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D4h molecular orbitals are comprised of bonding, antibonding, and nonbonding molecular orbitals that arise from the interaction of atomic orbitals. These interactions between the atomic orbitals can be further classified by the symmetry of the interaction between the resulting molecular orbitals.

D4h point group
Each point group has a unique character table that is comprised of a unique set of symmetry operations that are present within the respective point group. The character table of a point group is a collection of irreducible representations and the characters of the matrices associated with them. [PdCl4]2- has D4h symmetry and therefore, will be utilized as an example to construct the molecular orbitals corresponding to the D4h point group.

D4h character table
The D4h character table is comprised of representations that show the character of the matrix corresponding to each symmetry operation in the D4h point group.

The s orbitals are symmetric with respect to all symmetry operations and transform as the totally symmetric representation, listed first in the character table (in this point group, A1g). The p orbitals transform as the x, y, and z coordinates (in this point group, A2u and Eu). The d orbitals transform according to the species of their corresponding direct product (in this point group, A1g, B1g, B2g, and Eg).

Reducible representations for σ and π interactions
In order to construct the molecular orbital diagram for [PdCl4]2-, the reducible representations for both the σ bonding and π bonding interactions must be found first. The reducible representations for the σ and π bonding interactions can be found using the s, px and py orbitals of the four pendant chlorine atoms as the basis sets. The pz orbitals transform in the exact same manner as the s orbitals and thus, have the same reducible and irreducible representations. In order to generate a reducible representation, whether orbitals are shifted or non-shifted by each class of operations of the group must be noted. Each orbital shifted through space contributes 0 to the character for the class. Each non-shifted orbital contributes 1 to the character of the class. An orbital shifted into the negative of itself contributes -1 to the character for the class.

The reducible representations are found to be:

Irreducible representations for σ and π interactions
Each reducible representation gives rise to only one set of irreducible representations. From the reducible representations of the σ and π bonding interactions, the irreducible representations can be found via the reduction formula:

$$n_i = {1\over h}\sum N x_R x_I$$

where:


 * ni = the number of times the irreducible representation i occurs in the reducible representation
 * N = the coefficient in front of each symmetry element symbol
 * h = the order of the group (the sum of the coefficients N, h = ΣN)
 * XR and XI = the characters of the reducible and irreducible representations respectively

Using this reduction formula, the irreducible representations for the σ and π bonding interactions are found to be:

Γσ = A1g + B1g + Eu

Γπ = A2g + A2u +B2g + B2u + Eg + Eu

The metal orbital symmetries can be found from the last 2 columns of the character table and are as follows:

A1g: s, dz2

B1g: dx2-y2

B2g: dxy

Eg: (dxz, dyz)

A2u: pz

Eu: (px, py)

Symmetry adapted linear combinations (SALCs)
After finding the irreducible representations comprising the reducible representations for both the σ and π bonding interactions, the symmetry adapted linear combinations (SALCs) of the atomic orbitals of the ligand can be found by using the projector operator technique.

SALCs for s orbitals
In this [PdCl4]2- molecule, the SALCs for the s orbitals of Chlorine atoms can be found via applying the projector operator technique on the Chlorine atoms and finding the transformations of the Cla and Clb s orbitals and multiplying them by the characters from each irreducible representation obtained from the reduction formula.

The P denotes the performance of the projector operator technique and the subscript letter represents which Chlorine atom the projector operator was performed on and the superscript irreducible representation indicates the characters of that respective irreducible representation that was multiplied. The parentheses within the boxes represent the multiplication of each character to the respective transformation.

Because Eu is doubly degenerate another equation that is orthogonal to the first one must be found via performing the exact same projector operator technique on a different Chlorine atom (Clb in this case) and obtaining the respective transformations for all the symmetry elements and multiplying said transformations with the characters of the Eu representation. The SALC for an irreducible representation can be obtained by addition of all the transformations in each row (each irreducible representation) and normalization via the formula:

$$N = {1 \over \sqrt { \sum c_i ^2}} $$

where:


 * N = normalizing factor
 * c = coefficient of each respective transformation

The SALCs for the s orbitals of the Chlorine atoms (denoted as ψ) are as follows:

ΣPaA1g Transformations = 4a + 4b + 4c + 4d

= a + b + c + d

=1/√4(a + b + c + d)

ΨΑ 1g = 1/2 (a + b + c + d)

ΣPaB1g Transformations = 2a - 2b + 2c - 2d

= a - b + c - d = 1/√4 (a - b + c - d) ΨB 1g = 1/2 (a - b + c - d)

ΣPaEu Transformations = 4a - 4c

= a - c ΨE u (a) = 1/√2 (a - c)

ΣPbEu Transformations = 4b - 4d

= b - d ΨE u (b) = 1/√2 (b - d)

SALCs for p orbitals
Once the SALCs for the s orbitals of Chlorine atoms are found, the SALCs for the p orbitals of the Chlorine atoms can be found by applying the exact same projector operator technique and finding the transformations of the Cla and Clb p orbitals (px and py) and multiplying them by the characters from each irreducible representation obtained from the reduction formula.

The SALCs for the p orbitals of the Chlorine atoms (denoted as ψ) are as follows:

ΣPp xa A2g Transformations = 4pxa + 4pxb + 4pxc + 4pxd

= pxa + pxb + pxc + pxd

= 1/√4 (pxa + pxb + pxc + pxd)

ΨA 2g = 1/2 (pxa + pxb + pxc + pxd)

ΣPp ya A2u Transformations = 4pya + 4pyb + 4pyc + 4pyd

= pya + pyb + pyc + pyd

= 1/√4 (pya + pyb + pyc + pyd)

ΨA 2u = 1/2 (pya + pyb + pyc + pyd)

ΣPp xa B2g Transformations = 2pxa - 2pxb + 2pxc - 2pxd

= pxa - pxb + pxc - pxd

= 1/√4 (pxa - pxb + pxc - pxd)

ΨB 2g = 1/2 (pxa - pxb + pxc - pxd)

ΣPp ya B2u Transformations = 2pya - 2pyb + 2pyc - 2pyd

= pya - pyb + pyc - pyd

= 1/√4 (pya - pyb + pyc - pyd)

ΨB 2u = 1/2 (pya - pyb + pyc - pyd)

ΣPp ya Eg Transformations = 4pya - 4pyc

= pya - pyc

ΨE g (a) = 1/√2 (pya - pyc)

ΣPp yb Eg Transformations = 4pyb - 4pyd

= pyb - pyd

ΨE g (b) = 1/√2 (pyb - pyd)

ΣPp xa Eu Transformations = 4pxa - 4pxc

= pxa - pxc

ΨE u (a) = 1/√2 (pxa - pxc)

ΣPp xb Eu Transformations = 4pxb - 4pxd

= pxb - pxd

ΨE u (b) = 1/√2 (pxb - pxd)

D4h molecular orbital diagram
It is important to note that when constructing molecular orbital diagrams:


 * bonding molecular orbitals always lie lower in energy than the antibonding molecular orbitals formed from the same atomic orbitals
 * non-bonding molecular orbitals tend to have energies between those of bonding and antibonding molecular orbitals formed from similar atomic orbitals
 * π interactions tend to have less effective overlap than σ interactions and therefore, π bonding molecular orbitals tend to have higher energies than σ bonding molecular orbitals formed from similar atomic orbitals
 * molecular orbitals energies tend to rise as the number of nodes increases and therefore, molecular orbitals with no nodes tend to lie lowest in energy and those with the greatest number of nodes tend to lie the highest in energy
 * among σ bonding molecular orbitals, those belonging to the totally symmetric representation tend to lie the lowest

[PdCl4]2- molecular orbital diagram


Within [PdCl4]2-, each Cl- atom has 6 valence electrons in its p orbitals. The 4 Cl- atoms contribute a total of 24 valence electrons and the Pd2+ atom has 8 valence electrons in its d orbitals and therefore, the [PdCl4]2- molecule has 32 valence electrons total. The highest occupied molecular orbital (HOMO) is the eg with π antibonding symmetry. The lowest unoccupied molecular orbital (LUMO) is the b1g with σ antibonding symmetry. It is important to note that B2u is lower in energy than Eu because the interaction with the 4p is very weak whereas the generic D4h molecular orbital diagram has B2u higher in energy than Eu.

[PdCl4]2- Molecular Orbitals

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