PROGRAM RCE MOD 20
LEAST-SQUARES FITTING OF ATOMIC ENERGY LEVELS
Robert D. Cowan
Los Alamos National Laboratory
February 1978
(revised August l986, January 1991, and August 1993)
CONTENTS
I. Introduction 2
II. Source Programs 2
III. Input (TAPE2E) 3
IV. Input (INE20) 4
V. Dimensions 8
VI. Running RCG and RCE (single parity) 8
VII. Output 8
VIII. Running RCN, RCN2, RCG, and RCE (single parity) 9
IX. Formats for ELEMID and CONFIG 10
X. Units 10
XI. Rerunning RCE or RCG (output file PARVALS) 11
XII. Running both parities together 11
XIII. RCE output files LEVELS1, LEVELS2, and LEVELS3 12
XIV. Notes regarding convergence problems 13
XV. Notes regarding Tables I and II 15
I. Introduction
RCE is a FORTRAN 77 program for CRAY-1 or CYBER 205 or
similar computers. Similarly to RCN, RCN2, and RCG, it has been
adapted to run also on VAXs, SUNs, IBM RISCs, and Macintosh
Centris computers, and should be easily adaptable also to Apollos,
Hewlett-Packards, PCs, etc.
The basic purpose of RCE is to adjust the values of various
theoretical parameters so as to produce computed atomic energy
levels in the best possible (least-squares-wise) agreement with
experimentally known level values. Levels whose energies are not
known experimentally can be omitted from consideration. Any of
the parameters can be held fixed at specified values, or groups of
parameters can be forced to vary in such a way that the ratios of
the values within a group remain fixed relative to each other.
The fitting process is carried out by an automatic iterative
procedure until the parameter values no longer change from one
iteration cycle to the next (by more than 0.03), or for a
specified maximum number of cycles. The iteration can be carried
out in any one of the seven angular-momentum coupling schemes
available in program RCG; final eigenvectors are printed in this
representation, and also in either the LS or JJ representation
(see Sec. VII).
The basic theory involved is discussed in R. D. Cowan, The
Theory of Atomic Structure and Spectra (University of California
Press, Berkeley, 1981), especially Sections 16-3 to 16-5,
hereinafter referred to as TASS.
II. Source Programs
The program assumes that the coefficient matrices for each
value of J and for each parameter are contained in a binary input
file TAPE2E, precomputed by program RCG. The experimental levels
(term values) T, together with (optional) estimated parameter
values and various control information must be provided as a BCD
input file INE20 (read internally as unit 10); this file can,
except for the experimental level values, be obtained in basic
form as output from RCG (a file named OUTGINE--short for
OUTrcgINrce).
The following brief discussion of the function of each
subroutine provides a rough outline of the basic calculational
procedure.
MAIN Handles all input (including, for each J, transfer of
the [LS-JJ or other] transformation matrix from TAPE2 to TAPE5,
and transfer of coefficient matrices from TAPE2 to TAPE4). It
also handles the iteration control and most output.
EIGEN Called by MAIN. Reads coefficient matrices from
TAPE4, calculates the energy matrix (for each J) for current
values of the parameters, and calls subroutine TRED2 to
diagonalize the energy matrix and thereby obtain computed energy
levels
ORDER Called by EIGEN (and also by MAIN) to arrange the
computed eigenvalues and vectors in the order of largest
eigenvector component (i.e., so as to make the eigenvector matrix
such that the eigenvector components of largest magnitude tend to
lie on the diagonal); vectors not having clearly dominant
components are arranged so that the corresponding eigenvalues lie
in the same numerical order as do the remaining energy levels. [A
test number CRIT can be chosen to be zero, so that no vector will
be considered to have a dominant component. Then all eigenvalues
will be correlated with experimental level values in numerical
order.]
CALCX Called by MAIN to modify the parameter values X so as
to improve the agreement between eigenvalues (W) and energy levels (T).
LSS Linear system solver called by CALCX.
TRED2/TQL2 Suite of matrix-diagonalization programs.
WR Program to compact a two-dimensional array (dimensions
IA x IA) containing a matrix of size IM (IM.LE.IA) into a one-
dimensional array of size IM*IM or IM*(IM+1)/2.
RD Program to accomplish the inverse of WR.
CLOCK Program to obtain CPU time in minutes (for
information only). This program contains sections appropriate for
a CRAY, VAX or Macintosh, SUN, or IBM RISC; a section appropriate
to another type of computer can be added, or the output time
simply set to zero.
OUT8 Writes on TAPE8 (external file name LEVELS1) a summary
of the final results: experimental level, eigenvalue, Lande g-
value, dominant configuration, and the percentage composition of
the level (three largest components if an input constant CRIT1 is
zero, or a unique component plus the two largest remaining
components if CRIT1 is greater than zero) in each of two coupling
representations. Levels are arranged in the order of increasing J
value, and for given J in order of increasing energy.
SORT8TO7 and SORT Sort the information on TAPE8 in the
order of increasing eigenvalues, and write it on the OUTPUT file
and on a file TAPE7 (external file name LEVELS2). Also, sort the
information on TAPE7 in the order of dominant configuration, and
write it on a file TAPE12 (external file name LEVELS3).
SORT2/ORDERI Sort numbers in one table in numerically
increasing order, and similarly rearrange numbers in other tables.
III. Input (TAPE2E)
Details of the TAPE2E binary file will not be given here,
but can be deduced from subroutine RCEINP of program RCG, where
this file is written, or from RCE MAIN, where the file is read.
Briefly, this file contains (for each of one or more "CSETS"
--i. e., for each of one or more sets of electron configurations
of interest):
(1) A record including a serial number for the CSET, the
number of values of J, the total number of parameters, the number
of configurations, and the number of parameters for each
configuration and for each pair of interacting configurations.
For each J-value:
(2) The value of J, the matrix size, the transformation
matrix from the primary to the secondary coupling representation,
the serial number of the configuration for each matrix row, and
brief basis-state labels in both of the representations.
(3) The energy-coefficient matrix for each parameter, and
the Lande g-value matrix, all in the primary representation.
IV. Input (INE20)
The input file INE20 (internally read as unit 10) must
contain the following information. [Input items (2) through (10)
may be repeated as many times as desired.]
(1) NOCYPR, NOCYCR, NOCYCE, I216, IW6 (Format 5I5).
The least-squares iteration will proceed for at most NOCYCE
cycles, output being printed every NOCYPR cycles, with input
levels being reordered (into the same numerical order as the
computed eigenvalues) every NOCYCR cycles. [If NOCYCR ³ NOCYCE,
this reordering will never take place]. If I216 2 (normal value
is I216=0 or blank), then I216 is set to 16. If IW6 < 0, then
abbreviated output will be sent to the monitor screen to show the
real-time progress of the RCE calculation. (If the input file
INE20 is prepared by using the output file OUTGINE of an RCG run,
then the value of IW6 will be the same as that in columns 73-74 of
the RCG control card; if this control card has not been changed
from that prepared by RCN2 in the file OUT2ING, then IW6 will be
the same as the value in columns 71-75 of the RCN control card.)
(2) NCDES, IDENOM, IPRNV, IPRNA, WMAX, IPRNX, IPRNSQ, S
(Format 4I5, F10.5, 2I5, F10.5).
(a) If NCDES < 0, this was originally a signal to exit
from RCE. Now this card is simply ignored, and exit is
accomplished instead by an EOF test.
(b) If NCDES = 0, use CSET (on files 4 and 5) obtained
from file 2 on the preceding least-squares calculation. Also, if
IDENOM < 0, set IDENOM=0; if IDENOM=0, use previous values of IDENOM
and DENOM; if IDENOM > 0, read new set of DENOM [item (3) below].
(c) If NCDES > 0, find desired CSET (serial number
NOCSET = NCDES) from file 2 and transfer to files 4 and 5.
(d) If IPRNV.NE.0, do not print D, V, B matrices [vkl =
¶wk/¶Xl; B = VV; D = V*(T-W); equation to be solved is B*(DX) = D;
see TASS, reference above].
(e) If IPRNA.NE.0, do not print A (energy) matrices.
(f) If IPRNX.NE.0, do not print new parameter values
between calls of CALCX and EIGEN.
(g) Do not print results for levels with eigenvalue
greater than WMAX. (If zero is read in, WMAX is set to 1010).
(h) If IPRNSQ 0, do not print the squares of the
eigenvector components.
(i) The scale factor S affects only results on files
LEVELS1, LEVELS2, and LEVELS3 [Sec. XIII].
[At this point, one record of tape 2 is read to provide MMAX
= number of matrices (J values), LMAX = number of parameters, and
NMAX = number of configurations].
(3) If on card (2) NCDES=0 and IDENOM > 0, or if NCDES > 0
and IDENOM.NE.0, then
Read (DENOM(L), L=1,LMAX), (Format 7F10.5).
These are denominators to convert from Fk to Fk , etc; the
latter are printed (if IDENOM.NE.0) after the least-squares iteration
has been completed. [In practice this option is seldom used, so
that IDENOM is made zero on card (2), and the card(s) (3)
omitted.]
(4) One or more cards containing element identification and
configuration name for each of the configurations:
Read ELIDEN, (CONFIG(I), I=1,NMAX) [Format 7(A8,2X)].
(5) Read (XIDEN(L), L=1,LMAX)
These are parameter names [the format is 7(A6,A4)--i.e., A6,A4 for
each parameter, seven parameters per card].
(6) For each J-matrix (of size IM, read earlier from
TAPE2):
(6a) Read experimental level values (term values)
(T(M), M=1,IM) (Format 7F10.5)
(6b) Read level flags
(NF(M), M=1,IM) (Format 7I10)
The order of the term values may be in increasing numerical value,
or may be such that their nominal designations are arranged in the
same order as the matrix basis-state labels in program RCG for the
primary representation being used in RCE [see under item (10)
below]. The value of NF(M)=M if the Mth level is to be included
in the least-squares fit, or NF(M)=-M if it is to be excluded. If
in item (6a) the last value of M on a full card is less than the
matrix size IM, and a positive number TEMP (format F10.5) is
punched in columns 71-80, then no more values of T (nor NF) will
be read for this J; instead, the value TEMP will be used for all
remaining term values of this J, and the corresponding values of
NF(M) will be set to -M. [Actually, all term values are first
read into an array W(M), and then these are moved to the T(M)
array via T(M) = W(|NF(M)|), so that using values of |NF(M)| other
than M provides a means of rearranging the term values that is
easier than retyping the term values in a revised order, if such
should prove to be necessary].
(7) Read parameter flags
(LF(L), L=1,LMAX) (Format 7I10)
If |LF(L)| = 100, the Lth parameter is held fixed. If LF(L) < -50,
the value of the Lth parameter is allowed to go negative
(physically permissible only for the configuration-interaction
parameters Rk, or for effective-operator parameters such as a, §,
g). In least-squares mode minus one (LSQM=-1), parameters to be
linked together (ratios of their values held fixed) are to be
given equal negative values of LF between -1 and -98 inclusive.
Successive values -1 or -51, -2 or -52, -3 or -53, ... must be
used (though not necessarily in that order) if RCE is to correctly
compute the effective total number of free parameters. [The value
-99 can be used for a negative parameter not linked to others.
After the code has determined those parameters that are to be
allowed to go negative (-51.GE.LF.GE.-99), it changes each -99 to 0,
adds 50 to all values -51.GE.LF.GE.-98 (see the appended notes to
Tables I and II), and takes the number of parameters with LF=0
minus the most negative value of LF from -1 to -50 as the total
number of free parameters.]
(8) Read (XMAX(L), L=1,LMAX) (Format 7F10.5)
The value of parameter X(L) is restricted to the range 0 to XMAX
[for LF(L) = -50], or to the range -XMAX to +XMAX [for LF(L) < -50];
if XMAX=0 is read, a default value of 1010 is used.
(9) Read initial estimates of the parameter values
(X(L), L=1,LMAX) (Format 7F10.5)
(10) A control card containing the following:
Variable Format Normal Value
-------- ------ ------------------------
X(L) 0 X(L) = 0
----------- -----------
F0 F10.5 1.0 0.2
FM F10.0 0.0 0.0
F1 F10.5 -1.0 -1.0
DELF F5.3 0.0 0.2
NOFMAX I5 1000 2 or 3
GCOE F10.3 1.0 0.8
DXMAX E10.3 10.0 1.0 to 5.0
CRIT1 F5.3 0.3 0.3
CRIT F5.3 0 (or 0.85) 0 (or 0.85)
CRIT2 F2.0 -1.0 -1.0
The significance of the control-card quantities is the following
(except see Sec. XIII regarding CRIT1):
If all of the initial parameter estimates X(L) are zero
(except for those held fixed by specifying |LF(L)| = 100), then the
iteration is started by setting the eigenvectors equal
approximately to the basis vectors (1000...), (0100...), etc.
(See TASS, page 470). To give energy matrices with eigenvectors
that initially are approximately basis vectors, the off-diagonal
matrix elements are multiplied by a fudge factor F, which is
initially small (F = F0) but is gradually increased to its proper
value of unity, with NOFMAX cycles being run at each value of F.
The value of F is increased in the following manner: Starting
with F = F0, then F is multiplied successively by a factor FM until
F.GE.F1, then increased by an additive amount DELF until F.GE.1, at
which point F is set to unity and NOFMAX to 1000. (As indicated
by the suggested value of F1=-1 in the table above, the multiplier
stage can usually be omitted).
Normally, a calculation will be started using non-zero
estimates of the parameters X(L) from an RCG run, and the
calculation will be begun using F0=1.
In either case, the code attempts to increase the speed of
convergence by letting the parameter values change by a factor G
times the predicted changes. Initially G=1, but its value is
automatically recomputed each cycle (G < 1 if the convergence is
oscillatory, G > 1 if it is monotonic). The degree by which G is
allowed to depart from unity is proportional to GCOE, but G is
always kept small enough that no X(L) is allowed to change on any
one iteration cycle by more than 2*DXMAX. [The units of X and
DXMAX are the same as those of the input term values T(M); the
values of DXMAX listed in the table assume units of 1000 cm-1.]
On each iteration cycle, the computed eigenvalues are
arranged in the order of the dominant component of the
corresponding eigenvectors--a dominant component cd being one such
that
(ci)**2 < CRIT*(cd)**2
for all components i.NE.d. Eigenvalues for which the vectors have no
dominant component are arranged in the same order as the
"unassigned" term values. [It is therefore necessary that unknown
term values, deleted from the least-squares fit via negative
values of NF, nevertheless be given values of appropriate
magnitude as best they can be estimated.] Alternatively, CRIT can
be set to zero, NOCYCR made equal to or greater than NOCYCE, and
all input term values (including estimated values for unknown
levels) arranged in increasing numerical order; then all
eigenvalues will likewise be arranged in numerical order (the same
as in RCG) in all iteration cycles.
If a linked-parameter calculation has been made (LSQM=-1,
defined by reading in two or more equal values of LF(L) lying in
the range -51 to -98), then a second calculation with all
parameters independently free will automatically be made (LSQM=0)
unless CRIT2 < +1.
The rather complicated changes that the code makes in values
of XMAX, XMIN, and LF at various stages of the calculation are
given in Tables I and II. [Ignore the case LSQM= +1, which is not
included in RCE Mod 20.]
V. Dimensions
The present version of RCE has been compiled with array
dimensions that limit it to use on problems involving at most 20
configurations, 280 parameters, and 12 matrices (J-values), each
no bigger than 75 x 75 in size; the total number of levels of all J
must be no greater than 4*75=300. The dimensions can be readily
increased via the parameter-statement variables IC, IB, IX, IA,
and IV, respectively. These dimensions and the array sizes
actually required for the current run are printed near the
beginning of the output file; if any one of the array sizes
exceeds the corresponding dimension, the program will
automatically stop. (It is possible that the program may bomb on
an exceeded dimension before the printout occurs.)
VI. Running RCG and RCE (single parity)
The program RCG must be run in order to produce the RCE
binary input file TAPE2E. The procedure will be illustrated by
supposing we are going to want to make a least-squares fit of
levels of Cu XIV 3s 3p5 + 3s2 3p3 3d. The RCG input file shown in
the upper portion of Table III can be easily constructed by hand.
[The value -1 for NOCSET has been assumed.]
Execution of RCG produces not only TAPE2E, but also a file
named OUTGINE (written internally as TAPE11) shown in the bottom
portion of Table III. This file may if desired be used as a
template for constructing the RCE input file. It is necessary to
replace the zero values of T(M) by the known experimental values,
and to make NF(M) negative for unknown levels; also, to insert
estimated values of X(L) if desired, together with any appropriate
non-zero values of LF(L), to insert identifying names for the
element and configurations (of which only the first fifteen will
appear on the printed output file OUTE20), and to make any other
desired changes (in NOCYCR, IW6, XIDEN, XMAX, etc).
In inserting the values of T(M), these should (for each
value of J) be arranged in numerically increasing order (assuming
CRIT=0.0). Alternatively, they can be arranged with their
designations in the order of the basis states given in the
quantum-number-listing portion of the RCG output (for the primary
representation being used in RCE), provided CRIT is changed to
about 0.85 and NOCYCR made equal to or less than NOCYCE, as one
judges to be appropriate (depending on the confidence one has in
the empirical-analysis level designations)--see the discussion
regarding CRIT near the end of Section IV.
VII. Output
The representation in which RCE will carry out the least-
squares fitting process is determined by the information on the
RCG control card.
If, as on the first card in the upper portion of Table III,
KCPL (column 5) is one and NLSMAX (columns 23-25) is zero, then
the coefficient matrices that RCG writes on TAPE2 will be in the
LS representation, and RCE will therefore use LS as its primary
representation; eigenvectors will be printed each iteration cycle
on the output file OUTE20 in the LS representation, and on the
final cycle will be printed also in the secondary representation,
JJ.
If KCPL=2 and NLSMAX=0, then in RCE the primary
representation will be JJ and the secondary will be LS.
If NLSMAX > 0 and if the KCPLD(I), 3.LE.I.LE.7, in columns
33-37 are non-zero for all but one of these values, say I=M, then
RCG will write coefficient matrices on TAPE2E in representation M;
the primary representation in RCE will be M and the secondary
representation will be LS or JJ according as KCPL is 1 or 2,
respectively.
[Obviously the transformation matrix mentioned under item
III(2) above is the LS-JJ, JJ-LS, M-LS, or M-JJ matrix in the
above four respective cases.]
On the final RCE iteration cycle there are printed not only
eigenvectors, but if IPRNSQ=0 then also "vectors" consisting of
the squares of the eigenvector components (for each of the two
representations).
Along with each vector there is also printed (in a
horizontal format) not only the corresponding eigenvalue W and
experimental term value T (appended with an asterisk if omitted
from the fitting process), but also T-W, the Lande g-value G, and
a quantity GKK. This last quantity is the diagonal element of the
Lande g-matrix, and hence would be the g-value if the coupling
were pure in the representation in question. [In all cases, the
values of GKK are printed in the order of the basis states for the
representation in question, and will be roughly equal to the
intermediate-coupling g-value G in the same output line only if
the eigenvectors have been sorted in the order of dominant
component (via CRIT=~0.85 rather than zero) and if the eigenvectors
are fairly close to pure basis vectors.]
VIII. Running RCN, RCN2, RCG, and RCE (single parity)
An input file for RCE containing appropriate starting
estimates for the parameters X(L) can be obtained by running RCG
using as input an output file from RCN2 rather than one
constructed by hand.
Appropriate input files for RCN and RCN2 are shown in the
upper two sections of Table IV. Note that on the G5INP input card
for RCN2, a non-zero value (namely 1) has been used for NOCSET in
column 10, and an estimated value of Eav (=450 kK) for the first
configuration (3s 3p5) has been used for EAV11 in columns 14-20.
Inclusion of the non-zero value of NOCSET causes RCN2 to
include a negative number (-|NOCSET|, = -1 in this case) in columns
9-10 of the first card of the OUT2ING output file, and also a
"-55555555." card before the "-99999999." card. This file is
ready to be used with little or no change as input to RCG.
[Values of NLSMAX and KCPLD can be set on the RCN2 control card
for automatic carry-through to the RCG control card, but a value
KCPL=2 can be obtained only by hand alteration after RCN2 has been
executed.]
The output file OUTGINE from RCG is the same as that in
Table III, except that labels are now obtained for the element and
configurations, the eigenvalues computed by RCG appear for both
the initial theoretical values W and for the term values T, and
the (scaled HF) parameters used in computing said eigenvalues
appear as the starting parameter values X(L).
If this file were used unchanged as input to RCE,
convergence should of course be obtained more-or-less immediately,
with essentially no change in the parameter values.
Normally, of course, one will make changes in the same way
as described in Section VI, except that there is now no need to
insert element and configuration names nor estimates for the
parameters X(L).
IX. Formats for ELEMID and CONFIG
When an input file for RCE is to be produced by an
RCN/RCN2/RCG run as just described, some care should be exercised
in the typing of the RCN input cards: the element and spectrum
identification (e.g., CuXIV in the Sec. VIII example) should be
restricted to columns 11-16 of the RCN input card, and the
configuration identification should be limited to columns 17-24
(preferably 17-22). This will ensure that the element and configura-
tion identifications in both RCG and RCE will turn out satisfactorily.
If RCE output (Sec. XIII) is to be used as input to program
INTENSITY [a special-purpose program not of general interest],
then the configuration identifications should be right-adjusted in
columns 17-22, because only the information in columns 20-22 will
be included in the INTENSITY output files.
X. Units
When starting estimates of the parameter values X(L) are
obtained as described in Section VIII, the units of the X(L)
produced by RCN2/RCG are normally kK (thousands of cm-1). Thus
one either has to insert the experimental term values in these
same units, or (via an editor) he has to convert the X(L) to the
units he uses for the term values. Alternatively, he can add a
rescaling card (see RCG writeup, page 36) at the front of the RCG
input file. To obtain the X(L) in cm-1, for example, this card
would contain blanks in columns 1-6 except a zero in column 5, and
1.0 in columns 51-60. [However, if values of X(L) written to the
RCE output file PARVALS are used as input back into RCG, in the
manner described below, the units printed on the RCG output file
("PARAMETER VALUES IN xxxx CM-1"), together with oscillator
strengths and transition probabilities, will be incorrect if
anything other than kK was used throughout.]
XI. Rerunning RCE or RCG (output file PARVALS)
Because of setup errors of one sort or another, RCE may fail
to converge properly, and it may have to be rerun with altered
control variables (or simply for additional iteration cycles). To
facilitate setting up starting parameter values for the rerun, RCE
writes on an output file PARVALS (internally called TAPE11) the
parameter values for each cycle; values from the desired cycle can
be copied into the RCE input file to replace the starting values
used on the previous RCE run.
When final least-squares parameter values have been
obtained, one may wish to make an RCG run to compute oscillator
strengths and transition probabilities. To facilitate this, the
final parameter values are added to the end of the RCE PARVALS
output file in the format required for RCG input, so that these
can be edited into an RCG input file in place of the previously
used values. An example of this PARVALS file is given in Table
VIII.
*************IMPORTANT**************
If least-squares parameters for both parities are to be put back
into RCG as mentioned above, then it is highly advisable to produce
the required TAPE2E input file for RCE via two-parity RCN/RCN2/RCG runs
(the CSET files for both parities then being both in a single TAPE2E,
rather than in two separate ones). This is because RCN2 will probably
set up the configuration subshells differently in a two-parity run than
in two separate one-parity runs; in the latter case, Rydberg-series
configuration-interaction-coefficient and R0-parameter modifications
may be made as discussed in my book on page 394, whereas these
modifications may not be made in the two-parity case. It is still
permissible to make the least-squares fittings one parity at a time
(by breaking the file outgine into two separate files)--the single two-
parity file TAPE2E serves without change as input for both parities.
************************************
XII. Running both parities together
The examples of Sections VI-VIII were for only a single
parity. However, it is permissible to run RCG/RCE or
RCN/RCN2/RCG/RCE for both parities together. An example for
Si III 3s2 + 3p2 + 3s 3d - 3s 3p + 3p 3d
is shown in Tables V and VI. On the G5INP control card of the
RCN2 input file in Table V, an estimated value EAV11=4.0 has been
used in order to allow for downward perturbation of the 3s2
configuration by 3p2 and 3s3d. (As the result E(3s2 1S) = -1.8308
of the RCG run in Table VI shows, a better estimate would have
been EAV11 =~ 6.0.) Also on the G5INP card, NLSMAX has been set to
250, and all KCPLD(I) for I > 3 have been made non-zero, so that the
primary representation in RCE will be JK; see Sec. VII. (This is
for illustrative purposes only, as the secondary representation LS
is much closer to the physical coupling conditions.)
In Table VII the OUTGINE output file from RCG (Table VI) has
been modified by hand in that the RCG eigenvalues have been
replaced by the corresponding experimental energy levels. In
lines 2 and 23, the scale factor S has been changed from 1 to 1000
so as to give energies in cm-1 in output files LEVELS1, LEVELS2,
and LEVELS3. Also, in line 15 of the file, the value LF(9) = -52
has been changed to -51 so as to force the two configuration-
interaction parameter values to retain a fixed ratio; this has
been done in order to fit the ten even-parity levels with only
eight parameters instead of nine and make convergence more stable.
[In the odd-parity half of the OUTGINE file, the LF values
for the two CI parameters are both equal to -51 already in Table
VI. This is the result of the way RCG has been coded: RCG sets
LF = -51 for all CI parameters involving configuration number one;
LF = -52 for all interactions of configuration 2 with configurations
3, 4, 5, ...; etc. This arbitrary scheme has been employed in
order to keep the total number of parameters from becoming much
greater than the total number of levels when there are many
interacting configurations, but it can of course be altered by
hand.]
The file in Table VII can be divided in half following the
first card containing "-1" in columns 4-5, and RCE run separately
for the two parities. However, the file can (as in the present
example) be used intact to make a single RCE run for both
parities, one after the other. [Note that the "-1" card in the
middle of the input file, formerly the NCDES < 0 exit card (see Sec.
IV(2)(a) on page 4), need not be removed.] The resulting RCE
output file PARVALS (see Sec. XI) is shown in Table VIII.
XIII. RCE output files LEVELS1, LEVELS2, and LEVELS3
As noted at the end of Section II, RCE produces output files
LEVELS1 and LEVELS2 (internally, TAPE8 and TAPE7), shown for the
Si III example in Tables IX and X; also an output file LEVELS3
(internal name TAPE12), which is not shown. The energy units for
T, W, and T-W are the same as in the output file OUTE20, except
multiplied by the scale factor S read from the input card IV(2);
thus, with units of kK used in the main body of the code, units of
cm-1 have been obtained here as the result of setting S=1000 (see
Sec. XII above).
For each level, the percentage composition is listed for the
three largest eigenvector components (first in the primary
representation JK, and then in the secondary representation LS),
provided CRIT1 = 0.0. (If CRIT1 is greater than zero, the program
chooses the first component so that no two levels have a first
component belonging to the same basis state; in cases of strong
basis-state mixing, the first component for some levels may be
rather small. The remaining two printed components for a given
level are then the two largest ones other than the first printed
component.) Also listed is the dominant electron configuration,
determined by summing over all basis states of each configuration;
note that this dominant configuration is not necessarily the same
as the configuration to which the leading eigenvector component
belongs.
Unfortunately, the quantum-number designations for the three
eigenvector components are frequently incomplete, except in the
case of two-subshell configurations in the LS representation.
Complete basis-state designations for the LS and JJ representa-
tions can be obtained from the quantum-number-listing section of
the RCG output file; for other representations, they can only be
inferred from the coding details in RCG subroutine CPL37.
Correlation of complete basis-state designations with the
eigenvector-component labels in the LEVELS1-LEVELS3 files can, in
ambiguous cases, be obtained only by working back via the
eigenvalues to the complete eigenvector printout in the main RCE
output file OUTE20.
It should be noted that the LEVELS1-LEVELS3 output
corresponds to the final cycle of the RCE run. If, through some
quirk of instability in the RCE iteration, the final cycle is no
good and one wishes results for an earlier cycle, one can make an
RCE rerun with NOCYCE=0, using input parameters obtained from the
appropriate section of the PARVALS file. This will likewise
provide (on the new PARVALS file) the proper set of parameter
values for making an RCG rerun (Sec. XI).
XIV. Notes regarding convergence problems
Obtaining satisfactory convergence of the least-squares
level fitting can be a tricky business requiring several reruns,
especially when several interacting configurations are present.
Suggested items to watch out for are the following:
(1) The number of free parameters (counting each set of
linked parameters as one, and excluding parameters held fixed via
|LF(L)| = 100) must of course be no greater than the number of
levels being fit (not counting levels for which NF < 0); in complex
multi-configuration cases, the former number should preferably be
considerably less than the latter.
(2) When coupling conditions are close to pure LS coupling,
the number of free Coulomb parameters (including Eav) should be no
greater than the number of LS terms having levels included in the
fit.
(3) If the number of parameters needs to be reduced in
order to keep the iteration from wandering aimlessly, the prime
candidates for parameters to be held fixed or linked to other
parameters are those that oscillate erratically from one cycle to the
next, or ones that show large values of the standard deviation SDX on
the OUTE20 file (especially a value that is large compared with
the value of the parameter itself). Such parameters are likely to be
spin-orbit parameters for excited electrons (especially of large l),
or CI parameters that produce only minor perturbations.
(3a) In configurations like 2p3p or 3d4d, only the sum
of the two spin-orbit parameters zi is well determined, and it is
usually necessary to fix the z for the outer electron or to link
the two z's together.
(3b) In configurations ln where levels are known for
only one LS term, or configurations lnl2 where levels based on
only one ln parent are known, then all Fk(ll) must be held fixed,
or all Fk(ll) linked together and Eav held fixed. [This is just a
special case, for each configuration or for each parent, of item
(2) above.] Violation of such requirements will usually lead to
very wild parameter-value changes and huge values of SDX!
(4) In some cases, one may wish to include a configuration
for which no levels are known experimentally--because, for
example, this configuration is suspected of significantly
perturbing known levels. One approach is to include the RCG
theoretical levels in the fit, as though they were experimental
ones, and leave the associated parameter values free (to
compensate for shifts in eigenvalues of other configurations,
which then result in changes in CI effects on the configuration in
question). However, probably a better approach is to delete the
unknown levels from the fit, and to hold fixed at their
theoretical values all parameters that involve the unknown
configuration. In either case, it is of course important that one
properly correct for any systematic (Eav) error in the unknown
configuration, as indicated by similar errors (for known
configurations) in the RCG calculation that provided the starting
values of the X(L). Otherwise, computed CI perturbations will be
inaccurate, and perhaps even in the wrong direction. The
situation is even worse when some, but not all, levels are known.
If the theoretical values for the unknown levels are not properly
corrected, then they may (for example) lie below known levels of
other configurations when they should lie above. RCE will then
probably make incorrect correlations of computed eigenvalues with
input term values, throwing off the path of the parameter-value
iteration, even though the unknown levels have nominally been
omitted from the fitting process; this is particularly likely to
happen if correlation is done solely by numerical order of term
value and eigenvalue (CRIT = 0.0) rather than by dominant
eigenvector component (CRIT @ 0.85).
(5) As noted in Sec. XII, RCG initially links together the
various CI parameters in groups as follows: all parameters
involving configuration 1 are in the first group (LF = -51), all
remaining parameters involving configuration 2 are in the second
group (LF = -52), all remaining parameters involving configuration 3
are in the third group (LF = -53), etc. On early RCE runs in
complicated cases it is probably best to link all CI parameters
into a single group. On later runs one may then experiment with
combinations of greater flexibility involving more than one group.
Notes regarding Tables I and II
XMAX(L) and XMIN(L)
On cards (8), page 6 above, quantities XMAX(L) may be read
in, zeroes being replaced by 10**10. Quantities XMIN(L) are
calculated by the code, and the parameters restricted during the
least-squares-fitting iteration to the range
XMIN(L).le.X(L).le.XMAX(L) .
The quantities XMAX are determined by
( max[X(L), XMAX(L)], XMAX(L).NE.0
XMAX(L) = (
( 10**10 , XMAX(L) = 0
and the quantities XMIN by
( max[10**-6, X(L)-(XMAX(L)-X(L))], LF(L) > -50
XMIN(L) = (
( -XMAX(L) , LF(L) < -49
These relations, plus changes in LF and in XMAX and XMIN when F
reaches unity, are summarized in Table I.
Alterations in LF before each type of least-squares
calculation, and the effect of LF(L) on X(L), are summarized in
Table II. Values of LF(L) are printed out each cycle so that one
can tell which LSQM is being used, which parameters are linked or
held fixed, etc.
TABLE I
Change in LF Changes in LF, XMAX,
Input value after setting XMIN when F reaches 1
of LF(L) Initial values of XMAX(L), XMIN(L) XMAX, XMIN (LSQM = 0 or +1 only)
----------- -------------------------------------- ------------- ------------------------------------
XMAX =0 XMAX.ne.0 XMAX=10**10 XMAX.lt.10**10
--------- -------------------- ------------ ---------------------
/ / /
LF.ge.100 _| XMAX=10**10 _| XMAX=max(X,XMAX) none none _| LF=0 (LSQM=0)
| XMIN=10**-6 | XMIN=max(10**-6,2*X-XMAX) | LF=1 or 2 (LSQM=1)
\ \ \
\ /
0.lt.LF.lt.100 " " LF=0 | | XMAX=10**10
|> <| If XMIN.lt.0, set XMIN=-10**10
0 " " none | | If XMIN.gt.0, set XMIN=10**-6
/ \
-50.lt.LF.lt.0 " " none " "
/ /
-98.le.LF.le.-50 _| XMAX=10**10 _| XMAX=max(X,MMAX) LF=LF+50 " "
| XMIN=-10**10 | XMIN=-XMAX
\ \
LF=-99 " " LF=0 " "
/
LF.le.-100 " " none none _| LF=0 (LSQM=0)
| LF=1 or 2 (LSQM=1)
\
TABLE II
Value of LF
before first Changes before Changes before Changes before
least squares LSQM=-1 Effect of LF(L) LSQM=0 LSQM=+1 Effect of LF(L)
calculation calculation on X(L) calculation calculation on X(L)
------------- ---------------------------- --------------- -------------- -------------- ---------------
XMAX=10**10 XMAX.lt.10**10
------------ --------------
LF.ge.100 none same as LF=0 hold fixed none none hold fixed
/ /
| Change to successively more | LF=2 Fit gross structure
LF=0 <| neg. values than those already " none <| or
| in the range -1 to -49 | LF=1 Fit fine structure
\ \
/
| Change X(L) with
-50.lt.LF.lt.0 none none <| equal LF by a LF=0 (LF has been
| common factor changed to zero.)
\
LF.le.-100 none same as LF=0 hold fixed none none hold fixed
TABLE III. Input for, and Monitor-screen and OUTGINE files from, RCG
1 -13 2 20 0 00 01 00 1000.0000 0.00 01-6 0 0
s 1 p 5 d 0
s 2 p 3 d 1
-55555555.
-99999999.
-1
rcg mod 11 ls coupling nsconf= 3 2 2 0 0 0 iabg=0 iv=0 119 010000000 99.0 99.0 print=00000 500 1 0 0 0
finished lncuv at 0.005 min
finished plev at 0.006 min
finished pfgd at 0.013 min
finished prk at 0.016 min
finished calcfc at 0.039 min
start rceinp at time= 0.041 min
end calc of matrix 1 of 6; time= 0.053, size= 5, npars= 10
end calc of matrix 2 of 6; time= 0.060, size= 11, npars= 10
end calc of matrix 3 of 6; time= 0.067, size= 11, npars= 10
end calc of matrix 4 of 6; time= 0.074, size= 9, npars= 10
end calc of matrix 5 of 6; time= 0.081, size= 5, npars= 10
end calc of matrix 6 of 6; time= 0.088, size= 1, npars= 10
cset number 1 written on tape 2 at time= 0.089 for couplings ls and jj
npars,noic,noicmup,nord= 10 0 0 0 0 0
STOP (normal exit)
1 30 5 -6 * NOCYPR, etc.
1 0 1 1 0.000 1 1 1.0000 * NCDES, etc.
* ELEMID & confs.
eav zeta 2 g1(12) eav f2(22) zeta 2 zeta 3 ** XID
f2(23) g1(23) g3(23) 1 21d2213 **
0.0000 0.0000 0.0000 0.0000 0.0000 * T * J=0
1 2 3 4 5 * NF * "
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 * J=1
0.0000 0.0000 0.0000 0.0000 * "
1 2 3 4 5 6 7 * "
8 9 10 11 * "
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 * J=2
0.0000 0.0000 0.0000 0.0000 * "
1 2 3 4 5 6 7 * "
8 9 10 11 * "
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 * J=3
0.0000 0.0000 * "
1 2 3 4 5 6 7 * "
8 9 * "
0.0000 0.0000 0.0000 0.0000 0.0000 * J=4
1 2 3 4 5 * "
0.0000 * J=5
1 "
0 0 0 0 0 0 0 * LF
0 0 0 -51 * ""
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 * XMAX
0.0000 0.0000 0.0000 0.0000 * "
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 * X
0.0000 0.0000 0.0000 0.0000 * "
1.0 0.0 -1.0 0.00 1000 1.000 10000.00 0.30 0.00-1 * F0, etc.
-1 * exit card
TABLE IV. Input for RCN and RCN2, output file OUT2ING from RCN2
(which is also the input file ING11 for RCG), and
output file OUTGINE from RCG
2 -5 2 10 1.0 5.e-08 1.e-11-2 090 1.0 0.65 0.0 0.00 -6
29 14CuXIV 3s 3p5 3s 3p5
29 14CuXIV 3p3 3d 3s2 3p3 3d
-1
g5inp 1000 450.00 00 009099909090 0.00 01229
-1
1 -13 2 20 0 00 00 00 1000.0000 0.00 01-6 0 0
s 1 p 5 d 0 s 0 s 0 s 0 s 0 s 0 CuXIV 3s 3p5 -345915.172 0.0000
s 2 p 3 d 1 s 0 s 0 s 0 s 0 s 0 CuXIV 3p3 3d -345783.197 0.0000
CuXIV 3s 3p5 3 45000000 1878932 18821734 00 00hf90999090
CuXIV 3p3 3d 7 58197480 14568911 1878402 195472 14486193hf90999090
16454234 10712534
3s 3p5 -3p3 3d 1 172.24795 0.00005 0.00005 0.00005 0.00005hf90999090
-55555555.
-99999999.
-1
1 30 5 -6
1 0 1 1 0.000 1 1 1.0000
CuXIV 3s 3p5 3p3 3d
eav 3s 3p5zeta 2 g1(12) eav 3p3 3df2(22) zeta 2 zeta 3
f2(23) g1(23) g3(23) 1 21d2213
382.3404 477.2642 532.5568 593.5412 678.2602
1 2 3 4 5
371.5288 450.3499 477.7646 518.5799 586.6236 596.5524 645.1191
665.0452 679.0597 699.3160 769.2944
1 2 3 4 5 6 7
8 9 10 11
355.7872 478.1020 508.2362 522.8549 576.1982 598.7570 602.7810
614.2886 659.6079 690.8297 710.5173
1 2 3 4 5 6 7
8 9 10 11
478.7353 515.5883 527.2983 548.7192 599.1607 613.6588 648.7939
678.8232 733.9648
1 2 3 4 5 6 7
8 9
481.0237 534.3001 552.5638 563.9309 604.1965
1 2 3 4 5
556.9239
1
0 0 0 0 0 0 0
0 0 0 -51
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000
450.0000 18.7893 188.2173 581.9748 145.6891 18.7840 1.9547
144.8619 164.5423 107.1253 172.2479
1.0 0.0 -1.0 0.00 1000 1.000 10000.00 0.30 0.00-1
-1
TABLE V. Input for RCN and RCN2, and output file OUT2ING from RCN2
(which is also the input file ING11 for RCG). Note that
on the g5inp and RCG control cards, NLSMAX has been set
to 250 and KCPLD to 0101111, so that in RCE the primary
and secondary coupling representations will be jK and LS.
2 -5 2 10 1.0 5.e-08 1.e-11-2 090 1.0 0.65 0.0 0.00 -6
14 3Si III 3s2 3s2
14 3Si III 3p2 3p2
14 3Si III 3s 3d 3s 3d
14 3Si III 3s 3p 3s 3p
14 3Si III 3p 3d 3p 3d
-1
g5inp 1000 4.0000 250 0101111 008299828282 0.00 01229
-1
1 -13 3 33 2 20 250 0101111 00 1000.0000 0.00 01-6 0 0
s 2 p 0 d 0 s 0 s 0 s 0 s 0 s 0 Si III 3s2 -63429.454 0.0000
s 0 p 2 d 0 s 0 s 0 s 0 s 0 s 0 Si III 3p2 -63301.407 0.0000
s 1 p 0 d 1 s 0 s 0 s 0 s 0 s 0 Si III 3s 3d -63290.803 0.0000
s 1 p 1 d 0 s 0 s 0 s 0 s 0 s 0 Si III 3s 3p -63374.942 0.0000
s 0 p 1 d 1 s 0 s 0 s 0 s 0 s 0 Si III 3p 3d -63223.246 0.0000
Si III 3s2 1 400000 00 00 00 00hf82998282
Si III 3p2 3 13204700 3762581 21402 00 00hf82998282
Si III 3s 3d 3 14265140 282 2625164 00 00hf82998282
3s2 - 3p2 1 51.54765 0.00005 0.00005 0.00005 0.00005hf82998282
3p2 - 3s 3d 1 42.57735 0.00005 0.00005 0.00005 0.00005hf82998282
Si III 3s 3p 3 5851180 21122 5148764 00 00hf82998282
Si III 3p 3d 6 21020840 22972 322 3095933 3864384hf82998282
2334854
3s 3p - 3p 3d 2 44.02455 32.46835 0.00005 0.00005 0.00005hf82998282
Si III 3s2 Si III 3s 3p 2.07471( 3s//r1// 3p)-1.000hf -98-100
Si III 3s2 Si III 3p 3d 0.00000( //r1// )0.0000hf 0 0
Si III 3p2 Si III 3s 3p -2.04899( 3p//r1// 3s)-1.000hf -98-100
Si III 3p2 Si III 3p 3d 3.34581( 3p//r1// 3d)-0.997hf -92 -98
Si III 3s 3d Si III 3s 3p -3.35083( 3d//r1// 3p)-0.997hf -91 -98
Si III 3s 3d Si III 3p 3d 2.00905( 3s//r1// 3p)-1.000hf -98-100
-55555555.
-99999999.
-1
TABLE VI. OUTGINE output file from RCG, using the file
from Table V as input file ING11.
1 30 5 -6
1 0 1 1 0.000 1 1 1.0000
Si III 3s2 3p2 3s 3d
eav 3s2 eav 3p2 f2(22) zeta 2 eav 3s 3dzeta 3 g2(13)
1 21d1122 2 31d2213
-1.8295 127.3148 155.9400
1 2 3
127.4249 140.0220
1 2
120.4735 127.6408 140.0248 166.6136
1 2 3 4
140.0290
1
0 0 0 0 0 0 0
-51 -52
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000
4.0000 132.0470 37.6258 0.2140 142.6514 0.0028 26.2516
51.5476 42.5773
1.0 0.0 -1.0 0.00 1000 1.000 10000.00 0.30 0.00-1
-1
1 30 5 -6
2 0 1 1 0.000 1 1 1.0000
Si III 3s 3p 3p 3d
eav 3s 3pzeta 2 g1(12) eav 3p 3dzeta 2 zeta 3 f2(23)
g1(23) g3(23) 1 21d1223 1 22e1223
48.8961 212.3303
1 2
49.0002 78.2922 212.2660 213.2683 234.5227
1 2 3 4 5
49.2105 199.6473 201.8703 212.1539 213.3161
1 2 3 4 5
199.7703 213.3622 231.2989
1 2 3
199.9290
1
0 0 0 0 0 0 0
0 0 -51 -51
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000
58.5118 0.2112 51.4876 210.2084 0.2297 0.0032 30.9593
38.6438 23.3485 44.0245 32.4683
1.0 0.0 -1.0 0.00 1000 1.000 10000.00 0.30 0.00-1
-1
[Note: When this file is used without change as input file INE20 to RCE,
appreciable changes in the values of Eav(3s2) and R1(3s2,3p2)
will be seen in the first-parity calculation, because these two
parameters are the only ones of importance for the two 1S0 levels,
and effects of their changes can offset each other; thus these
parameter changes do not indicate any error in the RCE code.
No similar peculiarities will be seen in the second parity.]
TABLE VII. File of Table VI, as modified to be input file INE20 to
RCE, including replacement of theoretical energy levels
by experimental values, change of scale factor S from
1.0 to 1000.0 (so as to give energies in cm-1 instead
of kK in output files LEVELS1, LEVELS2, and LEVELS3),
and change of LF(9) in the even parity from -52 to -51
(so as to link together the two CI parameters).
1 30 5 -6
1 0 1 1 0.000 1 1 1000.0
Si III 3s2 3p2 3s 3d
eav 3s2 eav 3p2 f2(22) zeta 2 eav 3s 3dzeta 3 g2(13)
1 21d1122 2 31d2213
0.0000 129.7085 153.4442
1 2 3
129.8420 142.9483
1 2
122.2145 130.1005 142.9458 165.7650
1 2 3 4
142.9437
1
0 0 0 0 0 0 0
-51 -51
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000
4.0000 132.0470 37.6258 0.2140 142.6514 0.0028 26.2516
51.5476 42.5773
1.0 0.0 -1.0 0.00 1000 1.000 10000.00 0.30 0.00-1
-1
1 30 5 -6
2 0 1 1 0.000 1 1 1000.0
Si III 3s 3p 3p 3d
eav 3s 3pzeta 2 g1(12) eav 3p 3dzeta 2 zeta 3 f2(23)
g1(23) g3(23) 1 21d1223 1 22e1223
52.7247 216.3503
1 2
52.8533 82.8844 216.2887 217.3858 228.6998
1 2 3 4 5
53.1150 198.9232 205.0291 216.1902 217.4399
1 2 3 4 5
199.0265 217.4895 235.4139
1 2 3
199.1641
1
0 0 0 0 0 0 0
0 0 -51 -51
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000
58.5118 0.2112 51.4876 210.2084 0.2297 0.0032 30.9593
38.6438 23.3485 44.0245 32.4683
1.0 0.0 -1.0 0.00 1000 1.000 10000.00 0.30 0.00-1
-1
TABLE VIII. RCE output file PARVALS produced by the input file
INE20 of Table VII.
Si III 3s2 3p2 3s 3d
parameter values, cycle number 0, avdev= 2.378
4.0000 132.0470 37.6258 0.2140 142.6514 0.0028 26.2516
51.5476 42.5773
parameter values, cycle number 1, avdev= 0.003
5.2934 133.6052 30.2933 0.2565 144.8944 0.0000 19.4906
48.5343 40.0884
parameter values, cycle number 2, avdev= 0.001
5.2996 133.6039 30.2821 0.2562 144.8950 0.0000 19.4971
48.5306 40.0853
parameter values, cycle number 3, avdev= 0.001
5.2996 133.6039 30.2821 0.2562 144.8950 0.0000 19.4971
48.5306 40.0853
parameter values for rcg input
Si III 3s2 5.2996
Si III 3p2 133.6039 30.2821 0.2562
Si III 3s 3d 144.8950 0.0000 19.4971
3s2 - 3p2 48.5306
3p2 - 3s 3d 40.0853
Si III 3s 3p 3p 3d
parameter values, cycle number 0, avdev= 3.774
58.5118 0.2112 51.4876 210.2084 0.2297 0.0032 30.9593
38.6438 23.3485 44.0245 32.4683
parameter values, cycle number 1, avdev= 0.824
60.9787 0.2621 46.8715 212.4723 0.5776 0.0437 27.8896
43.7723 11.5244 26.8414 19.7956
parameter values, cycle number 2, avdev= 0.775
60.9813 0.2610 46.7864 212.4745 0.6015 0.0519 27.8891
43.7709 11.5902 22.0929 16.2936
parameter values, cycle number 3, avdev= 0.774
60.9802 0.2610 46.7823 212.4746 0.6110 0.0509 27.8939
43.7693 11.5950 21.5831 15.9177
parameter values, cycle number 4, avdev= 0.774
60.9803 0.2610 46.7828 212.4747 0.6098 0.0513 27.8933
43.7694 11.5940 21.5825 15.9172
parameter values, cycle number 5, avdev= 0.774
60.9804 0.2610 46.7829 212.4747 0.6095 0.0515 27.8931
43.7695 11.5937 21.5815 15.9165
parameter values for rcg input
Si III 3s 3p 60.9804 0.2610 46.7829
Si III 3p 3d 212.4747 0.6095 0.0515 27.8931 43.7695 ls99999999
11.5937
3s 3p - 3p 3d 21.5815 15.9165
TABLE IX. RCE output file LEVELS1 produced by the input file of Table VII.
The columns are T, W, T-W (all in cm-1), Lande g-value, dominent
configuration, J-value, and percent contribution of the three
largest eigenvector components, sorted primarily by parity and
secondarily by J-value, in the jK- and LS-coupling representations.
Note that for both parities, LS is much the better representation.
Si III 3s2 3p2 3s 3d
0.000 0. 0.000 0.000 3s2 j= 0.0 96.5 3s2 0.0k 0.0 3.5 3p2 0.0k 0.0 0.0 3p2 0.0k 1.0
129708.500 129709. -0.042 0.000 3p2 j= 0.0 100.0 3p2 0.0k 1.0 0.0 3p2 0.0k 0.0 0.0 3s2 0.0k 0.0
153444.200 153444. 0.000 0.000 3p2 j= 0.0 96.5 3p2 0.0k 0.0 3.5 3s2 0.0k 0.0 0.0 3p2 0.0k 1.0
129842.000 129842. 0.064 1.501 3p2 j= 1.0 100.0 3p2 0.0k 1.0 0.0 3s 3d 0.5k 1.5
142948.300 142945. 2.970 0.499 3s 3d j= 1.0 100.0 3s 3d 0.5k 1.5 0.0 3p2 0.0k 1.0
122214.500 122214. 0.000 1.000 3p2 j= 2.0 65.5 3p2 0.0k 2.0 20.7 3s 3d 0.5k 2.5 13.8 3s 3d 0.5k 1.5
130100.500 130101. -0.022 1.501 3p2 j= 2.0 100.0 3p2 0.0k 1.0 0.0 3p2 0.0k 2.0 0.0 3s 3d 0.5k 2.5
142945.800 142945. 0.469 1.167 3s 3d j= 2.0 60.0 3s 3d 0.5k 1.5 40.0 3s 3d 0.5k 2.5 0.0 3p2 0.0k 2.0
165765.000 165765. 0.000 1.000 3s 3d j= 2.0 39.3 3s 3d 0.5k 2.5 34.5 3p2 0.0k 2.0 26.2 3s 3d 0.5k 1.5
142943.700 142945. -1.633 1.334 3s 3d j= 3.0 100.0 3s 3d (2s) 3d
Si III 3s2 3p2 3s 3d
0.000 0. 0.000 0.000 3s2 j= 0.0 96.5 3s2 (1s) 1s 3.5 3p2 (1s) 1s 0.0 3p2 (3p) 3p
129708.500 129709. -0.042 0.000 3p2 j= 0.0 100.0 3p2 (3p) 3p 0.0 3p2 (1s) 1s 0.0 3s2 (1s) 1s
153444.200 153444. 0.000 0.000 3p2 j= 0.0 96.5 3p2 (1s) 1s 3.5 3s2 (1s) 1s 0.0 3p2 (3p) 3p
129842.000 129842. 0.064 0.000 3p2 j= 1.0 100.0 3p2 (3p) 3p 0.0 3s 3d (2s) 3d
142948.300 142945. 2.970 0.000 3s 3d j= 1.0 100.0 3s 3d (2s) 3d 0.0 3p2 (3p) 3p
122214.500 122214. 0.000 0.000 3p2 j= 2.0 65.5 3p2 (1d) 1d 34.5 3s 3d (2s) 1d 0.0 3p2 (3p) 3p
130100.500 130101. -0.022 0.000 3p2 j= 2.0 100.0 3p2 (3p) 3p 0.0 3s 3d (2s) 1d 0.0 3p2 (1d) 1d
142945.800 142945. 0.469 0.000 3s 3d j= 2.0 100.0 3s 3d (2s) 3d 0.0 3p2 (1d) 1d 0.0 3s 3d (2s) 1d
165765.000 165765. 0.000 0.000 3s 3d j= 2.0 65.5 3s 3d (2s) 1d 34.5 3p2 (1d) 1d 0.0 3p2 (3p) 3p
142943.700 142945. -1.633 0.000 3s 3d j= 3.0 100.0 3s 3d (2s) 3d
Si III 3s 3p 3p 3d
52724.700 52725. -0.293 0.000 3s 3p j= 0.0 99.9 3s 3p 0.5k 0.5 0.1 3p 3d 1.5k 0.5
216350.300 215994. 356.552 0.000 3p 3d j= 0.0 99.9 3p 3d 1.5k 0.5 0.1 3s 3p 0.5k 0.5
52853.300 52854. -0.741 1.501 3s 3p j= 1.0 66.0 3s 3p 0.5k 0.5 33.9 3s 3p 0.5k 1.5 0.1 3p 3d 1.5k 0.5
82884.400 82882. 2.054 1.000 3s 3p j= 1.0 65.4 3s 3p 0.5k 1.5 33.6 3s 3p 0.5k 0.5 0.3 3p 3d 0.5k 1.5
216288.700 215843. 445.209 1.485 3p 3d j= 1.0 63.3 3p 3d 1.5k 0.5 25.6 3p 3d 0.5k 1.5 11.0 3p 3d 1.5k 1.5
217385.800 217871. -485.429 0.515 3p 3d j= 1.0 53.7 3p 3d 1.5k 1.5 44.8 3p 3d 0.5k 1.5 1.5 3p 3d 1.5k 0.5
228699.800 228878. -178.072 1.000 3p 3d j= 1.0 29.2 3p 3d 0.5k 1.5 35.0 3p 3d 1.5k 1.5 34.8 3p 3d 1.5k 0.5
53115.000 53116. -0.533 1.501 3s 3p j= 2.0 99.9 3s 3p 0.5k 1.5 0.1 3p 3d 0.5k 1.5 0.1 3p 3d 1.5k 1.5
198923.200 199325. -401.450 0.669 3p 3d j= 2.0 84.1 3p 3d 0.5k 2.5 15.6 3p 3d 1.5k 2.5 0.2 3p 3d 0.5k 1.5
205029.100 202574. 2454.821 0.997 3p 3d j= 2.0 52.2 3p 3d 1.5k 2.5 21.7 3p 3d 0.5k 1.5 18.6 3p 3d 1.5k 1.5
216190.200 215617. 572.853 1.495 3p 3d j= 2.0 58.1 3p 3d 0.5k 1.5 40.7 3p 3d 1.5k 1.5 0.9 3p 3d 1.5k 2.5
217439.900 218045. -605.465 1.173 3p 3d j= 2.0 40.5 3p 3d 1.5k 1.5 31.3 3p 3d 1.5k 2.5 20.0 3p 3d 0.5k 1.5
199026.500 199709. -682.209 1.084 3p 3d j= 3.0 46.0 3p 3d 0.5k 2.5 42.0 3p 3d 1.5k 3.5 12.0 3p 3d 1.5k 2.5
217489.500 218215. -725.599 1.334 3p 3d j= 3.0 77.8 3p 3d 1.5k 2.5 22.2 3p 3d 0.5k 2.5 0.0 3p 3d 1.5k 3.5
235413.900 235140. 274.071 1.000 3p 3d j= 3.0 57.9 3p 3d 1.5k 3.5 31.9 3p 3d 0.5k 2.5 10.2 3p 3d 1.5k 2.5
199164.100 200190. -1025.768 1.251 3p 3d j= 4.0 100.0 3p 3d (2p) 3f
Si III 3s 3p 3p 3d
52724.700 52725. -0.293 0.000 3s 3p j= 0.0 99.9 3s 3p (2s) 3p 0.1 3p 3d (2p) 3p
216350.300 215994. 356.552 0.000 3p 3d j= 0.0 99.9 3p 3d (2p) 3p 0.1 3s 3p (2s) 3p
52853.300 52854. -0.741 0.000 3s 3p j= 1.0 99.9 3s 3p (2s) 3p 0.1 3p 3d (2p) 3p 0.0 3s 3p (2s) 1p
82884.400 82882. 2.054 0.000 3s 3p j= 1.0 99.0 3s 3p (2s) 1p 1.0 3p 3d (2p) 1p 0.0 3s 3p (2s) 3p
216288.700 215843. 445.209 0.000 3p 3d j= 1.0 98.3 3p 3d (2p) 3p 1.5 3p 3d (2p) 3d 0.1 3s 3p (2s) 3p
217385.800 217871. -485.429 0.000 3p 3d j= 1.0 98.3 3p 3d (2p) 3d 1.6 3p 3d (2p) 3p 0.1 3p 3d (2p) 1p
228699.800 228878. -178.072 0.000 3p 3d j= 1.0 98.8 3p 3d (2p) 1p 1.0 3s 3p (2s) 1p 0.1 3p 3d (2p) 3d
53115.000 53116. -0.533 0.000 3s 3p j= 2.0 99.9 3s 3p (2s) 3p 0.1 3p 3d (2p) 3p 0.0 3p 3d (2p) 1d
198923.200 199325. -401.450 0.000 3p 3d j= 2.0 99.0 3p 3d (2p) 3f 1.0 3p 3d (2p) 1d 0.0 3p 3d (2p) 3d
205029.100 202574. 2454.821 0.000 3p 3d j= 2.0 99.0 3p 3d (2p) 1d 1.0 3p 3d (2p) 3f 0.1 3p 3d (2p) 3p
216190.200 215617. 572.853 0.000 3p 3d j= 2.0 98.0 3p 3d (2p) 3p 1.8 3p 3d (2p) 3d 0.1 3s 3p (2s) 3p
217439.900 218045. -605.465 0.000 3p 3d j= 2.0 98.2 3p 3d (2p) 3d 1.8 3p 3d (2p) 3p 0.0 3p 3d (2p) 3f
199026.500 199709. -682.209 0.000 3p 3d j= 3.0 100.0 3p 3d (2p) 3f 0.0 3p 3d (2p) 3d 0.0 3p 3d (2p) 1f
217489.500 218215. -725.599 0.000 3p 3d j= 3.0 100.0 3p 3d (2p) 3d 0.0 3p 3d (2p) 1f 0.0 3p 3d (2p) 3f
235413.900 235140. 274.071 0.000 3p 3d j= 3.0 100.0 3p 3d (2p) 1f 0.0 3p 3d (2p) 3d 0.0 3p 3d (2p) 3f
199164.100 200190. -1025.768 0.000 3p 3d j= 4.0 100.0 3p 3d (2p) 3f
TABLE X. RCE output file LEVELS2 produced by the input file of Table VII,
sorted primarily by parity and secondarily by energy, for both
the jK- and LS-coupling representations.
Si III 3s2 3p2 3s 3d
0.000 0. 0.000 0.000 3s2 j= 0.0 96.5 3s2 0.0k 0.0 3.5 3p2 0.0k 0.0 0.0 3p2 0.0k 1.0
122214.500 122214. 0.000 1.000 3p2 j= 2.0 65.5 3p2 0.0k 2.0 20.7 3s 3d 0.5k 2.5 13.8 3s 3d 0.5k 1.5
129708.500 129709. -0.042 0.000 3p2 j= 0.0 100.0 3p2 0.0k 1.0 0.0 3p2 0.0k 0.0 0.0 3s2 0.0k 0.0
129842.000 129842. 0.064 1.501 3p2 j= 1.0 100.0 3p2 0.0k 1.0 0.0 3s 3d 0.5k 1.5
130100.500 130101. -0.022 1.501 3p2 j= 2.0 100.0 3p2 0.0k 1.0 0.0 3p2 0.0k 2.0 0.0 3s 3d 0.5k 2.5
142948.300 142945. 2.970 0.499 3s 3d j= 1.0 100.0 3s 3d 0.5k 1.5 0.0 3p2 0.0k 1.0
142945.800 142945. 0.469 1.167 3s 3d j= 2.0 60.0 3s 3d 0.5k 1.5 40.0 3s 3d 0.5k 2.5 0.0 3p2 0.0k 2.0
142943.700 142945. -1.633 1.334 3s 3d j= 3.0 100.0 3s 3d (2s) 3d
153444.200 153444. 0.000 0.000 3p2 j= 0.0 96.5 3p2 0.0k 0.0 3.5 3s2 0.0k 0.0 0.0 3p2 0.0k 1.0
165765.000 165765. 0.000 1.000 3s 3d j= 2.0 39.3 3s 3d 0.5k 2.5 34.5 3p2 0.0k 2.0 26.2 3s 3d 0.5k 1.5
Si III 3s2 3p2 3s 3d
0.000 0. 0.000 0.000 3s2 j= 0.0 96.5 3s2 (1s) 1s 3.5 3p2 (1s) 1s 0.0 3p2 (3p) 3p
122214.500 122214. 0.000 0.000 3p2 j= 2.0 65.5 3p2 (1d) 1d 34.5 3s 3d (2s) 1d 0.0 3p2 (3p) 3p
129708.500 129709. -0.042 0.000 3p2 j= 0.0 100.0 3p2 (3p) 3p 0.0 3p2 (1s) 1s 0.0 3s2 (1s) 1s
129842.000 129842. 0.064 0.000 3p2 j= 1.0 100.0 3p2 (3p) 3p 0.0 3s 3d (2s) 3d
130100.500 130101. -0.022 0.000 3p2 j= 2.0 100.0 3p2 (3p) 3p 0.0 3s 3d (2s) 1d 0.0 3p2 (1d) 1d
142948.300 142945. 2.970 0.000 3s 3d j= 1.0 100.0 3s 3d (2s) 3d 0.0 3p2 (3p) 3p
142945.800 142945. 0.469 0.000 3s 3d j= 2.0 100.0 3s 3d (2s) 3d 0.0 3p2 (1d) 1d 0.0 3s 3d (2s) 1d
142943.700 142945. -1.633 0.000 3s 3d j= 3.0 100.0 3s 3d (2s) 3d
153444.200 153444. 0.000 0.000 3p2 j= 0.0 96.5 3p2 (1s) 1s 3.5 3s2 (1s) 1s 0.0 3p2 (3p) 3p
165765.000 165765. 0.000 0.000 3s 3d j= 2.0 65.5 3s 3d (2s) 1d 34.5 3p2 (1d) 1d 0.0 3p2 (3p) 3p
Si III 3s 3p 3p 3d
52724.700 52725. -0.293 0.000 3s 3p j= 0.0 99.9 3s 3p 0.5k 0.5 0.1 3p 3d 1.5k 0.5
52853.300 52854. -0.741 1.501 3s 3p j= 1.0 66.0 3s 3p 0.5k 0.5 33.9 3s 3p 0.5k 1.5 0.1 3p 3d 1.5k 0.5
53115.000 53116. -0.533 1.501 3s 3p j= 2.0 99.9 3s 3p 0.5k 1.5 0.1 3p 3d 0.5k 1.5 0.1 3p 3d 1.5k 1.5
82884.400 82882. 2.054 1.000 3s 3p j= 1.0 65.4 3s 3p 0.5k 1.5 33.6 3s 3p 0.5k 0.5 0.3 3p 3d 0.5k 1.5
198923.200 199325. -401.450 0.669 3p 3d j= 2.0 84.1 3p 3d 0.5k 2.5 15.6 3p 3d 1.5k 2.5 0.2 3p 3d 0.5k 1.5
199026.500 199709. -682.209 1.084 3p 3d j= 3.0 46.0 3p 3d 0.5k 2.5 42.0 3p 3d 1.5k 3.5 12.0 3p 3d 1.5k 2.5
199164.100 200190. -1025.768 1.251 3p 3d j= 4.0 100.0 3p 3d (2p) 3f
205029.100 202574. 2454.821 0.997 3p 3d j= 2.0 52.2 3p 3d 1.5k 2.5 21.7 3p 3d 0.5k 1.5 18.6 3p 3d 1.5k 1.5
216190.200 215617. 572.853 1.495 3p 3d j= 2.0 58.1 3p 3d 0.5k 1.5 40.7 3p 3d 1.5k 1.5 0.9 3p 3d 1.5k 2.5
216288.700 215843. 445.209 1.485 3p 3d j= 1.0 63.3 3p 3d 1.5k 0.5 25.6 3p 3d 0.5k 1.5 11.0 3p 3d 1.5k 1.5
216350.300 215994. 356.552 0.000 3p 3d j= 0.0 99.9 3p 3d 1.5k 0.5 0.1 3s 3p 0.5k 0.5
217385.800 217871. -485.429 0.515 3p 3d j= 1.0 53.7 3p 3d 1.5k 1.5 44.8 3p 3d 0.5k 1.5 1.5 3p 3d 1.5k 0.5
217439.900 218045. -605.465 1.173 3p 3d j= 2.0 40.5 3p 3d 1.5k 1.5 31.3 3p 3d 1.5k 2.5 20.0 3p 3d 0.5k 1.5
217489.500 218215. -725.599 1.334 3p 3d j= 3.0 77.8 3p 3d 1.5k 2.5 22.2 3p 3d 0.5k 2.5 0.0 3p 3d 1.5k 3.5
228699.800 228878. -178.072 1.000 3p 3d j= 1.0 29.2 3p 3d 0.5k 1.5 35.0 3p 3d 1.5k 1.5 34.8 3p 3d 1.5k 0.5
235413.900 235140. 274.071 1.000 3p 3d j= 3.0 57.9 3p 3d 1.5k 3.5 31.9 3p 3d 0.5k 2.5 10.2 3p 3d 1.5k 2.5
Si III 3s 3p 3p 3d
52724.700 52725. -0.293 0.000 3s 3p j= 0.0 99.9 3s 3p (2s) 3p 0.1 3p 3d (2p) 3p
52853.300 52854. -0.741 0.000 3s 3p j= 1.0 99.9 3s 3p (2s) 3p 0.1 3p 3d (2p) 3p 0.0 3s 3p (2s) 1p
53115.000 53116. -0.533 0.000 3s 3p j= 2.0 99.9 3s 3p (2s) 3p 0.1 3p 3d (2p) 3p 0.0 3p 3d (2p) 1d
82884.400 82882. 2.054 0.000 3s 3p j= 1.0 99.0 3s 3p (2s) 1p 1.0 3p 3d (2p) 1p 0.0 3s 3p (2s) 3p
198923.200 199325. -401.450 0.000 3p 3d j= 2.0 99.0 3p 3d (2p) 3f 1.0 3p 3d (2p) 1d 0.0 3p 3d (2p) 3d
199026.500 199709. -682.209 0.000 3p 3d j= 3.0 100.0 3p 3d (2p) 3f 0.0 3p 3d (2p) 3d 0.0 3p 3d (2p) 1f
199164.100 200190. -1025.768 0.000 3p 3d j= 4.0 100.0 3p 3d (2p) 3f
205029.100 202574. 2454.821 0.000 3p 3d j= 2.0 99.0 3p 3d (2p) 1d 1.0 3p 3d (2p) 3f 0.1 3p 3d (2p) 3p
216190.200 215617. 572.853 0.000 3p 3d j= 2.0 98.0 3p 3d (2p) 3p 1.8 3p 3d (2p) 3d 0.1 3s 3p (2s) 3p
216288.700 215843. 445.209 0.000 3p 3d j= 1.0 98.3 3p 3d (2p) 3p 1.5 3p 3d (2p) 3d 0.1 3s 3p (2s) 3p
216350.300 215994. 356.552 0.000 3p 3d j= 0.0 99.9 3p 3d (2p) 3p 0.1 3s 3p (2s) 3p
217385.800 217871. -485.429 0.000 3p 3d j= 1.0 98.3 3p 3d (2p) 3d 1.6 3p 3d (2p) 3p 0.1 3p 3d (2p) 1p
217439.900 218045. -605.465 0.000 3p 3d j= 2.0 98.2 3p 3d (2p) 3d 1.8 3p 3d (2p) 3p 0.0 3p 3d (2p) 3f
217489.500 218215. -725.599 0.000 3p 3d j= 3.0 100.0 3p 3d (2p) 3d 0.0 3p 3d (2p) 1f 0.0 3p 3d (2p) 3f
228699.800 228878. -178.072 0.000 3p 3d j= 1.0 98.8 3p 3d (2p) 1p 1.0 3s 3p (2s) 1p 0.1 3p 3d (2p) 3d
235413.900 235140. 274.071 0.000 3p 3d j= 3.0 100.0 3p 3d (2p) 1f 0.0 3p 3d (2p) 3d 0.0 3p 3d (2p) 3f