ni2.rac

The file ni2.rac specifies an Hamiltonian ACTOR which includes the atomic hamiltonian (OPER HAMILTONIAN) and the cubic crystal field (OPER SHELL2) acting on the initial state and also on the final state. Additionally a dipole transition ACTOR is specified.


Y
RACER O3
TO OH
ENDCHAIN
  ACTOR 0+ HAMILTONIAN GROUND   PRINTEIG
  OPER  HAMILTONIAN
    BRANCH 0+ > 0 0+   1.00
  OPER SHELL2
    BRANCH 4+ > 0 0+   2.50
  ACTOR 0+ HAMILTONIAN EXCITE   PRINTEIG
  OPER  HAMILTONIAN
    BRANCH 0+ > 0 0+   1.00
  OPER SHELL2
    BRANCH 4+ > 0 0+   2.50 
  ACTOR 1-  DIPOLE   TRANSI    PRINTTRANS
  OPER MULTIPOLE
    BRANCH 1- > 0 1-   1.00

OUTPUT FILE

Y tells the program that you want an output-file ni2.rac_out.

BRANCHINGS

RACER O3
TO OH
ENDCHAIN

The lines starting with the command RACER and ending with ENDCHAIN define the changes in symmetry. The starting symmetry is spherical (O3), from which the symmetry is changed to Cubic (OH).

Instead of RACER also the command BUTLER can be given.

RACER implies tailoring of the inormation to be used in BANDER

BUTLER implies more complete output information (in case of a crystal field multiplet calculation)

The branching from spherical symmetry to tetragonal symmetry as used within a crystal field multiplet calculation is given as:

BUTLER O3
TO OH
TO D4H
ENDCHAIN

ACTOR

The operators to use in the calculation are specified by lines starting with ACTOR and include all following lines until the end of file or another ACTOR is begun.

The ACTOR command is followed by:

a label for the lowest group in the chain that specifies the symmetry of the operator, for example 0+ or 1-.
a string that gives a name to the operator, for example HAMILTONIAN or DIPOLE.
a word that specifies the states it operates on, either GROUND, EXCITE or TRANSI.
and a word that specifies what information should be printed on ni2.rac_out, either PRINTRAW, PRINTEIG or PRINTTRANS.

For example,

ACTOR 0+ HAMILTONIAN GROUND PRINTEIG

print the eigenvalues of the diagonalized matrix of the operator HAMILTONIAN of symmetry 0+ acting on the ground state.

ACTOR: printing options

Within ACTOR different types of print statements can be given.

There are at least 3 print options that are accepted PRINTRAW, PRINTEIG and PRINTTRANS. These may be followed by a number which must be separated from the option by spaces.

PRINTRAW just prints the raw matrix without any processing.
PRINTEIG requests that the matrix be diagonalised and the eigenvalues printed. If the option is followed by a number between 1 and 9 then it will also print the eigenvectors. This option is only appropriate for operators like the Hamiltonian.
PRINTTRANS will print the magnitude of the transition matrix transformed into the eigenbasis (so there must also be a Hamiltonian operator to provide this information). If it is followed by a number then the number of rows of the matrix that is printed is truncated to this value.

OPER

Each ACTOR commands must be followed by one or more lines starting with OPER. The label that follows the OPER command must be the name of a reduced matrix element found in ni2.rcg_rme, for example

HAMILTONIAN, the atomic hamiltonian,

SHELL2, the crystal field hamiltonian acting on shell number 2,

MULTIPOLE, matrix elements of the dipole transition operator.

The OPER command is followed by one or more lines specifying branchings.

    BRANCH 0+ > 0 0+   1.00

These lines start with BRANCH then contain the representation labels in the original spherical symmetry (0+), the branching sign (>) and the label of the final octahedral symmetry (0 0+). A multiplicity label must also be specified if the branching can result in multiple copies of a representation, hence the 0 in front of the OH labels. The line finishes with a number that specifies the strength of the operator.

Definition of the Crystal Field parameters

The program TT-RAC uses its own definition of crystal field operators. In order to compare these to other definitions like Dq, Ds, Dt, we compare their effects on the set of d-functions.

Generally, the best way to specify the strength of the crystal field is to give the energy separations of the d-functions. The parameter values are often artificial and uninformative. The unit crystal field operators given by RCG are scaled such that <d||U(k)||d>=1 for the allowed values of k. For k equal to 2 and 4, in Butler's notation <2||2||2>=1 and <2||4||2>=1.

In the octahedral group we have for the irreps E (2) and T2 (^~1):

The irrep E has dimension 2, implying that the reduced matrix elements must be divided by sqrt(2). The energy of the two d-orbitals dz2 and dx2-y2 is 1/sqrt(30).

Crystal Field parameters in Tetragonal Symmetry

In tetragonal symmetry (D4h) the crystal field is given by three parameters, X⁴⁰⁰, X⁴²⁰ and X²²⁰. An equivalent description is to use the parameters Dq, Ds and Dt. This table writes the energies of the 3d-orbitals as a function of the parameters:

From this table we can relate both notations and write X⁴⁰⁰, etc as a function of Dq, ds and Dt.

and