Question

Are the localized bonding orbitals in Equation (24.13) defined by $$\begin{array}{l} \sigma^{\prime}=2 c_{1} \phi_{H 1 s A}+\left(c_{2} \phi_{B e 2 s}-\frac{c_{1} c_{4}}{c_{3}} \phi_{B e 2 p_{z}}\right) \text { and } \\ \sigma^{\prime \prime}=2 c_{1} \phi_{H 1 s B}+\left(c_{2} \phi_{B e 2 s}+\frac{c_{1} c_{4}}{c_{3}} \phi_{B e 2 p_{z}}\right) \end{array}$$ orthogonal? Answer this question by evaluating the integral \(\int\left(\sigma^{\prime}\right)^{*} \sigma^{\prime \prime} d \tau\).

Short answer

The two given localized bonding orbitals \(\sigma^{\prime}\) and \(\sigma^{\prime\prime}\) are orthogonal, as the integral \(\int\left(\sigma^{\prime}\right)^{*} \sigma^{\prime \prime} d \tau\) evaluates to zero.

Step-by-step solution

Write down the expressions for the localized bonding orbitals

We are given the expressions for \(\sigma^{\prime}\) and \(\sigma^{\prime\prime}\): \[ \sigma^{\prime} = 2 c_{1} \phi_{H 1 s A}+\left(c_{2} \phi_{B e 2 s}-\frac{c_{1} c_{4}}{c_{3}} \phi_{B e 2 p_{z}}\right) \] and \[ \sigma^{\prime\prime} = 2 c_{1} \phi_{H 1 s B}+\left(c_{2} \phi_{B e 2 s}+\frac{c_{1} c_{4}}{c_{3}} \phi_{B e 2 p_{z}}\right) \]

Write down the expression for the integral

Now we need to evaluate the following integral: \[ \int\left(\sigma^{\prime}\right)^{*} \sigma^{\prime \prime} d \tau \]

Calculate the complex conjugate of \(\sigma^{\prime}\)

Since the wave functions are real, the complex conjugate of \(\sigma^{\prime}\) is itself: \[ \left(\sigma^{\prime}\right)^{*} = \sigma^{\prime} \]

Multiply \(\left(\sigma^{\prime}\right)^{*}\) by \(\sigma^{\prime \prime}\)

Next, we multiply \(\sigma^{\prime}\) by \(\sigma^{\prime\prime}\): \[ \sigma^{\prime} \sigma^{\prime\prime} = \left(2 c_{1} \phi_{H 1 s A}+\left(c_{2} \phi_{B e 2 s}-\frac{c_{1} c_{4}}{c_{3}} \phi_{B e 2 p_{z}}\right)\right)\left(2 c_{1} \phi_{H 1 s B}+\left(c_{2} \phi_{B e 2 s}+\frac{c_{1} c_{4}}{c_{3}} \phi_{B e 2 p_{z}}\right)\right) \]

Evaluate the integral

Now, we evaluate the integral: \[ \int\sigma^{\prime} \sigma^{\prime\prime} d \tau = \int\left( 4 c_{1}^{2} \phi_{H 1 s A} \phi_{H 1 s B} + \left( 2c_{1} c_{2} \phi_{H 1 s A} \phi_{B e 2 s} - 2 c_{1}^{2} c_{4} \phi_{H 1 s A} \phi_{B e 2 p_{z}} \right) + \left( 2c_{1} c_{2} \phi_{H 1 s B} \phi_{B e 2 s} + 2 c_{1}^{2} c_{4} \phi_{H 1 s B} \phi_{B e 2 p_{z}} \right) + \left( c_{2}^{2} \phi_{B e 2 s}^{2} - c_{4}^{2} \phi_{B e 2 p_{z}}^{2}\right)\right)d\tau \] Considering these factors: 1. In both \(\sigma^{\prime}\) and \(\sigma^{\prime\prime}\), the orbitals \(\phi_{B e 2 s}\) and \(\phi_{B e 2 p_{z}}\) are atomic orbitals of the same atom, their integral is zero. 2. The orbitals \(\phi_{H 1 s A}\) and \(\phi_{H 1 s B}\) are atomic orbitals of different atoms, their integral is zero. Using these observations, many terms of the integral vanish as their integrals go to zero. As a result, we have: \[ \int\sigma^{\prime} \sigma^{\prime\prime} d \tau = 0 \]

Conclusion

Since the integral evaluates to zero, the two given localized bonding orbitals \(\sigma^{\prime}\) and \(\sigma^{\prime\prime}\) are orthogonal.

Key concepts

Localized Bonding Orbitals

Localized bonding orbitals, as shown in the provided exercise, are specific wave functions that aim to describe the bonding characteristics between atoms in a molecule. Here, the localized bonding orbitals \( \sigma^{\prime} \) and \( \sigma^{\prime\prime} \) are expressions that combine different atomic orbitals, like \( \phi_{H 1 s A} \) and \( \phi_{Be 2 s} \), to form new, localized wave functions. These combinations represent the probability distribution of electrons in the bond and can reveal the strength and nature of the bond.

• The power of localized bonding orbitals is in their ability to simplify the understanding of complex atomic interactions by breaking them down into more tangible entities.
• They help chemists visualize how electrons are shared or transferred in a molecule, which is crucial for determining molecular geometry, bond strength, and reactivity.

Understanding localized bonding orbitals requires one to consider the symmetry and overlap of the atomic orbitals involved. These overlaps are mathematical operations that lead to the conclusion whether the function predicts a significant electron density between two nuclei, indicating a bond, or not.

Integral Evaluation

Integral evaluation is a mathematical process used in quantum chemistry to determine various properties of wave functions, such as orthogonality. In the context of the exercise, this involves evaluating the integral \( \int\left(\sigma^{\prime}\right)^{*} \sigma^{\prime\prime} d \tau \), where the primary goal is to determine if the localized bonding orbitals \( \sigma^{\prime} \) and \( \sigma^{\prime\prime} \) are orthogonal.

• Evaluating integrals in quantum chemistry often provides quantitative information about the overlap between different wave functions. If the resulting value is zero, it suggests that the wave functions are orthogonal.
• This evaluation is crucial because orthogonality usually means that the wave functions describe non-overlapping, independent states.

To perform the evaluation, one must consider the characteristics of the individual atomic orbitals and how they interact:

• The integral includes terms that vanish due to orthogonality and normalization conditions specific to atomic orbitals.
• The terms related to orbitals from different atoms, like \( \phi_{H 1 s A} \) and \( \phi_{H 1 s B} \), contribute nothing when integrated over all space, as their overlap is zero.

Understanding the results of integral evaluation helps to comprehend the chemical and physical interactions presented in molecular and atomic orbital theories.

Orthogonality in Quantum Chemistry

Orthogonality is a fundamental concept in quantum chemistry and refers to the condition where two orbitals have an inner product (integral of their product over space) equal to zero. In simpler terms, when orbitals are orthogonal, they do not share any electron density, indicating independent electron placements in space.

• Orthogonality is vital when solving the Schrödinger equation for multi-electron atoms because it ensures that solutions, or orbitals, are independent of each other.
• In quantum chemistry, using orthogonal orbitals simplifies calculations and interpretations of electronic structures.

In the exercise provided, orthogonality was determined through integral evaluation. The conclusion that \( \int\sigma^{\prime} \sigma^{\prime\prime} d \tau = 0 \) indicates that the orbitals \( \sigma^{\prime} \) and \( \sigma^{\prime\prime} \) are orthogonal.

• This orthogonality implies that these orbitals cover distinct regions in space, allowing chemists to describe multi-electron systems without the complexity of electron-electron interactions overlapping in the calculations.
• It is an essential property for ensuring that approximations in molecular orbital theory do not lead to misrepresentations of the true electronic structure.

Understanding orthogonality assists students and chemists alike in predicting and explaining molecular behavior and interactions in complex systems.