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A linear combination involves adding or subtracting constant multiples of functions. For the given normalized functions, it's the process of combining two specific functions, \(G_1(x, y, z; \alpha)\) and \(G_2(x, y, z; \alpha)\). By subtracting \(G_2\) from \(G_1\), we create a new function. In our specific problem, the linear combination is:

• \(G_1(x, y, z; \alpha) - G_2(x, y, z; \alpha)\)

This results in: \( \left( \frac{2048 \alpha^7}{9 \pi^3} \right)^{1/4} (x^2 e^{-\alpha r^2} - y^2 e^{-\alpha r^2}) \)Notice how both functions \(x^2 e^{-\alpha r^2}\) and \(y^2 e^{-\alpha r^2}\) are present with similar coefficients before subtraction, emphasizing that linear combinations preserve relationships in similar functions. This combination gives us insight into characteristics or behaviors of new functions, such as identifying particular orbitals.

Normalized functions in quantum mechanics are adjusted so that their overall magnitude is consistent and measurable. Simply put, these functions are scaled and adjusted to reflect a total probability of 1. In the context of the problem, the normalization factor is represented now as the common term: \(\left( \frac{2048 \alpha^7}{9 \pi^3} \right)^{1/4} e^{-\alpha r^2} \) This factor ensures the functions \(G_1\), \(G_2\), and \(G_3\) have the same scale when performing operations like linear combination. When working with atomic orbitals, maintaining this normalization helps us accurately depict electron probabilities and spatial distributions. This consistency is essential in fields like quantum chemistry, where predicting electron behavior is key.

In the context of functions and vectors, orthogonality means being perpendicular or independent. For hydrogen atomic orbitals, orthogonality indicates that combinations of orbitals won't interfere with each other, preserving distinct directional qualities. The function \(x^2 - y^2\) derived from linear combination \( \left( \frac{2048 \alpha^7}{9 \pi^3} \right)^{1/4} e^{-\alpha r^2} (x^2 - y^2)  \)is inherently tied to the concept of orthogonality. This specific form is recognized as matching the orbital \(d_{x^2-y^2}\), known for its orientation along the x and y axes, demonstrating zero overlap with each axis separately.Similarly, understanding orthogonality helps distinguish different functions or orbitals from one another, especially when predicting how they might combine or interact. In chemistry and physics, leveraging orthogonal properties simplifies computation and understanding of multi-component systems.