The orbital energy in a multi-electron atom is determined by several factors, including the following:
Principal Quantum Number (n): The principal quantum number (n) determines the energy level or shell of an orbital. As the value of n increases, the energy of the orbital also increases. Orbitals with the same value of n are said to belong to the same energy level.
Azimuthal Quantum Number (l): The azimuthal quantum number (l) determines the shape and subshell of an orbital. Orbitals with different values of l have different energies. The energy increases with increasing values of l within the same energy level. For example, in a given energy level (n), the s orbital has the lowest energy (l = 0), followed by the p orbitals (l = 1), then the d orbitals (l = 2), and so on.
Magnetic Quantum Number (m): The magnetic quantum number (m) specifies the orientation of the orbital in space. For a given value of l, there are (2l + 1) possible values of m, which represent the different spatial orientations of the orbital. Orbitals with the same values of n and l but different values of m have the same energy.
Electron-Electron Repulsion: In multi-electron atoms, the presence of multiple electrons leads to electron-electron repulsion. Electrons have negative charge, and they repel each other. This repulsion affects the energy of the orbitals. Electrons in the same orbital experience stronger repulsion compared to electrons in different orbitals. This repulsion can lead to splitting of energy levels and sublevels in a phenomenon known as electron-electron repulsion or electron-electron correlation.
Effective Nuclear Charge (Zeff): The effective nuclear charge experienced by an electron is the net positive charge felt by the electron due to the attractive force of the nucleus and the shielding effect of other electrons. The value of Zeff affects the energy of the orbitals. Higher values of Zeff result in greater attraction between the nucleus and the electron, leading to lower energies of the orbitals.
These factors collectively determine the energy of the orbitals in a multi-electron atom. The specific arrangement and distribution of electrons within these orbitals follow the principles of electron configuration and the Aufbau principle, which dictate the filling of orbitals in order of increasing energy.