Example: Multiplet States: [Cr(NH3)6]3+

Download SD_CrNH3_6.run

The computation of multiplet states corresponding to an open-shell system can be carried out with ADF by first computing the ‘Average-of-Configuration’ (aoc) state, where all orbitals in the open shell are degenerate and equally occupied. This computation is spin-restricted and serves as a fragment file for the multiplet run, where then different occupation numbers are assigned to the various orbitals in the open shell. The corresponding energies are computed in the field of the aoc, which is achieved by not iterating the self-consistency equations to convergence but only computing the orbitals in the initial field.

Since ADF requires that all symmetry-partners in an irreducible representation (irrep) have equal occupations, the multiplet calculation, where such orbitals are not equally occupied, must be carried out in a formally lower point group symmetry. The point group to select and the appropriate occupation numbers to apply must be worked out by the user ‘on paper’ in advance. An auxiliary program asf, developed by the group of Claude Daul in Fribourg can be used to determine which calculations are needed, and how to compute the multiplet energies from the results. See the discussion of Multiplet energies in the Theory document.

The script starts with the ‘creation’ of the required basic atoms, N, H, Cr using a fair basis set quality.

The next step is the computation of the ammonia fragment NH3 . This is not a crucial step here: the multiplet state computation can equally well be carried out by not using any intermediate compound fragments. However, it illustrates once more how a bigger molecule can be built up from smaller, but not trivial fragments.

$ADFBIN/adf <<eor
title AMMONIA
NOPRINT sfo,frag,functions

define
xH=0.95522523
yH=xH*sqrt(3)/2
zH=0.3711068
end

atoms
N    0      0    0
H   -xH     0    zH
H    xH/2  -yH   zH
H    xH/2   yH   zH
end

Basis
 Type TZP
 Core Small
End

symmetry  C(3V)

endinput
eor

mv TAPE21 t21.NH3

The input of the atomic coordinates uses expressions, in this case to enforce exact symmetry relations that would otherwise require 14-digit input values or some inaccuracy. The symmetry specification is redundant: the program would also find it by itself.

Average-of-Configuration

The next step is to compute the reference state, with respect to which we will later compute the multiplet states. The reference state is the so-called ‘Average-of-configuration’ (aoc) state. The result file (TAPE21) of this calculation will be used as a fragment file.

$ADFBIN/adf <<eor
title Cr(NH3)6 : Average-of-Configuration run

COMMENT
using NH3-fragments
END

symmetry D(3d)

scf
iterations  25
mix   0.15
end

atoms
 Cr      0.000000     0.000000     0.000000
  N      0.000000     1.714643     1.212436  f=NH3/1
  H      0.000000     1.466154     2.206635  f=NH3/1
  H     -0.827250     2.293404     1.036727  f=NH3/1
  H      0.827250     2.293404     1.036727  f=NH3/1
  N     -1.484924    -0.857321     1.212436  f=NH3/2
  H     -1.269726    -0.733077     2.206635  f=NH3/2
  H     -1.572521    -1.863121     1.036727  f=NH3/2
  H     -2.399771    -0.430282     1.036727  f=NH3/2
  N      1.484924    -0.857321     1.212436  f=NH3/3
  H      1.269726    -0.733077     2.206635  f=NH3/3
  H      2.399771    -0.430282     1.036727  f=NH3/3
  H      1.572521    -1.863121     1.036727  f=NH3/3
  N      0.000000    -1.714643    -1.212436  f=NH3/4
  H      0.000000    -1.466154    -2.206635  f=NH3/4
  H      0.827250    -2.293404    -1.036727  f=NH3/4
  H     -0.827250    -2.293404    -1.036727  f=NH3/4
  N      1.484924     0.857321    -1.212436  f=NH3/5
  H      1.269726     0.733077    -2.206635  f=NH3/5
  H      1.572521     1.863121    -1.036727  f=NH3/5
  H      2.399771     0.430282    -1.036727  f=NH3/5
  N     -1.484924     0.857321    -1.212436  f=NH3/6
  H     -1.269726     0.733077    -2.206635  f=NH3/6
  H     -2.399771     0.430282    -1.036727  f=NH3/6
  H     -1.572521     1.863121    -1.036727  f=NH3/6
  H     -1.572521     1.863121    -1.036727  f=NH3/6
end

fragments
Cr  t21.Cr
NH3 t21.NH3
end

occupations
 A1.G 8.75
 A2.G 2
 E1.G 16 1.5 0.75
 A1.U 2
 A2.U 8
 E1.U 20
END

end input
eor

mv TAPE21 t21.CrA6ES

Occupation numbers are specified, to make certain what the reference state is that we will start from in the subsequent calculations. The result file TAPE21 is saved to serve as fragment file in the subsequent calculations.

One-determinant states

Now, we proceed with the multiplet calculations. In the example they are combined in one single run, but they could also be evaluated in separate runs. For each calculation it is required to:

  1. Use the aoc TAPE21 file as fragment file
  2. Choose which molecular orbitals in the open shell to occupy: select the appropriate point group symmetry and the UnRestricted key if necessary and specify the occupation numbers, using the irreducible representations of the selected point group.

The results are one-determinant calculations, which must then, later, be combined analytically to obtain the required multiplet energy values.

$ADFBIN/adf <<eor
title Cr(NH3)6 : SlaterDeterminants run
NOPRINT frag

symmetry C(I)  !  lower symmetry

scf
iterations 0
end

atoms
   Cr        0.000000       0.000000       0.000000  f=CrA6
    N        0.000000       1.714643       1.212436  f=CrA6
    H        0.000000       1.466154       2.206635  f=CrA6
    H       -0.827250       2.293404       1.036727  f=CrA6
    H        0.827250       2.293404       1.036727  f=CrA6
    N       -1.484924      -0.857321       1.212436  f=CrA6
    H       -1.269726      -0.733077       2.206635  f=CrA6
    H       -1.572521      -1.863121       1.036727  f=CrA6
    H       -2.399771      -0.430282       1.036727  f=CrA6
    N        1.484924      -0.857321       1.212436  f=CrA6
    H        1.269726      -0.733077       2.206635  f=CrA6
    H        2.399771      -0.430282       1.036727  f=CrA6
    H        1.572521      -1.863121       1.036727  f=CrA6
    N        0.000000      -1.714643      -1.212436  f=CrA6
    H        0.000000      -1.466154      -2.206635  f=CrA6
    H        0.827250      -2.293404      -1.036727  f=CrA6
    H       -0.827250      -2.293404      -1.036727  f=CrA6
    N        1.484924       0.857321      -1.212436  f=CrA6
    H        1.269726       0.733077      -2.206635  f=CrA6
    H        1.572521       1.863121      -1.036727  f=CrA6
    H        2.399771       0.430282      -1.036727  f=CrA6
    N       -1.484924       0.857321      -1.212436  f=CrA6
    H       -1.269726       0.733077      -2.206635  f=CrA6
    H       -2.399771       0.430282      -1.036727  f=CrA6
    H       -1.572521       1.863121      -1.036727  f=CrA6
end

fragments
  CrA6  t21.CrA6ES
end

UnRestricted

SlaterDeterminants
  Check AOC
    A1.g   4 0.375         // 4 0.375
    A2.g   1               // 1
    E1.g:1 4 0.375 0.1875  // 4 0.375  0.1875
    E1.g:2 4 0.375 0.1875  // 4 0.375  0.1875
    A1.u   1//1
    A2.u   4//4
    E1.u:1 5//5
    E1.u:2 5//5
  SUBEND
  State1
    A1.g   4 1             // 4 1
    A2.g   1               // 1
    E1.g:1 4 0     0       // 4 0      1
    E1.g:2 4 0     0       // 4 0      0
    A1.u   1//1
    A2.u   4//4
    E1.u:1 5//5
    E1.u:2 5//5
  SUBEND
  State2
    A1.g   4 1             // 4 1
    A2.g   1               // 1
    E1.g:1 4 0     0       // 4 1      0
    E1.g:2 4 0     0       // 4 0      0
    A1.u   1//1
    A2.u   4//4
    E1.u:1 5//5
    E1.u:2 5//5
  SUBEND
  State3
    A1.g   4 1             // 4 1
    A2.g   1               // 1
    E1.g:1 4 0     1       // 4 0      0
    E1.g:2 4 0     0       // 4 0      0
    A1.u   1//1
    A2.u   4//4
    E1.u:1 5//5
    E1.u:2 5//5
  SUBEND
end

end input
eor

The SlaterDeterminants block may contain any number of sub blocks, each starting with an (arbitrary) title record, followed by a set of occupation numbers and closed by a SubEnd record. Each such subkey block specifies a single one-determinant-state calculation. All occupation numbers must reference the irreps of the specified point group symmetry, C(I) in the example, and must be just a reassignment of the electrons that are equally distributed over the corresponding degenerate irreps in the reference aoc calculation.

The so-obtained energies of the one-determinant states can now be combined to calculate the desired multiplet energies. See the Theory document and the adf User’s Guide.

Note carefully that in the calculation of the SingleDeterminants, the scf procedure is prevented to cycle to convergence by setting the subkey Iterations to zero in the SCF data block.