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#1 2018-02-21 11:35:07

Registered: 2018-02-09
Posts: 5

Density and inegreated density at nuclei positions

Dear Tian and other forum members,

I have a question related to the one posted in the other thread.

Looking at density plots one finds very large contribution at nuclei position.
Grid integration doesn't change the picture, as it is just multiplication with the differential volume, which is uniform and non-zero for discrete integration.
However, for the radial distribution and its integral contribution at nuclei is zero.
The later is obvious because for polar integration there is the r^2 term.

The density should be zero because of Coulomb repulsion with nuclei, but this volume is probably too small to be resolved in practical calculation.
Moreover, such contribution looks reasonable only for s orbitals (maybe even only 1s), is this the case?

What is the physical reality?
Any comments are welcome!


#2 2018-02-22 05:02:35

Tian Lu (Multiwfn developer)
From: Beijing
Registered: 2017-09-11
Posts: 362

Re: Density and inegreated density at nuclei positions


Electron density at nuclei is not zero, but shows sharp peak character, this is because electron has negative charge and thus has strong Coulomb attractive interaction with nuclei, which has positive charge. You can consult the book "A Chemist's Guide to Density Functional Theory 2ed" written by Koch, in Section 2.1 there is nice brief introduction about electron density distribution.

Atomic orbitals with angular moment higher than s has exactly zero contribution to electron density at nuclei, because these orbitals have nodal character at that position. All occupied s orbitals contribute to electron density at nucleic, this is not limited to 1s, because 2s, 3s... orbitals have penetration effect.

The contribution to the total number of electron from integration points at various place is another topic, it seems that you have interesting about this, so I explain more. When Multiwfn conducts atomic-center integration, the radial part and angular part are treated separately. The second kind Gauss-Chebyshev method is used for generating position and weight for radial part, while Lebedev method is used for the angular part, and then the two parts are combined together to form the final integration points and weights.

For 20 radial points case, the radial distances (Bohr) and integration weights are shown below
178.064275    142426688.497378
44.017473      1087888.306620
19.195669        61292.430818
10.510048         7748.707839
  6.492092         1512.534405
  4.311941          385.545595
  3.000000          117.220500
  2.151299           40.214419
  1.572417           14.993421
  1.161531            5.908530
  0.860932            2.405989
  0.635964            0.991962
  0.464835            0.405674
  0.333333            0.160796
  0.231914            0.059984
  0.154034            0.020202
  0.095147            0.005749
  0.052095            0.001225
  0.022718            0.000150
  0.005616            0.000004

As you can see, when r is close to zero (very close to nuclei), the weight is quite small. Therefore, despite the fact that at nuclei the electron density is rather large, in practical situation, the region very close to nucleic only has marginal contribution to total number of electrons.

Best wishes,



#3 2018-02-26 10:58:43

Registered: 2018-02-09
Posts: 5

Re: Density and inegreated density at nuclei positions

Thank you for detailed explanation!


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