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(is this correct … ?)
electrons "orbiting" a single atom obey the Schrodinger equation, whose solutions are linear combinations of an orthogonal basis of solutions of the form R(r)Θ(θ)Φ(φ)
where r θ and φ are the usual spherical coordinates: θ = 0 is the usual z axis
these "orbitals" can be visualised most easily according to the numbers of their nodes (strictly, nodal surfaces, ie the surfaces along which they are zero): n-1 m and l-m …
i shall use "sector" to refer to a region (other than a sphere) extending to infinity without any hole, "collar" to refer to a region (other than a sphere) extending to infinity with one hole, and "spherical zone" to refer to a sphere or a spherical shell
Θ(θ)Φ(φ) is also written Ylm(θ,φ)
R(r) is a Laguerre polynomial of order n-l-1 (with n-l-1 roots) times a negative exponential factor, times rl
n = total nodes + 1 (principal quantum number):
the total number of nodes is called n-1
m = plane nodes (magnetic quantum number):
Φ(φ) has nodes with φ = constant
so Φ(φ) only has plane nodes, all through the principal (z) axis
they divide space into 2m equal "vertical" sectors
m is the number of values of φ for which F(φ) = 0
l-m = conical nodes (orbital quantum number minus magnetic quantum number):
Θ(θ) has nodes with θ = constant
so Θ(θ) only has conical nodes, all with axes along the principal (z) axis
they divide space into l-m+1 cylindrically symmetric regions: two sectors and l-m-1 collars
we have to count each cone as two nodes, except the "flat cone" (along the "equatorial plane", θ = π/2), which we count as one node … this is because each cone (except the "flat cone") corresponds to two values of θ
diagrams such as this show the intersections of the nodes with a sphere as circles of latitude … unfortunately, although correct, this obscures the conical nature of the nodes!
l-m is the number of values of θ for which P(θ) = 0
n-l-1 = spherical nodes
R(r) has nodes with r = constant
so R(r) only has spherical nodes
they divide space into n-l regions
n-l-1 is the number of values of r for which R(r) = 0
total nodes and regions:
by adding, or multiplying, the above numbers …
the total number of nodes is m + (l-m) + (n - l - 1) = n-1
the total number of regions is 2m(l-m+1)(n - l)
this is the number of "blobs" shown in diagrams of orbitals …
in those diagrams, each "blob" represents a region in which the electron is likely to be: since it cannot be at a node, the "blobs" must fit between the nodes
special cases:
l = m = 0 (the s orbitals): there are n-1 spherical nodes …
the electron is likely to be in a sphere round the centre, or in one of n-2 spherical shells
this is the only case where the origin is not on a node (because all other types of node are planes or cones through the centre), and therefore is the only case in which the electron is likely to be near the centre
this is the only case where the origin is not on a node (because all other types of node are planes or cones through the centre), and therefore is the only case in which the electron is likely to be near the centre
l = 1 (the p orbitals): there is one plane node, and n-2 spherical nodes
this is because m and l-m can only be 0 or 1, so there are m "vertical" plane nodes and l-m conical nodes, ie 1"vertical" plane node and 0 conical nodes or 0 "vertical" plane nodes and 1 conical node (the "flat" cone, so it actually is another plane!)
the electron is likely to be on either side of a plane, in one of n-1 spherical zones
l = 2 (the d orbitals): there are two perpendicular plane nodes (m = 2 or 1), or one ("genuine", non-"flat") conical node (m = 0), and n-3 spherical nodes
the electron is likely to be in one of four parallel sectors, or in one of two sectors and a collar separating them, divided into n-3 spherical zones
l = 3 (the f orbitals): there are three plane nodes at 60° (m = 3), or three perpendicular plane nodes (m = 2), or one plane node and one ("genuine", non-"flat") conical node (m = 1), or two (one "genuine", and one "flat") conical nodes (m = 0), and n-4 spherical nodes
the electron is likely to be in one of six parallel sectors, or in one of 8 "octahedral" sectors, or in one of 6 sectors in "asterisk" configuration, or in one of two sectors and two collars separating them, divided into n-3 spherical zones
l = 4 (the g orbitals): there are four plane nodes at 45° (m = 4), or three plane nodes at 60° and one perpendicular plane node (m = 3), or two perpendicular plane nodes and one ("genuine", non-"flat") conical node (m = 2), or one plane node and two (one "genuine", and one "flat") conical nodes (m = 1), or two ("genuine", non-"flat") conical nodes (m = 0), and n-5 spherical nodes
and so on …
here, from http://en.wikipedia.org/wiki/Spherical_harmonics, is a diagram of the orbitals for l = n-1 = 0 to 3 (click for a larger version) …
… for l < n-1, each region shown must be divided into n-l-1 regions by n-l-2 invisible spheres: see http://chemlinks.beloit.edu/Stars/pages/orbitals.html for a diagram: as you go along each row, each region (yes, including the collar) is further divided
(see also http://en.wikipedia.org/wiki/Atomic_orbital)
remember that these diagrams show contours inside which there is a (say) 90% probability of finding the electron: the electron can be found outside the regions shown, and indeed anywhere except on the actual nodes