Shape of Slater type orbitals

[ziand]

In the ATK Reference (http://quantumwise.com/documents/manuals/latest/ReferenceManual/index.html/chap.atkse.html#sect1.atkse.intro) the shape of the orbitals used within the Hückel-Method is given as



Therein n, l and m are the principal, angular and magnetic qunatum numbers, respectivley:

1s :  n=1, l=0, m=0
2s :  n=2, l=0, m=0
2p :  n=2, l=1, m=-1,0,1
3s :  n=3, l=0, m=0
3p :  n=3, l=1, m=-1,0,1
3d :  n=3, l=2, m=-2,-1,0,1,2
...


The radial part R_nl(r) is given as a superposition of two (double zeta) Slater type orbitals (STOs):



The constants C1, C2, eta1, eta2 are the orbital weights and slater coefficients fitted by different authors and tabulated within the ATK-manual.

My question is now: why does the radial part depend on the angular momentum? In literature I often see STOs (here only single zeta for simplicity) of the form:

R_n(r) = N*r^(n-1)*exp(-eta*r)

with

N={ ( [2eta]^[2n+1] ) / (2n)! }^(1/2) .

The difference is the missing r^-l and a square root on some prefactors (N={...}^(1/2)).

By using the Cerda carbon graphite parameters of the 2p orbitals as an example (as given in a paper of Cerda and also in the ATK reference), I calculated the integral over Conjugate[phi]*phi  (phi being the whole orbital, including the spherical harmonics) and found out, that only by dropping the angular dependency and including the mentioned square roots, the orbitals are normalized.

Any comments would be highly recommended, as I want to plot the orbitals I'm using and I want to be sure that everything is OK.
Thanks.

[Anders Blom]

I think you may be right about the square root, we should check if that's a typo in the manual.

Regarding the other point, we should be clearer to specify that "n" in this formula is the principal quantum number, not the radial quantum number (which is equal to the number of nodes in the radial wave function). See http://en.wikipedia.org/wiki/Principal_quantum_number regarding the difference of these two (also cf. http://en.wikipedia.org/wiki/Hydrogen_wavefunction#Wavefunction).

[ziand]

Sorry for asking again.
Just to make it clear (for me). The relation between principal and radial and angular quantum numbers is
n = n_r + l + 1
n ... principal quantum number
n_r ... radial quantum number (number of radial nodes)
l ... angular quantum number

Now let C1 = 0.27152; eta1 = 1.62412; C2 = 0.73886; eta2 = 2.17687; (2p orbital of Carda carbon graphite) and



Which n and l (numbers) should be used to get the right orbitals? For me it would be logical to use n=2 and l=1 (that is 2p). However the whole orbital is not normalized in this case.

[kstokbro]

There is an error in the manual,
for the right definition see the attachment

[Anders Blom]

Clearly I (and the manual author ) were both looking at hydrogen-like orbitals. We'll fix the manual for 11.8.