# Advanced Inorganic Chemistry/NH3 Molecular Orbitals
## Point Group of NH3

The symmetry elements of NH3 are E, 2C3, and 3 sigma-v. To elaborate, the molecule is of C3v symmetry with a C3 principal axis of rotation and 3 vertical planes of symmetry. The image of the ammonia molecule (NH3) is depicted in Figure 1 and the following character table is displayed below [2]:
| C3v | E | 2C3v | 3 σ$_v$ | | |
| A1 | 1 | 1 | 1 | z | x2+y2, z2 |
| A2 | 1 | 1 | -1 | Rz | |
| E | 2 | -1 | 0 | (x,y)(Rx,Ry) | (x2-y2,xy)(xz,yz) |

## The Construction of Molecular Orbitals of NH3

The Molecular Orbital Theory (MO) is used to predict the electronic structure of a molecule. Molecular orbitals are formed from the interaction of 2 or more atomic orbitals, and the interactions between atomic orbitals can be bonding, anti-bonding, or non-bonding. A bonding orbital is the interaction of two atomic/group orbitals in phase while an anti-bonding orbital is formed by out-of-phase combinations.
In general, the energy level of molecular orbitals increases from bonding, to non-bonding, and anti-bonding molecular orbitals. Pi-bonding molecular orbitals generally have greater energies than sigma-bonding molecular orbitals because the pi interactions are less effective than sigma interactions. The energy of molecular orbitals increases when the number of nodes also increases, and vice versa [6]. Within bonding molecular orbitals of the same symmetry, the lowest energy are from completely symmetrical sigma bonding molecular orbitals.

### Projection Operator Methode:

The Projection Operator Methode can be used to determine MO of NH3, the next steps can be used:
1) Determine the point group of molecular;
2) Label S orbital of H;
3) Generate a reducible representation (ᒥ) for H;
4) Reduce reducible representation to irreducible representation;
5) Generate the symmetry adapted linear combinations (SALCs) of orbitals that arise from these irreducible representations;
6) Drawing group orbital combinations and determine the atomic orbitals of the centeral atom;
7) MO

### Example

To proceed in constructing the molecular orbitals of NH3, one must first identify the symmetry adapted linear combinations (SALCs) of the 3 hydrogen 1s orbitals. This concept is depicted in Figure 2.
The reduced reducible representation (methodology and concepts further elaborated in Representations section of Advanced Inorganic Chemistry) of the sigma bonds is then written into linear combinations of irreducible representations of Γ$_{SH}$= a1 + e, as shown in the table. The irreducible representations are determined by the equation n$_i$ = 1/h Σ N$_{XR}$$_{XI}$ where n$_i$ is the coefficient of the ith reducible representation, X$_R$ and X$_I$ are characters of reducible and irreducible representations respectively, N is the coefficient in front of each symmetry elements found on the top row of the character table, and h is the order of the group and sum of the coefficients of the symmetry element symbols.
| C3v | E | 2C3 | 3 σ$_v$ |
| A1 | 1 | 1 | 1 |
| A2 | 1 | 1 | -1 |
| E | 2 | -1 | 0 |
| Γ$_{SH}$ | 3 | 0 | 1 |
From the a1 and e symmetry adapted linear combinations, the properties of transformation of the H orbitals are retained in rotational C3 subgroup and so, the C3 is then dropped as shown below:
| C3v | E | C3 | C3$^2$ | σ$_v$ | σ$_v$$^2$ | σ$_v$$^3$ |
| S$_1$ | S$_1$ | S$_2$ | S$_3$ | S$_1$ | S$_3$ | S$_2$ |
Under σ$_v$'s, S$_1$-S$_3$ are already accounted for in the rotation operators.
The applications and concepts of Symmetry Adapted Linear Combinations (SALCs), using projection operators can be found in SALCs and the projection operator technique section in Advanced Inorganic Chemistry. The ligands in Figure 2 are labeled σ$_1$, σ$_2$, and σ$_3$ for S$_1$, S$_2$, and S$_3$, and the normalized SALCs of the sigma orbitals of the ligand groups (Figure 3) can be determined from Γ$_{SH}$= a1 + e:
| C3v | E | C3 | C3$^2$ | σ$_v$ | σ$_v$$^2$ | σ$_v$$^3$ |
| S$_1$ | S$_1$ | S$_2$ | S$_3$ | S$_1$ | S$_3$ | S$_2$ |
From the table above and character table, the irreducible representation found (a1 and e) is taken and the SALCs of the respective ligands are derived:
1. SALC(A1)$_{S1}$ = S$_1$ + S$_2$ + S$_3$ + S$_1$ + S$_2$ + S$_3$
2. SALC(E)$_{S1}$ = 2S$_1$ - S$_2$ - S$_3$
3. SALC(E)$_{S2}$ = -S$_1$ + 2S$_2$ - S$_3$
4. SALC(E)$_{S3}$ = -S$_1$ - S$_2$ + 2S$_3$
Line A below is obtained by summing up S$_1$, S$_2$, S$_3$ of Line 1, resulting in 2S$_1$ + 2S$_2$ + 2S$_3$. Then, 2S$_1$ + 2S$_2$ + 2S$_3$ was reduced by 2 to give S$_1$ + S$_2$ + S$_3$. Line C below is achieved by subtracting line 3 from line 4, resulting in 0 + 3S$_2$ - 3S$_3$ before reducing it be 3 to give S$_2$ - S$_3$.
A. SALC(A1)$_{S1}$ = 2S$_1$ + 2S$_2$ + 2S$_3$ ≈ S$_1$ + S$_2$ + S$_3$
B. SALC(E)$_{S1}$ = 2S$_1$ - S$_2$ - S$_3$
C. SALC(E)$_{S2}$ - $_{S3}$ = 3S$_2$ - 3S$_3$ ≈ S$_2$ - S$_3$
The projection operator method is useful in visualizing group orbitals, as shown in Figure 4.

### NH3 MO Diagram

Molecular orbital diagram is useful in displaying and explaining the chemical bonds of molecules in conjunction with the molecular orbital theory. Formation of molecular orbitals involve interactions between atomic orbitals if their symmetries are compatible with each other, based on group theory. Linear combination of atomic orbitals can only occur with orbitals of similar symmetry. The overlapping orbitals of similar symmetry contributes to the construction of molecular orbitals. When there is a big enough overlap, splitting of the resulting molecular orbitals occurs.
In regards to NH3, the a1 SALC of the 3 H(1s) orbitals can only merge with a1 orbitals of nitrogen (such as the 2s and 2pz). The 2py nitrogen orbital can combine with the e1 SALC and 2px is merged with e2. The linear combination of atomic orbitals is drawn and detailed in Figure 4.
Ammonia or NH3 has 8 valence electrons, consisting of a lone pair on its nitrogen and 3 N-H sigma bonds. The molecular orbital diagram of NH3 is presented in Figure 5 and will be elaborated in regards to its interactions. The s orbitals for the 3 hydrogens are used to set up the sigma and anti bonding combinations of N sp$^3$ orbitals and the H 1s orbitals.
The H$_3$ 1s orbitals form an a1 and e combination. Bonding and anti-bonding interactions are made with N orbitals of similar symmetry. The a1 orbital that remained on N is higher in energy above the level of atomic orbital as the orbital interacts with a lower energy a1 bonding orbital. Thus, this orbital is considered non-bonding.
To take note, the a molecular orbitals are non-degenerate and symmetrical with respect to the rotation around z in an x, y, z axis. However, the bonding e orbitals are doubly degenerate orbitals of p$_x$ and p$_y$ as shown in Figure 6 [4].
The two frontier molecular orbitals of concern are the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO). Taking a look at the molecular orbital diagram of NH3, note that there are 2 electrons in the 2a1 orbital, making it the HOMO. On the other hand, the 3a1 is an anti-bonding molecular orbital and contains no electrons, making it the LUMO. Thus, these 2 orbitals are at the frontiers of the NH3 molecule [1].
In essence, it is valuable to understand and know what molecular orbitals are in terms of their structure, shape, and relative energies because they will ultimately determine the chemistry of the molecule. This helps chemists study and see how a molecule of interest, such as ammonia, reacts.
References
1. Adapted Linear Combinations. Chemistry LibreTexts. National Science Foundation. Sep. 02, 2017. https://chem.libretexts.org/Core/Physical_and_Theoretical_Chemistry/Spectroscopy/Vibrational_Spectroscopy/Vibrational_Modes/Symmetry_Adapted_Linear_Combinations
2. Greeves, N. SALC Ammonia - Molecular Orbitals. ChemTube3D. University of Liverpool. http://www.chemtube3d.com/SALC-ammonia-MO.htm
3. Raj, G.; Bhagi, A.; Jain, V. Group Theory and Symmetry in Chemistry, 3rd ed.; Satyendra Rastogi, 2010. pp 113-114. https://books.google.com/books?id=mTe07xK9f5gC&dq=Group+Theory+and+Symmetry+in+Chemistry,+3rd+ed.%3B+Satyendra+Rastogi&source=gbs_navlinks_s
4. Locke, W. Introduction to Molecular Orbital Theory. ICSTM Department of Chemistry. 1996-1997. http://www.ch.ic.ac.uk/vchemlib/course/mo_theory/
5. SALCS for Common Geometry. The State University of New York. 2018. http://employees.oneonta.edu/viningwj/chem342/SigmaMOdiagramsforTMs.pdf
6. James. The Pi Molecular Orbitals of Benzene. Master Organic Chemistry. 2018. https://www.masterorganicchemistry.com/2017/05/05/the-pi-molecular-orbitals-of-benzene/