Question

Given that the atomic orbitals used to form hybrids are normalized to 1 and mutually orthogonal, (a) show that the two tetrahedral hybrids \(h_{1}=\mathrm{s}+\mathrm{p}_{x}+\mathrm{p}_{y}+\mathrm{p}_{z}\) and \(h_{3}=\mathrm{s}-\mathrm{p}_{x}+\) \(\mathrm{p}_{y}-\mathrm{p}_{z}\) are orthogonal. (b) Construct the remaining two tetrahedral hybrids that are orthogonal to these two hybrids. Hint: Two wavefunctions are orthogonal if \(\int \psi_{1} \psi_{2} \mathrm{~d} \tau=0\), where \(\int \ldots \mathrm{d} \tau\) means "integrate over all space."

Short answer

The overlap integral of \(h_1\) and \(h_3\) is zero, proving their orthogonality. The other two orthogonal tetrahedral hybrids are \(h_2 = \mathrm{s} - \mathrm{p}_x - \mathrm{p}_y + \mathrm{p}_z\) and \(h_4 = \mathrm{s} + \mathrm{p}_x - \mathrm{p}_y - \mathrm{p}_z\).

Step-by-step solution

Understand Orthogonality

Two wavefunctions, or in this context, two hybrid orbitals, are considered orthogonal if their overlap integral over all space is equal to zero. Mathematically, for two hybrids \(h_1\) and \(h_3\), they are orthogonal if \(\int h_1 h_3 \, \mathrm{d} \tau = 0\). The integral represents the sum of the product of their values at all points in space.

Represent the Tetrahedral Hybrids

Write out the expressions for the tetrahedral hybrids \(h_1\) and \(h_3\) as given by: \[h_1 = \mathrm{s} + \mathrm{p}_x + \mathrm{p}_y + \mathrm{p}_z\] and \[h_3 = \mathrm{s} - \mathrm{p}_x + \mathrm{p}_y - \mathrm{p}_z\].

Calculate the Overlap Integral

Calculate the overlap integral, which involves multiplying the two hybrids and integrating over all space. \[\int h_1 h_3 \, \mathrm{d} \tau = \int (\mathrm{s} + \mathrm{p}_x + \mathrm{p}_y + \mathrm{p}_z)(\mathrm{s} - \mathrm{p}_x + \mathrm{p}_y - \mathrm{p}_z) \, \mathrm{d} \tau\].

Expand the Product

Expand the product in the integrand to get: \[\int (\mathrm{s}^2 - \mathrm{s}\mathrm{p}_x + \mathrm{s}\mathrm{p}_y - \mathrm{s}\mathrm{p}_z + \mathrm{p}_x\mathrm{s} - \mathrm{p}_x^2 + \mathrm{p}_x\mathrm{p}_y - \mathrm{p}_x\mathrm{p}_z + \mathrm{p}_y\mathrm{s} + \mathrm{p}_y\mathrm{p}_x - \mathrm{p}_y^2 + \mathrm{p}_y\mathrm{p}_z + \mathrm{p}_z\mathrm{s} - \mathrm{p}_z\mathrm{p}_x + \mathrm{p}_z\mathrm{p}_y - \mathrm{p}_z^2) \, \mathrm{d} \tau\].

Utilize Orthogonality and Normalization

Given the atomic orbitals are normalized and mutually orthogonal, terms where different atomic orbitals are multiplied (like \(\mathrm{s}\mathrm{p}_x, \mathrm{p}_y\mathrm{p}_z\), etc.) will integrate to zero, and terms where the same atomic orbitals are multiplied (like \(\mathrm{s}^2, \mathrm{p}_x^2\)) will integrate to one. Simplify the integral by eliminating the cross terms and adding up the same orbital terms.

Verify Orthogonality of \(h_1\) and \(h_3\)

After simplifying, it is apparent that the integral over all space will equal zero since the positive and negative terms will cancel each out. This verifies that \(h_1\) and \(h_3\) are orthogonal.

Construct Remaining Hybrids

To construct the remaining two tetrahedral hybrids that are orthogonal to \(h_1\) and \(h_3\), maintain the same magnitudes but change signs to ensure orthogonality. The two additional hybrids could be: \[ h_2 = \mathrm{s} - \mathrm{p}_x - \mathrm{p}_y + \mathrm{p}_z \] and \[ h_4 = \mathrm{s} + \mathrm{p}_x - \mathrm{p}_y - \mathrm{p}_z \].

Key concepts

Tetrahedral Hybrid Orbitals

The concept of tetrahedral hybrid orbitals fundamentally originates from the need to explain how carbon, with its four equivalent bonds, forms substances with equally angled bond geometries, such as in methane (CH₄). At the heart of this concept is the process of hybridization, where atomic orbitals within an atom, typically the s and p orbitals, mix to create new hybrid orbitals.

The mixing of one s orbital and three p orbitals (p_x, p_y, and p_z) yields four sp³ hybrid orbitals. Each of these orbitals points towards the corners of a tetrahedron and is equivalent in energy. This spatial arrangement allows for the maximum separation between electron pairs according to VSEPR theory, explaining the observed 109.5-degree bond angles in a methane molecule.

However, it is essential to understand that these hybrid orbitals are orthogonal to one another, ensuring that there is no overlap in space and thus no reduction in stability due to electron repulsion between the orbitals. In a solved exercise, the task of proving orthogonality might involve showing that the overlap integral of any two different hybrid orbitals is zero.

Orthogonality in Quantum Chemistry

Orthogonality is a core principle in quantum chemistry that applies to wavefunctions, such as atomic and hybrid orbitals. In essence, two wavefunctions are orthogonal if they do not share common space, a condition quantified by their overlap integral being zero.

To assess this condition, consider the integral \[ \int \psi_1 \psi_2 \, \mathrm{d} \tau = 0 \. \] This expression determines whether two orbitals, \(\psi_1\) and \(\psi_2\), can occupy the same quantum state or not. In context with tetrahedral hybrid orbitals, this principle of orthogonality guarantees that electrons within a single atom can occupy discrete energy states without interfering with each other, contributing to atomic stability.

In the exercise, formulated as part (a), the students are asked to show that the hybrids \( h_1 \) and \( h_3 \) do not occupy the same space and thus have no spatial overlap. Through a process of multiplication and integration, the students demonstrate orthogonality, ensuring the wavefunctions can occupy the same atom without violation of the Pauli Exclusion Principle.

Atomic Orbitals Integration

The process of atomic orbitals integration involves mathematically combining atomic orbitals to describe the electron distribution in molecules more accurately. During the hybridization process, where new hybrid orbitals are formed, an integral over space is performed to verify the orthogonality of the resulting orbitals.

Within the exercise, atomic orbitals integration is vital for step 3, where students calculate the overlap integral. To do this, they multiply hybrid orbitals (which are linear combinations of atomic orbitals), and integrate the product across all space. This uses the mathematical property of orthogonality, assuming atomic orbitals involved are normalized, meaning that each orbital has an integral of one over space. The exercise efficiently utilizes this concept to simplify the computational process, showing that any term with different orbitals goes to zero, while those with the same orbital sum to one.

Ultimately, atomic orbitals integration confirms that individual hybrid orbitals do not interfere, reflecting the physical reality of electron distributions within the atom and maintaining the principles of quantum mechanics.