P for the purpose of making
parameter studies.
(4) If configurations of both parities are included
[NSCONF(2,2) > 0], then cards for the second parity analogous to
(1), (2), and (if IQUAD > 1) (3); also,
(5) A set of radial dipole-integral cards (optional), one
card for each pair of configurations of opposite parity, arranged
in the following order:
Serial number of configuration
------------------------------
First parity Second Parity
------------ -------------
1 1
1 2
1 3
. .
. .
. .
2 1
2 2
2 3
. .
. .
. .
(These cards are read in subroutine SPECTR, statements 104-110.
(5a) Additional sets of dipole cards (5), if desired for
parameter studies.
(6) Additional complete sets of cards (1)-(5) if desired
(for parameter studies, isoelectronic-sequence calculations,
etc.).
(7) A card containing "-99999999." in columns 21-30. This
is a new first-configuration parameter card with a fictitious
value of the parameter Eav; it is a signal that there are no more
sets of input data (1)-(6), and that the program is to read a new
control card, etc. [cards (A)-(D), page 12].
(8) A card containing a negative integer in columns 1-5.
This is a new control card (A) with "illegal" value of KCPL, and
is a signal for RCG to make a normal exit.
V. Output
The amount of output sent to the print file IW (internal disk
unit 9; external name OUTG11) is controlled by quantities punched
in columns 6-7 of the ID2 control card, and in columns 6-7, 21-50,
and 61-72 of the calculational-deck control card, as discussed in
the preceding section. In most cases, all these columns are left
blank, except that (a) for configurations in light elements where
LS coupling is a good approximation, column 32 may be non-zero to
delete printing of JJ-representation eigenvectors, and (b) if
columns 23-25 are non-zero to activate calculation of eigenvectors
in still other representations (numbers 3 to 7), then non-zero
punches can be used in columns 31 to 37 to delete printing of
eigenvectors in the corresponding undesired representations.
(Note: When using these options, any number in column 33 must be
less than 5 to avoid activating a plane-wave-Born calculation; see
Sec. XIII.)
Information routinely printed in the normal case (columns 21-
50 and 61-71 blank or zero) consists of:
(1) In CUVFD (if called as the result of inclusion of cfp
decks), a list of subshells for which cfp decks were included.
(2) In PLEV, a list of dimensions actually used for certain
arrays. (If the array sizes specified in DIMENSION statements are
exceeded, a fatal-error STOP results.)
(3) No output from PFGD and PRK unless ICPC > 0, in which
case a list of parameters for which coefficients have been computed is
printed, with array sizes and (for PRK) the configuration-
interaction class. In the parameter names "FK(i,j)" and
"GK(i,j)", the values of i and j refer to the serial numbers
of the subshells liwi and ljwj involved; "ZETA(i)" is to be
interpreted similarly for zi; and "mnkDiji'j'" and "mnkEiji'j'"
refer to Rdk(ij,i'j') and Rek(ij,i'j'), where ij are subshell
numbers for the configuration with serial number m, and i'j' are
subshell numbers for configuration n.
(4) In CALCFC if ICFC > 0, for each J-matrix a list of
quantum numbers for each row (or column) of the matrix in both the LS
and JJ representations. In each case, symbols in parentheses refer to
quantum numbers aiLiSiJi for subshell liwi, and symbols not in
parentheses refer to coupled quantum numbers LiSiJi as accumulated
for coupling of successive subshells from left to right [TASS,
Eqs. (12.1) and (12.2), or as listed under "NLSMAX" in Sec. IV
above (page 15)].
(5) In ENERGY, a list of the parameter values read as input;
for each J, the energy matrix, eigenvalues, Land g-values,
eigenvectors in both the LS and JJ representations (except as
described in the first paragraph of this section, V, above), and
eigenvector purities (square of largest eigenvector component) in
each representation. Each eigenvector is tabulated vertically
beneath its corresponding eigenvalue and g-value, in sets of
eleven eigenvectors horizontally, with abbreviated basis-state
labels at the left side of the page, and (immediately above each
eigenvector) the configuration and basis-state label of the
largest component; for complete basis-state definitions it may be
necessary to consult the quantum-number listings in (4) above.
(6) In SPECTR, a list of the input radial multipole
integrals, the number of spectrum lines for each J-J', and lists of
the spectrum lines themselves. For each spectrum line there is
tabulated (after a serial number), the level value, the J value,
the serial number of the dominant configuration, and the dominant
eigenvector basis-state--first for the "first-parity" level, and
then for the "second-parity" level (assuming dipole transitions
between levels of opposite parity). Then comes the wavenumber in
the same units as for the energy levels, the wavelength in , the
line strength divided by the largest P2 for any of the transition
arrays present, the weighted oscillator strength gf and its common
logarithm (for absorption oscillator strength f, g = statistical
weight of the lower energy level, equal to 2J+1 or 2J'+1 as the
case may be), the weighted Einstein transition probability gA in
sec-1 (g = statistical weight of the higher energy level), and in
the final column either an F-format number =< 1 [representing a
cancellation factor in the calculation of line strength; TASS, Eq.
(14.107)] or an E-format number usually of the order of 10**8 to
10**14 [representing the quantity BRNCH involved in dielectronic-
recombination problems--see Sec. XII(a)]. The line list is
printed with lines sorted in the order of increasing energy of the
"first-parity" energy levels, and/or in the order of increasing
energy of the "second-parity" levels, and/or in the order of
increasing wavenumber (decreasing wavelength), depending on the
value of ISPECC (column 72 of the control card); in the first two
cases, values are tabulated for the sum of oscillator strengths
for all upward and for all downward transitions involving a given
level, as are the transition-probability sum for all downward
transitions and the radiative-decay lifetime corresponding to this
sum. [Note that these sums and lifetimes will not be correct if
the line-list storage dimension KLAM is not large enough to store
and process all lines in a single pass--see the following
section.]
(7) In ELECDEN, values at intervals of 40 RCN radial mesh
points of the pure single-configuration electron density for each
configuration; then for each eigenstate (energy level), the
intermediate-coupling CI electron density for the diagonal terms
only and, if KCPLD(3) = 4, with cross-term contributions included.
Sample output is included in Sec. XV at the end of this
document, and it is suggested that readers skip to that point.
The material in Secs. VI-XIV is highly specialized, and the reader
need refer to it only when and if he has need for information on
these special topics.
VI. Array Dimensions
It has already been mentioned that RCG allows up to 8
subshells. The limitation is only partly a matter of array
dimensions; a change would also involve some coding differences in
PLEV, and numerous do-loop-limit and format differences throughout
the program. [Note: 12-subshell versions of RCN2 and RCG are available.]
Required dimensions for many array variables depend very
strongly on the complexity of the calculations that one wishes to
handle. Some dimensionally imposed limitations in MOD 11 as
currently provided are as follows:
Array Dimensional
Upper limit on: variables parameter Limit
----------------------------- ---------- ----------- --------
Complexity of subshell 1 U1,V1,CFP1 KLS1 dw or f4
Complexity of subshell i=2,3 Ui,Vi,CFP2 KLSi dw or f3
Complexity of subshell i=4,6 Ui,Vi,CFP2 KLSi d3 or g2
Complexity of subshell i=7,8 Ui,Vi,CFP2 KLS6 d3 or g2
No. of configs (first parity) SOPI2,PMUP KC 50
No. of configs (second parity) NIJK,PMUP KC 50
No. of parameters (each parity PARNAM,VPAR KPR 2100
Matrix size C,CT4,TMX KMX 250
No. of spectrum lines T,TP,FLAM KLAM 10000
It should be noted that spectrum lines are processed in
batches, and that the dimension KLAM of T, TP, FLAM, etc. limits
only the maximum number of lines that can be processed in each
batch [ultimately, the number of lines arising from any one J-J'
matrix, excluding weak lines deleted by a non-zero value of DMIN
and short- and long-wavelength lines deleted by FLBDN and FLBDX
(page 10)]. However, if the total number of retained lines is so
great that processing is done in more than one batch, then one has
to manually combine multiple line lists to obtain (for example)
all lines involving a given upper level, and thereby obtain a
total downward transition probability and a correct radiative
lifetime. But note: If DMIN > 0, or NTKEV > 0, then DMIN is
automatically increased (in steps of 0.1 till DMIN=4.0, and then
by a factor of 2 at a time) until the number of spectrum lines is
small enough to be processed in one batch. For plane-wave-Born
calculations, DMIN must be zero, and is so set by the code
regardless of the value punched on the control card. Normally, a
non-zero value of DMIN deletes lines with S/X < DMIN, see page 15).
However, if a non-zero value of TEXCIT has been read in on a
rescale card (see page 30), then the cutoff is not on inherent
line strength, but rather on relative intensities in a light
source with effective excitation temperature TEXCIT (in the same
units as eigenvalues): If DMIN=0, DMIN is set initially to 10**4
and increased automatically by a factor 2 until the number of
retained lines is no more than KLAM; the cutoff is on
gA*exp(-(max(EIG(L), EIGP(LP))-EMEAN)/TEXCIT)
less than DMIN, where EIG(L) and EIGP(LP) are the eigenvalues of
the levels involved in the transition, and EMEAN is the average of
the largest of all eigenvalues and the larger of the minimum
energies for the two parities.
For most purposes, the dimensions can be considerably
reduced. If subshell 1 contains f electrons, subshell 2 is not
likely to be more complicated than d2, and the other subshells no
more complex than pw. Thus dimensions of U2, V2, CFP2 can be
reduced from 17 to 5 (there being only 5 LS-terms in d2), and
those of U3 ... V6 from 8 to 3. Except when computing detailed
Fano profiles in photoionization spectra, the number of
configurations usually need be at most 10 of each parity, the
number of parameters 100, matrix sizes about 75, and number of
spectrum lines 1000 or so. Other dimensions can also be decreased
greatly; e.g., those of ISER, PC, and PCI to 500, NOPCCC
to 1200, etc.
Many required dimensions are very difficult to estimate, and
have to be ascertained more or less by trial and error (increasing
them whenever a run bombs because of exceeded dimensions).
Nowadays, available fast memory on most computers is quite
large, and the dimensions in the current code are large enough to
handle most cases of interest, without any necessity for either
reducing them or having to increase them. However, if changes are
required, guides to estimating a few required values are as
follows:
Dimensions for f5 to f10 subshells: Dimension changes
required to be able to handle all fiw subshells (for i=1) are
given by comment cards near the beginning of the main program: one
needs to make the following changes in parameter statements
throughout the program:
KLSI=119
KJP=350
KMX=360 (or greater)
KTRN=3200 (or greater),
the last two values being required to keep certain calculated
dimensions from being negative. Even with these changes, fwp or
fwd configurations may be so complex as to overflow many other
dimensions, and be completely impractical to compute unless the
number of terms kept for fw is truncated (see page 11 above).
Dimension of PCI: Given a set of terms liwi aiLiSi, the
required dimension is
Sum (1/2) (no. of values of ai) (1 + no. of values of ai)
LiSi
A dimension of 21 will handle all dw and f3 and f11. [PCI is also
used for spin-orbit parameters--see under PC.]
Dimension of ISER: The dimension of ISER is computed
similarly to that for PCI, except that now we are considering a
sum over the different possible values of the total quantum
numbers LS, and in place of a we have the total number of different
sets of quantum numbers aiLiSiLiSi having the given LS. If the
configuration involves only one open subshell, the result is the
same as for PCI; with more than one open subshell, the result is
greater--for example:
p2 : terms 1S 3P 1D ; PCI(3), ISER(3)
d : terms 2D ; PCI(1), ISER(1)
p2d: terms 4PDF, 2SPPDDDFFG,; PCI(3), ISER(17),
where the 17 comes from
(1/2) (1*2 + 1*2 + 1*2 + 1*2 + 2*3 + 3*4 + 2*3 + 1*2)
.
Dimension of PC: Must be at least as large as that of ISER.
However, PC is also used to store spin-orbit coefficients, and
this may require a larger dimension: for the most complex subshell
liwi, set up a table of all possible aiLiSiJi. Then the required
dimension is
Sum (1/2) (no. of levels with Ji=J) (1 + no. of levels with Ji=J)
J
For example, p2 : 1S0 3P012 1D2 ,
PC(7), where 7 = (1/2) (2*3 + 1*2 + 2*3) .
Dimensions of CC(i,j): The dimension j must be as great as
the maximum number of different parameters Rdk and Rek (for any
one pair of configurations, and any one set of four interacting
electrons). The dimension i must be inferred from the code of
PRK, statements 705-780; it is equal to the number of coefficients
of Rk that are not inherently zero because of the LS selection
rule, because of zero values of the cfp involved, because of
incompatibility of intermediate quantum numbers, etc. The
simplest and safest way to set the dimension i would be to
consider only the LS rule; i.e., to find the maximum value, for all
pairs of interacting configurations, of
Sum (no. of terms LS in first conf.) (no. of terms LS in
LS second conf.)
For example, s p4 - s2 p2 d, terms of s p4 = 4P 2SPD, terms of s2 p2 d
= 4PDF 2SPPDDDFFG;
dimension = 1*1 + 1*1 + 1*2 + 1*3 = 7 .
4P 2S 2P 2D
Of course, except for CC, the above sample dimensions become
much greater when the presence of several configurations increases
the number of terms of each LS or the number of levels of each J,
dimensions going up roughly as the square of the number of
configurations of given parity.
In modifying dimensions it is important to note that the
labeled common block /C1/ is identical in all subroutines, but
that blank common and several other labeled commons differ from
one subroutine to another. This may lead to complications in the
case of linker/loaders that require the first subroutine
containing a given common to have the longest version of that
common or that require all to be of the same length, though in
RCG11 commons have been reorganized to minimize this problem.
VII. Memory Requirements
Even with the code extensively overlayed, it is difficult to
fit RCG into a 64K-word computer (200K words, octal) with anything
but rather small dimensions. Memory requirements for a CRAY with
present dimensions are about 640K octal 64-bit words, plus I/O
buffer space. On a Macintosh or PC, the executable file is about
one Mbyte in size, and execution requires about 3.4 Mbytes of RAM.
VIII. Execution Times
Like required array dimensions, execution times vary
tremendously, depending on the complexity of the problem--as
measured especially by the number of parameters and the matrix
sizes. Some examples are given in TASS, Tables 16-1 and 16-2.
Generally speaking, CRAY Y-MP execution times are less than half a
minute for problems involving no more than 40 parameters, matrices
up to 50 x 50, and no more than a couple of thousand spectrum lines.
SUN, Power Mac, and PC times are only a few times greater for state-
of-the-art desk computers.
IX. Modifications for Other Computers
As noted in the Introduction, modifications for other
computers should be minor provided sufficient memory is available
or a virtual-memory system is in use, and this is generally true
of even modest desktop computers these days. However, if memory is a
problem, the following modifications are possible.
(1) Code storage space can be reduced by overlaying.
Suggested groupings of subroutines are indicated throughout the
RCG source program by commented OVERLAY and CALL OVERLAY cards.
If this feature is used, duplicate copies of the routines RDIJ and
REIJ in the PFDG overlay should be added to the PRK overlay.
(2) Other possibilities in addition to reducing array
dimensions are:
(a) If least-squares fitting of experimental energy
levels is of no interest, delete subroutine RCEINP and the call
thereof in ENERGY.
(b) Delete CPL37, SPRN37, LVDIST, and WNDIST and calls
thereof.
(c) If plane-wave-Born calculations are of no interest,
BORN, AKNINT, RCOEFF, E1, and CSEVL can be deleted, as can the
variable GOSS and several others.
(d) The code could be broken into a chain of three
programs, containing essentially (i) CUVFD to prepare disks 72,
73, 74; (ii) LNCUV through MUPOLE to prepare disk 41; (iii) LNCUV
plus ENERGY and SPECTR to use disk 41 in the calculation of energy
levels and spectra.
X. Rescaling of Input Data
Most frequently, input data for an energy-level/spectrum
calculation will have been obtained via an RCN/RCN2 calculation.
The data will then be such that (1) the center-of-gravity energy
Eav of the first configuration will be zero, (2) all parameter
values (and hence all computed eigenvalues) will be in kilo-
kaysers (units of 1000 cm-1), and (3) energy parameter values
other than Eav (excepting perhaps spin-orbit parameters) will have
been scaled down by factors less than unity to allow for omitted
weak configuration-interaction effects (TASS, Sec. 16-2); these
scale factors appear in columns 73-80 of the parameter cards (see
Sec. IV-D above, pages 20-21), and the factor that has been used
for a given parameter is indicated by an integer JPAR in the tenth
column of the parameter-value field, as follows:
JPAR kind of parameter scale factor
---- ----------------- -------------
0 Eav -
1 Fk(ii) columns 73-74
2 z 75-76
3 Fk(ij) 77-78
4 Gk(ij) 79-80
5 Rk 79-80
It is frequently desirable to shift all energy levels
upward (by adding a constant to all Eav) to make the calculated
ground level of the atom zero (as this is the convention used in
tabulating experimental energy levels), to change the eigenvalue
unit (to cm-1, eV, or rydbergs, for example), or to modify the
parameter scale factors to obtain better agreement with
experiment. This can be done by using an editor to modify the
data in the input file by hand, but it can be done much more
easily with the aid of a rescaling card of the form:
cols. format variable action
----- --------- -------- -----------------------------------
1-4 - (blanks)
5 I1 (zero) defines a rescaling card
6 - (blank)
21-30 F10.5 DELEAV DELEAV (same units as on input para-
(do not punch meter cards) added to all Eav
decimal point)
31-40 5I2 IFACT0(5) New scale factors, in percent
(except 99 = 100 %, 01 = 0.1 %)
51-60 F10.5 UENRGY New energy unit in cm-1 (8065.47 for
eV, 109737.3 for Ry, etc.), applied
after the addition of DELEAV
61-65 F5.2 TEXCIT Effective light-source excitation
temperature (in eigenvalue energy
units) (see Sec. VI, pages 25-26)
Default values are zero for DELEAV, no rescaling for any IFACT0(I)
that is zero or blank, and the unit of energy specified on the
normal control card (A) if UENRGY is zero or blank.
Note that (1) the new scale factors represent exactly that,
and not additional scaling over and above the scaling already
present in the input parameter values; (2) old scale factors
(columns 73-80 of the parameter cards) equal to zero are assumed
to be unity; (3) non-zero values of IFACT0(I), I = 1 to 4, will
not have the intended effect if the correct value of JPAR does not
appear in the tenth column of each parameter field; (4) if scale
factors are changed via IFACT0 0, the value of DELEAV required to
give zero energy for the ground level will in general be different
from the value inferred from a previous run made with the old
scale factors.
A rescale card may be placed in front of any normal control
card (A) of a calculational deck; it is read at statement 100 of
the main program as a normal control card with KCPL=0, and
reinterpreted as a rescale card. Non-default values read from a
rescale card remain in effect throughout the remainder of a
calculation until modified by another rescale card; a rescale card
containing default values (zeroes or blanks) for one or more
quantities returns program control to use of values of these
quantities specified in the individual calculational decks.
XI. Photoionization Cross Sections Q
A. Neglecting resonances, and for cases in which continuum-
continuum (intra-channel) configuration-interaction effects are
unimportant, photoionization-cross-section calculations are most
easily made with programs RCN/RCN2/RCG in the following way. (The
procedure will be illustrated using the special case of Ar XVII
1s2 - 1s ep, e = kinetic energy of the free electron in rydbergs.)
(1) Estimate the threshold value for photoionization, or
calculate it by making RCN runs for Ar XVII 1s2 and Ar XVIII 1s
and differencing values of Eav. In this case, a simple estimate
is: binding energy of a 1s electron = Zc2 = 17.52 = 306 Ry, and a
calculation gives Eav(1s) - Eav(1s2) = -325.397 + 628.459 = 303
Ry.
(2) Choose values of e at which the cross section is to be
computed. If one wishes a threshold value of Q, include a
continuum configuration with e = 1 to 5 % of the threshold energy.
In the argon case, e = 5 or 10 Ry is appropriate.
(3) Make an RCN/RCN2 calculation. In order to be able to
interpolate across threshold, one can include bound configurations
1s np, say for n = 10 to 15. Larger n values get one closer to
threshold, but are more likely not to converge, so one might
choose n = 8 and 12. The input to RCN is then
cols. 4-5 9-10 11-23 27 (or greater) on
--------- ---- ------------- ------------------
18 17 Ar 17 1s2 1s2
18 17 Ar 17 1s 8p 1s 8p
18 17 Ar 17 1s 12p 1s 12p
18 17 Ar 17 1s 5.p 1s 99p 5.0
18 17 Ar 17 1s 10.p 1s 99p 10.0
18 17 Ar 17 1s 25.p 1s 99p 25.0
etc.
On the RCN control card, EMX (columns 61-65) should be chosen such
that all values of e for which calculations are to be made lie in
the range from about EMX/10 to EMX. If the value of EMX is
inappropriate, or n for the bound states too large, the RCN runs
may fail to converge or bomb on an overflow. [If EMX is left zero
on the control card, RCN will set EMX equal to the largest value
of e on any input configuration card.]
On the G5INP control card that is input to RCN2, columns 51-
60 [the empirical scale factors for Fk(ii), zi, Fk(ij), Gk(ij), and
Rk(ij,i'j')] should have appropriate values, such as (for a highly
ionized atom like Ar XVII) 9599959595. Column 72 of this card
(and thereby column 72 of the RCG control card) should contain a
one (for ease in using the RCG line-list output), and column 75
must be equal to 8.
(4) Run RCG as usual, using as input the unmodified output
file OUT2ING from RCN2 (with name changed to ING11).
(5) Values of "f" in the RCG line-list output are oscillator
strengths for bound-bound transitions (1s2 - 1s 8p and 1s2 - 1s 12p
in this example), and are values of df/de (e in rydbergs) for
bound-free transitions (1s2 - 1s 5.0p, etc.). For interpolation
purposes, the bound-bound oscillator strengths can be converted to
averaged (smeared-out-lines) values of df/de = (df/dn) / (de/dn) by
taking
de de d (-Zc**2/(n*)**2)
-- = --- = ------------------ = 2Zc**2/(n*)**3 (7)
dn dn* dn*
with Zc the spectrum number (1 for neutral) and n* the effective
quantum number (printed on the last page of the RCN listing for
each configuration, under the heading "n*rc"), where it was
calculated from
E = - Zc**2/(n*)**2
with E = the binding energy ("eps fgr," with correlation) of the
electron in Ry.
Then
Q = 8.067 * 10-18 df/de cm2 . (8)
Values of Q (correct only for the bound-free transitions) will be
printed in the spectrum line list in place of gA, provided
NSCONF(3,2) is negative, columns 51-64 are identical on the last
two dipole-integral cards, and each of the dipole cards contains a
number equal to 1.0 in columns 65-70.
(B) When resonances (bound-free interactions) or free-free
(intra-channel) configuration interactions are important, the set-
up for RCG runs is much more complex, requiring introduction of
many values of e--in order to resolve the detailed shape of the
resonance, for example. Calculations are then practical only for
rather simple cases, such as those of Al I, Cl I, and Ba I in
TASS, pages 539ff.
One approach is the following (TASS, Sec. 18-8), though it
involves a great deal of hand work.
(1) Make an RCN/RCN2 run including all essential bound
configurations, and continuum configurations in which the kinetic
energy e covers the range of interest or importance; if values of e
cover a range of more than 10:1, it may be necessary to make two
or three runs, each of which includes the important bound
configurations but only one of several overlapping subsets of the
continua, each covering a 10:1 energy range (with appropriate
value of EMX in each case).
(2) Draw graphs like those in TASS, Fig.18-3, giving Rk and
radial dipole integrals as functions of e.
(3) Divide the energy range of the continuum into a number
of segments, the ith segment having a width (del)i rydbergs. [For an
example in the case of neutral chlorine, see J. Opt. Soc. Am. 64,
1474 (1974).]
(4) Read values of the radial integrals from the graphs at
points ei = center of each segment and multiply by (del)i**1/2, or by
(del)i**1/2 (del)j**1/2 for free-free Rk. [Note that the values of
del must be in rydbergs; values of Rk tabulated by RCN2 in "kK" have
in all cases been converted from values in "Ry" by multiplying by
109.737, even though Rk has dimensions of (energy)**1/2 for bound-
free interactions and has no energy dimension at all for free-free
interactions (see TASS, Sec. 18-4)].
(5) Construct an appropriate RCG calculational deck, using
the results (4), and using single-configuration parameter values
for each "pseudo-discrete" configuration appropriate to the
corresponding ei, and run in the usual way.
If NSCONF(3,2) is less than zero, the last two radial-dipole
input cards contain the same information in columns 51 to 64, and
each bound-free radial-dipole card contains the appropriate value
of (del)i**1/2 in columns 65-70, RCG will calculate and print
photoionization cross sections SIGMA = Q in place of weighted
transition probabilities gA. These values will have been computed
from Eq. (8) assuming de = (del)i for all spectrum lines involving
upper levels belonging to the configuration "i". This will be
inappropriate if perturbations appreciably alter the mean energy
spacing between adjacent configurations, and will be particlarly
incorrect for bound-state autoionizing levels strongly mixed with
continuum states.
An alternative method is available that involves no handwork
in the preparation of the RCG input deck. This alternative
requires that all desired bound and continuum configurations be
included in a single RCN/RCN2 run, and that IDIP = 7 on the RCN2
control card; this may involve considerably more computer time
than the hand method, and will not work if the desired values of
principal quantum number n for bound functions and values of e for
continuum functions cannot all be handled with a single value of
EMX. (If this last is a problem, sometimes a compromise can be
made, using a value of EMX smaller than the maximum e, as the
requirement EMX ³ largest e is somewhat conservative.)
The alternative method will be described by means of an
example for neutral silicon, where the unperturbed position of
3s3p3 3Po lies between 3s23p nd 3Po for n = 4 and 5, but
interactions of sp3 are significant throughout the entire discrete
3p nd Rydberg series and well into the 3p e d continuum (which
thereby distorts the energy dependence of the photoionization
cross section). An appropriate set of input configurations might
be
Si I 3p2 - 3s 3p3 + 3p 3d + 3p 4d +...+ 3p 9d + 3p 10d + 3p 12d
+ 3p 0.036d + 3p 0.22d + 3p 0.58d .
RCN2 recognizes the discontinuity between 10d and 12d as well as
the presence of continuum configurations, and sets up four pseudo-
discrete configurations with unmodified Eav, but with appropriate
values of Di to cover the energy range from just above 10d to well
beyond 0.58d. Values of Rk and radial dipole integrals are
automatically modified by appropriate "(del)i**1/2" factors, which are
called "DEL" within RCN2, but printed out labeled "SQRTDEL".
RCG can then be run using as input the unmodified output deck
from RCN2 provided IDIP on the RCN2 control card has the value 7.
The correct value of gf for 12d is obtained by dividing the listed
value of gf by (SQRTDEL)2; i.e., "Di" for bound configurations is
here not an energy width in rydbergs, but rather a weighting
factor related to the effective number of bound configurations
represented by the pseudo configuration. Photoionization cross
sections for the pseudo configurations derived from continuum
configurations will be printed explicitly, or may be obtained as
before from (8) by dividing the f obtained from the gf column by
(SQRTDEL)2; i.e., "Di" is here both a weighting factor and an
energy width in rydbergs.
Note: The energies of the above set of three continuum
configurations were chosen because of primary interest in the low
bound configurations. For photoionization calculations, a larger
number of more closely spaced continua would be called for.
XII. Autoionization Transition Probabilities
RCG has the capability of computing autoionization transition
probabilities Aa (designated AA in the FORTARN program, or GAA for
the weighted transition probability gAa). The basic theory is
discussed in TASS, Secs. 18-7 and 18-11, and in Appendix B of LA-
6220 by Merts, Cowan, and Magee, which also contains numerous
numerical examples.
The calculation of values of Aa will be described by means of
an example in Fe XXI, which has the ground configuration 2s2 2p2.
Suppose we wish to calculate Aa for the levels of Fe XXI 2s 2p2 15p,
which all lie above the ionization limit Fe XXII 2s2 2p.
(1) First, a preliminary RCN run is made for Fe XXI 2s 2p2 15p
and Fe XXII 2s2 2p. From the computed total binding energies it
will be found that Eav(2s 2p2 15p) lies 3.4 Ry above Eav(2s2 2p).
[See Table B-III of the above report LA-6220.]
(2) The approximation is made that all levels of 2s 2p2 15p
lie 3.4 Ry above the ionization limit, and a final RCN/RCN2 run is
made for
Fe XXI 2s2p215p
Fe XXI 2s22p 3.4s (9)
Fe XXI 2s22p 3.4d ,
parity and other selection rules permitting autoionization into
only the s and d continua. [Preparation of the RCN input
configuration cards involves punching "99s" or "99d" in the
configuration-definition part of the card (not the configuration-
label part), followed after at least one space by "3.4".] The
G5INP control card in the RCN2 input deck should have an 8 or 9 in
column 72 (ISPECC for the subsequent RCG run), and must have a 2
in column 73 and a 9 in column 75; these punches in columns 73 and
75 produce certain appropriate modifications in the RCN2 output
for RCG input. [ISPECC can also be 1 in this case, or 2 if the
continua are in the second parity, but the output is longer.]
(3) Output from the RCN/RCN2 run will include the usual
punched-card RCG-input deck for the three interacting
configurations (9). The three printed values of Eav will be equal
(to within 0.005 Ry @ 5 kK). However, in the punched-card output
(file out2ing) that is input for RCG, the values of Eav for the
continuum configurations will have been changed to -9500 and -8500
(as a result of the 9 in column 75 of the G5INP control card).
(4) Running RCG with these modified values of Eav will
result in some eigenvalues being less than -4000 and hence a
signal to the computer program (ENERGY, statement 362+) that these
belong to continuum states. After saving the bound-continuum CI
matrix elements, the program zeroes these elements of the matrix
so that diagonalization produces zero eigenvector mixing of
discrete and continuum states. Values of Aa are computed in
ENERGY, statements 360-380, using the perturbation theory
expression
Ajia = (4pi**2/h) |< j|H|i >|**2
= (4pi**2/h) | Sum < j|b> < b|H|b' > < b'|i > |**2 . (10)
bb'
The intermediate-coupling eigenvector components < j|b > for the
pure-discrete (but potentially autoionizing) state j and the
components < b'|i > for the pure continuum state i are obtained from
the energy-matrix diagonalization; the basis-state configuration-
interaction matrix elements < b|H|b' > prior to diagonalization have
been saved in the block CI at statement 255. [Note: At statement
363+, Aa is calculated in units of 10**13 sec-1 by using h/2pi =
(10-13/2066) Ry-sec, and using the factor 1/109.735 to convert
< b|H|b' > from the incorrect units put out by RCN2 (see pages 32-33
above) to units of Ry1/2.]
It is essential in the RCG input deck (and hence in the RCN
input deck also) that all discrete configurations come first, and
all continuum configurations come last. For any given value of
the total-angular-momentum quantum number J, let the matrix size be
NLS, and let LX (< NLS) be the number of discrete levels and JX-1 =
NLS-LX be the number of continuum levels. Then the minimum
permissable block size for the variable CI is LX by JX-1, for the
variable CII is NLS by NLS, and for AA is NLS by JX+1. After
completion of the matrix diagonalization (which leaves the
eigenvalues in the order of increasing value), the continuum
eigenvalues will be numbers 1 through JX-1 and the discrete
eigenvalues will be numbers JX through NLS. The value of Ajia is
stored in AA(j,i) [JX.LE.j.LE.NLS, 1.LE.i.LE.JX-1]. The value of
Si AA(j,i) is in AA(j,JX) for all Ei < -8000 and in AA(j,JX+1) for
all Ei < -4000; the distinction between these two sums, Aja1 and
Aja2, respectively, is for special purposes having to do with
dielectronic recombination. All results are written on disk unit
IE (= 31 or 32, depending on parity) for use in SPECTR.
(a) Kinetic energies of autoionized electrons
Kinetic energies of the free electron (energy of the auto-
ionizing level minus the ionization energy) are calculated and
printed following the eigenvalue print, relative to each of the
possible levels of the ion core. (Calculation of the ionization
limits involves use of the total configuration-average binding
energies on the configuration cards that follow the RCG control
card.) Negative values of kinetic energy are of course not physically
meaningful, but are printed in case one is interested in knowing
how far below the ionization limit the level lies. Autoionization
rates are also calculated for these bound levels, but again are
physically meaningless.
(b) Branching ratios and dielectronic recombination
If configurations of both parity are included, for example
Fe XXI 2s2 2p 15p
Fe XXI 2s 2p2 15p (11)
Fe XXI 2s2 2p 3.4s
Fe XXI 2s2 2p 3.4d ,
then spectral transitions are computed as usual, but only between
levels with E ³ -4000 kK. Provided ISPECC (column 72 of the RCG
control card) is 6,7,8, or 9, then for each spectrum line L = j®k,
a quantity
gj Aja1 Ajkr
Bjkr = BRNCH(L) = ---------------- (12)
Aja2 + Sum Ajk'r
k'
is computed in SPECTR, and printed in the final column of the line
list (in place of the cancellation factor normally printed there).
The factor gj Aja1 is proportional to the total rate of
dielectronic capture of free electrons by ions initially in states
having Eav less than -8000 (in practice, ions in the ground
configuration), and the remaining factor in (12) is the branching
ratio for radiative decay to stable (non-autoionizing) levels,
including the effect in Aja2 of possible autoionization to excited
ion states defined by -8000 < Eav < -4000. The quantity Bjkr is
equal to gj times the final fraction in TASS, Eq. (18.120) [or
equal to GmFjk in Eq. (B.21) of LA-6200, where Gm = Sum gm is the
total statistical weight of the ground configuration of the
recombining ion], and the contribution of the levels j and k to
the dielectronic-recombination rate coefficient is given by
(18.120).
Also computed in SPECTR (called BRNCHR in the code, and
printed as "GM*FRBAR) is the quantity
Br = Sum Bjkr , (13)
jk
which provides the total contribution of all levels j and k to the
dielectronic-recombination rate coefficient via TASS, Eq.
(18.116).
In equation (18.120) mentioned above, there has been assumed
a mean kinetic energy Es (i.e., a mean energy of autoionizing
levels above the ionization limit). When the energy spread of
autoionizing levels j is not small compared with the plasma
electron temperature of interest, this is a poor approximation.
The appropriate correction factor is computed for each of NTKEV
(£5) temperatures TKEV (in units of keV) read in on the type-b
optional control card (page 10). If NPTKEV (read in on this card)
is 1 or greater than 2, then there will be written on unit 13 the
value of Br/Gm together with the temperature correction factors,
and also the values of Sj Bjkr/Gm for each lower level k with the
corresponding correction factor. If NPTKEV is greater than 1,
then there will be written on the normal print file IW the value
of Br and of Br times each correction factor. EIONRY is the
ionization energy (in rydbergs) from the ground level of the
recombining ion; this is used to compute (and write on 13) the
center-of-gravity energy of the autoionizing levels (with Gm and
Br/Gm), and the energies of the individual levels k (with Jk and
Sum(j) Bjkr/Gm), all relative to the ground level of the recombined
ion. In order for this to work correctly, it is essential that
the (final) continuum input parameter card contain the kinetic
energy in rydbergs in columns 7-12 (format F6.2), and that
therefore the corresponding RCN input configuration card contain
this information in columns 17-22.
Note that the DIEL feature in RCN (Sec. II.L
of the RCN writeup) makes it possible to set up
input for dielectronic-recombination calculations
such as that in (11) without having to explicitly
make a preliminary calculation to obtain the
free-electron kinetic energy nor figure out by
hand the possible values of the free-electron
angular momentum. At the same time, RCN auto-
matically puts the kinetic energy in columns 17-22
of the continuum-configuration cards to satisfy
the above requirement.
A non-zero value of EMINA is intended for handling problems
in which some levels of a configuration have too low an energy to
lie above the bottom of a continuum to which they might otherwise
autoionize. Consider, for example levels of 2s2p(1P)nl that lie
above 2s2p(3P), and therefore can autoionize to 2s2p(3P)el' but not
to 2s2p(1P)el'. If EMIN.NE.0, RCG allows autoionization only if the
bound level has energy greater than EMINA, and the continuum level
either has energy less than -8000 or has an energy less than or
equal to the center-of-gravity energy of the configuration to
which it belongs. In the above example, EMINA should be the
energy of 2s2p(3P) [determined by means of a preliminary
calculation], Eav should be greater than -8000 for each 2s 2p el'
configuration , and Eav should be less than -8000 for 2s2 el".
(Obviously the code is not general; one would run into difficulty
if the spin-orbit splitting of the 3P were large, or in more
complex cases such as 2s 2p2 el'.
If ISPECC (column 72 of the control card) is 8 or 9, SPECTR
calls WNDIST: the latter calculates and punches (on disk unit 11),
in the form of a histogram, the normalized [(12) divided by (13)]
energy distribution of radiation resulting from excited levels j
produced by dielectronic recombination in low-density plasmas.
The quantity DELEKEV (read in the main program from columns 21-30
of the type-b optional control card, page 10) specifies the width
of the histogram bins. (If no Aa calculations are being
performed, the histogram gives the normalized distribution of gf
rather than of Bjkr, representing approximately the energy
distribution of radiation resulting from collisional excitations
in low-density plasmas, assuming the optical approximation
(excitation rates proportional to oscillator strengths). Both
types of punched deck constitute input for a computer program
RADRATE; for sample output of this program, see TASS, Figs. 19-12
and 19-13.
It should be noted that calculation of Bjkr and related
quantities will be incorrect if there are too many spectrum lines
for SPECTR to store and process them all in one pass (i.e., if
statement 560 is reached from the IF statement following statement
511 or 550 rather than from the IF that follows 200), because the
summation over k' in (12) cannot be correctly evaluated. [This is
not a program bug, but just an inherent storage limitation.] If
DMIN > 0 or if NTKEV > 0, DMIN is automatically increased until
enough weak lines have been deleted so that this storage
limitation is not encountered (see pages 25-26).
(c) Autoionization contributions to collisional ionization
Electron-impact ionization may take place either by direct
ejection of an electron from an atom (or ion), or by collisional
excitation of an inner-subshell electron to a level j lying above
the ionization limit, followed by autoionization. For this second
(indirect) process, one needs to know excitation-rate coefficients
to all possible levles j, together with the branching ratios
Sum(m) Ajma
Bja = ---------------------------
Sum(m) Ajma + Sum(k') Ajk'r
gj Aja2
= --------------------------- (14)
gj Aja2 + gj Sum(k') Ajk'r
for autoionization to all possible states of the ion; see, for
example, R. D. Cowan and J. B. Mann, Astrophys. J. 232, 940
(1979).
If an RCG run is made as described in the preceding
subsection, values of gj Sum Ajk'r are printed as "SUMGA" in the
spectrum line list sorted by second-parity levels, and gj Aja2 and
Bja are printed as "GAATOT" and "BRION" in the following line.
[Note that these prints will be obtained only if ISPECC =
2,3,6,7,8, or 9, and that if equal to 9 then the lines themselves
are not printed.]
(d) Autoionization contributions to collisional excitation
Similarly to collisional ionization, excitation (or de-
excitation) can take place either directly, or indirectly via
dielectronic capture of the impacting electron into a highly
excited state, followed by autoionization into a state of the
target atom different from the original one. Computation of the
indirect process involves much the same quantity as that (12) for
dielectronic recombination, except that the required branching
ratio is for autoionization instead of radiative decay:
(gj Ajma) (gj Ajia)
Bmjia = -------------------------- (15)
gj Aja2 + gj Sum(k') Ajk'r
unlike (12), we have not summed over states m involving the ground
configuration of the target atom to give Aja1, and the quantities
Ajma and Ajia in the numerator of (15) are the values printed by
subroutine ENERGY as "AA" in all but the last two lines (which are
Aja1 and Aja2).
States m and i of the (N+1)-electron system --target atom
plus free electron--will be basically JJ coupled, and we denote
them by
m = [(gJt)m,(Jl)m]J
and (16)
i = [(gJt)i,(Jl)i]J ,
where gJt defines the target-atom state with total angular momentum
Jt, the free electron el has orbital angular momentum l and total
angular momentum (l+s) of Jl, and the total (N+1)-electron angular
momentum (Jm or Ji) is necessarily equal to the value J º Jj of the
resonance-capture state j. If we sum (15) over the two possible
values Jl = l =/- 1/2, and redefine m and i to represent the sum of
the two states:
m = [(gJt)m, lm]J
and (17)
i = [(gJt)i,li]J ,
then values of the quantities in parentheses in the numerator of
(15) are printed by ENERGY as an array of numbers "GAAXC" ("gAa
excitation"), labeled on the left by the serial number of the
free-electron configuration target+el and the energy "EXC" of the
target level gJt (relative to Eav).
In SPECTR, the above values are combined with the denominator
of (15) and summed over all levels j [including thereby a
summation over the quantum number J in (17)] to obtain values
"BRNCHX(m,i)" = Sum Sum Sum Bmjia (18)
j (Jl)m (Jl)i
printed in a square array, labeled similarly to the GAAXC array
(as to both row and column), except with rows and columns sorted
in order of increasing configuration serial number and increasing
value of EXC.
This array (which of course forms a symmetric matrix)
gives maximum physically significant detail for the collisional
excitation/de-excitation problem, as it provides the total
contribution for all levels j and for all possible couplings of
the free electron (with given l) to the target state. (Diagonal
elements pertain to resonant-state contributions to elastic
scattering.)
Actually, the l of the free electron is really only of
mathematical (not physical) interest, so in a second array,
columns are combined to give the summation of (18) over the
possible values of li. [This array is printed only if it is
actually narrower than the first one.] It is left to the user to
perform the corresponding sum over rows (lm) to give total values
from one target level to another; the result is the quantity
gmFmi (19)
involved in Eqs. (2) and (3) of R. D. Cowan, J. Phys. B. 13, 1471
(1980). [There, m and i have been furter redefined to refer only
to the levels gJt of the N-electron target.]
Still a third array is printed (if narrower than the second)
in which a summation has been carried out over levels i of each
target configuration. The user may manually sum rows over lm to
obtain the total excitation rate from a given target level to all
levels of the excited configuration. Or he may also sum over the
levels of the initial configuration to obtain the quantity
_
Sum(m) Sum(i) gmFmi = GmFa . (20)
If this is divided by the total statistical weight Gm º Sm gm of
all levels of the ground configuration of the target, one has a
quantity proportional to the mean total excitation rate, summed
over all levels i and averaged over all levels m; this is
appropriate for application to moderate-density plasmas, where the
(metastable) levels of the ground configuration may be more or
less statistically populated. The quantity (20) is printed as
gj Aja1 (Aja2 - Aja1)
"GM*FABAR" = Sum(j) _____________________ . (21)
Aja2 + Sum(k') Ajk'r
In some cases (as for example the O IV case discussed in the
above JPB paper), not all levels j of the excited configuration
will lie above the ionization limit, and the summations over j in
(18) and (21) must correspondingly be limited appropriately. (RCG
does not inherently have enough information to set Ajma to zero
for levels lying below the limit.) The necessary modifications
may be accomplished (as described earlier) by introducing the
quantity EMINA (format F10.5) read from columns 71-80 of the
type-b optional control card (pages 10 and 37). If EMINA.NE.0 and
the continuum level has energy greater than -8000, then Aa will be
set to zero if the level j has energy less than EMINA, or if the
continuum level has energy greater than Eav for the continuum
configuration in question. The second of these two restrictions
was introduced in order to compute excitations from 2s2el to
2s2p(3P)e'l' via autoionizing levels 2s2p(1P)n"l" [i.e., in order
to exclude continuum levels 2s2p(1P)e'l'], and may not always be an
appropriate restriction--in which case the code will have to be
changed accordingly.
XIII. Plane-Wave-Born Collision Strengths
If KCPLD(3) (column 33 of the calculational-deck control
card) is greater than 4, then columns 33 to 37 are interpreted not
as KCPLD(3) to KCPLD(7), but rather as
column format variable
------ ------ ---------------
33 I1 IGEN
34 I1 IRNQ (.GE.0)
35 I1 IRXQ (.GE.IRNQ)
36 I1 IRND (.GE.1)
37 I1 IRXD (.GE.IRND)
and plane-wave-Born collision strengths will be computed [from
levels of only the first configuration of the first parity unless
NCK(1)--column 6 of the control card--is greater than one], via
calls near the end of ENERGY to subroutine BORN.
The amount of printed output is smaller the larger the value
of IGEN from 5 to 9 (9 is recommended). If collision strengths
for optically forbidden excitations are desired for the first,
second, or both parities, then IQUAD must be 1, 2, or 3,
respectively, and IRNQ and IRXQ (even integers) must specify the
minimum and maximum values of t for the Bessel-function matrix
elements
< li||jt(Kr) C(t)||lk > (22)
that will be involved for the jumping electron in any of the
excitations specified by NCK(1) and NCK(2). Similarly, if
optically allowed (electric-dipole allowed) excitations are to be
calculated, then IRND and IRXD (odd integers) must be specified
appropriately. [See TASS, Secs. 18-12 and 18-13 for details.]
This same set of five integers (and IQUAD) must be punched on the
G5INP control card for RCN2 in order that the latter compute the
matrix elements (22) in place of the normal electric multipole
matrix elements
< li||rt C(t)||lk > , (t = 2 or 1) , (23)
and also to set up a card containing various other required
quantities (SPECTR, format 8).
The basic printed output consists of:
(i) a table of values of the momentum transfer K;
(ii) for each spectrum line (or rather, each J - J'
excitation), a table of values of the weighted generalized
oscillator strength gfJJ'(K), and a table containing X º kinetic
energy of the impacting electron (e) in units of the excitation
energy (DE), the kinetic energy e in rydbergs, the unmodified
collision strength W, and two modifications of W that should be
physically more accurate at small X--specifically,
( W(3) F(X) , X < 3
WM1(X) = (
( W(X) F(X) , X > 2
where F(X) = 1 - 0.2 exp[0.07702(1 - X)] ,
and
WM2(X) = W(X + 3/(1+X)) ,
with WM2 generally being more accurate than WM1. Also printed is
a table of excitation-rate coefficients computed from WM2 by
integration over a Maxwellian distribution, at electron
temperatures ranging from T=1 to 2000 eV for low excitation
energies (low ionization stages) or from T=5 to 10000 eV for high
excitation energies (highly ionized atoms). (The quantity labeled
"CORR" is the percentage of the printed rate coefficient
contributed by extrapolating WM2(X) from the largest tabulated
value of X to infinity.)
(iii) for each transition array, the array-average optical
oscillator strength fa [TASS, Eq. (14.97), evaluated by numerical
summation of the individual weighted oscillator strengths] if
IGEN = 5, three alternative values for the array-average excitation
energy (the first being DEav, the second being Eav for the higher
configuration minus the lowest energy of the lower configuration,
and the third being an average of the individual excitation
energies weighted with the unmodified collision strength at the
largest X value being calculated), and a table of values of X and
the unmodified and modified values of collision strength (summed
over all levels of both the lower and the upper configuration).
Rate coefficients are tabulated as in (ii).
XIV. List of Principal Variables
Sections of code: A = all
C = cfp-deck calculations (CUVFD)
P = preliminary calcs. (LNCUV,PLEV,PFGD,PRK)
F = final J matrices (CALCFC,CPL37)
M = line-strength matrix elements (MUPOLE)
E = energy diag.(ENERGY,CALCV,LVDIST)
S = spectrum calculation (SPECTR,WNDIST)
B = plane-wave-Born (BORN)
Variable Sections Significance
AA(L,J) E autoionization transition probability from
bound state L to continuum level J
AA(L,JX) E total Aa from L to all continuum levels
with E < -8000 (= Aa1)
AA(L,JX+1) E total Aa from L to all continuum levels
with E < -4000 (= Aa2)
AA(L) S Aa1 for level L of first parity
AAP(LP) S Aa1 for level LP of second parity
AAT(L) S Aa2 for level L of first parity
AAPT(LP) S Aa2 for level LP of second parity
ALF CP alpha in aiLiSi (BCD serial number for terms
of lw with given LS)
_ ___
ALFBR C a in aLS (BCD serial number for terms
__
of lw-1 with given LS)
ALF3 E skewness parameter a3 in skewed Gaussian
AVCG(K) E (2J + 1)-weighted average of Eav for all
levels of parity K
AVEIG(K) E (2J + 1)-weighted average of all eigenvalues
of parity K [should equal AVCG(K)]
AVP(M,K) E average purity of all eigenvectors of
parity K for coupling M
BIGKS SB ln K, where K = momentum transfer
BRION S branching ratio (14) for autoionization
BRNCH(L) S quantity (12) for contribution of line L
to dielectronic recombination
BRNCHA S quantity (21) for total contribution to
collisional excitation
BRNCHR S quantity (13) for total contribution to
dielectronic recombination
BRNCHX S collisional-excitation quantity (18), and
partial sums thereof
C FMES coefficient (fk, etc.), multipole, energy,
or eigenvector matrix
CAVE P correction to diagonal coeff. matrix
elements to give zero contribution to Eav
CC,CCC P temporary storage for config.-interaction
matrix elements
CFGP,CFGPT C coefficient of fractional grandparentage
CFGP1 PM coef. of frac. grantparentage (subshell 1)
CFGP2 PM coef. of frac. grantparentage (subshells 2-6)
CFP CPM coef. of fractional parentage (lw)
CFPM1 C coef. of fractional parentage (lw-1)
CFP1 CPM coef. of fractional parentage (subshell 1)
CFP2 CPM coef. of fractional parentage (subshells 2-6)
CIJK P < li||C(k)||lj >
COUPL A BCD label for coupling type (LS,JJ,etc.)
CPURTY(L,J) E purity of configuration J in level L
CS(I,J) B collision strength at Xi for transition array J
CSM B collision strength (modification one)
CSM2 B collision strength (modification two)
CTA B temporary storage for matrix products, etc.
CT4 all temporary storage for matrix products, etc.
C1(I) B collision strength at Xi for specific transition
C2(I) B collision strength at Xi (modification one)
D S line strength (dipole, etc.) matrix, D = S**1/2
EIG ES eigenvalue
EIGP S eigenvalue, second parity ("eigenvalue prime")
ELEM ES BCD name of element and configuration
(from parameter-value card)
FJ(L,I) F Ji for Lth row of coefficient matrix
FJS1 F Ji for Lth row of coefficient matrix (maybe not
literally the first subshell, but rather the
first occupied subshell with 2.le.w.le.2l)
EXC ES excitation energy of target (relative to Eav)
FJT(I),FJTP(I) S Jq,Jq' for Ith spectrum line
FK F quantum number K for LK coupling
FKJ F quantum number K for JK coupling
FK6 F quantum number K for coupling number 6 (LSJLKS)
FL CPF Li (preliminary set of LS quantum numbers)
FLBR C Li bar (in parent term alpha bar, Lbar, Sbar)
FLAM S wavelength of spectrum line (floating-point lambda)
FLL(I) CP li (floating-point little el) for the subshell I
and the parity under consideration
FLLIK(I,K) P li for subshell I and parity K
FLLIJK(I,J,K) P li for subshell I and configuration J of parity K
FNU(I) S wavenumber nu (= sigma) of Ith spectrum line
GA S weighted radiative transition probability gAr
GAA(I) S weighted autoionization transition probability
gAa1 for level of first parity in spectrum line I
GAAP(I) S same, for level of second parity in line I
GAAT(I) S gAa2 for level of first parity in line I
GAAPT(I) S gAa2 for level of second parity in line I
GAAXC(J,I) ES gjAjia for bound level J to continuum level I
(summed over Jl of continuum electron)
GF SB weighted oscillator strength gf
GOSS(I,IT) B weighted generalized oscillator strength for
transition I and momentum transfer IT
IBK SB number of values of momentum transfer K
ICS B number of transition arrays
IC A disk unit number (=41) for coefficient matrices
(and transf. mxs. for cpls. 3-7 and mupole mxs.)
IE ES disk unit number (=30+K) for eigenvalues and
vectors, parity K)
IL F disk unit number (=30+K) for quantum numbers
and LS-JJ transformation matrix, parity K
ISER(M) P serial number of the term of subshell i for the
Mth preliminary basis function
ISER(I) S serial number of level of second parity for
spectrum line I
IPNT(J) C serial number I of parent for cfp of Jth term;
(alphaiLiSi bar|}alphajLjSj)
ITRM(M) C serial number of Ith input term of lw for Mth
truncated term
JEXC(I) E serial number of configuration for target level
with energy EXC(I)
K or KK A K=1 for first parity, K=2 for second parity
KCPL A kind of coupling (1=LS, 2=JJ, etc.)
KCPLD(J) A If KCPLD(J) > 0, transformations to this kind
of coupling are to be deleted
KPAR A kind of parameter (or matrix--see comment cards
in subroutine SPRIN)
LBCD(M,I) P BCD symbol for Li for Mth set of preliminary
quantum numbers [SPDFG, etc.]
LBRBCD C BCD symbol for Lbar [SPDFG, etc.]
LBCDI(M) P L (BCD) for Mth term of lw
LCDLT(L) ES serial no. of dominant configuration for level L
LCDLTP(LP) S serial no. of dominant configuration for level LP
LHS1 FE script L1 (Hollarith=BCD, used for matrix and
eigenvector labeling, subshell 1--maybe not
literally the first subshell, but rather the
first occupied subshell with 2.le.w.le.2l)
LHS4 FE script L (Hollarith=BCD, used for matrix and
eigenvector labeling, for the final subshell)
LHS1J FE script Li (same as LHS1, except for labeling
in JJ representation)
LHS4J FE not used
LHQQ FE script L (BCD label used in coupling number 6)
LL A BCD symbol (spdfg, etc.) analogous to FLL
LLIK A BCD symbol (spdfg, etc.) analogous to FLLIK
LLIJK A BCD symbol (spdfg, etc.) analogous to FLLIJK
MULSI(M) P multiplicity of Mth term of lw
MULS1 FE multiplicity 2(script S1) + 1 for subshell 1
(cf. LHS1)
MULS4 FE multiplicity 2(script S) + 1 for final subshell
(cf. LHS4)
MULS1J FE multiplicity 2(script S1) + 1 for subshell 1
(cf. LHS1J)
MULT(M,I) PF multiplicity 2Si + 1 for Mth set of preliminary
quantum numbers
MULTBR C multiplicity 2 Sbar + 1 for parent term
alpha bar, Lbar, Sbar
MULTQQ F multiplicity used in label for cplg. #6 (cf. LHQQ)
NALS(L,I) F serial number of term aiLiSi for Lth matrix row
(LS coupling)
NALSJI(M) CP serial number of term aiLiSi for Mth term
aiLiSiJi of liwi (JJ coupling)
NALSJ(L,I) F serial number of term aiLiSi for Lth matrix row
(JJ coupling)
NALSJP(M,I) PF serial number of term aiLiSi for Mth set of
preliminary quantum numbers (JJ coupling)
NALSP(M,I) PF serial number of term aiLiSi for Mth set of
preliminary quantum numbers (LS coupling)
NBIGKS SB number of values of momentum transfer BIGKS
NCFG(L) F configuration serial number for matrix row L
NCFGJP(M) PF configuration serial number for Mth set of
preliminary quantum numbers (JJ)
NCFGP(M) PF configuration serial number for Mth set of
preliminary quantum numbers (LS)
NCSER(I) S serial number of dominent configuration,
level of first parity for spectrum line I
NCSERP(I) S same, second parity
NDIFFJ(I) P number of different values of Ji for subshell i
(considering diff. configs. to imply diff. Ji)
NDIFFT CP number of different LS terms (i.e., number of
different possible values of script LqSq,
(considering diff. configs. to imply diff. LqSq)
NDIFSJ F total number of sets of values of {Ji} (all i)
for given script Jq
NENRGS SB number of energies at which to calc. coll. strength
NI(I) CP occupation number wi of subshell i (liwi)
NIJK(I,J,K) A occupation number wi of subshell i for
configuration J of parity K
NIJKP(I,J) P similar to NIJK, for subshell I of configuration J
(used in PRK)
NJJ F number of basis states for given script Jq
(JJ representation)
NJK(K,I) P number of basis states for Kth value of Ji
(see NDIFFJ)
NLASTT(I) P serial number of last term of liwi retained in
setting up basis states
NLS F number of basis states for given script Jq (LS
representation)--hence matrix size for that Jq
NOPC CP number of coefficients (fk, gk, etc.) in
preliminary tables
NOPCCC P serial number of rk coefficient (PRK 755)
NOSUBC A no. of subshells ("subconfigurations")=NSCONF(1,K)
NOTOTJ(I) P total number of basis functions for subshell I
(all configurations)
NTOTJJ(I,J) P total number of basis functions for subshell I,
summed through configuration J-1
NOTOTT P total number of basis functions (all q subshells
and all configurations)
NTOTTJ(J) P total number of basis functions, all subshells,
summed through configuration J-1
NOTSJ1(I,J) P number of LS terms for subshell I,
summed through configuration J-1
NOTSJP M number of LS terms for subshell I,
summed through configuration J-1 (second parity)
NPAR PFE total number of energy parameters (including Eav's)
for current parity
NPARJ(I,J) PE number of single-configuration parameters for
configuration I (=J), or number of configuration-
interaction parameters for interaction I-J
NPARK(K) E value of NPAR for parity K
NPAV(J) E serial number of parameter Eav for Jth config.
NSCONF A see writeup Section IV. A
NSCRJ8 F-S number of different values of script Jq
for current parity
NTI P same as NTRMK
NTRMK(L) CPF number of LS terms with given value of script LqSq
(serial number L) for any given configuration
PC(M) C cfp
PC(M) PF preliminary coefficient (fk, gk, etc.)--M=serial
number--including off-diagonal coefficients
(also used for assorted tempoary storage)
PCI P similar to PC
PJ, PJI P similar to PC, except for zeta coeffs. in JJ repr.
PMUP(I,J) S electric dipole or quadrupole reduced matrix
element,
PSCRL(L,I) P script Li for Lth LS term
PSCRS(L,I) P script Si for Lth LS term
S(L,I) CPF Si for Lth LS term of ith subshell, liwi (cf. FL)
SBR C Si bar for Lth LS term (cf. FLBR)
S2(I) S line strength (except for factor P**2) for Ith line
SCRJ(L,I) F script Ji for Lth row of matrix (JJ representation)
SCRJ8 F script Jq (total J value for given matrix)
SCRJ8P MS script Jq (total J value for given matrix), 2nd parity
SCRL(L,I) F script Li for Lth row of matrix (LS representation)
SCRS(L,I) F script Si for Lth row of matrix (LS representation)
SCRL6 F script L for coupling representation number 6
SCRS6 F script S for coupling representation number 6
SOABSS(I) S cancellation factor (S/|S|) with sign of S**1/2
for spectrum line I
SOPI2(I,J) MS sum of line strengths (divided by P**2) for array I-J
SUMGA S sum of radiative gAr for given energy level
SUMGAA E sum of gAa1 for all levels up to current J value
SUMGAAT E same, for gAa2
SUMGAR S sum of gAr for all levels
T(I) S term (energy level, first parity) for Ith line
TP(I) S term (energy level, second parity) for Ith line
TMX F transformation matrix (LS to JJ, or LS to ...)
TMXP M transformation matrix, second parity
U1,U2,...U6 P matrix elements of U(k) for 1st,2nd,...subshells
V1,V2,...V6 P matrix elements of V(k1) for 1st,2nd,...subshells
V S eigenvector matrix
V E V matrix (derivatives of eigenvalues with respect
to parameter values
VECT E energy matrix; eigenvector matrix
VPAR(I,K) E energy parameter values (Eav,Fk,etc.) for Kth parity
X S temporary storage
XV. Program Usage and Example
The primary storage location for the various programs is an anonymous
FTP directory on the Los Alamos t4 network. To obtain the files using a
Web browser such as Netscape or Internet Explorer, go to
http://www.t4.lanl.gov
and follow links to my programs, or go directly to
ftp://aphysics.lanl.gov/pub/cowan
The FORTRAN files are named rcn.f, rcn2.f, rcg.f, and rce.f . Various
sample input files have fairly obvious names starting with "in". The
files rcng.UNIX and rcng.VMS are procedure files for running rcn, rcn2,
and rcg in succession for a sample test run in five-fold-ionized potassium,
and also contain sample input; output from such a run is given in file
OutputK+5. The files diel.UNIX and diel.VMS are similar procedure files
for making an autoionization/dielectronic-recombination run for Se+24, and
output is contained in file OutputSe+24. Of course, the three programs
can also be run one at a time with appropriate input files, instead of
using these procedure files. The file "readme" contains further
information regarding compilation, file sizes, and execution times.
The file rcg.f uses dimensions adequate for all s, p, and d subshells,
and for all f subshells except f5 to f10, and the file ing11k contains cfp
decks for all such subshells; when renamed ing11, it can be used for a
first RCG run to calculate the binary files tape72, tape73, and tape74.
The file rcglg.f is a source file for RCG with dimensions large enough
to make runs with all f subshells, and cfp decks for calculation of tapes
72-74 for all subshells are contained in the file "cfp". [However, rcglg.f
is based on an older version of rcg.f, so it might best be used only as a
guide to making appropriate dimension changes in the current rcg.f.
Likewise, included 12-subshell versions of rcg and rcn2 are not up to date.]
XVI. Sample Monitor Screen Output (from RCG run for K VI)
Times are for a Macintosh Centris 650 and are about a factor 10 or 20
longer than pertinent for current Power Macs and PCs.
rcg mod 11 ls coupling nsconf= 3 2 2 3 1 1 iabg=0 iv=0 119
finished lncuv at 0.010 min
finished plev at 0.011 min
finished pfgd at 0.015 min
finished prk at 0.017 min
finished calcfc at 0.024 min
finished plev at 0.025 min
finished pfgd at 0.027 min
finished calcfc at 0.035 min
finished mupole matr for j=0.0 and jp=1.0 at time= 0.038 min, matr size= 4 by 3
finished mupole matr for j=1.0 and jp=0.0 at time= 0.039 min, matr size= 2 by 1
finished mupole matr for j=1.0 and jp=1.0 at time= 0.041 min, matr size= 2 by 3
finished mupole matr for j=1.0 and jp=2.0 at time= 0.041 min, matr size= 2 by 4
finished mupole matr for j=2.0 and jp=1.0 at time= 0.042 min, matr size= 4 by 3
finished mupole matr for j=2.0 and jp=2.0 at time= 0.043 min, matr size= 4 by 4
finished mupole matr for j=2.0 and jp=3.0 at time= 0.043 min, matr size= 4 by 3
finished energy for j=0.0 at time= 0.049 min, matrix size= 4
finished energy for j=1.0 at time= 0.051 min, matrix size= 2
finished energy for j=2.0 at time= 0.052 min, matrix size= 4
finished energy for j=0.0 at time= 0.054 min, matrix size= 1
finished energy for j=1.0 at time= 0.055 min, matrix size= 3
finished energy for j=2.0 at time= 0.057 min, matrix size= 4
finished energy for j=3.0 at time= 0.058 min, matrix size= 3
finished energy for j=4.0 at time= 0.059 min, matrix size= 1
finished lower-level sort at time= 0.069 min
finished upper-level sort at time= 0.077 min
finished wavelength sort at time= 0.084 min
time= 0.084 min (abs time= 121.658)
for pmax= 1.70752, 10.0000 10.00000
sums2,sumgf= 10.0000 10.00000 22.3585
sumf= 7.3938 sumgar= 9.6958E+11
s2min= 0.00000
0 lines omitted, with max s2= 0.00000
0 lines omitted, with lambda.gt. 500000.0000 Angstroms
0 lines omitted, with lambda.lt. 0.0010 "
0 lines omitted, insuff storage
0 lines omitted, conf serial nos .gt. 50, 50
STOP (normal exit)
Appendix--Notes on basis-state and level designations
Some details regarding basis-state labeling and level designations
will be given, using an example from singly ionized argon, containing
two configurations of odd parity and five configurations of even parity.
At the beginning of the output from subroutine spectr, there is
a list of radial dipole integrals ( nl//r1// n'l'), together with
the names of the corresponding configurations and the serial numbers
of the configurations:
serial no. conf. serial no. conf. dipole integral
1 Ar II 3p5 1 Ar II 3p4 4s ( 3p//r1// 4s)=
1 Ar II 3p5 2 Ar II 3p4 5s ( 3p//r1// 5s)=
1 Ar II 3p5 3 Ar II 3p4 3d ( 3p//r1// 3d)=
1 Ar II 3p5 4 Ar II 3p4 4d ( 3p//r1// 4d)=
1 Ar II 3p5 5 Ar II 3p4 5d ( 3p//r1// 5d)=
2 Ar II 3p4 4p 1 Ar II 3p4 4s ( 4p//r1// 4s)=
2 Ar II 3p4 4p 2 Ar II 3p4 5s ( 4p//r1// 5s)=
2 Ar II 3p4 4p 3 Ar II 3p4 3d ( 4p//r1// 3d)=
2 Ar II 3p4 4p 4 Ar II 3p4 4d ( 4p//r1// 4d)=
2 Ar II 3p4 4p 5 Ar II 3p4 5d ( 4p//r1// 5d)=
Then a bit later, the list of (the first five) configurations is
repeated, together with configuration serial number:
serial no. conf. conf.
1 Ar II 3p5 --- Ar II 3p4 4s
2 Ar II 3p4 4p --- Ar II 3p4 5s
3 --- Ar II 3p4 3d
4 --- Ar II 3p4 4d
5 --- Ar II 3p4 5d
There follows the line list itself, of which I give a few of the lines below:
0 E J Conf Ep Jp Confp delta e
1 153.4484 2.5 2 (3P) 4P 131.5956 3.5 3 (3P) 4D 21.8528
2 155.2788 3.5 2 (3P) 4D 131.5956 3.5 3 (3P) 4D 23.6832
3 155.7404 2.5 2 (3P) 4D 131.5956 3.5 3 (3P) 4D 24.1448
8 -0.4039 1.5 1 (2P) 2P 131.7381 2.5 3 (3P) 4D 132.1420
9 153.4484 2.5 2 (3P) 4P 131.7381 2.5 3 (3P) 4D 21.7103
61 -0.4039 1.5 1 (2P) 2P 132.7190 2.5 1 (3P) 4P 133.1228
62 153.4484 2.5 2 (3P) 4P 132.7190 2.5 1 (3P) 4P 20.7295
174 172.7695 1.5 2 (1D) 2P 141.4790 2.5 3 (3P) 4F 31.2905
175 192.6821 1.5 2 (1S) 2P 141.4790 2.5 3 (3P) 4F 51.2031
For each line, the first column is a line serial number, which has no
physical significance--it is given only so that one can see how many lines
there are. The next two columns give the energy and total J value for the
level of first parity (odd parity, in this example), and the next column
is the configuration serial number--1 means 3p5 and 2 means 3p4 4p. Now
comes the complicated part. As an example, in (3P) 4P, the 3 and 4 should
be read as superscripts; the part in () being the term of 3p4, and the
final part, 4P, being the overall LS term--that is, the total multiplicity
2S+1 (=4) and the total L (P=one unit of angular momentum).
The part (3P) is in the present example always the multiplicity and
angular momentum of the first subshell of the configuration; for 3p4, the
only possible values are 3P, 1D, and 1S, as in lines 62, 174, and 175.
For configuration 1 (3p5), there is only one subshell, and so the
quantity in () is always the same as the total LS.
For cases in which there are more than two subshells, the quantity in
() is the LS for the outermost (last) subshell that contains at least two
electrons and at least two holes; if there is no such subshell, then the
LS in () is the total LS (vectorially) for the next-to-last subshell.
[The computer code involved is that in subroutine calcfc, statements
350 to 420 (or to 460 for jj coupling).]
The above information for the first-parity level is followed by the
corresponding information for the second-parity level of the spectrum line,
and thus includes the serial number of the second-parity configuration.
Clearly, the amount of information given in the line list, limited by
space considerations to only one intermediate LS (for a single subshell,
or for the vector sum of all but the final subshell), is not generally
sufficient to specify the level designation completely. In such cases,
the only way of obtaining complete information is the following:
(1) On the g5inp control card for RCN2 or on the RCG control card, a
value of ICFC=1 in column 69 will result in all details of the basis-state
quantum numbers for each state being written to the file outg11.
(2) For the eigenvalue and J value of interest, one goes to the
output of eigenvalues and eigenvectors for this J value. The desired
eigenvector lies below the desired eigenvalue, in the same column; the
configuration serial number and (abbreviated) level designation lie in
this same column in between the eigenvalue and eigenvector.
(3) In the eigenvector, find the eigenvector component that corresponds
to the term label in the line list--this is usually (but not always) the
component with the largest magnitude, but in any case, the configuration
serial number and name and the basis-state label are given in the left-hand
portion of the eigenvector-component row. Note the row number of this
eigenvector component.
(4) Going back in the file outg11 to the place where basis-state
quantum numbers are listed for the J value in question, the desired full
set of quantum numbers is that for the row number just mentioned.
[Note, however, that for a given energy level, the correct row number will
in general not be the same for the jj-representation eigenvector as it
will be for the LS-representation eigenvector, because the dominant
eigenvector component will in general lie in a different row.]
Two levels may sometimes appear to have the same level designation;
this is not supposed to happen, but the computer code may not be infallible.
However, such apparent duplication may be because the levels actually
belong to two different configurations, or may result because of the missing
parts of the level designations in the case of more than two subshells.