Antiferromagnetic coupling is an effect that is often important for molecules with high spin multiplicity like this bridged manganese complex:

This is a typical transition metal system in which antiferromagnetic coupling is of interest: Mn₂O₂(NH₃)₈. In the netural molecule, in which the Mn atoms are nominally MnII, there are 5 d electrons on each Mn, and the antiferromagnetic singlet (5 alpha d electrons on one Mn and 5 beta d electrons on the other) may be the ground state. The cations Mn₂O₂(NH₃)₈ⁿ⁺ (n=1-4) are also of interest and may also have antiferromagnetic ground states.

In doing computations on this set of molecules, it is easiest to start with the system having half-filled d shells (in this case, the neutral). The rationale behind this approach is that because the charged systems with partially filled d shells give rise to additional issues of symmetry breaking in both the wavefunction and in the geometry. So it is best to do the filled case first, and then use its orbitals as the initial guess for the other cases.

This molecule can have D2h symmetry, so it is best to start the optimization from this high symmetry configuration. First, we sketch the molecule in GaussView, starting with an octahedral Mn atom, selected from the Element panel:

We next replace two hydrogens with oxygens and then add the other Mn atom as a separate fragment. We then move it into its approximate position by holding down the Alt key while dragging it.

Next, we check the Mn-O bond distance (before deleting the hydrogens) using Inquire mode:

The bond distance is display at the bottom of the window. We note that is it 1.83 Angstroms.

We go on to delete the two unneeded hydrogens on the second Mn atom, and then create the bonds between this atom and each oxygen atom. Once this is completed, we use the Clean function to regularize the geometry.

At this point, it is helpful to use GaussView's Edit Point Group dialog to ensure that the resulting Mn₂O₂H₈ system has D2h symmetry. We open the dialog (via the Edit=>Point Group menu path), and check Enable Point Group Symmetry. We change the tolerance to Very loose, and select D2h from the popup menu in the Approximate higher-order point groups section. Then we click the Symmetrize button to impose this point group on the structure:

If D2h is not a selection listed in the popup menu when you build this molecule, symmetrize the molecule to the highest listed point group and them examine it in Inquire mode. You may need to adjust some bond or dihedral angles manually in order for D2h symmetry to be recognized.

The next step in building the molecule is to replace the hydrogens with CH₃ groups. First, we replace the four hydrogens which are above and below the plane of the heavy atoms. After doing so, we again use the Point Group Symmetry dialog to return the molecule to D2h symmetry.

If you have trouble at this step, it may help to replace the hydrogens in pairs. After adding the first pair (e.g., on the hydrogens above the left Mn and below the right Mn), adjust the dihedrals and used the Point Group Symmetry dialog to impose C2h symmetry. Then, add the other pair, and impose D2h symmetry.

Once this is completed, go on to replace the other four hydrogen atoms with methyl groups. In this case, you will almost certainly need to adjust dihedral atoms to achieve D2h symmetry. For example, the dialog below shows the dihedral angle for the hydrogen atom pointing out of the plane of the screen/paper being set to 0 degrees:

Note that you must be sure to set the action for Atom 1 to Fixed to constrain movement in the molecule to rotation of the methyl group.

The final step is to replace the carbon atoms with nitrogens, and then impose D2h symmetry one final time to produce the final input structure.

We ran sequence of calculations to find the optimal geometry and wavefunction for the high spin state (multiplicity 11), followed
by use of the resulting wavefunction as the initial guess for the antiferromagnetic singlet.

Job 1: Here we optimize the structure for the high spin state from the input generated with GaussView. We can use the GaussView Gaussian Calculation Setup dialog to perform most of the work:

We specify a geometry optimization using the UB3LYP/3-21G model chemistry. We leave the Spin set to Singlet since GaussView is currently limited to 9 as its highest spin multiplicity (this will be fixed in a future release). We add the SCF=Tight keyword in the Additional Keywords field.

Once we have made all of the appropriate selections, we click on the Edit button. After saving the Gaussian input file, GaussView opens it in a text editor:

We manually change the spin multiplicity to 11 and save the file. We can then submit it to Gaussian.

Here is the beginning of the completed input file:

%chk=afc1
# opt ub3lyp/3-21g geom=connectivity scf=tight

Mn2O2(NH3)8 High Spin (antiferromagnetic coupling)

0 11
Mn
O   1   B1 
geometry specification continues ...

Note: When we actually ran this job, we also included additional directives in the route section specifying memory and the desired number of processors. Be aware that it requires nontrivial computational resources.

Job 2: Next, we check the stability of the wavefunction at the final geometry. Here is the input file:

%chk=afc2

#p b3lyp/3-21g scf=(tight) guess=read geom=check stable=opt             

Mn2O2(NH3)8 High Spin (antiferromagnetic coupling)

0,11

Note: We copied the checkpoint file from the first job to the one named afc2 prior to running this job.

As we suspected would be the case, a lower energy high spin state was found by the Stable=Opt job. The presence of an unstable wavefunction is indicated by lines like these in the output file:

 ***********************************************************************
 Stability analysis using <AA,BB:AA,BB> singles matrix:
 ***********************************************************************

 Eigenvectors of the stability matrix:
                       
   Eigenvector 1: ?Spin -B2G Eigenvalue=-0.0129755
   75A -> 79A 0.68583
   76A -> 80A 0.70288
   
   Eigenvector 2: ?Spin -B1U Eigenvalue=-0.0066257
   70A -> 79A -0.10926
   75A -> 80A 0.37604
   76A -> 79A 0.89115
   
   Eigenvector 3: ?Spin -B3U Eigenvalue= 0.0058150
   77A -> 80A -0.34398
   78A -> 79A 0.92278
   The wavefunction has an internal instability.

The job succeeds in finding a lower energy stable wavefunction.

Job 3. The next step is to reoptimize the high spin geometry using the better high spin wavefunction, using the following input file:

%chk=afc3
#p b3lyp/3-21g scf=tight opt=readfc geom=allcheck guess=read

The optimization succeeded.

Job 4: Check the stability of the high-spin wavefunction at the final geometry of the second optimization. Here is the input file:

%chk=afc4
# b3lyp/3-21g scf=(tight) guess=read geom=allcheck stable=opt 

The wavefunction was confirmed to be stable.

Job 5: If the molecule did not have high symmetry, we would now be ready to use the high-spin wavefunction as input for the antiferromagnetic singlet. However, in this case, the two Mn atoms are equivalent in D2h symmetry, and so the d orbitals produced by the high-spin SCF are sums and differences of the d functions on each Mn.

We generated a broken symmetry solution with partial localization on each Mn by doing a single point calculation with a small electric field applied in the x direction, which is the Mn-Mn bond direction in the standard orientation. Here is the input file:

%chk=afc5
#p b3lyp/3-21g scf=(tight) guess=read geom=check symm=noint field=x+50

The resulting checkpoint file will be used to examine and select MOs for the antiferromagnetic single initial guess.

Selecting, Populating and Reordering Orbitals in GaussView. The next step is to generate an initial guess for the antiferromagnetic singlet, using the 5 unpaired orbitals from the high spin calculation which are primarily on one Mn atom as the highest five alpha orbitals in the guess, and the other 5 unpaired orbitals from the high spin as the highest five beta spin orbitals in the guess. This requires use
of the Guess=Alpha option to Gaussian 03 -- to use the alpha spin orbitals for the initial guess -- plus some rearrangement of the orbitals so that the correct ones are occupied for each spin.

The Edit MOs dialog in GaussView makes this task very easy. We open the afc5 checkpoint file in GaussView, and then open the Edit MOs dialog using the Edit=>MOs menu path.

Initially, GaussView is a bit confused by having 11 open shells, so it starts out showing singlet as the Spin in the dialog. However, you must select a different spin-state and then change the setting back to Singlet in order for GaussView to function properly (this will be fixed in a future release). After doing this, specify Unrestricted (alpha) as the Wavefunction, which says that the alpha orbitals will be used for both the alpha and beta guess orbitals. At this point, the orbital diagram displays properly for the singlet, with 73 orbitals occupied for each spin case:

The 5 highest occupied orbitals from the high spin calculation are now the lowest 5 virtuals for the singlet.

We need to move electrons around in both spin cases, so that each d orbital was occupied in exactly one spin case. We displayed the orbitals numbered 63-78, using a contour level (Isovalue) of 0.04, using the Visualize tab in the dialog:

The default contour level in GaussView is appropriate for valence, orbitals but is too small a value to pick up the d orbitals well. For example, the 0.02 and 0.04 orbital plots are displayed for orbital 65 below:

We needed to look at 16 orbitals. Most of the d orbitals were clear from inspection, but a couple are mixed with other occupied orbitals and so some care is required in inspecting the orbitals. Orbitals exhibiting d character on both Mn atoms were assigned to one or other by determining which had the larger values.

Here are the orbitals in consideration, label by MO number. Ones selected for single occupation are followed by alpha or beta (corresponding to the left and right Mn atoms in the illustrations, respectively):

Click anywhere in the image to open a larger view in another window.

Once the orbital selection was complete, we used the dialog to drag electrons to the proper orbitals in both the alpha and beta lists. When complete, we used the Diagram panel to rearrange the MOs by occupancy, moving unoccupied ones above occupieds:

Job 6: Next, we generated the Gaussian input file for a single-point SCF calculation, adding Guess=Alpha to the Additional Keywords field. GaussView does not put the blank line in the input which is required to separate the permutations of the alpha and beta orbitals, so Iwe once again used the Edit button edit the input file and add this blank line before running the job (indicated by the red line in the following illustration):

Note that this job is to be performed for the singlet, so we do not alter the spin multiplicity in the input file.

%chk=afc6
# ub3lyp/3-21g guess=(alpha,permute,read) scf=(tight) geom=check

Mn2O2(NH3)8 Singlet (antiferromagnetic coupling)

0 1

1-62,64-68,71-73,76-78,63,69-70,74-75,79-196

1-63,65-67,69-71,74-75,77-78,64,68,72-73,76,79-196

The job produced a wavefunction which had a reasonable energy and which had spin densities of about +4 on one Mn and -4 on the other. The spin densities aren't expected to be exactly 5 on each metal, because some of the spin density delocalizes onto the ligands, so this is a reasonable result for antiferromagnetic coupling of 10 electrons.

Here is the relevant part of the output file:

Mulliken atomic spin densities:
              1
     1  Mn   3.931283
     2  O   -0.000000
     3  O   -0.000000
     4  Mn  -3.931283
     5  H    0.005293
     ...

Job 7: We did a stabilty calculation on the resulting wavefunction and confirmed that it was stable:

%chk=afc7
# ub3lyp/3-21g scf=(tight,nosymm) guess=read geom=allcheck stable=opt

As a check, we also did a Stable=Opt calculation starting from the closed-shell singlet. The closed-shell singlet was unstable, as expected, and a subsequent Stable=Opt found a wavefunction with Sz=0 which was lower in energy than the closed shell but not as low as the antiferromagnetic one, consistent with the antiferromagnetic state being the ground state.

The following table summarizes the energies of the various states we studied: