Classifying electron orbitals by nodes

[tiny-tim]

(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 Y_l^m(θ,φ)

R(r) is a Laguerre polynomial of order n-l-1 (with n-l-1 roots) times a negative exponential factor, times r^l

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​

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

 

[DrDu]

Orbitals with a fixed magnetic quantum number m are eigenstates of the Hamiltonian also when the rotational degeneracy is split by an applied magnetic field along the z axis. That's why it is called "magnetic" quantum number. However, these eigenstates are complex orbitals which have no plane nodes. The real orbitals you are describing are superpositions of two complex orbitals with +|m| and -|m|. Hence the real orbitals are labeled by |m| rather than by m. Furthermore as long as m not equal 0 there are two real orbitals with the same |m|, e.g. p_x and p_y, d_{xz} and d_{yz} (with |m|=1) or d_{xy} and d_{x^2-y^2} (with |m|=2).

 

[DrDu]

(is this correct … ?)
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) …​

I can't see any difference in the orbitals on the left and the ones on the right. They should be rotated against each other.​

 

[tiny-tim]

Hi DrDu!

Thanks for your help.

Yes, I've oversimplified by not using |m|, and ignoring the complex nature of the solutions.

I'll put the (hopefully ) corrected version in the next post.

I can't see any difference in the orbitals on the left and the ones on the right. They should be rotated against each other.

ah, they are … the red and green colours are interchanged, indicating a relative rotation

i find it difficult to imagine the correct description of the orbitals in the complex (non-zero magnetic field) case

how would i complete this sentence? …

"these diagrams show the probabilities (or amplitudes) in a zero magnetic field: in a non-zero magnetic field, the probabilities are found by combining … … … … "​

 

[tiny-tim]

(is this correct … ?)​

electrons "orbiting" a single atom obey the Schrodinger equation, whose solutions are linear combinations of an orthogonal basis of complex solutions of the form e^imφ times a real function R(r)Θ_l^m(cosθ)

where r θ and φ are the usual spherical coordinates: θ = 0 is the usual z axis

the real parts of 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

R(r) is a Laguerre polynomial of order n-l-1 (with n-l-1 roots) times a negative exponential factor, times r^l

Θ_l^m(x) is a multiple of the associated Legendre polynomial P_l^m(x), a polynomial of degree l in x and/or √(1 - x²) … so P_l^m(cosθ) is a polynomial of degree l in cosθ and sinθ

P_l^m is a multiple of P_l^-m, and so the complex orbitals e^imφR(r)Θ_l^m (for -l ≤ m ≤ l) can be visualised in terms of the shapes of the real and imaginary parts of e^imφR(r)Θ_l^|m| (for 0 ≤ m ≤ l) …

these are the shapes that are shown in diagrams! ​

​

n = total nodes + 1 (principal quantum number):

the total number of nodes is called n-1

|m| = plane nodes (m = magnetic quantum number):

Re(e^imφ) and Im(e^imφ) have nodes with φ = constant …

obviously, they are |m| "vertical" plane nodes (through the principal (z) axis),

and they divide space into 2|m| equal "vertical" sectors

l-|m| = conical nodes (l = orbital quantum number):

Θ_l^m(cosθ) has nodes with θ = constant

so Θ_l^m(cosθ) 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 Θ_l^m(cosθ) = 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 2|m|(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​

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