The LCAO MO Approximation method, or Linear Combination of Atomic Orbitals, is a quantum mechanical method to approximate molecular orbitals. It states that as an electron approaches a certain nucleus, its wave function behaves as if it were an atomic orbital. This method simplifies the Schrödinger equation for molecules. The LCAO method approximates the wave function of a molecule as a linear combination of the wave functions of its constituent atoms.
What is the LCAO MO Approximation in Quantum Chemistry?
LCAO MO Approximation is the mathematical basis for the development of the modern form of the Molecular Orbital Theory. In quantum chemistry, the Schrödinger equation is extremely difficult to solve for a molecule that has more than one electron due to the repulsions that exist between the negatively charged particles. This is why the LCAO MO Approximation is a “workaround” for the problem. The LCAO MO Approximation is a hypothesis that the molecular orbitals that form a molecule are a product of the “constructive and destructive interference” of the atomic orbitals that combine to form the molecule.
“The LCAO MO hypothesis is that the atomic orbitals from individual atoms combine linearly when the atoms bond to form a molecule. If the electron is close to a nucleus labeled A, it is best described by the atomic orbital on A. If the electron is close to a nucleus labeled B, it is best described by the atomic orbital on B. Therefore, the molecular wave function is a weighted sum or linear combination of these individual parts.”
Essential LCAO Conditions for Effective Bonding
In order for the result from the LCAO MO approximation to be stable, certain criteria regarding the interacting atomic orbitals must be satisfied. Not all atomic orbitals are able to combine and form a bond; the result from the LCAO MO approximation is heavily dependent upon three important factors: the energy, symmetry, and overlapping capabilities of the orbitals.
1. Symmetry Rules Compatibility
The combining atomic orbitals must have the same symmetry properties relative to the molecular axis. For example, a $p_z$ orbital of one atom can combine with a $p_z$ orbital of another (assuming $z$ is the internuclear axis) because they share cylindrical symmetry. However, a $p_z$ orbital cannot combine with a $p_y$ orbital effectively under the LCAO MO approximation because their positive and negative lobes would cancel out, leading to zero net overlap.
2. Comparable Energy Levels
The LCAO MO approximation dictates that atomic orbitals can only combine effectively if their energies are similar. A $1s$ orbital from one atom will readily combine with a $1s$ orbital from another identical atom because their energies are identical. However, a $1s$ orbital will not combine significantly with a $3s$ orbital of the same atom or a much higher energy orbital of a different atom. A large energy gap prevents effective mixing, rendering the LCAO MO approximation ineffective for such pairings.
3. Maximum Extent of Overlap
The strength of the bond formed under the LCAO MO approximation is directly proportional to the extent of the overlap between the atomic orbitals. Greater spatial overlap results in a greater decrease in potential energy, leading to a more stable bond. This condition emphasizes that atoms must be close enough for their wavefunctions to interpenetrate significantly.
Derivation of the LCAO Method
The mathematical derivation of the LCAO MO approximation uses the variation method to find the coefficients that minimize the energy of the system. Consider a simple diatomic molecule AB. We approximate the molecular orbital ($\Psi$) as a linear combination of the atomic orbital of atom A ($\phi_A$) and atom B ($\phi_B$):
$$\Psi = c_A\phi_A + c_B\phi_B$$
Here, $c_A$ and $c_B$ are the MO coefficients (weighting factors) that determine the contribution of each atomic orbital to the molecular orbital. To find the energy $E$ associated with $\Psi$, we use the Schrödinger equation $H\Psi = E\Psi$. Multiplying by $\Psi$ and integrating over all space gives:
$$E = \frac{\int \Psi H \Psi \, d\tau}{\int \Psi^2 \, d\tau}$$
Substituting the expansion for $\Psi$:
$$E = \frac{\int (c_A\phi_A + c_B\phi_B) H (c_A\phi_A + c_B\phi_B) \, d\tau}{\int (c_A\phi_A + c_B\phi_B)^2 \, d\tau}$$
To solve this within the LCAO MO approximation, we use the Variation Principle, which states that we must select coefficients $c_A$ and $c_B$ such that the energy $E$ is minimized ($\partial E / \partial c_A = 0$ and $\partial E / \partial c_B = 0$). This leads to a system of secular equations involving Coulomb integrals ($\alpha$), Resonance integrals ($\beta$), and the overlap integral ($S$).
The Role of the Overlap Integral in LCAO
The overlap integral ($S$) is a critical component in the LCAO MO approximation that quantifies how much two atomic orbitals share the same region of space. Mathematically, it is defined as:
$$S = \int \phi_A \phi_B \, d\tau$$
The value of the overlap integral determines the nature of the interaction in the LCAO MO approximation:
- Positive Overlap ($S > 0$): Occurs when lobes of the same sign (phase) overlap. This leads to an accumulation of electron density between nuclei, resulting in a bonding molecular orbital.
- Negative Overlap ($S < 0$): Occurs when lobes of opposite signs overlap. This creates a node (zero electron density) between nuclei, resulting in an antibonding molecular orbital.
- Zero Overlap ($S = 0$): Occurs when there is no spatial interaction or when symmetry forbids interaction (e.g., $s$ orbital and $p_x$ orbital). No bond is formed.
In many simplified applications of the LCAO MO approximation (like basic Hückel theory), the overlap integral is often approximated as zero for neighbors ($S_{ij} = \delta_{ij}$), but in accurate calculations, explicitly calculating $S$ is vital for correct energy levels.
Constructing Bonding and Antibonding Orbitals
Using the LCAO MO approximation, we generate two distinct types of molecular orbitals from two atomic orbitals: Bonding and Antibonding. This is a direct consequence of the wave nature of electrons.
Bonding Molecular Orbitals (BMO)
When atomic wavefunctions are added ($\Psi_+ = N(\phi_A + \phi_B)$), constructive interference occurs. The electron probability density increases between the nuclei. According to the LCAO MO approximation, this stabilizes the system because the electrons are attracted to both nuclei simultaneously, lowering the total energy relative to the isolated atoms.
Antibonding Molecular Orbitals (ABMO)
When atomic wavefunctions are subtracted ($\Psi_- = N(\phi_A – \phi_B)$), destructive interference occurs. The LCAO MO approximation predicts a nodal plane between the nuclei where the probability of finding an electron is zero. The electrons are excluded from the internuclear region, leading to repulsion between the nuclei and a higher energy state compared to the isolated atoms.
Application: The H2+ Molecule Example
The hydrogen molecular ion ($H_2^+$) is the simplest system to apply the LCAO MO approximation because it consists of two protons and only one electron. It serves as the standard “test bed” for checking the validity of molecular quantum mechanics.
In the H2+ example, the LCAO MO approximation constructs the wavefunction using the $1s$ orbitals of two hydrogen atoms ($1s_A$ and $1s_B$). The trial function is $\Psi = c_1 1s_A + c_2 1s_B$. Since the atoms are identical, symmetry dictates that $c_1^2 = c_2^2$.
This leads to two normalized solutions:
- Bonding: $\Psi_g = \frac{1}{\sqrt{2(1+S)}} (1s_A + 1s_B)$
- Antibonding: $\Psi_u = \frac{1}{\sqrt{2(1-S)}} (1s_A – 1s_B)$
The LCAO MO approximation successfully predicts the bond length and binding energy of $H_2^+$, although the calculated energy is slightly higher (less negative) than the exact experimental value. This discrepancy highlights the approximate nature of the method—it restricts the electron to looking like an atomic $1s$ orbital, whereas in reality, the orbital shape distorts (polarizes) as the nuclei approach.
Limitations and Critical Analysis of LCAO
Although the LCAO MO approximation is a cornerstone in chemistry, it can be considered an oversimplification in a technical sense. If a critical analysis is made, it can be found that the basic assumption that molecular orbitals are a combination of unperturbed atomic orbitals does not really apply.
The major problem associated with the basic LCAO MO approximation is that it does not take into account the contraction and polarization that occur in orbitals during bond formation. During bond formation, the electron density changes, and hence the shapes of the orbitals also change. The basic LCAO MO approximation does not take this change into account and hence leads to inaccurate results.
To solve this problem, modern computational chemistry uses an advanced form of the LCAO MO approximation, which includes “polarization functions” and “diffuse functions” in the basic set. This allows the mathematical model to be more flexible and hence corrects the basic problem associated with the LCAO MO approximation, which is its rigidity. Thus, it can be said that although is perfect in a conceptual sense, it needs to be augmented significantly to obtain results that can be considered “chemically accurate.”
10 GATE Style Questions on LCAO MO Theory
For students preparing for competitive exams, mastering the LCAO MO approximation requires solving numerical and conceptual problems. These questions cover MO coefficients, normalization, and energy levels.
Q1. In the LCAO MO approximation for the $H_2^+$ ion, if the overlap integral $S=0$, what is the normalization constant for the bonding orbital $\Psi = N(\phi_A + \phi_B)$?
- Answer: $1/\sqrt{2}$
Q2. Which quantum mechanical principle is used to determine the mixing coefficients in the LCAO MO approximation?
- Answer: The Variation Principle.
Q3. According to the LCAO MO approximation, what is the value of the resonance integral ($\beta$) for a stable bond?
- Answer: $\beta$ is negative ($\beta < 0$).
Q4. Identify the condition that renders the LCAO MO approximation invalid for combining a $1s$ and a $2s$ orbital of the same atom.
- Answer: Large energy difference (Energy mismatch).
Q5. In the Secular Determinant for ethene derived via the LCAO MO approximation, what does the term $(\alpha – E)$ represent?
- Answer: The Coulomb integral minus the orbital energy.
Q6. For a heteronuclear diatomic molecule AB where atom B is more electronegative, how does the LCAO MO approximation distribute the coefficients $c_A$ and $c_B$ in the bonding MO?
- Answer: $c_B > c_A$ (The electron spends more time near the more electronegative atom).
Q7. What is the bond order of the $He_2$ molecule according to the LCAO MO approximation?
- Answer: Zero (Bonding and antibonding effects cancel out).
Q8. In the LCAO MO approximation, how does the energy of the antibonding orbital compare to the bonding orbital relative to the atomic energy levels?
- Answer: The antibonding orbital is raised in energy by an amount greater than the bonding orbital is lowered (due to nuclear repulsion and $S > 0$).
Q9. Calculate the normalization constant $N$ for $\Psi = N(\phi_A – \phi_B)$ if $S=0.2$.
- Answer: $N = 1/\sqrt{2(1-0.2)} = 1/\sqrt{1.6} \approx 0.79$.
Q10. Why does the LCAO MO approximation predict that $p_x$ and $p_y$ orbitals do not overlap to form a bond along the z-axis?
- Answer: The overlap integral $S$ is zero due to orthogonal symmetry.
