Example: Multiplet States: [Cr(NH3)6]3+¶
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:
- Use the aoc TAPE21 file as fragment file
- 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.