[Wien] About plot figure

[Corsin Battaglia]

Dear Lo Achd

I pasted the input, I suggested to you, in my MATHEMATICA notebook and it
worked. I do not really know, what went wrong in your case.

The reason, why I suggested you to use

SphericalHarmonicY[2,0,theta,phi]*Conjugate[SphericalHarmonicY[2,0,theta,phi
]]

is the following: In general, spherical harmonics are complex valued
functions, so it is not possible to represent them in 3D space. The function
SphericalHarmonicY[2,0,theta,phi], which is the orbital with quantum numbers
(l,m)=(2,0) is real valued. But check out SphericalHarmonicY[2,1,theta,phi]
for instance, it is complex valued. When I try to plot this function,
MATHEMATICA gives me exactly the errors you pasted in your message.
One possibility is to represent to modulus of the function, which is real
valued. This is what I suggested to you.

Another possibility is to use a a real valued superposition of spherical
harmonics (I think, it is just a change of basis in the sens of vector
algebra). I think, this is the meaning of the characters in WIEN2k. Maybe
someone more experienced could confirm that.
Real valued combinations are given below. If you express the x,y and z's in
spherical coordinates, you will see, that they agree with the expression in
terms of spherical harmonics.
(now in LATEX style)

s corresponds to 1=2\sqrt{\pi}Y_{0,0}

p_z corresponds to z=2\sqrt{\frac{\pi}{3}}Y_{1,0}
p_x corresponds to x=\sqrt{\frac{2\pi}{3}}i( Y_{1,-1}-Y_{1,1})
p_y corresponds to y=i\sqrt{\frac{2\pi}{3}}(Y_{1,-1}+Y_{1,1})

d_{z^2}corresponds to 3z^2-1=4\sqrt{\frac{\pi}{5}}Y_{2,0}
d_{xz} corresponds to xz=\sqrt{\frac{2\pi}{15}}(Y_{2,-1}-Y_{2,1})
d_{yz} corresponds to yz=i\sqrt{\frac{2\pi}{15}} (Y_{2,-1}+Y_{2,1})
d_{xy} corresponds to xy=i\sqrt{\frac{2\pi}{15}}(Y_{2,-2}-Y_{2,2})
d_{x^2-y^2} corresponds to x^2-y^2=2\sqrt{\frac{2\pi}{15}}(Y_{2,2}+Y_{2,-2})


Now back to your question, the plotting of the d-eg orbital.
The d stands for l=3. The e_g thing, comes from group theory and designates
a irreducible representation of a group. I am not quite sure, if this is the
space group of your crystal or maybe some conventional notation in the DFT
community. (Could someone help us out here?) So when you tell me, that you
want to plot the d e_g orbitals, I understand that you want to plot the d
orbital, which transforms according to the e_g irreducible representation of
the group. This is usually one of the above real-valued linear combination
of spherical harmonics given above. So let us say you want to plot the
d_{xy} orbital, then just plot

i\sqrt{\frac{2\pi}{15}}(Y_{2,-2}-Y_{2,2})

Now without the Conjugate[] thing, since it is real valued. The i is
necessary to make the whole function real valued.

I am just Diploma student, so do not take everything for granted!!!

I hope this helps anyway

Corsin

----- Original Message ----- 
From: "Lo Achd" <wien_lo_7 at hotmail.com>
To: <wien at zeus.theochem.tuwien.ac.at>
Sent: Friday, April 16, 2004 11:01 AM
Subject: Re: [Wien] About plot figure


> Dear Dr. Corsin Battaglia
>
> Thank you very much for your reply and good suggestions.
>
> Following your suggestion I  input following
>
> -----------------------
> <<Graphics`ParametricPlot3D`
>
>
SphericalPlot3D[SphericalHarmonicY[2,0,theta,phi]*Conjugate[SphericalHarmoni
> cY[2,0,theta,phi]],{theta,0,Pi},{phi,0,2Pi}]
>
>
> ------------------------
>
> But I not get the figure of Dz2, and the following is appear at the
screen:
>
> --------------------
>
> ParametricPlot3D::"pplr":
> (<<1>>) {Sin[theta] Cos[phi],Sin[theta]Sin[phi], Cos[theta]} is not a list
> starting with three real numbers at {theta,phi}={0.0,0.0}.
>
> ParametricPlot3D::"pplr" :
> (<<1>>) {Sin[theta] Cos[phi],Sin[theta]Sin[phi], Cos[theta]} is not a list
> starting with three real numbers at {theta,phi}={0.165347,0.0}.
>
> ParametricPlot3D::"pplr" :
> (<<1>>) {Sin[theta] Cos[phi],Sin[theta]Sin[phi], Cos[theta]} is not a list
> starting with three real numbers at {theta,phi}={0.330694,0.0}.
>
> General::"stop": "Further output of \!\(ParametricPlot3D :: \"pplr\"\)
will
> \
> be suppressed during this calculation."
>
> ---------------------------------------
>
> But if I input the
>
> <<Graphics`ParametricPlot3D`
>
>
SphericalPlot3D[SphericalHarmonicY[2,0,theta,phi],{theta,0,Pi},{phi,0,2Pi}]
>
> I can get a figure which is very similar with the Dz2-orbital.
>
> Could you please tell me how I can plot the d-eg orbital?
>
> Thank you
>
> Lo
>
>
> >From: "Corsin Battaglia" <corsin.battaglia at unine.ch>
> >Reply-To: wien at zeus.theochem.tuwien.ac.at
> >To: <wien at zeus.theochem.tuwien.ac.at>
> >Subject: Re: [Wien] About plot figure
> >Date: Thu, 8 Apr 2004 18:35:29 +0200
> >
> >Dear Lo Achd
> >
> >I guess, this is not a job for Wien2k.
> >However, if you have MATHEMATICA, try this
> >
> ><<Graphics`ParametricPlot3D`
> >
>
>SphericalPlot3D[SphericalHarmonicY[2,0,theta,phi]*Conjugate[SphericalHarmon
i
> >cY[2,0,theta,phi]],{theta,0,Pi},{phi,0,2Pi}]
> >
> >You can make your own linear combinations.
> >
> >Best regards and good luck
> >
> >Corsin
> >
> >----- Original Message -----
> >From: "Lo Achd" <wien_lo_7 at hotmail.com>
> >To: <wien at zeus.theochem.tuwien.ac.at>
> >Sent: Wednesday, April 07, 2004 2:38 PM
> >Subject: [Wien] About plot figure
> >
> >
> > > Dear Wien users,
> > >
> > > I try to plot the 3D-figure for the d orbital, such as d_{3z2-r2},
> > > d_{x2-y2}. Who can tell me how I can do it?
> > >
> > > Thank you!
> > >
> > > Lo
> > >
> > > _________________________________________________________________
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