<P> Many different models have been proposed throughout the history of quantum mechanics, but the most prominent system of nomenclature spawned from the Hund - Mulliken molecular orbital theory of Friedrich Hund, Robert S. Mulliken, and contributions from Schrödinger, Slater and John Lennard - Jones . This system of nomenclature incorporated Bohr energy levels, Hund - Mulliken orbital theory, and observations on electron spin based on spectroscopy and Hund's rules . </P> <P> This model describes electrons using four quantum numbers, n, l, m, m, given below . It is also the common nomenclature in the classical description of nuclear particle states (e.g. protons and neutrons). A quantum description of molecular orbitals require different quantum numbers, because the Hamiltonian and its symmetries are quite different . </P> <Ol> <Li> The principal quantum number (n) describes the electron shell, or energy level, of an electron . The value of n ranges from 1 to the shell containing the outermost electron of that atom, i.e. <Dl> <Dd> n = 1, 2,...</Dd> </Dl> <P> For example, in caesium (Cs), the outermost valence electron is in the shell with energy level 6, so an electron in caesium can have an n value from 1 to 6 . </P> For particles in a time - independent potential (see Schrödinger equation), it also labels the nth eigenvalue of Hamiltonian (H), i.e. the energy, E with the contribution due to angular momentum (the term involving J) left out . This number therefore has a dependence only on the distance between the electron and the nucleus (i.e., the radial coordinate, r). The average distance increases with n, and hence quantum states with different principal quantum numbers are said to belong to different shells . </Li> <Li> The azimuthal quantum number (l) (also known as the angular quantum number or orbital quantum number) describes the subshell, and gives the magnitude of the orbital angular momentum through the relation <Dl> <Dd> L = ħ l (l + 1). </Dd> </Dl> <P> In chemistry and spectroscopy, "l = 0" is called an s orbital, "l = 1" ap orbital, "l = 2" ad orbital, and "l = 3" an f orbital . </P> <P> The value of l ranges from 0 to n − 1, so the first p orbital (l = 1) appears in the second electron shell (n = 2), the first d orbital (l = 2) appears in the third shell (n = 3), and so on: </P> <Dl> <Dd> l = 0, 1, 2,..., n − 1 . </Dd> </Dl> A quantum number beginning in 3, 0,...describes an electron in the s orbital of the third electron shell of an atom . In chemistry, this quantum number is very important, since it specifies the shape of an atomic orbital and strongly influences chemical bonds and bond angles . </Li> <Li> The magnetic quantum number (m) describes the specific orbital (or "cloud") within that subshell, and yields the projection of the orbital angular momentum along a specified axis: <Dl> <Dd> L = m ħ . </Dd> </Dl> <P> The values of m range from − l to l, with integer intervals: </P> The s subshell (l = 0) contains only one orbital, and therefore the m of an electron in an s orbital will always be 0 . The p subshell (l = 1) contains three orbitals (in some systems, depicted as three "dumbbell - shaped" clouds), so the m of an electron in ap orbital will be − 1, 0, or 1 . The d subshell (l = 2) contains five orbitals, with m values of − 2, − 1, 0, 1, and 2 . </Li> <Li> The spin projection quantum number (m) describes the spin (intrinsic angular momentum) of the electron within that orbital, and gives the projection of the spin angular momentum S along the specified axis: <Dl> <Dd> S = m ħ . </Dd> </Dl> <P> In general, the values of m range from − s to s, where s is the spin quantum number, an intrinsic property of particles: </P> <Dl> <Dd> m = − s, − s + 1, − s + 2,..., s − 2, s − 1, s . </Dd> </Dl> An electron has spin number s = 1⁄2, consequently m will be ± 1⁄2, referring to "spin up" and "spin down" states . Each electron in any individual orbital must have different quantum numbers because of the Pauli exclusion principle, therefore an orbital never contains more than two electrons . </Li> </Ol> <Li> The principal quantum number (n) describes the electron shell, or energy level, of an electron . The value of n ranges from 1 to the shell containing the outermost electron of that atom, i.e. <Dl> <Dd> n = 1, 2,...</Dd> </Dl> <P> For example, in caesium (Cs), the outermost valence electron is in the shell with energy level 6, so an electron in caesium can have an n value from 1 to 6 . </P> For particles in a time - independent potential (see Schrödinger equation), it also labels the nth eigenvalue of Hamiltonian (H), i.e. the energy, E with the contribution due to angular momentum (the term involving J) left out . This number therefore has a dependence only on the distance between the electron and the nucleus (i.e., the radial coordinate, r). The average distance increases with n, and hence quantum states with different principal quantum numbers are said to belong to different shells . </Li>

What properties of an orbital are defined by each of the three quantum numbers n l and ml