[Big-Daddy]

When relating the physical orbitals we know of to the Schrodinger wavefunctions that make them I have a problem. I know how the orbitals are combined for p-orbitals (indeed Wikipedia shows it: http://en.wikipedia.org/wiki/Atomic_orbital "Real orbitals").

I know which wavefunctions are needed for each orbital. d_z² is the orbital corresponding to uncombined wavefunction l=2,ml=0; d_xy and d_yz correspond to a combination of the l=2,ml=1 and l=2,ml=-1 wavefunctions; and d_xy and d_x²-y² correspond to a combination of the l=2,ml=2 and l=2,ml=-2 wavefunctions.

But how are they combined to reach each orbital? Wikipedia shows the two combinations needed for the px and py orbitals (see http://en.wikipedia.org/wiki/Atomic_orbital "Real orbitals") but doesn't show how the wavefunctions are combined for d or f orbitals?

[Corribus]

Orbitals in real space are always some kind of linear combination of the simple Schrodinger wavefunctions.  This works because it can easily be shown that linear combinations of wavefunctions have the same energy value as the wavefunctions themselves.

p orbitals are fairly easy to specify in real space because there are only a few wavefunctions involved.  d and f orbitals are generally prepared the same way.  The m = 0 case is always real, so can be plotted in real space with no combination necessary.  The other orbitals are prepared by taking linear combinations of the +/- l values, as you've specified. So you already have the answer to your question for the d orbitals. Doing the mathematical combination to arrive at a real solution is a fairly straightforward exercise, and answers can be found in most physical chemistry textbooks. 

Do you want a list of them?

[Big-Daddy]

Orbitals in real space are always some kind of linear combination of the simple Schrodinger wavefunctions.  This works because it can easily be shown that linear combinations of wavefunctions have the same energy value as the wavefunctions themselves.

p orbitals are fairly easy to specify in real space because there are only a few wavefunctions involved.  d and f orbitals are generally prepared the same way.  The m = 0 case is always real, so can be plotted in real space with no combination necessary.  The other orbitals are prepared by taking linear combinations of the +/- l values, as you've specified. So you already have the answer to your question for the d orbitals. Doing the mathematical combination to arrive at a real solution is a fairly straightforward exercise, and answers can be found in most physical chemistry textbooks. 

Do you want a list of them?

Thank you very much for clearing this up.

Yes, I would very much appreciate a list of the mathematical combinations required (or a link to somewhere which lists them)!

[Corribus]

Ok, here goes. 

These are only the angular portions of the wavefunctions, which are identical regardless of the value of n.  These are determined essentially by taking normalized linear combinations (+ and - combinations) of the spherical harmonic functions.  The radial portion, not included here, will differ based on what the n value is (e.g., 3d vs 4d orbitals).  Because the radial portion is missing, these representations will show the dimensional character of each orbital but not necessarily the true shape. 

Also note that these are hydrogenic  (1 electron) wavefunctions only.  You start putting more electrons in, things get more complicated in a hurry.  

Finally, keep in mind that there is nothing special, per se, about these specific combinations.  They are typically chosen because they conveniently along the various axes in a common x-y-z coordinate system.  But any orientation would be equivalent, though the formulas would be significantly more complicated.

[tex]d_{z^2}=\sqrt{\frac5{16\pi}}(3\cos^2\theta-1)[/tex]
[tex]d_{xz}=\sqrt{\frac{15}{4\pi}}\sin\theta\cos\theta\cos\phi[/tex]
[tex]d_{yz}=\sqrt{\frac{15}{4\pi}}\sin\theta\cos\theta\sin\phi[/tex]
[tex]d_{x^2-y^2}=\sqrt{\frac{15}{16\pi}}\sin^2\theta\cos2\phi[/tex]
[tex]d_{xy}=\sqrt{\frac{15}{16\pi}}\sin^2\theta\sin2\phi[/tex]

f-orbitals are more difficult, and they're not listed in any of my physical chemistry books. However thanks to the power of the internet, you can find them with a little sleuthing - such as the list of spherical harmonics functions here:

http://en.wikipedia.org/wiki/Table_of_spherical_harmonics

About 30% the way down the page, you will see a section entitled "Spherical harmonics with l = 3" and below that a subsection "Real spherical harmonics with l = 3".  That's basically your f-orbital angular wavefunctions.  To convert to polar coordinates like the d-orbital representations I gave above, there are formulae for theta and phi in terms of x, y, z and r at the top of the page.  It is not too cumbersome to make the conversions - just some tiresome algebra.  If you need help, though, let me know.  You can check to make sure you are doing it right be seeing if you can reproduce the formulae for the d-orbitals I provided above.

[Big-Daddy]

Ok, here goes. 

These are only the angular portions of the wavefunctions, which are identical regardless of the value of n.  These are determined essentially by taking normalized linear combinations (+ and - combinations) of the spherical harmonic functions.  The radial portion, not included here, will differ based on what the n value is (e.g., 3d vs 4d orbitals).  Because the radial portion is missing, these representations will show the dimensional character of each orbital but not necessarily the true shape. 

Also note that these are hydrogenic  (1 electron) wavefunctions only.  You start putting more electrons in, things get more complicated in a hurry.  

Finally, keep in mind that there is nothing special, per se, about these specific combinations.  They are typically chosen because they conveniently along the various axes in a common x-y-z coordinate system.  But any orientation would be equivalent, though the formulas would be significantly more complicated.

[tex]d_{z^2}=\sqrt{\frac5{16\pi}}(3\cos^2\theta-1)[/tex]
[tex]d_{xz}=\sqrt{\frac{15}{4\pi}}\sin\theta\cos\theta\cos\phi[/tex]
[tex]d_{yz}=\sqrt{\frac{15}{4\pi}}\sin\theta\cos\theta\sin\phi[/tex]
[tex]d_{x^2-y^2}=\sqrt{\frac{15}{16\pi}}\sin^2\theta\cos2\phi[/tex]
[tex]d_{xy}=\sqrt{\frac{15}{16\pi}}\sin^2\theta\sin2\phi[/tex]

f-orbitals are more difficult, and they're not listed in any of my physical chemistry books. However thanks to the power of the internet, you can find them with a little sleuthing - such as the list of spherical harmonics functions here:

http://en.wikipedia.org/wiki/Table_of_spherical_harmonics

About 30% the way down the page, you will see a section entitled "Spherical harmonics with l = 3" and below that a subsection "Real spherical harmonics with l = 3".  That's basically your f-orbital angular wavefunctions.  To convert to polar coordinates like the d-orbital representations I gave above, there are formulae for theta and phi in terms of x, y, z and r at the top of the page.  It is not too cumbersome to make the conversions - just some tiresome algebra.  If you need help, though, let me know.  You can check to make sure you are doing it right be seeing if you can reproduce the formulae for the d-orbitals I provided above.

Wow, thanks a lot!

There are formulae for theta and phi in terms of x, y, z and r. I had previously found some (at first apparently different) formulae for theta and phi in terms of x, y, z and r on the Cambridge natural sciences website, which read cosθ=z/r and tanφ=y/x. This enabled me to express the spherical harmonic in terms of x, y, z and r. (The same website previously gave a general formula for the spherical harmonic in terms of φ, θ, r and the second and third quantum numbers.) These seem to match up with the expressions on the Wikipedia page. This means you have two choices of ways to express the spherical harmonic (and the whole wavefunction): in terms of φ, θ, and r, or in terms of x, y, z and r.

However, what I was wondering is, shouldn't it conceptually be possible to express the harmonic just in terms of x, y and z? If these are coordinates representing a specific point, then the value of the wavefunction at that point should be specified by the coordinates alone, right, which raises the question of what the need is for the radius r. In fact, here is a deeper way of asking the same thing: why is r not equal to (x²+y²+z²)^1/2? Or is it?

[Corribus]

You are right that it technically doesn't matter if you express the wavefunction in terms of x,y,z (Cartesian) coordinates or r, theta, phi (spherical polar) coordinates.  Usually the latter is more convenient because the wavefunctions have some kind of radial symmetry and because it makes it easy to separate the radial from the angular portion of the wavefunction, whereby they can solved independently.  This is the way it's usually taught in physical chemistry courses, much to the consternation of students, who have to learn how to work with spherical polar coordinates.

Being honest, I was also a little confused by the way the authors of the wikipedia article chose to express their spherical harmonics (angular wavefunctions).  I've not come across them expressed in terms of x,y,z and r before and I had to do a few of the simple conversions with the d-orbital functions to make sure I was looking at the right thing.  I'd have to think about why r appears in those cartesian expressions, but my best quick guess is that it has something to do with the fact that they are trying to express an angular wavefunction in terms of a coordinate system that intrinsically includes distance from the origin.  The r in those functions "divides out" that radial dependency so that it is a purely angular wavefunction, but the result is a hybrid between two coordinate systems that frankly just makes the expressions a whole lot more cumbersome. 

You'll notice that when converted to spherical polar coordinates, the angular functions include no radial dependency.  This is expected - the radial dependency is wholly separable in sphereical polar coordinate systems and can be expressed as a radial wavefunction.  This is also convenient because only the radial portion depends on the principle quantum number.  So by usuing spherical polar coordinates we can talk about the general angular shape of, say, p-orbitals without worrying about whether it is 2p or 3p, etc.  I don't believe this is possible in a Cartesian coordinate system, though it's been a long time since I tried. 

Basically my guess is that if you combined those expressions in that wikipedia article with radial wavefunctions in x,y,z coordinates - which would probably also includes some r, theta and phi components - you'd end up with all the radial portions cancelling out, leaving you with a total wavefunction in pure x-y-z coordinates.  What they've done is try to split up that x-y-z total wavefunction into radial and angular portions, when that coordinate system isn't really compatible with such an approach.  They should have just done it right and done the whole, proper conversion rather than taking it halfway.  I'm not sure why they didn't.  Spherical harmonic functions have other uses than atomic wavefunctions, so maybe there's some other application I'm not aware of where it makes sense to express them as they have. 

Sorry for causing confusion by linking to that site.  I just saw that they worked when substituting in all the thetas and phis, so I thought this would save me the trouble of having to type it all in here.  That's what I get for taking short cuts.

[Big-Daddy]

You are right that it technically doesn't matter if you express the wavefunction in terms of x,y,z (Cartesian) coordinates or r, theta, phi (spherical polar) coordinates.  Usually the latter is more convenient because the wavefunctions have some kind of radial symmetry and because it makes it easy to separate the radial from the angular portion of the wavefunction, whereby they can solved independently.  This is the way it's usually taught in physical chemistry courses, much to the consternation of students, who have to learn how to work with spherical polar coordinates.

Being honest, I was also a little confused by the way the authors of the wikipedia article chose to express their spherical harmonics (angular wavefunctions).  I've not come across them expressed in terms of x,y,z and r before and I had to do a few of the simple conversions with the d-orbital functions to make sure I was looking at the right thing.  I'd have to think about why r appears in those cartesian expressions, but my best quick guess is that it has something to do with the fact that they are trying to express an angular wavefunction in terms of a coordinate system that intrinsically includes distance from the origin.  The r in those functions "divides out" that radial dependency so that it is a purely angular wavefunction, but the result is a hybrid between two coordinate systems that frankly just makes the expressions a whole lot more cumbersome. 

You'll notice that when converted to spherical polar coordinates, the angular functions include no radial dependency.  This is expected - the radial dependency is wholly separable in sphereical polar coordinate systems and can be expressed as a radial wavefunction.  This is also convenient because only the radial portion depends on the principle quantum number.  So by usuing spherical polar coordinates we can talk about the general angular shape of, say, p-orbitals without worrying about whether it is 2p or 3p, etc.  I don't believe this is possible in a Cartesian coordinate system, though it's been a long time since I tried. 

Basically my guess is that if you combined those expressions in that wikipedia article with radial wavefunctions in x,y,z coordinates - which would probably also includes some r, theta and phi components - you'd end up with all the radial portions cancelling out, leaving you with a total wavefunction in pure x-y-z coordinates.  What they've done is try to split up that x-y-z total wavefunction into radial and angular portions, when that coordinate system isn't really compatible with such an approach.  They should have just done it right and done the whole, proper conversion rather than taking it halfway.  I'm not sure why they didn't.  Spherical harmonic functions have other uses than atomic wavefunctions, so maybe there's some other application I'm not aware of where it makes sense to express them as they have. 

Sorry for causing confusion by linking to that site.  I just saw that they worked when substituting in all the thetas and phis, so I thought this would save me the trouble of having to type it all in here.  That's what I get for taking short cuts.

Ah, I see, and thanks again.

How would I go about finding expressions to reduce the wavefunctions from being in x, y, z and r into x, y and z? Do I really need to substitute in r=(x²+y²+z²)^1/2, or is there a quicker way?

(Or even from φ, θ and r into x, y and z - bearing in mind that all my current expressions for the polar coordinates in terms of the spherical ones still contain r!)

[Corribus]

I apologize for the delay in my reply, but replacing r for the expression you mentioned is really the only way to do it.  If you do, you'll see how much more cumbersome it is to work with Cartesian coordinates, so I'm not sure why you would feel the inclination to do so.  For hydrogenic wavefunctions, spherical polar coordinates are used almost exclusively, just because it's so much more convenient - not only in writing the dang things out but also doing mathematical exercises (e.g., integration) to find eigenvalues and so forth.

[Big-Daddy]

I apologize for the delay in my reply, but replacing r for the expression you mentioned is really the only way to do it.  If you do, you'll see how much more cumbersome it is to work with Cartesian coordinates, so I'm not sure why you would feel the inclination to do so.  For hydrogenic wavefunctions, spherical polar coordinates are used almost exclusively, just because it's so much more convenient - not only in writing the dang things out but also doing mathematical exercises (e.g., integration) to find eigenvalues and so forth.

I see, thank you very much for your *delete me*

[Enthalpy]

d_z² is the orbital corresponding to uncombined wavefunction l=2,ml=0; d_xy and d_yz correspond to a combination of the l=2,ml=1 and l=2,ml=-1 wavefunctions; and d_xy and d_x²-y² correspond to a combination of the l=2,ml=2 and l=2,ml=-2 wavefunctions.

Beware there are several sets of "basic" wavefunctions. None is more basic than the other, and each can be written as a combination of the other set.

Take the P orbitals: the doughnuts around x, y and z make one set, where the phase of the wave turns by 360° when the geometrical angle around the nucleus makes 360°.

But if you add two doughnut orbitals around x (or y or z) turning opposite ways, you get peanut orbitals centered on y or z or some other direction in this plane (depending on the relative phase you take for the two doughnuts around x) with one lobe positive and the other negative. This set of peanut orbitals is just as good a solution, and as stationary as, the doughnut set. And a doughnut is just a sum, with 90° phase shift, of two peanuts.

This situation is absolutely similar to light, where a circular polarization is a sum of two linear ones, and a linear is a sum of two circular ones. And similar to a propagating wave being the sum of two standing waves and a standing wave being the sum of two propagating ones.

Nice pictures there: http://winter.group.shef.ac.uk/orbitron/

Also, you can write any P wave as a (complex) weighted sum of this small set of P functions, but a D or F is not a sum of P. And any state where the electron is bound in the atom is a weighted sum of all (=infinite) the stationary solutions (=orbitals), but a sum of orbitals of different energies is not a stationary solution; it's a state that is evolving, for instance by absorbing or emitting an EM wave.