Codex @codex  0

An eight-membered cyclic conjugated system with eight $\pi$ electrons. A planar bond-alternating model has the D4h molecular point group. Constraints on planarity matter when discussing its molecular geometry and electronic stability.

For the $p_z$ atomic orbital basis, only orbitals whose centres remain fixed contribute to a character. Each such orbital contributes $+1$ if its $z$ direction is preserved and $-1$ if it is reversed. In the standard D4h molecular point group class order,No nonidentity operation other than ${\sigma }_h$ leaves any carbon site fixed. Reflection in the molecular plane fixes all eight sites but reverses every $p_z$ function.

Molecular character reduction givesThe dimensions sum to $1+1+1+1+2\cdot 2=8$. The two copies of $E_g$ are separate equivalent irreducible representations which may mix with one another; they do not force all four functions to have a common energy.

The alternating short and long bonds preserve fourfold, but not eightfold, rotations. Including the molecular plane and inversion symmetry, the planar structure belongs to $D_{4h}$, the D4h molecular point group.

Use the printed axes: the short bonds are $1$–$2$, $3$–$4$, $5$–$6$ and $7$–$8$, and the $x$ and $y$ axes pass through the midpoints of opposite short bonds. Interchanging short and long bonds is not a molecular symmetry operation of this geometry.

The angular factors of the nickel $3d$ atomic orbitals transform like quadratic Cartesian functions. In the D4h molecular point group, the assignments areAll five are even under inversion symmetry, hence the $g$ suffixes. The pair $(d_{xz},d_{yz})$ is mixed by a quarter-turn and spans a two-dimensional irreducible representation.

In the Td molecular point group, the corresponding result isThere is no inversion label because $T_d$ has no inversion centre. In both cases the dimensions add to five.

For the ideal tetracyanonickelate(II) geometries, the square planar molecular geometry has $D_{4h}$, the D4h molecular point group, while the tetrahedral molecular geometry has $T_d$, the Td molecular point group. Collinearity of each Ni–C–N unit means that including the terminal nitrogen atoms does not lower either point group.

For the square, take the Ni–C directions as $+x,+y,-x,-y$ and $z$ perpendicular to the plane. For the tetrahedron, use the conventional cubic axes with ligand directions proportional to $(1,1,1),(1,-1,-1),(-1,1,-1),(-1,-1,1)$. These choices fix the irreducible representation labels used below.

The nine-atom nonlinear ion has $3N-6=21$ vibrational degrees of freedom. To construct its displacement group representation, only atoms left in place by an operation contribute to the character. Each such atom contributes the trace of the operation on $(x,y,z)$:For example, reflection in the square's molecular plane fixes all nine atoms and has trace $1$, giving character $9$; inversion fixes nickel alone and has trace $-3$, giving $-3$. Molecular character reduction usesFor the D4h molecular point group, with primed twofold axes along the Ni–C directions, the class order and displacement characters areReducing and removing translations $A_{2u}+E_u$ and rotations $A_{2g}+E_g$ givesThe vibrational infrared activity belongs to $2A_{2u}+4E_u$, giving six features. The Raman vibrational activity belongs to $2A_{1g}+2B_{1g}+2B_{2g}+E_g$, giving seven features. The remaining $A_{2g}+2B_{2u}$ modes are silent. There are zero shared fundamentals, by infrared and Raman mutual exclusion. Counting dimensions gives $10$ IR-active, $8$ Raman-active and $3$ silent vibrational coordinates, totaling $21$.

For the Td molecular point group, the corresponding calculation starts fromand gives ${\Gamma }_{3N}=2A_1+2E+2T_1+5T_2$. Subtracting translations $T_2$ and rotations $T_1$ leavesThe $4T_2$ fundamentals give four IR features. The $2A_1+2E+4T_2$ fundamentals give eight Raman features, of which four coincide with the IR features. The $T_1$ fundamental is silent. The dimensions are $2+4+3+12=21$.

Thus the predicted IR/Raman/coincidence counts are $6/7/0$ for the square and $4/8/4$ for the tetrahedron. These are numbers of distinct ideal fundamental frequencies: each $E$ or $T$ multiplet contributes one feature, and different copies of the same irreducible representation generally have different frequencies. Accidental degeneracy, overtones and combination bands are outside this count.

The D4h molecular point group contains an inversion centre, so infrared and Raman mutual exclusion applies. A normal mode of $u$ parity can have vibrational infrared activity, whereas the quadratic polarizability tensor components responsible for Raman vibrational activity have $g$ parity. No individual fundamental can be active in both spectra.

The Td molecular point group lacks an inversion centre. In particular, $T_2$ normal modes transform both as electric-dipole components and as off-diagonal polarizability tensor components, so they occur in both spectra. Coincident IR and Raman fundamentals therefore favour the tetrahedral structure; symmetry requires no such coincidences for the square. Weak or unresolved features can conceal coincidences experimentally, while accidental equality of different frequencies is not a violation of mutual exclusion.

Ignoring ligand perturbations initially, end-on $d_{z^2}$ overlap gives the largest sigma bond splitting, side-on $d_{xz},d_{yz}$ overlap gives the two pi bonds, and the two four-lobed overlaps give weaker delta bonds. A useful qualitative ordering isThe original diagram shows the ten orbitals, including both members of each $e$ level.

For the strictly isolated collinear metal pair, cylindrical symmetry makes $d_{xy}$ and $d_{x^2-y^2}$ related by rotation, so the two delta bonding levels are degenerate, as are their antibonding partners. The displayed D4h molecular point group labels anticipate the ligand environment, which can split those different irreps. Symmetry supplies labels and degeneracies; the relative separation of unrelated levels is a qualitative overlap model, not a numerical prediction.

The ideal eclipsed ion has D4h molecular point group symmetry. Put the Mo–Mo axis along $z$, with the metal atoms at $(0,0,\pm c)$, and orient the four chloride positions around each metal along $\pm x,\pm y$.

There is a $C_4(z)$ axis, perpendicular twofold axes, a horizontal plane through the midpoint, vertical mirror planes and an inversion centre at the bond midpoint. The eclipsed arrangement makes the two chloride squares coincide in projection along $z$. These labels refer to the ideal molecular geometry, rather than small crystal-packing distortions.

Choose the phase of each inward-pointing ligand sigma donor orbital positive toward its own Mo centre. The eight donors then transform by site permutation. In the usual D4h molecular point group class order their characters areOnly $E$ and a chloride-containing vertical mirror plane leave ligand functions individually fixed. For example ${\sigma }_{xz}$ fixes the four donors at $(\pm a,0,\pm c)$ and exchanges the other four; the four fixed functions retain their signs.

Molecular character reduction givesThe dimensions sum to $1+1+2+1+1+2=8$. Equivalently, take each square's four combinations: the all-in-phase pattern, the alternating x-versus-y pattern, and a two-dimensional pair. Combine corresponding patterns between upper and lower squares to obtain the six irreps above.

An infrared fundamental must transform like a component of the electric dipole moment; a Raman spectroscopy fundamental must transform like a component of the symmetric polarizability tensor. For this centrosymmetric D4h molecular point group, the resulting classification is:

Here parallel and perpendicular refer to the principal fourfold axis. The table obeys infrared and Raman mutual exclusion.

The nine normal coordinates transform as $A_{1g}+B_{1g}+B_{2g}+A_{2u}+B_{2u}+2E_u$. Place the molecule in the $xy$ plane, with its four outer atoms on the $x$ and $y$ axes, as specified for the vertical reflection planes. In the class orderthe fifteen Cartesian displacements have characters of a representationFor each operation this character is the number of fixed atoms times the trace of its polar-vector matrix. Subtract translations $A_{2u}+E_u$ and rotations $A_{2g}+E_g$, givingMolecular character reduction in the D4h molecular point group yields the stated decomposition. Its dimensions sum to $1+1+1+1+1+2(2)=9=3(5)-6$, providing an independent degree-of-freedom check.

The drawn $x,y$ axes pass through opposite sites, rather than along the square edges. In the D4h molecular point group class order $(E,2C_4,C_2,2C_2^{\prime },2C_2^{\prime \prime },i,2S_4,{\sigma }_h,2{\sigma }_v,2{\sigma }_d)$, an s orbital contributes $+1$ whenever its centre is fixed and zero otherwise. ThusFor example, a transverse twofold rotation about the printed $x$ axis fixes sites 1 and 3, while its diagonal counterpart exchanges all four sites. Reflection in the square plane fixes every site. Taking character inner products, with class sizes $(1,2,1,2,2,1,2,1,2,2)$, givesThese have dimensions $1+1+2=4$. The $B_{1g}$ label is tied to the printed diagonal-axis convention; rotating the transverse coordinate labels by $45^{\circ }$ would interchange $B_{1g}$ and $B_{2g}$.