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Modelling the shape of an atom

  1. Dec 8, 2009 #1
    hi,

    when modelling the shape of a hydrogen atom orbital, is it the real part of the spherical harmonics that i plot?

    thanks
     
    Last edited: Dec 8, 2009
  2. jcsd
  3. Dec 8, 2009 #2

    alxm

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    Well the wave function (being time-independent and without an external field) is real, and so are the spherical harmonics (m being an integer).
     
  4. Dec 8, 2009 #3
    but this only works if [tex]\phi[/tex] is pi o 2pi? but [tex]\phi[/tex] can take any value? or am i misunderstanding it?
     
  5. Dec 8, 2009 #4

    alxm

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    Ah, I was a bit confused. Right, the harmonics have a complex phase, but it's physically insignficant here. So [tex]e^{i\phi m}[/tex] can be taken to be [tex]sin(\phi m)[/tex] if m > 0, [tex]cos(\phi m)[/tex] if m < 0 (or vice versa) and [tex]\frac{1}{\sqrt{2}}[/tex] if m=0
     
  6. Dec 8, 2009 #5
    You basically plot the probability-function, lYl2 = Y.Y*
     
  7. Dec 8, 2009 #6
    You want to plot the probability distribution for the probabilities of the electron.

    This includes both the radial part and the spherical harmonics.
     
  8. Dec 8, 2009 #7

    alxm

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    Why are you guys telling the OP what he wants?
    There's nothing wrong with plotting the wave function itself, it's done all the time (and there's a pedagogical point in doing so, for understanding MO theory).

    The question was how to handle the complex-valued phase of the spherical harmonics when doing so. If you're plotting the density then the question is moot; they disappear in the integration. The less-trivial point is that the phase is irrelevant, and not only for plotting; you can stick that real-valued wave function back into any equation and you'll still get all the same observables.

    Also, in numerical QM calculations, atomic wave functions are expressed as simple real-valued functions (namely gaussians) all the time.
     
  9. Dec 8, 2009 #8
    This is what the OP asked for: how to model the shape of a hydrogen atom. And you do so by plotting the square modulus (≡probability/electron density) of the wavefunction, which is real valued.
     
  10. Dec 8, 2009 #9
    When we plot wavefunctions(not its absolute square), we have two options. We can either plot real and imaginary parts separately, or choose a basis so that all basis functions are real-valued (We can do this whenever there is a time reversal symmetry). The latter is what they often do, especially in chemistry books.

    For l=1 case, Y_10 is already real, whereas Y_11 and Y_1,-1 are complex.
    Y_10 ~ cos(theta) = z/r
    Y_11 ~ sin(theta)exp(i*phi) = (x+iy)/r
    Y_1,-1 ~ sin(theta)exp(-i*phi) = (x-iy)/r

    Yet, if we take the linear combinations of Y_11 and Y_1,-1 we have real valued functions

    Y_10 ~ cos(theta) = z/r
    Y_11 + Y_1,-1 ~ sin(theta)cos(phi) = x/r
    Y_11 - Y_1,-1 ~ sin(theta)sin(phi) = y/r

    Now, we have all real orbitals! These orbitals are so called p_z, p_x and p_y orbitals.

    The rule of thumb is that you leave m=0 state alone, and take the sum of and difference between all m and -m states for m!=0.
    I guess you might want to do the same thing for l=2 and verify that we get x^2-y^2, 3z^2-r^2, xy, yz, zx states.
     
    Last edited: Dec 8, 2009
  11. Dec 11, 2009 #10
    I would try something like this:

    Inicialization:
    - define the number of points in cloud
    - define a maximum radial distance (since you cannot go as far as infinity)
    - define the radial discretization
    - define the azimuthal discretization
    - define the polar discretization
    - visit each point in the sphere with these discretization values
    - calculate the volume element
    - calculate psi modulus squared
    - multiply the two numbers
    - accumulate this value
    - end each
    - the accumulated value is the normalization factor

    Plotting:
    - for each point in cloud do
    - calculate a random value between 0 and 1 (collapse)
    - visit each point in the sphere with the discretization values
    - calculate the volume element
    - calculate psi modulus squared
    - multiply the two numbers
    - accumulate the value
    - divide this value by the normalization factor. If it is greater than collapse value, plot point and break each
    - end each
    - end each

    Obs.: the phase angle must be mapped to a specifc color
     
    Last edited: Dec 11, 2009
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