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1 The various types of orbitals

An orbital is uniquely identified by the values of the three quantum numbers, and each value of the three quantum numbers corresponds to exactly one orbital, but the quantum numbers only occur in certain combinations of values. The rules governing the possible values of the quantum numbers are as follows:

The quantum number n is always a positive integer. In fact, it can be any positive integer, but for reasons discussed below, large numbers are seldom encountered. Each atom has, in general, many orbitals associated with each value of n; these orbitals together are sometimes called a shell.

The quantum number is always an integer. Within a shell where n is some integer n₀, ranges across all (integer) values satisfying the relation . For instance, the n = 1 shell has only orbitals with , and the n = 2 shell has only orbitals with , , and . The set of orbitals associated with a particular value of are sometimes collectively called a subshell.

The quantum number is also always an integer. Within a subshell where is some integer , ranges thus: .

The above results may be summarized in the following table. Each cell represents a subshell, and lists the values of available in that subshell. Empty cells represent subshells that do not exist.

Subshells are usually identified by their - and -values. is represented by its numerical value, but is represented by a letter as follows: 0 is represented by 's', 1 by 'p', 2 by 'd', 3 by 'f', and 4 by 'g'. For instance, one may speak of the subshell with and as a '2s subshell'.

(Historical note: The names 's', 'p', 'd', and 'f' originate from a now-discredited system of categorizing spectral lines as "strong", "principal", "diffuse", or "fundamental". When the first four types of orbitals were described, they were associated with these spectral line types, but there were no other names. The designations 'g' and 'h' were derived by following alphabetical order.)

2 The shapes of orbitals

Any discussion of the shapes of electron orbitals is necessarily uncertain, because a given electron, regardless of which orbital it occupies, can at any moment be found at any distance from the nucleus and in any direction.

However, the electron is much more likely to be found in certain areas of the atom than in others. Given this, a boundary surface can be drawn so that all areas within the surface have high values of the probability density function and all areas outside the surface have low values. The precise placement of the surface is arbitrary, but any reasonably compact determination must follow a pattern specified by the behavior of . This boundary surface is what is meant when the "shape" of an orbital is mentioned.

Generally speaking, the number determines the size of the orbital: as increases, the size of the orbital increases. Also in general terms, determines an orbital's shape, and its orientation. However, since some orbitals are described by equations in complex numbers, the shape sometimes depends on also.

The relationship to is more complex. -orbitals () are shaped like spheres. -orbitals have the form of two ellipsoids with a point of tangency at the nucleus. The three -orbitals in each shell are oriented at right angles to each other, as determined by their respective values of .

Four of the five -orbitals are shaped like four somewhat irregular balls, each ball tangent to two others, and the centers of all four lying in one plane. Three of these planes are the -, -, and -planes, and the fourth is tilted at an angle of 45°. The fifth and final -orbital consists of three regions of high probability density: a torus with two roughly spherical regions placed symmetrically on its axis.

3 Orbital energy

In atoms with a single electron (essentially hydrogenhydrogen helium H Li Full table General Name, Symbol, NumberHydrogen, H, 1 Chemical series nonmetals Group, Period, Block 1 (IA), 1 , s Density, Hardness 0. 0899 kg/m3, NA Appearance colorless Atomic properties Atomic weight 1. 00794 amu Atomic radius (ca), the energy of an orbital (and, consequently, of any electrons in the orbital) is determined exclusively by . The orbital has the lowest possible energy in the atom. Each successively higher value of has a higher level of energy, but the difference decreases as increases. At a certain point, the level of energy becomes high enough that the electron escapes from the atom.

In atoms with multiple electrons, the energy of an electron depends not only on the intrinsic properties of its orbital, but also on its interactions with other electrons. These interactions depend on the orbital's shape. Consequently, the energy levels of orbitals depend not only on but also on . High values of are associated with higher values of energy; for instsance, the 2p state is higher than the 2s state. When becomes sufficiently large, the shape-induced variance in energy between orbitals in a single shell becomes so large as to push the energy of some of the shell's orbitals above the energy of some orbitals in the next higher shell.

The energy order of the first 24 subshells is given in the following table. Each cell represents a subshell with and given by its row and column indices, respectively. The number in the cell is the subshell's position in the sequence. Empty cells represent subshells that either do not exist or stand farther down in the sequence.

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