For transition probabilities the orbitals of the initial and final state need not be orthogonal. A biorthogonal transformation is used for the evaluation of matrix elements. For Breit-Pauli calculations, all operators may be included, spin-orbit, spin-other orbit, spin-spin, and orbit-orbit. In addition to transition rates of all types, isotope shifts and hyperfine constants can be computed as well as g_J factors.
set -x # compute the LSJ transitions using biotr. echo "computing *.j files" # set n to desired value (this script will work for up to n=7) n=4 #copy the files for initial and final state cp E4.c I.c cp O4.c F.c # the script computes for Z=8,9 but provided that the wavefunctions and # .j have already been computed for other Z's the line below can be modified # to reflect different desired Z's for Z in 8 9 do (echo # copy the wave function files from ../E1 into I.w cp ../E1/E1.${Z}_${n}.w I.w #copy the *.j file cp E1.${Z}.j I.j #copy the final wave function from O1 into F.w cp ../O1/O1.${Z}_${n}.w F.w cp O1.${Z}.j F.j # use input file in_biotr which is universal and it accepts as # file names for initial and final states as I (needs I.w, I.c I.j) # and final state F (needs F.w, F.c, F.j) ${ATSP}/bin/biotr<in_biotr # the LSJ transitions between groups E1 and O1 will be saved into # a file with a unique name, note that since Breit-Pauli is # performed only for the most accurate calculation and $n does # not need to be incorporated into the *.lsj file name. mv I.F.lsj E1.O1.${Z}.lsj echo) done # remove not needed files rm ?.? fort*
After completion biotr will leave the following *.lsj files in the directory LSJ.
-rw-r--r-- 1 georgio georgio 11738 Aug 19 19:15 LSJ/E1.O1.8.lsj -rw-r--r-- 1 georgio georgio 11738 Aug 19 19:15 LSJ/E1.O1.9.lsj
The script starts with copying the appropriate files and running biotr. The input file for biotr is shown below:
E1 # initial state O1 # final state n # do not give a full printout (only for debugging) y # relativistic calculation E1 # transition
The transitions between each two groups are saved in a .lsj file which contains essential data per transition:
1 -74.35652224 2s(2).2p(3)2P1_2P 1 -73.64006121 2s.2p(4)1S0_2S 157239.63 CM-1 635.97 ANGS(VAC) 635.97 ANGS(AIR) E1 S = 4.87252D-01 GF = 2.32723D-01 AKI = 1.91900D+09 4.71411D-01 2.25157D-01 1.85662D+09The first two lines are blank, the next line is the initial state including the J value, the energy and the label. The same information for the final state is shown on the next line. The first line for the transition data is computed in the length form, whereas the second is the velocity form. The Breit-Pauli methods have not modified the transition operator for the lowest order relativistic corrections in the velocity form. These are not important for the allowed transitions, but are important in spin-forbidden transitions. Generally, the accuracy of a transition depends on the accuracy of the length and velocity form in the non-relativistic approximation, and the accuracy of the Breit-Pauli transition energy, with the normalized length form value preferred. For intercombination transition, accuracy also depends on other factors, such as the accuracy of the separation of the two terms important for the transition.