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Subsections


BIOTR

Introduction

Given the two sets of configuration lists, wave functions and eigenvectors, biotr evaluates the transition data between the two CI expansions, ${\Psi_1\mid \sum_{i} \hat{O}(i) \mid \Psi_2}$, with ${\hat{O}(i)}$ representing a magnetic or electric transition probability operator of any order.

The calculation is non-relativistic when the eigenvectors were obtained using the mchf method. In this case the program expects to find the eigenvectors in a file with a .l suffix. If the user had applied the relativistic corrections using the Breit-Pauli method, the transitions determined by biotr will include the relativistic effects as well. In this case the program expects to read the eigenvectors from a file with .j suffix. In biotr, the initial and final states, which may not be orthogonal, are transformed in such a way that the orbitals are orthogonal between states so that orthogonal methods may be used to evaluate the transition matrix elements [#!mrg-bio!#].

Program Structure

Data is initialized in initr() and factr(), then the user selects the initial and final state, type of calculation (relativistic, or non-relativistic) and the type of transition E1, E2, M1, M2, .. or *'. The program then proceeds with the following steps:

  1. cfgin2() reads the two sets of configuration lists and determines the orthogonality conditions between the two sets. Next, rasin() checks if the CI expansions used for $\Psi_1$ and $\Psi_2$ satisfy the closure under de-excitation property.

  2. Calculate the CSF-coupling coefficient ${A^{\mu\nu}_{ij}}$ in the expression: ${a^{\dagger}_{i}\hat{a}_{j}\mid\Phi_{nu}\rangle =
\sum_{\nu}{A^{\mu\nu}_{ij}}\mid\Phi_{mu}}$, which are needed to perform the transformation of the CI-coefficients.

  3. Calculate the one-electron orbital overlap matrix and decompose it into block-triangular factors to find the new radial functions ${\{R^A_{\it nl}(r)\}}$ and ${\{R^B_{\it nl}(r)\}}$ which are biorthonormal.

  4. transform the configuration interaction matrix ${C^A_{\mu}}$ and ${C^B_{\mu}}$ by a sequence of single orbital replacements. main routine for employing biorthogonal rotations for RAS type wave functions, allowing the calculation of transition moments between two RAS states. The task of this part of the code is to change two sets of orbitals into biorthogonal orbitals and counter-rotate the CI coefficients. transform the configuration interaction matrix ${C^A_{\mu}}$ and ${C^B_{\mu}}$ by a sequence of single orbital replacements.

  5. Apply the orthogonal Racah algebra to transform the many electron amplitude into a sum of one-electron reduced matrix elements. the l.h.s and r.h.s orbital indices of the se now refer to the two different orbital biorthonormal sets.

Figure 6.35: Program structure.
\begin{figure}\begin{center}
\centerline{\psfig{figure=tex/fig/biotr_main.epsi}}\end{center}\end{figure}

The user is expected to provide information about which two sets of input files should be used and the type of calculation (E1, E2,..., M1, M2, ...). Note in the input below that E1 and O1 supplied by the user for Name of Initial State and Name of Final State, imply the existence of: E1.c E1.w E1.j and O1.c O1.w O1.j, if a relativistic calculation is being selected:

#  ........A parity changing transition calculation........
>biotr 
  Name of Initial State
E1
  Name of Final State
O1
  intermediate printing (y or n) ?
n 
  Relativistic calculation ? (y/n)
y
  Type of transition ? (E1, E2, M1, M2, .. or *)
E1
------------------------------------------------------------------------

The angular data is not saved and therefore, for calculations along an iso-electronic sequence, the angular calculations are repeated for each atom.

File IO

biotr requires two sets of input files: configuration list, wave function, and either non-relativistic eigenvector produced by mchf, or the relativistic counterpart, computed by bp_eiv. Note that when the calculation is nonrelativistic the user computes the LS transitions between the two states for a given set of two terms only. The relativistic calculations computes all LSJ transitions between two states and each state may contain a number of LS terms. The transition properties are saved in files. The filenames of the transition data are comprised by the strings which the user had provided for Initial and Final states, and additionally a .ls suffix is append in the case of non-relativistic calculation, or, .lsj is used for the relativistic case. The first line for the transition data is computed in the length form, whereas the second is the velocity form.

Figure 6.36: biotr IO files.
\begin{figure}\begin{center}
\centerline{\psfig{figure=tex/fig/biotr_io.eps}}\end{center}\end{figure}

Each .ls file contains a number of transition properties including: Atomic weight, principal quantum number, energies of initial and final states, transition energies, wavelength in vacuum, wavelength air, type of transition, line strengths, gf values, transition rates:

Format of an LS transition:
#####
  Transition between files:
  E
  O

 Z =   9 n =  7
   3  -97.50578137  2s(2).2p(3)2P1_2P
   3  -96.52277315  2s.2p(4)3P2_2P
  215739.13 CM-1       463.52 ANGS(VAC)       463.52 ANGS(AIR)
 E1  length:   S =  7.81694D-01   GF =  5.12259D-01   AKI =  2.65057D+09
    velocity:  S =  8.22227D-01   GF =  5.38822D-01   AKI =  2.78801D+09
#####
An LSJ transition:
#####


   1  -74.36649804  2s(2).2p(3)2P1_2P
   1  -73.65565658  2s.2p(4)1S0_2S
  156006.31 CM-1       641.00 ANGS(VAC)       641.00 ANGS(AIR)
 E1  S =  4.69243D-01   GF =  2.22364D-01   AKI =  1.80493D+09
          4.68123D-01         2.21833D-01          1.80062D+09
.....

The convergence of the length and velocity forms are important factor for estimating the accuracy of the model. 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.


next up previous contents
Next: HFS Up: ATSP2K manual Previous: BP_EIV   Contents
2001-10-11