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After my previous question (here), I'm now studying the orbitals actually used in Gaussian through calculating them myself using Basis Set Exchange data. However, I found that some orbital exponents of s-orbital are too large and correspondingly their normalization terms become too large. As a result, the scale of s-orbital and other orbitals (e.g., p- and d-orbitals) is very different.

For example, the following figure shows the orbital shapes on O-H bond of H2O molecule (note that the direction of spherical harmonic is along O-H bond, that is, O-H is the x-axis).

enter image description here

Of course, each primitive Gaussian function is normalized (e.g., six primitives for O_1s) so that the integral is one. Actually, Basis Set Exchange provides the following orbital exponent and contraction coefficient values for the primitive functions as follows.

SHELL TYPE PRIMITIVE        EXPONENT          CONTRACTION COEFFICIENT(S)
O     0
S    6   1.00
      0.5484671660D+04       0.1831074430D-02
      0.8252349460D+03       0.1395017220D-01
      0.1880469580D+03       0.6844507810D-01
      0.5296450000D+02       0.2327143360D+00
      0.1689757040D+02       0.4701928980D+00
      0.5799635340D+01       0.3585208530D+00
SP   3   1.00
      0.1553961625D+02      -0.1107775495D+00       0.7087426823D-01
      0.3599933586D+01      -0.1480262627D+00       0.3397528391D+00
      0.1013761750D+01       0.1130767015D+01       0.7271585773D+00
SP   1   1.00
      0.2700058226D+00       0.1000000000D+01       0.1000000000D+01
D    1   1.00
      0.8000000000D+00       1.0000000

Comparing p- and d-orbitals, the extreme value of s-orbital near O atom seems strange to me. Is this because the orbital exponent is too large (e.g., 0.5484671660D+04 and 0.8252349460D+03)? Or is this because multiple primitive Gaussian functions with large exponents actually represent a real Slater function? (Or this may be just a mistake of my implementation.)

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    $\begingroup$ I'm very confused. Are you asking why an s type orbital has a large value near the nucleus at which it is sited, while all other angular momenta do not? $\endgroup$
    – Ian Bush
    Feb 6, 2022 at 11:05
  • $\begingroup$ @IanBush Yes. After normalizing all orbitals, the scale or "max value" of the s-orbital and other orbitals is very different and this is strange to me, but is this not strange? If so, the electron density on the O-H bond will be very thin, but is this not strange? If this is normal, there is no problem. $\endgroup$
    – neco
    Feb 6, 2022 at 11:58

1 Answer 1

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Individual basis functions do not have to be physical. Only the superposition of basis functions is used to represent atomic or molecular orbitals. Several kinds of atomic basis functions can be used, as I have discussed in a recent open access review paper: Int. J. Quantum Chem. 119, e25968 (2019).

The only Gaussian basis functions that do have physical content are natural orbital basis sets, which represent components of the correlated electron density. Such basis sets are always generally contracted, i.e. each primitive basis function contributes to all contractions.

In contrast, segmented basis sets, such as the Pople sets, split the orbital description into pieces. They do not represent atomic orbitals any more.

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