(adsbygoogle = window.adsbygoogle || []).push({}); (is this correct … ?)

electrons "orbiting" a single atom obey theSchrodinger equation, whose solutions are linear combinations of anorthogonal basisof 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 bevisualisedmost easily according to thenumbers of their(strictly, nodal surfaces, ie the surfaces along which they are zero):nodesn-1 mandl-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 Y_{l}^{m}(θ,φ)

R(r) is aLaguerre polynomialof order n-l-1 (with n-l-1 roots) times a negative exponential factor, times r^{l}

n = total nodes + 1 (principalquantum number):

the total number of nodes is called n-1

m = plane nodes (magneticquantum number):

Φ(φ) has nodes with φ = constant

so Φ(φ) only hasplanenodes, 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 (orbitalquantum number minusmagneticquantum number):

Θ(θ) has nodes with θ = constant

so Θ(θ) only hasconicalnodes, 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 astwonodes, except the "flat cone" (along the "equatorial plane", θ = π/2), which we count asonenode … this is because each cone (except the "flat cone") corresponds totwovalues of θ

diagrams such as this show the intersections of the nodes with a sphereas circles of latitude… unfortunately, although correct, this obscures theconical natureof 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 hassphericalnodes

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 itcannotbe at a node, the "blobs" must fitbetween the nodes

special cases:

l = m = 0 (thes orbitals): there are n-1 spherical nodes …

the electron is likely to be ina sphere round the centre, or in one ofn-2 spherical shells

this is theonlycase where the origin isnoton a node (because all other types of node are planes or conesthroughthe centre), and therefore is theonlycase in which the electron is likely to be near the centre

l = 1 (thep 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 nodesor0 "vertical" plane nodes and 1 conical node (the "flat" cone, so it actuallyisanother plane!)

the electron is likely to be oneither side of a plane, in one ofn-1 spherical zones

l = 2 (thed orbitals): there are two perpendicular plane nodes (m = 2 or 1),orone ("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 inton-3 spherical zones

l = 3 (thef orbitals): there are three plane nodes at 60° (m = 3),orthree perpendicular plane nodes (m = 2),orone plane node and one ("genuine", non-"flat") conical node (m = 1),ortwo (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 inton-3 spherical zones

l = 4 (theg orbitals): there are four plane nodes at 45° (m = 4),orthree plane nodes at 60° and one perpendicular plane node (m = 3),ortwo perpendicular plane nodes and one ("genuine", non-"flat") conical node (m = 2),orone plane node and two (one "genuine", and one "flat") conical nodes (m = 1),ortwo ("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 showcontoursinside which there is a (say) 90% probability of finding the electron: the electroncanbe foundoutsidethe regions shown,and indeed anywhere except on the actual nodes

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# Classifying electron orbitals by nodes

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