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

Question

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?

Answer 1

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.