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Because electrons are fermions, the Pauli exclusion principle forbids these particles from having all the same quantum numbers. Therefore, for two electrons to occupy the same orbital, and thereby have the same orbital quantum number, they must have different spin quantum numbers. This also limits the number of electrons in the same orbital to two.
For example, if two electrons reside in the same orbital, then their values of n, ℓ, and m ℓ are equal. In that case, the two values of m s (spin) pair must be different. Since the only two possible values for the spin projection m s are +1/2 and −1/2, it follows that one electron must have m s = +1/2 and one m s = −1/2.
This phenomenon is only paradoxical if it is assumed that the energy order of atomic orbitals is fixed and unaffected by the nuclear charge or by the presence of electrons in other orbitals. If that were the case, the 3d-orbital would have the same energy as the 3p-orbital, as it does in hydrogen, yet it clearly does not.
The p z orbital is the same as the p 0 orbital, but the p x and p y are formed by taking linear combinations of the p +1 and p −1 orbitals (which is why they are listed under the m = ±1 label). Also, the p +1 and p −1 are not the same shape as the p 0, since they are pure spherical harmonics.
The third column is the maximum number of electrons that can be put into a subshell of that type. For example, the top row says that each s-type subshell (1s, 2s, etc.) can have at most two electrons in it. Each of the following subshells (p, d, f, g) can have 4 more electrons than the one preceding it.
The Slater determinant arises from the consideration of a wave function for a collection of electrons, each with a wave function known as the spin-orbital (), where denotes the position and spin of a single electron. A Slater determinant containing two electrons with the same spin orbital would correspond to a wave function that is zero everywhere.
Fermi heap and Fermi hole refer to two closely related quantum phenomena that occur in many-electron atoms. They arise due to the Pauli exclusion principle, according to which no two electrons can be in the same quantum state in a system (which, accounting for electrons' spin, means that there can be up to two electrons in the same orbital).
Distributing 8 electrons over 6 molecular orbitals leaves the final two electrons as a degenerate pair in the 2pπ* antibonding orbitals resulting in a bond order of 2. As in diboron, these two unpaired electrons have the same spin in the ground state, which is a paramagnetic diradical triplet oxygen.