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For a finite group $G$ represented unitarily by $U(g)$, the central projector onto the isotypic subspace of an irreducible representation $\lambda$ is $P_{\lambda }=(d_{\lambda }/|G|){\sum }_g{\chi }_{\lambda }(g{)}^{\ast }U(g)$, where $d_{\lambda }$ is its dimension. Character orthogonality implies that the operator is identity on copies of $\lambda$ and zero on other irreducibles. For a one-dimensional irreducible, applying it to an atomic orbital gives either zero or a symmetry-adapted linear combination of that symmetry, to be normalized afterward. A multidimensional central projector selects an entire isotypic component rather than one uniquely chosen component function.

Neglecting intersite orbital overlap, one normalized real set of symmetry-adapted linear combinations isThe coefficient vectors are orthonormal. The first has equal phase at all three sites; the second opposes site $A$ to $B,C$; the third has zero amplitude at $A$ and opposite phases at $B,C$. Any orthonormal rotation of the two $E$ functions is equally valid. With nonzero intersite orbital overlap, normalization must instead use the overlap matrix.

Write $p_j=p_{zj}$ and use the orthonormal basis of the Hückel method. The four normalized one-dimensional symmetry-adapted linear combinations areFor example, the $xz$ mirror plane exchanges sites $1\leftrightarrow 6$, $2\leftrightarrow 5$, $3\leftrightarrow 4$, $8\leftrightarrow 7$, without changing the sign of an individual $p_z$ orbital. The all-positive combination is unchanged by every vertical mirror but changes sign under ${\sigma }_h$, giving $A_{2u}$. The alternating combination changes sign under both sorts of vertical mirror, as required for $A_{1u}$. The two remaining sign patterns distinguish $B_{1u}$ from $B_{2u}$ in the printed axes. The signs refer to the upper $p_z$ lobes; the lower lobes have opposite quantum phases, and neither colour denotes an electric charge.

For a normalized coefficient vector $\mathbf{c}$, the energy isEvery site has one short-bond and one long-bond neighbour. Equivalently, apply the Hückel method matrix to any coefficient and read off its two neighbouring signs. The results areWith negative resonance integrals and $|{\beta }_S|>|{\beta }_L|$, the $A_{2u}$ and $B_{2u}$ levels are occupied, while $B_{1u}$ and $A_{1u}$ are unoccupied in the closed-shell alternating-bond state.

Choose the phase of each inward-pointing ligand sigma bond donor lobe consistently. An operation taking one donor to another then acts as a permutation on the four donor functions. For the square, label these functions $L_1,L_2,L_3,L_4$ cyclically along $+x,+y,-x,-y$. Normalized symmetry-adapted linear combinations areThus the ligand group representation is ${\Gamma }_{\sigma }=A_{1g}+B_{1g}+E_u$. Its irreducible representations allow the first two combinations to mix with the matching nickel $d$ atomic orbitals; $E_u$ instead matches nickel $p_x,p_y$.

For the tetrahedron in the vertex order specified above, takeThe three latter functions transform like $x,y,z$. Hence ${\Gamma }_{\sigma }=A_1+T_2$. These are four ligand symmetry-adapted linear combinations, rather than the five metal $d$ functions of part (ii).

Work with $M=(H-\alpha I)/\beta$. Its nearest-neighbour ring entries are $1$, and its diagonal entries are $1$ on the nitrogen sites and $0$ on the carbon sites. Because $M$ commutes with each molecular symmetry operation, it only mixes symmetry-adapted linear combinations with the same irreducible representation.

In the bases $(N_{+},C_0)$ and $(N_{-},C_y)$, the two nontrivial matrix blocks areTheir characteristic polynomials are $(1-\lambda {)}^2-2$ and ${\lambda }^2-3$. The remaining single-function blocks are $M_{B_{2g}}=1$ and $M_{A_u}=-1$. Thus, ordered from lowest to highest physical energy because $\beta <0$, the molecular orbitals areAll six levels are nondegenerate in this parametrization. The right-hand ladder of the diagram in part (ii) shows these levels and occupations. The occupied $B_{2g}$ level is the highest occupied molecular orbital, and the unoccupied $B_{1u}$ level immediately above it is the lowest unoccupied molecular orbital.

The nitrogen pair and the four-carbon set are separately invariant under the D2h point group. Construct symmetry-adapted linear combinations within these two sets; mixing equivalent representations across the sets is left for the Hückel method calculation.

For the nitrogen $p_z$ atomic orbitals, normalized combinations areFor the carbon $p_z$ atomic orbitals, takeThese six functions are orthonormal under the zero-overlap assumption. The suffixes $x,y,xy$ describe coefficient patterns across the carbon sites; the sign acquired by each underlying $p_z$ function must also be included when assigning an irreducible representation. In particular, reflection through the molecular plane changes the sign of every function, while inversion sends ${\chi }_1$ to $-{\chi }_4$. Ignoring that orbital sign would give incorrect $g/u$ labels.

The complete $\pi$ group representation is thereforeThe diagram shows all six ring symmetry-adapted linear combinations and the resulting energy ordering used in part (iii).

Let $d_{\mu ,A}$ and $d_{\mu ,B}$ be corresponding real 4d orbitals on the upper and lower metals, with identical global-axis phases. Their symmetry-adapted linear combinations arewith the normalization shown in the negligible-overlap approximation; including overlap changes the denominator but not the symmetry. Inversion exchanges the centres and each local d orbital has even intrinsic parity, so the plus combinations are gerade and the minus combinations ungerade.The reversal of bonding phase for $d_{xz},d_{yz}$ is important: the facing lobes have opposite signs when the two atomic orbitals share the same global phase convention, so their difference is bonding. Each $e$ irrep is two-dimensional. Thuswith total dimension ten. Rotating $x,y$ by $45^{\circ }$ interchanges the names of the two delta families; all labels here retain the chloride-aligned axes fixed above.

Write $p_i$ for the orthonormal $2p_z$ atomic orbital on numbered site $i$, with every positive lobe pointing along $+z$. A normalized set of symmetry-adapted linear combinations isEach has unit squared coefficient sum, and every distinct pair has zero inner product. The orbital representation therefore reduces to $\boxed{3B_{1u}+3B_{3g}+B_{2g}+A_u}$. For example, the fully symmetric site sum still has $B_{1u}$ orbital symmetry because a $p_z$ orbital itself changes sign under operations that reverse $z$. Ignoring that sign would assign the wrong representations.

Let $z$ be normal to the Cp ring through its centre. The five carbon $p_z$ orbitals form symmetry-adapted linear combinations with angular patterns $m=0$, $|m|=1$, and $|m|=2$. The all-in-phase $a_1$ combination has sigma symmetry about this axis and can interact with $d_{z^2}$. The $e_1$ pair matches $(d_{xz},d_{yz})$ by pi symmetry, while the $e_2$ pair matches $(d_{xy},d_{x^2-y^2})$ by delta symmetry. Rotating within either degenerate pair changes the drawing but not the physical interaction.

For the usual eta-five geometry, the metal $d$ interactions are ranked qualitatively pi > delta > sigma. The $d_{xz},d_{yz}$ lobes overlap effectively with the occupied ring combinations. The delta interaction is weaker and involves the higher, initially unoccupied Cp combinations; the $d_{z^2}$ interaction with the ring is particularly weak because its axial and equatorial parts contribute little net overlap in this geometry. Thus the statement that a sigma bond is always stronger than a pi bond cannot be transferred indiscriminately to this delocalized ring–metal overlap problem. metal $s$ and $p_z$ orbitals also have the symmetry of the $a_1$ coordination ligand combination and contribute to overall bonding, so weak $d_{z^2}$ overlap does not mean the entire sigma-symmetry channel is absent.

In ionic electron counting, Cp is ${\mathrm{C}}_5{\mathrm{H}}_5^{-}$ with six pi electrons in its three occupied ring combinations, hence a six-electron donor. The same coordination ligand is assigned five electrons in neutral counting.

Six inward-directed donor orbitals form symmetry-adapted linear combinations of $a_{1g}+e_g+t_{1u}$ symmetry. They mix respectively with metal $4s$, $(3d_{z^2},3d_{x^2-y^2})$ and $4p$ orbitals. Thus the sigma framework contains six occupied bonding molecular orbitals and six corresponding antibonding molecular orbitals. The metal $(d_{xy},d_{xz},d_{yz})$ orbitals have no sigma-compatible partner and remain nonbonding $t_{2g}$ orbitals in this model. The important metal-dominated octahedral splitting isThe $e_g^{\ast }$ orbitals point directly toward the donors; the $t_{2g}$ orbitals point between them. The original diagram shows the entire sigma framework, rather than treating the five metal d levels as the whole molecular-orbital diagram.

For the same donor set and approximately the same geometry, ${\Delta }_o(\mathrm{C}{\mathrm{o}}^{\mathrm{III}})>{\Delta }_o(\mathrm{C}{\mathrm{o}}^{\mathrm{II}})$. The higher metal charge usually shortens the metal–donor distances and strengthens the interaction that raises $e_g^{\ast }$. This comparison holds other chemical variables fixed: changing the ligand at the same time could reverse a comparison between two unrelated complexes.

Fix the positive lobes of the real atomic orbitals along the printed positive axes, and write ${\varphi }_1=p_x(1)$, ${\varphi }_4=p_x(4)$, ${\varphi }_5=p_y(5)$ and ${\varphi }_8=p_y(8)$. A coordinate model consistent with the figure is $r_1=(0,a,-b)$, $r_4=(0,-a,-b)$, $r_5=(a,0,b)$ and $r_8=(-a,0,b)$, with $a,b>0$. For the D2d molecular point group, take the two transverse twofold axes along $x+y$ and $x-y$, with vertical mirrors $xz$ and $yz$. The $A_2$ character table row on $(E,2S_4,C_2,2C_2^{\prime },2{\sigma }_d)$ is $(1,1,1,-1,-1)$. The character projection operator isThe eight signed images needed for this projection areso $P_{A_2}{\varphi }_1=({\varphi }_1-{\varphi }_4-{\varphi }_5+{\varphi }_8)/4$. Under the standard orthonormal atomic orbital approximation its squared norm is $4/16=1/4$. Therefore the normalized symmetry-adapted linear combination isAn overall minus sign describes the same quantum state; changing one local atomic orbital phase requires changing its coefficient too. If intersite orbital overlap were retained, normalization would instead use the overlap matrix.

Use a common Coulomb integral $\alpha$ on each site, a resonance integral $\beta$ on each adjacent pair, zero interaction across a diagonal, and the identity overlap matrix. In site order $(1,2,3,4)$ the Hückel method givesApplying it to the four symmetry-adapted linear combinations givesFor the conventional bonding sign $\beta <0$, the all-in-phase combination is lowest and the alternating combination is highest. The $E_u$ combinations have canceling nearest-neighbour interactions and are nonbonding within this model.

Treat the site atomic orbitals $s_1,\dots ,s_4$ as orthonormal, as in the Hückel method. A normalized set of symmetry-adapted linear combinations isThe first combination is unchanged by every operation; the alternating combination has the transformation signs of $x^2-y^2$ for the printed axes. The last two transform together like $(x,y)$. The squared coefficients sum to one in each case, and their pairwise inner products vanish. Any orthogonal rotation of the two $E_u$ combinations is an equally valid basis for that degenerate subspace.

Within each pair the in-phase and out-of-phase combinations have $A_1$ and $B_1$ symmetry respectively. One normalized choice isThe sign convention for $d$ is chosen to make the off-diagonal interaction positive in the corresponding two-by-two block. It does not change the physical predictions. These are pair symmetry-adapted linear combinations; the actual molecular orbitals are linear combinations within each of the two equal-symmetry pairs, found by diagonalizing the blocks.