# ORCA: meaning of numbers in ORCA.out of MOLECULAR ORBITALS for STO-3G Lithium

For Lithium STO-3G basis set

BASIS "ao basis" PRINT
#BASIS SET: (6s,3p) -> [2s,1p]
Li    S
0.1611957475E+02       0.1543289673E+00
0.2936200663E+01       0.5353281423E+00
0.7946504870E+00       0.4446345422E+00
Li    SP
0.6362897469E+00      -0.9996722919E-01       0.1559162750E+00
0.1478600533E+00       0.3995128261E+00       0.6076837186E+00
0.4808867840E-01       0.7001154689E+00       0.3919573931E+00
END


I see the for representing Lithium wave function (Single Slater's Determitatn) I need 3 basis STO-functions (STO(1s), STO(2s) and one STO(2p)) for compilation spin-orbitals. For Lithium we have $$1s^22s$$-eclectronic structure, so I need compile:

\begin{align} \chi_{1s} = \sum\limits_{j = 1}^{K} C_{(1s)_j} STO_j,\\ \phi_{1s} = \sum\limits_{j = 1}^{L} D_{(1s)_j} STO_j,\\ \chi_{2s} = \sum\limits_{j = 1}^{M} C_{(2s)_j} STO_j.\\ \ldots. \end{align} (As I understand)

So, I started to calculate with ORCA Lithium atom

! UHF STO-3G

%coords
CTyp xyz       # the type of coordinates = xyz or internal
Charge 0       # the total charge of the molecule
Mult 2         # the multiplicity = 2S+1
Units Angs     # the unit of length = angs or bohrs
coords
Li        0.000000       0.00000        0.00000
end
end
%output
Print[ P_Basis ] 2
Print[ P_MOs ] 1
end


and get information

------------------
MOLECULAR ORBITALS
------------------
0         1         2         3         4
-2.36917  -0.18012   0.13013   0.13013   0.13013
1.00000   1.00000   0.00000   0.00000   0.00000
--------  --------  --------  --------  --------
0Li  1s         0.992528 -0.276813  0.000000  0.000000 -0.000000
0Li  2s         0.029310  1.029989 -0.000000 -0.000000  0.000000
0Li  1pz       -0.000000  0.000000 -0.607531  0.786960 -0.107701
0Li  1px       -0.000000  0.000000 -0.176173 -0.265720 -0.947816
0Li  1py       -0.000000  0.000000  0.774512  0.556854 -0.300075

0         1         2         3         4
-2.33786   0.10225   0.19092   0.19092   0.19092
1.00000   0.00000   0.00000   0.00000   0.00000
--------  --------  --------  --------  --------
0Li  1s         0.991218 -0.281468  0.000000  0.000000  0.000000
0Li  2s         0.034144  1.029840 -0.000000 -0.000000 -0.000000
0Li  1pz       -0.000000 -0.000000  0.170191 -0.976799  0.129992
0Li  1px       -0.000000 -0.000000 -0.983678 -0.160587  0.081175
0Li  1py       -0.000000 -0.000000 -0.058416 -0.141685 -0.98818


So, how can I determine $$K$$, $$L$$ and $$M$$ numbers and $$C_{(1s)_j}$$, $$D_{(1s)_j}$$, $$C_{(2s)_j}$$ coefitients?

Firstly, note that the STO-3G basis set are not STOs, but approximations of STOs by the linear combination of three Gaussians.

In your case, you have 5 STO functions, of which two are s basis functions and three are p basis functions. Since p functions have zero contribution to the 1s and 2s orbitals, you may write $$K=L=M=2$$, or if you count zero coefficients as coefficients as well, you can write $$K=L=M=5$$ instead. The MO coefficients are read from the columns of the ORCA printout, but note that the 1s and 2s in the ORCA output file denote the first and second s functions, respectively, not the 1s and 2s orbitals (you may also notice this fact by observing that there are "1p" entries, which obviously can only refer to basis functions, not molecular orbitals). Keeping this in mind, you can read off the required coefficients as

$$C_{(1s)_1} = 0.992528, C_{(1s)_2} = 0.029310$$ $$D_{(1s)_1} = 0.991218, D_{(1s)_2} = 0.034144$$ $$C_{(2s)_1} = -0.276813, C_{(2s)_2} = 1.029989$$

Note that to actually use these coefficients, you have to use normalized basis functions instead of unnormalized ones.

• "but note that the 1s and 2s in the ORCA output file denote the first and second s functions, respectively, not the 1s and 2s orbitals" What number n do they belong to? (n = 1 or n = 2 or other) Commented Apr 18, 2022 at 8:38
• @Sergio Neither. Their certain linear combinations are the 1s and 2s orbitals, but they themselves do not have a well-defined quantum number n. This is a rather general feature of atomic orbital basis sets: the basis functions have well-defined l and m quantum numbers, but no well-defined n quantum numbers (however exceptions exist, e.g. the ANO basis sets). This is because having well-defined n quantum numbers does not benefit HF and DFT calculations in general, so basis set designers typically sacrifice this feature of the basis set in favor of other more desired properties. Commented Apr 18, 2022 at 9:09
• I mean, what is STO's r^(n-1)*exp(-zeta*r), what in n here? I need to plot it. Commented Apr 18, 2022 at 9:13
• @Sergio As I said, the basis functions of STO-3G are not STOs, but a linear combination of GTOs that is numerically similar, but analytically unrelated, to STOs. You have to expand each basis function to the linear combination of its constituent three GTOs, evaluate each GTO, and finally evaluate their linear combination. In no way is an expression like r^(n-1)*exp(-zeta*r) ever involved. Commented Apr 18, 2022 at 9:27
• Ok, I understand you (I think). I plotatomic orbital for helium drive.google.com/file/d/1nUkHl2WWqE-1vAuN7tCzBE-drhYrcXul/… Commented Apr 18, 2022 at 9:49