The wave function of hydrogen

[einstein1921]

HI,everyone.I have a problem. the angular portion of wavefunction of hydrogen,like 3d.
n=3,l=2,so m=2,1,0,-1,-2.I read some books that say dxy,dxz,dyz,dz2,dx2-y2,so what the
corresponding Relation between them. for example,dz2 corresponding what ?m=0?? and why?
any help will be highly appreciated!

 

[Amok]

You are correct. The cartesian coordinates subscripts usually indicate nodal planes. For example, the p_z function (l=1, m=0)can be written as:

$$<br /> \psi = \frac{1}{\sqrt{32 \pi}} \left( \frac{Z}{a_0} \right)^{5/2} z e^{\frac{-Zr}{2 a_0}}<br />$$

Where cartesian and polar coordinates are mixed together. However, when written in this form, it is easy to see that whenever z=0 then psi also = 0, which means that the xy plane is a nodal plane.

As for d functions, it becomes more complicated. I think that some of them will be complex functions, and you'll need to take linear combinations of them in order to get real functions that you can plot. The d_(x^2-y^2) function is a linear combination of the complex m=+2 and m=-2 functions. This will yield:

$$\psi = \frac{1}{81\sqrt{ 2\pi}} \left( \frac{Z}{a_0} \right)^{7/2} e^{\frac{-Zr}{3 a_0}} (x^2 - y^2)$$

So, in a way, this notation with the cartesian coordinates describes the geometry of the wavefunction.

 

[einstein1921]

You are correct. The cartesian coordinates subscripts usually indicate nodal planes. For example, the p_z function (l=1, m=0)can be written as:

$$<br /> \psi = \frac{1}{\sqrt{32 \pi}} \left( \frac{Z}{a_0} \right)^{5/2} z e^{\frac{-Zr}{2 a_0}}<br />$$

Where cartesian and polar coordinates are mixed together. However, when written in this form, it is easy to see that whenever z=0 then psi also = 0, which means that the xy plane is a nodal plane.

As for d functions, it becomes more complicated. I think that some of them will be complex functions, and you'll need to take linear combinations of them in order to get real functions that you can plot. The d_(x^2-y^2) function is a linear combination of the complex m=+2 and m=-2 functions. This will yield:

$$\psi = \frac{1}{81\sqrt{ 2\pi}} \left( \frac{Z}{a_0} \right)^{7/2} e^{\frac{-Zr}{3 a_0}} (x^2 - y^2)$$

So, in a way, this notation with the cartesian coordinates describes the geometry of the wavefunction.

Thank you,Amok.your answer is very helpful to me.would you please tell me that p_x corresponds to m=1 or m=-1 and why?
again thanks.

 

[cgk]

Thank you,Amok.your answer is very helpful to me.would you please tell me that p_x corresponds to m=1 or m=-1 and why?
again thanks.

It is more of a convention thing than anything; there are no strict relationships between the two. You can define whatever.

That being said, the most common convention in quantum chemistry has the real solid harmonics beeing approximated like this in terms of cartesians:

AngMom L = 0
S(0,m=+0) = + 1.00000
AngMom L = 1
S(1,m=-1) = + 1.00000 y
S(1,m=+0) = + 1.00000 z
S(1,m=+1) = + 1.00000 x
AngMom L = 2
S(2,m=-2) = + 1.73205 x y
S(2,m=-1) = + 1.73205 y z
S(2,m=+0) = + 1.00000 z^2 - 0.50000 y^2 - 0.50000 x^2
S(2,m=+1) = + 1.73205 x z
S(2,m=+2) = - 0.86603 y^2 + 0.86603 x^2
AngMom L = 3
S(3,m=-3) = - 0.79057 y^3 + 2.37171 x^2 y
S(3,m=-2) = + 3.87298 x y z
S(3,m=-1) = + 2.44949 y z^2 - 0.61237 y^3 - 0.61237 x^2 y
S(3,m=+0) = + 1.00000 z^3 - 1.50000 y^2 z - 1.50000 x^2 z
S(3,m=+1) = + 2.44949 x z^2 - 0.61237 x y^2 - 0.61237 x^3
S(3,m=+2) = - 1.93649 y^2 z + 1.93649 x^2 z
S(3,m=+3) = - 2.37171 x y^2 + 0.79057 x^3

This is what you will usually get when using "spherical" basis functions in a chemical electronic structure program, and correspondingly also what most plots and chemistry books refer to.

 

[Amok]

Exact same thing as p_z, except you the nodal plane will be yz, instead of xy. The fact that it is assigned to m=+1 is just a convention. Check this out:

http://en.wikipedia.org/wiki/Atomic_orbital#Orbitals_table

Also, if you want more details, check out QM textbooks. I use the Cohen-Tanoudji (because I'm a French speaker, but is also available in English). What you are asking about is explained in chapter VII, complement E (in some detail, but they give references as well). I think that if you look into good quantum chemistry books, you're bound to find explanations about this, since chemists use these concepts in order to explain chemical phenomena on a daily basis.