Multiple integrals: Applications of multiple integrals
Application of triple integrals in quantum chemistry
The table below gives the wave functions for different orbitals of a hydrogen-like atom with nuclear charge in atomic units (for hydrogen ) via spherical coordinates , where is the Bohr radius and and the following norms used are The probability of the electron being in a volume element at position is equal to for a given wave function .
This means that integration over the entire space must be equal to . In other words, the wave function is normalized.
Normalization of 1s orbital function We check normalization for the wave function of the orbital:
Average distance of an electron to the nucleus for the 1s orbital The average distance of an electron to the nucleus in the orbital of a hydrogen-like atom with nuclear charge is the expectation value of for the wave function . This can be calculated as follows: We do this in spherical coordinates: