Meta-Ligand Bonding for Oh Complexes only Sigma interaction

Metal Ligand Bonding 

Sigma bonding in octahedral complexes:

•Let us apply simple MO theory to the σ-bonding of a regular octahedral complex M-L belonging to point group.

•A four step process will illustrate how MO theory can be applied to these metal complexes qualitatively through group theories formalism developed until now. The primary concern of MO theory is orbital - overlap leading to a significant degree of covalency of metal-ligand bond unlike that of crystal field theory.

•In an ideal model of the octahedral complex M-L the metal ion M is surrounded by six identical ligands L and each of these ligands is considered to contribute one type orbital in the bonding direction.

•This orbital maybe a simple s or p orbital or may even be a suitable hybrid orbital (Sp, Sp2 or Sp3) whose major lobe is pointing towards the central metal ion.

•The atomic orbitals on the central metal ion may be ns, np and (n-1)d orbitals which together are called valence orbitals.

•An overlap between these metal orbitals and ligand, orbitals have to be considered in terms of group theoretical analysis.

STEP-1 Classification of metal valence orbitals into Sigma symmetry:

•We already know how to determine which AOs on the metal ion may be combined to produce a set of six equivalent hybrid orbitals which in turn may overlap with the orbitals on the ligands situated at six corners of an octahedron.

•We can recall the process once again using the Figure - 1 showing the coordinate system with appropriate designations of the σ-hybrid orbitals.

•It can be shown that the character system of the six vectors works out in Point group as:

•Of the nine valence orbitals, the orbitals required for σ - bonding hybridization are only six with a1g, eg and t1u symmetry as in :

•A1g : s 

•T1u : Px, Py, Pz

•Eg : dz2, dx2-y2

•Thus the hybrid combination required is d2sp3.

•The remaining three d-orbitals (dxy, dyz, dxz) symmetry remain as non-bonding when σ - only interaction is considered.

STEP-2 Formation of Ligand Group Orbitals (LGOs):

• The combination of ligand orbitals which have the same symmetry as the metal σ-hybrid orbitals can be worked out by 'symmetry adapted linear combination of Aos’ (SALC of AOs).
• Such combinations when normalized yield what are called "Ligand Group Orbitals" (LGOs).
• Alternately, they can be also called as SALCs.
• These LGOs can be constructed alternately and by a much simpler and non-mathematical method as illustrated below.
• The problem now is to construct a linear combination of the ligand orbitals which correspond to the same IRs as the metal ion hybrid orbitals.
• Assuming that the ligand σ - orbital (whether a simple p-orbital or a suitable hybrid orbital) is always oriented to the central metal ion with its +ve lobe, we can construct LGOs with various central metal ion AOs as shown in the following table.

dz2 orbitals is taken as d2z2-x2-y2. All the LGOs are Normalized.

A pictorial approach can be used to obtain the LGO  

combinations:

•The ligand orbital combination with the metal s orbital which has +ve sign everywhere interacting positively with σ1,σ2, σ3, σ4, σ5 and σ6 is pictured in above Figure.

•The wave function ψa_(1g,)corresponding to this proper linear combination can be given by :             ψa1g = 1/√6 (σ1+σ2+σ3+σ4 +σ5+σ6)

•The metal Px,  orbital has a lobe with the +ve sign (conventionally shown towards the +ve direction of the X-axis or L1 - L3, axis) and another lobe with a -ve sign [Figure (b)]

•A proper linear combination of this orbital with the appropriate L1 and L3 hybrid orbitals (whose major lobes point to the metal and are always taken as the) can be given by the wave function.        ψt1u = 1/√2 (σ1 - σ3)

•Similarly, we can show the wave functions corresponding to LCAOs of the ligand σ-hybrids overlapping with the other metal Py Pz. dx2-y2 [Figure (c)] and dz2 [Figure (d)] as shown under LGO column of the table.

STEP-3 Formation of molecular orbitals: 

•The metal ion valence hybrid orbitals and the ligand group orbitals of σ-symmetry have the right symmetry to further combine to give the corresponding molecular orbitals.

•Each of these combinations results in a pair of MOs, one of bonding and the other of anti-bonding in nature.

•The bonding MOs are formed when the σ-LGOS and the metal ion orbitals combine with a maximum positive overlap (addition) and the anti-bonding MOs are produced when the combination occurs with a maximum -ve overlap (subtraction).

•Mathematically this can be expressed, say for a1g-type, as:

•ψA_1g= N1[a1g(S) + 1/√6 (σ1+σ2+σ3+σ4 +σ5+σ6)]

•Ψ*A_1g= N1[a1g(S) - 1/√6 (σ1+σ2+σ3+σ4 +σ5+σ6)]

•Where ψA1g  and ψ*A_1g are the wave functions for bonding and antibonding MOs respectively.

• §Similarly, for MOs of t1u [ψt1u(Px), ψt1u(Py), ψt1u(Pz)] and
• §eg [ψEg(dx2 - y2), ψEg(dz2)] type, appropriate wave functions can be written as:
• §ψT1u(Px)/ ψ*T1u(Px) = N2[t1u(Px) ± 1/√2 (σ1 - σ3)]
• §ψT1u(Py)/ ψ*T1u(Py) = N3[t1u(Py) ± 1/√2 (σ2 – σ4)]
• §ψT1u(Pz)/ ψ*T1u(Pz) = N4[t1u(Pz) ± 1/√2 (σ5 – σ6)]
• §ψEg(dx2-y2)/ ψ*Eg(dx2-y2) = N5[eg(dx2-y2) ± 1/2 (-σ1 - σ2 + σ3 + σ4)]
• §ψEg(dz2)/ ψ*Eg(dz2) = N6[eg(dz2) ± 1/√12 (- σ1 - σ2 - σ3 - σ4 +2σ5 + 2σ6 )]
• 
  

STEP - 4 Construction of Molecular Orbital Energy Level Diagram (MOED):

• A qualitative energy level diagram (MOED) for σ-bonding only is shown in Figure 2 for an octahedral complex such as [Co(NH3)6]3+.
• But in order to do we must set up and solve the various secular determinants.
• The methods used to such secular equations are too much involved mathematically and hence are considered further.
• Thus, we can only build a qualitative energy level diagram showing the logical ordering of MOs as shown in the Figure 2, with the assumption that: σ-LGOg < (n-1)d < ns < np.
• In this case the T2g, set (dxy, dyz, dzx) of metal orbitals are shown as non-bonding.
• ns and np valence orbitals of the metal ion have been shown to lie above the (n-1)d orbitals because in a complex, the metal atom generally has +ve charge.
• In the free metal atom the ordering may not be same as that in the ions.
• Also, in a metal complex, the metal valence orbitals are energetically less stable than most of the ligand’s σ - valence orbitals (LGOs).
• The diagram presented in Figure 2 has been constructed incorporating all these criteria.
• However, we should remember a general thumb rule that when two orbitals of same symmetry with as large a difference in energy as 2-3 eV are combined, the two resulting MOs are quite different in character.
• The lower energy bonding MO assumes primarily the character of lower energy component orbital and the higher energy antibonding MOs have the character of higher energy component orbital.
• Hence in the octahedral complex the six bonding MOs of Figure 2 are primarily of ligand in character and thus can be filled with 12 donor electrons from the ligands.
• The remaining electrons from the metal ion can then be filled starting from the next highest non-bonding T2g  MO.
• Thus, we can attribute various degrees of metal and ligand character to the MOs. Though A1g, Eg and T1g MOS have largely ligand character, they have aquired some of the metal orbital character from which they are derived. 

•On the other hand, though the E*g, T*1u and A*1g MOs have mainly the metal orbital character, they are somewhat delocalized onto the ligands.

•Only the non-bonding T2g MO has pure metal d-orbital character.

•The energy gap 10D (or Δo) between T2g and Eg MOs is very important and the magnitude of this parameter directly reflects upon the ligand field strength.

•From the MOED presented in Figure 2, it is also clear that the various electronic transitions involving A1g, Eg,T1u, A*1g, T*1u, are also possible in addition to the lone T2g → Eg transition as implied from simple crystal field theory.

MOED OF OCTAHEDRAL COMPLEX ONLY SIGMA 

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