Problem in basic equalities for Huckel energy

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SUMMARY

The discussion centers on the equality of integrals in the context of Huckel energy calculations, specifically the proof of the symmetry property $$S_{ij}=S_{ji}$$. Participants seek clarification on the integral $$\int f_i f_j dg = \int f_j f_i dg$$, emphasizing the need for a foundational understanding of complex functions. The reference document from the University of Oxford introduces this concept on page 2 as $$$$. It is noted that the equality holds true only if the functions $$f_i$$ and $$f_j$$ are real-valued.

PREREQUISITES
  • Understanding of complex functions and their properties
  • Familiarity with integral calculus and its applications in quantum mechanics
  • Knowledge of the Huckel method in molecular orbital theory
  • Basic grasp of inner product notation in functional analysis
NEXT STEPS
  • Study the properties of inner products in Hilbert spaces
  • Learn about the Huckel method and its applications in computational chemistry
  • Explore the implications of complex conjugates in quantum mechanics
  • Review integral calculus with a focus on symmetry properties of integrals
USEFUL FOR

This discussion is beneficial for chemistry students, quantum mechanics researchers, and anyone involved in computational chemistry, particularly those working with Huckel theory and integrals of complex functions.

georg gill
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http://vallance.chem.ox.ac.uk/pdfs/VariationPrincipleNotes.pdf

In the proof above I need to understand why: $$S_{ij}=S_{ji}$$. Which is the same as proving

$$\int f_i f_j dg=\int f_j f_i dg$$ (I)
Not sure about what I should call the variable for so I called it g. Can someone prove this from the basics? From starting from definition of complex functions and start from left of (I) and get to the right of (I). I know that complex times complex conjugate gives probability. Thank you!In the link this is introduced at page 2 as $$<f_i|f_j>$$
 
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