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I was thinking about basis sets used in Quantum Chemistry programs and thought of why not try to visualize them.

I started with GTO from EMSL website for $\ce{H2}$, specifically cc-pVDZ basis which is the following,

! s functions
H    5    3
     13.0100000            0.0196850              0.0000000              0.0000000        
      1.9620000            0.1379770              0.0000000              0.0000000        
      0.4446000            0.4781480              0.0000000              0.0000000        
      0.1220000            0.0000000              1.0000000              0.0000000        
      0.0297400            0.0000000              0.0000000              1.0000000        
! p functions
H    2    2
      0.7270000           1.0000000              0.0000000        
      0.1410000           0.0000000              1.0000000  

I framed the basis functions for $s-$orbital as following, $$0.019685\times exp(-13.01 r^2) + 0.137977\times exp(-1.962 r^2) + 0.4446\times exp(-0.478148 r^2)$$ $$1\times exp(-0.12200 r^2)$$ $$1\times exp(-0.02974 r^2) $$

This is only the radial part (the associated spherical harmonic is not here), but still I can plot the above against $r$.

plotting of the function given above

The above plot is for the $s-$functions.


I am able to do the same thing for $p-$functions, and I get similar Gaussians (not shown here) with different widths, and centered at zero.

Shouldn't the lobe of $p-$ be centered larger than zero (which implies that the functional form I am using is not correct/incomplete for $p-$function). Or the lack of explicit spherical harmonic functions does not allow the $p-$functions to be similarly visualized. Could someone clarify.

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    $\begingroup$ Right now, this is unclear. Is the plot for s or p functions? What does "Shouldn't the lobe of p− be centered larger than zero" mean? Is it that the amplitude at $r=0$ should be zero and $r>0$ should be nonzero? $\endgroup$ – pentavalentcarbon Sep 11 '17 at 15:01
  • $\begingroup$ @pentavalentcarbon : The plot is for $s-$functions. The $p-$ function have almost zero amplitude near $r$ , while non zero probability at certain distance from $r$ (the dumbel shape in one axis, say towards arbitrary direction , described as +ve). I plan to achieve that. $\endgroup$ – ankit7540 Sep 12 '17 at 10:59
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Your suspicion is correct - you need the angular part, that is, a spherical harmonic (sometimes it may be cartesian instead) to get the shapes of the orbitals and thus, actual lobes of $p$-functions.

Also note that projection to $(x)$ from $(x,y,z)$ may not be straightforwardly the same for all angular momenta. Additionally, the angular part also has influence on the normalization of the basis functions, which may or may not be included in the basis set format at hand. The last point, however, can be checked by numerical integration.

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