LCAOs

Introduction

The covalent bonding in complexes is ignored in crystal field theory, which treats ligands as point-charged or dipoles. Rather, it considers the bonding to be solely ionic. The measurements like electron spin resonance or electron paramagnetic resonance, nuclear magnetic resonance, nuclear quadrupole resonance, and Racah parameters calculations using electronic spectra show that coordination compounds have covalent bonding.

The covalent bonding in complexes has been explained by ligand field theory (LFT). According to ligand field theory, covalent bonds between metal and ligands are formed by the linear combinations of the metal atomic orbitals and ligand group orbitals (LGOs). The ligand group orbitals’ symmetries must match the symmetries of the metal atomic orbitals, resulting in positive overlapping of LGOs with metal orbitals along the bonding axes. Based on orbital symmetry, predict which orbitals can mix to generate a molecule orbital and how many molecular orbitals will result from the interaction of one or more atomic orbitals.

What is the meaning of LCAOs?

A mathematical function that explains the wave-like motion of one or more electrons in an atom is known as an atomic orbital. This function may be used to compute the likelihood of finding any electron in any given area surrounding an atom’s nucleus. An orbital can also refer to the physical region in which the electron can be estimated to exist, as determined by the orbital’s specific mathematical shape.

Molecules are made up of two or more atoms that are bonded together. To characterise the electron orbitals of a molecule, the known orbitals of its constituent atoms can be combined. MOs are formed by merging atomic orbitals and signify locations in a molecule where an electron is likely to be located. An MO can define the electron configuration of a molecule, and it is most typically expressed as a linear combination of atomic orbitals (the LCAO-MO approach), especially in qualitative or approximate applications. Through molecular orbital theory, these models give a simplified description of molecule bonding.

Hydrogen’s Molecular Orbital Diagram An MO diagram effectively illustrates the energetics of the bond between the two atoms, whose AO unbonded energies are represented on the sides of a diatomic molecule. The unbonded energy levels are greater than those of the bound molecule, indicating the most energetically advantageous arrangement.

Linear Combination of Atomic Orbitals

A linear combination of atomic orbitals, or LCAO, is a quantum superposition of atomic orbitals and a method in quantum chemistry for computing molecular orbitals. In quantum mechanics, atomic electron configurations are represented as wave functions. In a mathematical sense, these wave functions are the fundamental functions that characterise the electrons of a specific atom. Chemical reactions modify orbital wave functions—the electron cloud changes—depending on the atoms involved in the chemical bond.

Sigma bonding in octahedral complexes

The ligands approach the metal cation in octahedral complexes on x-, y-, and z-axes. Therefore, LGOs will overlap with metal orbital along the octahedral bonding axes to form sigma bonds. In its valence shell, the metal cation has one ns, three np, and five (n-1)d-orbitals. The octahedral symmetry has transformed s-orbital into a1g, p-orbitals into t1u,dxy ,dyz, and dzx orbitals into t2g and dx2-y2and dz2 orbitals into eg sets.

π- Bonding in Octahedral Complexes

In addition to metal-ligand sigma interactions, many ligands with orbitals with -symmetry concerning octahedral axes are capable of -bonding interaction with the metal atom or cation. The -bonding will be significant if the metal and ligand orbital have proper -symmetry, comparable size, and energy. The metal and ligands orbitals involved in -bonding are perpendicular to the M-L axis. There are 12 ligands group orbitals in an octahedral complex capable of -interactions. These LGOs belong to four symmetry classes: t1g, t2g, t1u and t2u.

On the other hand, the transition metal cation in an octahedral complex has two types of orbitals (t1u and t2g) which are of correct symmetry for bonding. The t1g and t2u ligand group orbitals are nonbonding because there are no metal orbitals of the same symmetries. The metal orbitals of t1u symmetry are directed at the ligands and, therefore, involved in sigma bonding. Therefore, these orbitals are unavailable for -bonding. The t2g ligand group orbitals and the metal orbitals of the exact symmetry (i.e., t2g) can form metal-ligand – molecular orbitals (i.e., three bonding molecular orbitals t2g and three antibonding molecular orbitals t*2g ). The bonding molecular orbitals (t2g) are of lower energy, and the antibonding molecular orbitals (t*2g) are higher than the atomic orbitals. Both bonding and antibonding molecular orbitals are triply degenerate. In octahedral complexes, the ligand group orbitals corresponding to t2g symmetry may be p-, d-, *, or *. 

Sigma bonding in Tetrahedral complexes

The procedure for the constitution of molecular orbital diagrams for tetrahedral and square planar complexes is the same as that of octahedral complexes. In each case, the metal atom or ion uses its same nine valence orbitals available for bonding, but their symmetry properties are different for each geometry. The metal s and p-orbitals of a tetrahedral ML4 complex exhibit a1 and t2 symmetries, respectively. It is seen that the balances of p- orbitals and that of the first three-d-orbitals are identical, i.e., t2 symmetry.

Solving the Schrödinger wave equation for the molecule yields the wave function of the molecular orbital. Approximation methods are employed to get the wave function for molecular orbitals since solving the Schrödinger equation is too difficult. The linear combination of atomic orbitals is the most prevalent method (LCAO).

We already know that the wave function represents atomic orbitals. Assume that two atomic orbitals with equivalent energy, represented by the wave functions A and B, are merged to generate two molecular orbitals. The first is the bonding molecular orbital (bonding), while the second is the antibonding molecular orbital (antibonding). The linear combination of the wave functions for these two molecular orbitals yields the wave functions for these two molecular orbitals.

A and B atomic orbitals are shown below.

bonding = ψA + ψB

antibonding = ψA – ψB

The development of bonded molecular orbitals can be attributed to constructive interference of atomic orbitals. The destructive interference of atomic orbitals can result in the formation of an antibonding molecular orbital.

Conclusion

The atomic orbitals can be described by wave functions (‘s) that indicate the amplitude of electron waves. The solution to the Schrodinger wave equation yields these. However, because it cannot be solved for any system with more than one electron, obtaining molecular orbitals, which are one-electron wave functions for molecules, directly from the solution of the Schrodinger wave equation is problematic. An approximation known as a linear combination of atomic orbitals (LCAO) has been used to address this issue. It is worth noting that the energy of the antibonding orbital is more than the energy of the parent atomic orbitals that have merged, whereas the energy of the bonding orbital is less than that of the parent orbitals. However, the overall energy of two molecular orbitals is the same as that of two original atomic orbitals.