How Does Psi4 Handle Basis Set Cartesian vs. Spherical GTOs in Quantum Chemistry Calculations

1

Hi everyone,

I am trying to understand whether PSI4 performs its calculations using spherical Gaussian-type orbitals (GTOs) or Cartesian GTOs. Could someone with experience in this area help clarify?

Additionally, if possible, could you provide:

  1. The complete mathematical expressions for the p- and d-type basis functions in both Cartesian and spherical forms?
  2. Any relevant transformation rules between the two representations (e.g., how Cartesian d-orbitals map to spherical harmonics)?

For reference, I’ve checked sources like:

  • Schlegel & Frisch (Int. J. Quantum Chem. 1995)
  • Helgaker/Jørgensen/Olsen (Molecular Electronic-Structure Theory)

But a concise explanation or example would be incredibly helpful!

Thanks in advance!

2

PSI4 can either use spherical harmonic type functions, or use Cartesian type functions. As far as I know, PSI4 uses spherical harmonics by default. You can check this information in PSI4 output, which would look like

  -AO BASIS SET INFORMATION:
    Name                   = STO-3G
    Blend                  = STO-3G
    Total number of shells = 5
    Number of primitives   = 15
    Number of AO           = 7
    Number of SO           = 7
    Maximum AM             = 1
    Spherical Harmonics    = TRUE

When using user-defined basis set in the input file, one can choose which type of functions to be used, e.g.

memory 4 GB

molecule mymol {
symmetry C1
no_reorient
0 1
O1       0.00000000       0.00000000       0.06200700
H1       0.00000000      -0.78397600      -0.49205200
H2       0.00000000       0.78397600      -0.49205200
}

basis mybas {
 assign O1 gen1
 assign H1 gen2
 assign H2 gen3
[gen1]
spherical
****
O 0
S   6  1.00
   5.484671660E+03   1.831074430E-03
   8.252349460E+02   1.395017220E-02
   1.880469580E+02   6.844507810E-02
   5.296450000E+01   2.327143360E-01
   1.689757040E+01   4.701928980E-01
   5.799635340E+00   3.585208530E-01
SP   3  1.00
   1.553961620E+01  -1.107775500E-01   7.087426820E-02
   3.599933590E+00  -1.480262630E-01   3.397528390E-01
   1.013761750E+00   1.130767020E+00   7.271585770E-01
SP   1  1.00
   2.700058230E-01   1.000000000E+00   1.000000000E+00
D   1  1.00
   8.000000000E-01   1.000000000E+00
****
... (not shown)

Note the spherical above **** means spherical harmonics. If you modify it into cartesian, Cartesian type functions would be used.

For the relationship between two types, you can find that in this blog. It is written mainly in Chinese, but I guess you can read formulae and symbols therein. Please keep in mind that blog is written using conventions in the Gaussian software package, which might not be exactly the same as conventions in PSI4, but the technique detail or spirit is supposed to be very similar.

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3

Your post has several of the signatures of being written by AI. Do not use AI to generate posts for the forum.

As for mathematical expressions and transformation rules, what part of those references do you not understand?

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4

yes I had used Al to correct my statement. I will avoid that in the future.
Thanks everyone for your reply.
I want to write H2O example in detail with each vector and matrix ( H, F, Veff, I, C, …) and how their expression change while , Psi4 handle QC of DFT.
I think the details are useful for beginners so that they do not use a tool and benefit from its results without understanding its essence.

I looked to SCF source code :
“”"

Integral generation from Psi4’s MintsHelper

wfn = psi4.core.Wavefunction.build(mol, psi4.core.get_global_option(‘BASIS’))
t = time.time()
mints = psi4.core.MintsHelper(wfn.basisset())
S = np.asarray(mints.ao_overlap())

“”"

if i am not mistake, Psi4 use cartesian formula to get overlap integral even you specify spherical harmonic basis set,

“…AO (Atomic Orbitals): Cartesian Gaussians (6D, 10F, etc.), (L + 1)(L + 2)/2 functions per shell of angular momentum L. The ordering of Cartesian exponents for a given L is given by the standard ordering…”

the same situation for Kinetic and Columbic attraction matrix:
“”"

Compute required quantities for SCF

V = np.asarray(mints.ao_potential())
T = np.asarray(mints.ao_kinetic())
I = np.asarray(mints.ao_eri())

print(‘\nTotal time taken for integrals: %.3f seconds.’ % (time.time() - t))
t = time.time()

Build H_core: [Szabo:1996] Eqn. 3.153, pp. 141

H = T + V

“”"

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Using AI for editing is fine. Your post looked like it was generated by AI for content-scraping, which is not allowed. In future, please explain why you’re asking questions from the start.

I suggest you look at the Psi4Numpy tutorials. I suspect that what you want to do is already done.

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