Orthogonal condition of the HF orbitals

[Lexin Ding]

Hi everyone,

I have a rather silly question. I am studying the carbon dimer, and I found an inconsistency between the LCAO coefficients of the HF orbitals and the atomic overlap matrix. 

The canonical orbitals I chosen to form the active space is the following

Orbital  Occupation  Energy              Coefficients

 3.1     2.00000    -0.97790     1 1s    0.52851     2 1s    0.52851

 4.1     2.00000    -0.42404     1 1s    0.74698     1 2pz   0.26548     2 1s   -0.74698     2 2pz   0.26548

 7.1     0.00000     0.02997     1 1s    0.35594     1 2pz   0.62740     2 1s    0.35594     2 2pz  -0.62740

10.1     0.00000     1.17699     1 1s   -1.26422     1 2pz   1.24130     2 1s    1.26422     2 2pz   1.24130

And the atomic overlaps are the following

 Overlap matrix loaded to S

 Orbital matrix S               

   Orb      Coefficients

            1 1s      1 1s      1 2px     1 2py     1 2pz     2 1s      2 1s      2 2px     2 2py     2 2pz 

 

   1.1    1.000000  0.248362  0.000000  0.000000  0.000000  0.000016  0.056603  0.000000  0.000000  0.094479

   2.1    0.248362  1.000000  0.000000  0.000000  0.000000  0.056603  0.458262  0.000000  0.000000  0.447821

   3.1    0.000000  0.000000  1.000000  0.000000  0.000000  0.000000  0.000000  0.290270  0.000000  0.000000

   4.1    0.000000  0.000000  0.000000  1.000000  0.000000  0.000000  0.000000  0.000000  0.290270  0.000000

   5.1    0.000000  0.000000  0.000000  0.000000  1.000000 -0.094479 -0.447821  0.000000  0.000000 -0.315125

   6.1    0.000016  0.056603  0.000000  0.000000 -0.094479  1.000000  0.248362  0.000000  0.000000  0.000000

   7.1    0.056603  0.458262  0.000000  0.000000 -0.447821  0.248362  1.000000  0.000000  0.000000  0.000000

   8.1    0.000000  0.000000  0.290270  0.000000  0.000000  0.000000  0.000000  1.000000  0.000000  0.000000

   9.1    0.000000  0.000000  0.000000  0.290270  0.000000  0.000000  0.000000  0.000000  1.000000  0.000000

  10.1    0.094479  0.447821  0.000000  0.000000 -0.315125  0.000000  0.000000  0.000000  0.000000  1.000000

Surprisingly, I failed to confirm the orthonormal relations between the canonical orbitals. Take orbital 3.1 for example, its norm is 2*0.52851^2 + 2*0.52851^2*0.458262 = 0.81465. So orbital 3.1 is not normalized. This is not a big concern. However, if you take the overlap between orbital 3.1 and 7.1, you'd find it to be 0.52851*(0.35594*2*(1+0.458262)-0.62740*2*0.447821)=0.25167 and not 0.

I know that Molpro wouldn't have made such an obvious mistake that the HF orbitals are non-orthogonal. Could someone please let me know if I am missing something here? I have attached the complete output file here as well. Thank you very much in advance!

Best wishes,

Lexin

[andreas...@gmail.com]

Dear Lexin,

the output for the MO coefficients is incomplete (maybe skips smaller coefficients according to

some threshold?).

You can check the orthonormality condition of the MO's using:

{matrop
load,c,orb
load,s
print,c
tran,smo,s,c
print,smo}

which for your example prints:

 MATRIX C              

 SYMMETRY BLOCK 1.1
    0.70237936   0.70240344  -0.19485957  -0.18141425  -0.00000000  -0.00000000  -0.05239609  -0.00000000   0.00000000   0.13320127
    0.01432870   0.02822825   0.52850957   0.74697700   0.00000000   0.00000000   0.35594422   0.00000000  -0.00000000  -1.26421836
    0.00000000  -0.00000000   0.00000000  -0.00000000  -0.26636658   0.56264074   0.00000000   0.78625170  -0.29377143   0.00000000
   -0.00000000   0.00000000  -0.00000000  -0.00000000   0.56264074   0.26636658  -0.00000000   0.29377143   0.78625170  -0.00000000
   -0.00147544  -0.00922109  -0.17594477   0.26548326   0.00000000   0.00000000   0.62740164  -0.00000000   0.00000000   1.24130195
    0.70237936  -0.70240344  -0.19485957   0.18141425  -0.00000000   0.00000000  -0.05239609   0.00000000  -0.00000000  -0.13320127
    0.01432870  -0.02822825   0.52850957  -0.74697700   0.00000000  -0.00000000   0.35594422  -0.00000000   0.00000000   1.26421836
   -0.00000000   0.00000000   0.00000000  -0.00000000  -0.26636658   0.56264074   0.00000000  -0.78625170   0.29377143  -0.00000000
    0.00000000  -0.00000000  -0.00000000  -0.00000000   0.56264074   0.26636658  -0.00000000  -0.29377143  -0.78625170   0.00000000
    0.00147544  -0.00922109   0.17594477   0.26548326  -0.00000000   0.00000000  -0.62740164  -0.00000000   0.00000000   1.24130195

 SMO = C` S C

 MATRIX SMO            

 SYMMETRY BLOCK 1.1
    1.00000000   0.00000000  -0.00000000   0.00000000  -0.00000000   0.00000000   0.00000000   0.00000000  -0.00000000  -0.00000000
    0.00000000   1.00000000   0.00000000  -0.00000000   0.00000000  -0.00000000  -0.00000000  -0.00000000   0.00000000   0.00000000
   -0.00000000   0.00000000   1.00000000   0.00000000  -0.00000000   0.00000000   0.00000000   0.00000000  -0.00000000  -0.00000000
    0.00000000  -0.00000000   0.00000000   1.00000000  -0.00000000  -0.00000000   0.00000000  -0.00000000   0.00000000   0.00000000
   -0.00000000   0.00000000  -0.00000000  -0.00000000   1.00000000   0.00000000   0.00000000  -0.00000000   0.00000000  -0.00000000
    0.00000000  -0.00000000   0.00000000  -0.00000000   0.00000000   1.00000000  -0.00000000  -0.00000000   0.00000000   0.00000000
    0.00000000  -0.00000000   0.00000000   0.00000000   0.00000000  -0.00000000   1.00000000  -0.00000000  -0.00000000   0.00000000
    0.00000000  -0.00000000   0.00000000  -0.00000000  -0.00000000  -0.00000000  -0.00000000   1.00000000  -0.00000000   0.00000000
   -0.00000000   0.00000000  -0.00000000   0.00000000   0.00000000   0.00000000  -0.00000000  -0.00000000   1.00000000  -0.00000000
   -0.00000000   0.00000000  -0.00000000   0.00000000  -0.00000000   0.00000000   0.00000000   0.00000000  -0.00000000   1.00000000

The first matrix printed is the full MO coefficient matrix in case you needed it (MO's in columns).

Best wishes,

Andreas

[Lexin Ding]

Dear Andreas,

Thanks a lot for the prompt reply. It did solve my issue :).

Another small and perhaps also dumb question, how come the 1s and 2s orbitals on the same atom are not orthogonal to each other as demanded by their distinct quantum numbers? Is it due the this particular choice of basis namely sto-3g?

Best wishes,

Lexin

[andreas...@gmail.com]

Dear Lexin,

this is just because the AO basis functions are normalised but not orthogonal to each other. This

is generally the case and not a specific feature of the sto-3g basis set.