Start-MOs for broken symmetry treatments ("flip")

Broken-symmetry treatments suggested by e.g. Noodleman or Ruiz are a popular tool for the calculation of spin coupling parameters in the framework of DFT. As an example one might consider two coupled Cu$^{II}$ mathend000# centers, e.g. for a (hypothetical) arrangement like this:

$coord
 0.0  2.7 0.0 cu
 0.0 -2.7 0.0 cu
 0.0 -6.1 0.0 f
 0.0  6.1 0.0 f
 2.4  0.0 0.0 f
-2.4  0.0 0.0 f
$end

The high-spin case, a doublet with an excess alpha electron at each Cu atom, "aa" in an obvious notation, preserves $D_{2h}$ mathend000# symmetry, while the low spin state "ba" does not. For a broken-symmetry treatment it is advidsable to calculate the high-spin state first, and then broken-symmetry state(s); from the energy difference(s) one may calculate approximate values for the spin-spin coupling parameters as described by e.g. the above authors. Access to broken-symmetry states usually is possible by the choice of appropriate start-MOs, followed by an SCF-procedure. Start MOs may be obtained by first applying a localization procedure to the MOs of the high-spin state and then by "moving" localized alpha orbitals to the beta subset.

The preparation of broken-symmetry start-MOs can be done with DEFINE (semi-)automatically. Prerequisite is a converged wave function for the high-spin state in $C_1$ mathend000#-symmetry, that fulfills the $aufbau$ mathend000# principle.

If in this case one enters flip in the orbital definition menu, DEFINE selects the occupied valence orbitals of the system (by an orbital energy criterion, which one can usually accept, unless the system is highly charged and the orbital energies are shifted). Next a Boys orbital localization procedure is carried out, which - depending on the size of the problem - may take some time. Then the user is asked:

ENTER INDICES OF ATOMS OR ELEMENT TO BE MANIPULATED (example: 1,2-3 or "mn")

In case of our above example one may enter "cu", which immediately leads to the following output (a def-SV(P) basis and the B-P functional were used for the high-spin state):

 RELEVANT LMOS FOR ATOM   1 cu
 ALPHA:
index occupation "energy"     s    p    d    f   (dxx  dyy  dzz  dxy  dxz  dyz)
  15    1.000     -0.357     0.01 0.00 0.98       0.20 0.27 0.01 0.50 0.00 0.00
  18    1.000     -0.357     0.01 0.00 0.98       0.20 0.27 0.01 0.50 0.00 0.00
  20    1.000     -0.335     0.00 0.00 1.00       0.00 0.00 0.00 0.00 1.00 0.00
  22    1.000     -0.333     0.01 0.00 0.99       0.13 0.03 0.32 0.00 0.00 0.51
  23    1.000     -0.333     0.01 0.00 0.99       0.14 0.03 0.34 0.00 0.00 0.49

 BETA:
  39    1.000     -0.326     0.00 0.00 1.00       0.33 0.08 0.09 0.00 0.00 0.50
  41    1.000     -0.326     0.00 0.00 1.00       0.33 0.08 0.09 0.00 0.00 0.50
  43    1.000     -0.321     0.00 0.00 1.00       0.00 0.00 0.00 0.00 1.00 0.00
  46    1.001     -0.318     0.05 0.00 0.95       0.00 0.43 0.51 0.00 0.00 0.00

 RELEVANT LMOS FOR ATOM   2 cu
 ALPHA:
index occupation "energy"     s    p    d    f   (dxx  dyy  dzz  dxy  dxz  dyz)
  16    1.000     -0.357     0.01 0.00 0.98       0.20 0.27 0.01 0.50 0.00 0.00
  17    1.000     -0.357     0.01 0.00 0.98       0.20 0.27 0.01 0.50 0.00 0.00
  19    1.000     -0.335     0.00 0.00 1.00       0.00 0.00 0.00 0.00 1.00 0.00
  21    1.000     -0.333     0.01 0.00 0.99       0.13 0.03 0.32 0.00 0.00 0.51
  24    1.000     -0.333     0.01 0.00 0.99       0.14 0.03 0.34 0.00 0.00 0.49

 BETA:
  40    1.000     -0.326     0.00 0.00 1.00       0.33 0.08 0.09 0.00 0.00 0.50
  42    1.000     -0.326     0.00 0.00 1.00       0.33 0.08 0.09 0.00 0.00 0.50
  44    1.000     -0.321     0.00 0.00 1.00       0.00 0.00 0.00 0.00 1.00 0.00
  45    1.001     -0.318     0.05 0.00 0.95       0.00 0.43 0.51 0.00 0.00 0.00

 a2b    : FLIPPING ALPHA TO BETA (default)
 b2a    : FLIPPING BETA TO ALPHA
 r      : repeat atom choice

As evident from the second column, for each Cu atom five localized alpha and four localized beta orbitals were generated which are of d-type (the sixth column labelled "d" shows values close to 1, the other columns such close to 0). The six columns at the right show the individual contributions of the six cartesian d-functions.

What has to be done to generate start MOs for the "ba"-case? Obviously one of the five localized alpha spin orbitals from the first Cu atom (atom label 1 cu) has to become a beta spin orbital. These five orbitals have the indices 15, 18, 20, 22, 23. In order to avoid linear dependencies, it is advisable to take the orbital that has no beta-analogue. This can be found by comparing the contributions of the six d-functions. In the present example this is the case for the localized alpha orbitals 15 and 18: in contrast to all localized beta orbitals they show significant contributions from $d_{xy}$ mathend000#. One thus enters

a2b
15

and after confirming the replacement of original MOs with the generated start-MOs one is finally asked

 It is advisable to modify damping and orbital shift in the following way:

 $scfdamp   start=5.000  step=0.050  min=0.500
 $scforbitalshift  automatic=1.0
 $scfiterlimit 999

 Do you want to replace the corresponding entries in the control-file? (y)

which should be confirmed, as otherwise the prepared spin state might be destroyed during the SCF iterations.

From this input one may start the SCF(HF/DFT)-procedure. For recommended choices of DFT functionals and formulae to calculate the coupling parameters from these energy differences please consult the papers of the above-mentioned authors. For reasons of economy, a pre-optimization by a pure (non-hybrid) DFT-functional is reasonable.

Important: For the converged wave function one should check, whether the resulting state is really the desired one. This can quite reliably be done by a Mulliken population analysis. For this purpose, add $pop to the control file, type ridft -proper or dscf -proper, respectively, and check the signs of the calculated numbers of unpaired electrons in the output.