User:Stacylee14/D4h molecular orbitals

D_4h 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.

D_4h 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 D_4h symmetry and therefore, will be utilized as an example to construct the molecular orbitals corresponding to the D_4h point group.

D_4h character table

The D_4h character table is comprised of representations that show the character of the matrix corresponding to each symmetry operation in the D_4h point group.

D_4h	E	2C₄	C₂	2C₂'	2C₂"	i	2S₄	σ_h	2σ_v	2σ_d	linear, rotations	quadratic
A_1g	1	1	1	1	1	1	1	1	1	1		x² + y², z²
A_2g	1	1	1	-1	-1	1	1	1	-1	-1	R_z
B_1g	1	-1	1	1	-1	1	-1	1	1	-1		x² - y²
B_2g	1	-1	1	-1	1	1	-1	1	-1	1		xy
E_g	2	0	-2	0	0	2	0	-2	0	0	(R_x, R_y)	(xz, yz)
A_1u	1	1	1	1	1	-1	-1	-1	-1	-1
A_2u	1	1	1	-1	-1	-1	-1	-1	1	1	z
B_1u	1	-1	1	1	-1	-1	1	-1	-1	1
B_2u	1	-1	1	-1	1	-1	1	-1	1	-1
E_u	2	0	-2	0	0	-2	0	2	0	0	(x, y)

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, A_1g). The p orbitals transform as the x, y, and z coordinates (in this point group, A_2u and E_u). The d orbitals transform according to the species of their corresponding direct product (in this point group, A_1g, B_1g, B_2g, and E_g).

Reducible representations for σ and π interactions

The reducible representations for both the σ bonding and π bonding interactions for [PdCl₄]^2- are found first by using the s, p_x and p_y orbitals of the four pendant chlorine atoms as the basis sets. The p_z orbitals transform in the exact same manner as the s orbitals and thus, have the same reducible and irreducible representations.^[1] 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:

D_4h	E	2C₂'	σ_h	2σ_v
Γ_σ(s)	4	2	4	2
Γ_{π(p_z)}	4	2	4	2
Γ_{π(p_x, p_y)}	8	-4	0	0
Γ_{σ + π(p_x, p_y)}	12	-2	4	2

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^[2]:

$n_{i}={1 \over h}\sum Nx_{R}x_{I}$

where:

n_i = 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)
X_R and X_I = 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:

Γ_σ = A_1g + B_1g + E_u

Γ_π = A_2g + A_2u +B_2g + B_2u + E_g + E_u

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

A_1g: s, d_z²

B_1g: d_x²-y²

B_2g: d_xy

E_g: (d_xz, d_yz)

A_2u: p_z

E_u: (p_x, p_y)

Symmetry adapted linear combinations (SALCs)

Once the irreducible representations comprising the reducible representations for both the σ and π bonding interactions are found, the symmetry adapted linear combinations (SALCs) of the atomic orbitals of the ligand are determined by using the projector operator technique.^[2]

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 Cl_a and Cl_b s orbitals and multiplying them by the characters from each irreducible representation obtained from the reduction formula.

D_4h	E	C₄	C₄	C₂	C₂'	C₂'	C₂"	C₂"	i	S₄	S₄	σ_h	σ_v	σ_v	σ_d	σ_d
P_a^A_1g	a	b	d	c	b	d	a	c	c	b	d	a	a	c	b	d
P_a^B_1g	a	(-)b	(-)d	c	b	d	(-)a	(-)c	c	(-)b	(-)d	a	a	c	(-)b	(-)d
P_a^E_u	(2)a	(0)b	(0)d	(-2)c	(0)b	(0)d	(0)a	(0)c	(-2)c	(0)b	(0)d	(2)a	(0)a	(0)c	(0)b	(0)d
P_b^E_u	(2)b	(0)c	(0)a	(-2)d	(0)a	(0)c	(0)d	(0)b	(-2)d	(0)c	(0)a	(2)b	(0)d	(0)b	(0)a	(0)c

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 E_u 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 (Cl_b in this case) and obtaining the respective transformations for all the symmetry elements and multiplying said transformations with the characters of the E_u representation.^[2] 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:

ΣP_a^A_1g Transformations = 4a + 4b + 4c + 4d

= a + b + c + d

=1/√4(a + b + c + d)

Ψ_{Α_1g} = 1/2 (a + b + c + d)

ΣP_a^B_1g Transformations = 2a - 2b + 2c - 2d

= a - b + c - d

= 1/√4 (a - b + c - d)

Ψ_{B_1g} = 1/2 (a - b + c - d)

ΣP_a^E_u Transformations = 4a - 4c

= a - c

Ψ_{E_u}^(a) = 1/√2 (a - c)

ΣP_b^E_u Transformations = 4b - 4d

= b - d

Ψ_{E_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 Cl_a and Cl_b p orbitals (p_x and p_y) and multiplying them by the characters from each irreducible representation obtained from the reduction formula.

D_4h	E	C₄	C₄	C₂	C₂'	C₂'	C₂"	C₂"	i	S₄	S₄	σ_h	σ_v	σ_v	σ_d	σ_d
P_{p_x^a}^A_2g	p_x^a	p_x^b	p_x^d	p_x^c	(-)-p_x^b	(-)-p_x^d	(-)-p_x^a	(-)-p_x^c	p_x^c	p_x^b	p_x^d	p_x^a	(-)-p_x^a	(-)-p_x^c	(-)-p_x^b	(-)-p_x^d
P_{p_y^a}^A_2u	p_y^a	p_y^b	p_y^d	p_y^c	(-)-p_y^b	(-)-p_y^d	(-)-p_y^a	(-)-p_y^c	(-)-p_y^c	(-)-p_y^b	(-)-p_y^d	(-)-p_y^a	p_y^a	p_y^c	p_y^b	p_y^d
P_{p_x^a}^B_2g	p_x^a	(-)p_x^b	(-)p_x^d	p_x^c	(-)-p_x^b	(-)-p_x^d	-p_x^a	-p_x^c	p_x^c	(-)p_x^b	(-)p_x^d	p_x^a	(-)-p_x^a	(-)-p_x^c	-p_x^b	-p_x^d
P_{p_y^a}^B_2u	p_y^a	(-)p_y^b	(-)p_y^d	p_y^c	(-)-p_y^b	(-)-p_y^d	-p_y^a	-p_y^c	(-)-p_y^c	-p_y^b	-p_y^d	(-)-p_y^a	p_y^a	p_y^c	(-)p_y^b	(-)p_y^d
P_{p_y^a}^E_g	(2)p_y^a	(0)p_y^b	(0)p_y^d	(-2)p_y^c	(0)-p_y^b	(0)-p_y^d	(0)-p_y^a	(0)-p_y^c	(2)-p_y^c	(0)-p_y^b	(0)-p_y^d	(-2)-p_y^a	(0)p_y^a	(0)p_y^c	(0)p_y^b	(0)p_y^d
P_{p_y^b}^E_g	(2)p_y^b	(0)p_y^c	(0)p_y^a	(-2)p_y^d	(0)-p_y^a	(0)-p_y^d	(0)-p_y^d	(0)-p_y^b	(2)-p_y^d	(0)-p_y^c	(0)-p_y^a	(-2)-p_y^b	(0)p_y^d	(0)p_y^b	(0)p_y^a	(0)p_y^c
P_{p_x^a}^E_u	(2)p_x^a	(0)p_x^b	(0)p_x^d	(-2)p_x^c	(0)-p_x^b	(0)-p_x^d	(0)-p_x^a	(0)-p_x^c	(-2)p_x^c	(0)p_x^b	(0)p_x^d	(2)p_x^a	(0)-p_x^a	(0)-p_x^c	(0)-p_x^b	(0)-p_x^d
P_{p_x^b}^E_u	(2)p_x^b	(0)p_x^c	(0)p_x^a	(-2)p_x^d	(0)-p_x^a	(0)-p_x^d	(0)-p_x^d	(0)-p_x^b	(-2)p_x^d	(0)p_x^c	(0)p_x^a	(2)p_x^b	(0)-p_x^d	(0)-p_x^b	(0)-p_x^a	(0)-p_x^c

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

ΣP_{p_x^a}^A_2g Transformations = 4p_x^a + 4p_x^b + 4p_x^c + 4p_x^d

= p_x^a + p_x^b + p_x^c + p_x^d

= 1/√4 (p_x^a + p_x^b + p_x^c + p_x^d)

Ψ_{A_2g} = 1/2 (p_x^a + p_x^b + p_x^c + p_x^d)

ΣP_{p_y^a}^A_2u Transformations = 4p_y^a + 4p_y^b + 4p_y^c + 4p_y^d

= p_y^a + p_y^b + p_y^c + p_y^d

= 1/√4 (p_y^a + p_y^b + p_y^c + p_y^d)

Ψ_{A_2u} = 1/2 (p_y^a + p_y^b + p_y^c + p_y^d)

ΣP_{p_x^a}^B_2g Transformations = 2p_x^a - 2p_x^b + 2p_x^c - 2p_x^d

= p_x^a - p_x^b + p_x^c - p_x^d

= 1/√4 (p_x^a - p_x^b + p_x^c - p_x^d)

Ψ_{B_2g} = 1/2 (p_x^a - p_x^b + p_x^c - p_x^d)

ΣP_{p_y^a}^B_2u Transformations = 2p_y^a - 2p_y^b + 2p_y^c - 2p_y^d

= p_y^a - p_y^b + p_y^c - p_y^d

= 1/√4 (p_y^a - p_y^b + p_y^c - p_y^d)

Ψ_{B_2u} = 1/2 (p_y^a - p_y^b + p_y^c - p_y^d)

ΣP_{p_y^a}^E_g Transformations = 4p_y^a - 4p_y^c

= p_y^a - p_y^c

Ψ_{E_g}^(a) = 1/√2 (p_y^a - p_y^c)

ΣP_{p_y^b}^E_g Transformations = 4p_y^b - 4p_y^d

= p_y^b - p_y^d

Ψ_{E_g}^(b) = 1/√2 (p_y^b - p_y^d)

ΣP_{p_x^a}^E_u Transformations = 4p_x^a - 4p_x^c

= p_x^a - p_x^c

Ψ_{E_u}^(a) = 1/√2 (p_x^a - p_x^c)

ΣP_{p_x^b}^E_u Transformations = 4p_x^b - 4p_x^d

= p_x^b - p_x^d

Ψ_{E_u}^(b) = 1/√2 (p_x^b - p_x^d)

D_4h 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^[2]
non-bonding molecular orbitals tend to have energies between those of bonding and antibonding molecular orbitals formed from similar atomic orbitals^[2]
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^[2]
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^[2]
among σ bonding molecular orbitals, those belonging to the totally symmetric representation tend to lie the lowest^[2]

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 pendant Cl atoms and Pd atom both have σ bonding with A_1g, B_1g, and E_u symmetry and π bonding with A_2u, B_2g, E_g, and E_u symmetry. The pendant Cl atoms also have π bonding interactions with A_2g and B_2u symmetry; however, the Pd atom does not have π bonding interactions with these two symmetries and thus, the A_2g and B_2u will remain as nonbonding orbitals. The highest occupied molecular orbital (HOMO) is the e_g with π antibonding symmetry. The lowest unoccupied molecular orbital (LUMO) is the b_1g with σ antibonding symmetry. It is important to note that B_2u is lower in energy than E_u because the interaction with the 4p is very weak whereas the generic D_4h molecular orbital diagram has B_2u higher in energy than E_u.

[PdCl₄]^2- Molecular Orbitals

References

^[2]

Category:Chemical bonding Category:Ligand field theory

^ Miessler, Gary (2014). Inorganic chemistry (Fifth edition ed.). Upper Saddle River, New Jersey: Pearson. ISBN 9781269453219. {{cite book}}: |edition= has extra text (help)
^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ Pfennig, Brian (2015). Principles of Inorganic Chemistry. Hoboken, New Jersy: John Wiley & Sons, Inc. ISBN 9781118859100.

[Miessler-1] Miessler, Gary (2014). Inorganic chemistry (Fifth edition ed.). Upper Saddle River, New Jersey: Pearson. ISBN 9781269453219. {{cite book}}: |edition= has extra text (help)

[Pfennig-2] ^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ Pfennig, Brian (2015). Principles of Inorganic Chemistry. Hoboken, New Jersy: John Wiley & Sons, Inc. ISBN 9781118859100.

[1]

[2]

D4h point group

D4h character table