Information on Molecular Orbitals

in the Gaussian output file

How much information on molecular orbitals is written to the Gaussian output file depends on the value of the Population or pop keyword. If no particular value is given, only the orbital energies (in atomic units) are printed. This is equivalent to pop=minimal. Selected information on the actual orbitals is obtained using pop=regular. Using this choice, information on the five highest occupied and the lowest five uncoccupied (virtual) orbitals are printed. This is, of course, only possible if there actually ARE at least five occupied and five virtual orbitals. In the following example of the HF/STO-3G orbitals of formaldehyde (CH2O, C2v point group):

#P HF/STO-3G scf=tight pop=regular

HF/STO-3G//HF/STO-3G sp formaldehyde

0 1
C1
O2  1  r2
H3  1  r3  2  a3
H4  1  r3  2  a3  3  180.0

r2=1.21672286
r3=1.10137241
a3=122.73666566
           

there are only four virtual orbitals and only those are printed. The orbital coefficients are given with respect to the molecule in its "Standard orientation" given at the beginning of the output file. The system is oriented such that the principal axis of the systems runs along the z-axis and that all atoms of the system are located in the yz-plane:

                         Standard orientation:                    
 ---------------------------------------------------------------------
 Center     Atomic     Atomic              Coordinates (Angstroms)
 Number     Number      Type              X           Y           Z
 ---------------------------------------------------------------------
    1          6             0        0.000000    0.000000   -0.533912
    2          8             0        0.000000    0.000000    0.682811
    3          1             0        0.000000    0.926436   -1.129510
    4          1             0        0.000000   -0.926436   -1.129510
 ---------------------------------------------------------------------
           

The output starts with a listing of the orbital symmetries of all orbitals of the system. In this particular example, the highest occupied molecular orbital (HOMO) is orbital No. 8 and belongs to the B2 irreducible representation (antisymmetric with respect to the principal C2 axis). Following this information, the orbital energies of all orbitals of the system are given. For open shell systems the energies of the alpha electrons are given first, followed by the energies of the beta orbitals:

 Orbital symmetries:
       Occupied  (A1) (A1) (A1) (A1) (B2) (A1) (B1) (B2)
       Virtual   (B1) (A1) (B2) (A1)
 The electronic state is 1-A1.
 Alpha  occ. eigenvalues --  -20.31271 -11.12507  -1.33744  -0.80775  -0.63291
 Alpha  occ. eigenvalues --   -0.54553  -0.44319  -0.35438
 Alpha virt. eigenvalues --    0.28199   0.62863   0.73441   0.91294
     Molecular Orbital Coefficients
                           4         5         6         7         8
                        (A1)--O   (B2)--O   (A1)--O   (B1)--O   (B2)--O
     EIGENVALUES --    -0.80775  -0.63291  -0.54553  -0.44319  -0.35438
   1 1   C  1S         -0.18562   0.00000   0.03301   0.00000   0.00000
   2        2S          0.57741   0.00000  -0.10669   0.00000   0.00000
   3        2PX         0.00000   0.00000   0.00000   0.60936   0.00000
   4        2PY         0.00000   0.53318   0.00000   0.00000  -0.18209
   5        2PZ        -0.22623   0.00000  -0.44751   0.00000   0.00000
   6 2   O  1S          0.09884   0.00000  -0.09381   0.00000   0.00000
   7        2S         -0.42913   0.00000   0.49909   0.00000   0.00000
   8        2PX         0.00000   0.00000   0.00000   0.67586   0.00000
   9        2PY         0.00000   0.44231   0.00000   0.00000   0.86991
  10        2PZ        -0.16466   0.00000   0.67688   0.00000   0.00000
  11 3   H  1S          0.26455   0.30023   0.15895   0.00000  -0.35921
  12 4   H  1S          0.26455  -0.30023   0.15895   0.00000   0.35921
                           9        10        11        12
                        (B1)--V   (A1)--V   (B2)--V   (A1)--V
     EIGENVALUES --     0.28199   0.62863   0.73441   0.91294
   1 1   C  1S          0.00000  -0.20803   0.00000  -0.09478
   2        2S          0.00000   1.30304   0.00000   0.63168
   3        2PX         0.82111   0.00000   0.00000   0.00000
   4        2PY         0.00000   0.00000   1.14840   0.00000
   5        2PZ         0.00000  -0.44498   0.00000   1.17315
   6 2   O  1S          0.00000   0.02810   0.00000   0.11577
   7        2S          0.00000  -0.16145   0.00000  -0.86387
   8        2PX        -0.76729   0.00000   0.00000   0.00000
   9        2PY         0.00000   0.00000  -0.31860   0.00000
  10        2PZ         0.00000   0.24606   0.00000   0.92393
  11 3   H  1S          0.00000  -0.88929  -0.83986   0.15471
  12 4   H  1S          0.00000  -0.88929   0.83986   0.15471
           

In the next section of output, the orbital coefficients are given first for the five highest occupied molecular orbitals. The header for each orbital includes the orbital symmetry, the label "O" for occupied or "V" for virtual orbitals, and the orbital energies in atomic units. The molecular orbital coefficients for the basis functions are then given in the order in which they appear in the basis set description (as obtained, for example, with the gfinput keyword). Orbital 7 listed in the formaldehyde example with an orbital energy of -0.44319 au is of B1 symmetry and composed exclusively of the 2px basis functions on carbon atom 1 and oxygen atom 2. This corresponds to the occupied pi-type molecular orbital of the C-O double bond. It is interesting to note that, in contrast to many textbook examples, this is not the highest occupied molecular orbital (HOMO) of this system. The latter (orbital 8 with an orbital energy of -0.35438 au) is mainly composed of the 2py basis functions on oxygen atom 2 and can best be thought of as an oxygen centered lone pair. The lowest unoccupied molecular orbital (LUMO) of the system is orbital 9 with an orbital energy of +0.28199 au and main contributions from the 2px basis functions on carbon atom 1 and oxygen atom 2, now with opposite sign. This is in line with expectations for the unoccupied pi* orbital of the C-O double bond.

Information on all orbitals of the system can be obtained with the pop=full keyword. Remember that the number of occupied orbitals is independent, but the number of virtual orbitals is strongly dependent on the number of basis functions. A full listing of all molecular orbitals can therefore become quite extensive for large basis set calculations on larger systems. This is particularly relevant for Gaussian as pop=full also forces a full Mulliken population analysis.

When moving away from minimal basis sets, the analysis of molecular orbital coefficients becomes more difficult, as there are many more basis functions (and therefore MO coefficients) than classical atomic orbitals. This is readily demonstrated using the HF/6-311G(d,p) orbitals of formaldehyde (HF/STO-3G structure) as an example:

Orbital symmetries:
       Occupied  (A1) (A1) (A1) (A1) (B2) (A1) (B1) (B2)
       Virtual   (B1) (A1) (B2) (A1) (A1) (B2) (B1) (B2) (A1) (A1)
                 (A1) (B1) (B2) (A2) (B1) (A1) (A1) (B2) (B2) (A1)
                 (B1) (A2) (A1) (B2) (B2) (A1) (A1) (B1) (A2) (A1)
                 (B1) (A1) (B2) (B2) (A1) (B1) (B2) (A1) (A1) (A1)
 The electronic state is 1-A1.
 Alpha  occ. eigenvalues --  -20.57126 -11.34192  -1.39677  -0.87126  -0.68760
 Alpha  occ. eigenvalues --   -0.64756  -0.52789  -0.44085
 Alpha virt. eigenvalues --    0.12516   0.15625   0.21077   0.33039   0.47943
 Alpha virt. eigenvalues --    0.55866   0.57437   0.74085   0.80110   0.85841
 Alpha virt. eigenvalues --    0.98667   1.09706   1.15224   1.24623   1.50263
 Alpha virt. eigenvalues --    1.55040   1.64320   1.65415   1.84157   1.88722
 Alpha virt. eigenvalues --    2.02403   2.11540   2.40847   2.60635   2.73802
 Alpha virt. eigenvalues --    2.75352   2.83229   2.95131   3.25505   3.26831
 Alpha virt. eigenvalues --    3.69981   3.73483   3.80946   4.10219   4.21669
 Alpha virt. eigenvalues --    5.39516   5.50403   6.06721  24.98550  51.59020
     Molecular Orbital Coefficients
                           4         5         6         7         8
                        (A1)--O   (B2)--O   (A1)--O   (B1)--O   (B2)--O
     EIGENVALUES --    -0.87126  -0.68760  -0.64756  -0.52789  -0.44085
   1 1   C  1S         -0.08789   0.00000   0.01245   0.00000   0.00000
   2        2S         -0.14713   0.00000   0.02091   0.00000   0.00000
   3        2PX         0.00000   0.00000   0.00000   0.14230   0.00000
   4        2PY         0.00000   0.17912   0.00000   0.00000  -0.08342
   5        2PZ        -0.08046   0.00000  -0.16684   0.00000   0.00000
   6        3S          0.40144   0.00000  -0.07991   0.00000   0.00000
   7        3PX         0.00000   0.00000   0.00000   0.23992   0.00000
   8        3PY         0.00000   0.27845   0.00000   0.00000  -0.14032
   9        3PZ        -0.11437   0.00000  -0.26911   0.00000   0.00000
  10        4S          0.25737   0.00000   0.04616   0.00000   0.00000
  11        4PX         0.00000   0.00000   0.00000   0.19312   0.00000
  12        4PY         0.00000   0.13653   0.00000   0.00000  -0.02694
  13        4PZ        -0.06713   0.00000  -0.06445   0.00000   0.00000
  14        5D 0       -0.00393   0.00000  -0.03003   0.00000   0.00000
  15        5D+1        0.00000   0.00000   0.00000   0.05175   0.00000
  16        5D-1        0.00000   0.00256   0.00000   0.00000   0.06960
  17        5D+2       -0.01407   0.00000  -0.01274   0.00000   0.00000
  18        5D-2        0.00000   0.00000   0.00000   0.00000   0.00000
  19 2   O  1S          0.04656   0.00000  -0.03712   0.00000   0.00000
  20        2S          0.07868   0.00000  -0.06286   0.00000   0.00000
  21        2PX         0.00000   0.00000   0.00000   0.22120   0.00000
  22        2PY         0.00000   0.15228   0.00000   0.00000   0.25178
  23        2PZ        -0.06544   0.00000   0.23348   0.00000   0.00000
  24        3S         -0.23278   0.00000   0.19078   0.00000   0.00000
  25        3PX         0.00000   0.00000   0.00000   0.34693   0.00000
  26        3PY         0.00000   0.23957   0.00000   0.00000   0.38028
  27        3PZ        -0.09897   0.00000   0.34271   0.00000   0.00000
  28        4S         -0.24010   0.00000   0.29517   0.00000   0.00000
  29        4PX         0.00000   0.00000   0.00000   0.31869   0.00000
  30        4PY         0.00000   0.17597   0.00000   0.00000   0.39654
  31        4PZ        -0.06410   0.00000   0.25624   0.00000   0.00000
  32        5D 0        0.00611   0.00000  -0.02418   0.00000   0.00000
  33        5D+1        0.00000   0.00000   0.00000  -0.02843   0.00000
  34        5D-1        0.00000  -0.01934   0.00000   0.00000  -0.00994
  35        5D+2        0.00124   0.00000  -0.00092   0.00000   0.00000
  36        5D-2        0.00000   0.00000   0.00000   0.00000   0.00000
  37 3   H  1S          0.10699   0.10980   0.05586   0.00000  -0.09884
  38        2S          0.15467   0.16948   0.08015   0.00000  -0.21991
  39        3S          0.00544   0.04175   0.03555   0.00000  -0.12090
  40        4PX         0.00000   0.00000   0.00000   0.00794   0.00000
  41        4PY        -0.02106  -0.01486  -0.00908   0.00000   0.00414
  42        4PZ         0.00944   0.01004  -0.00358   0.00000  -0.00727
  43 4   H  1S          0.10699  -0.10980   0.05586   0.00000   0.09884
  44        2S          0.15467  -0.16948   0.08015   0.00000   0.21991
  45        3S          0.00544  -0.04175   0.03555   0.00000   0.12090
  46        4PX         0.00000   0.00000   0.00000   0.00794   0.00000
  47        4PY         0.02106  -0.01486   0.00908   0.00000   0.00414
  48        4PZ         0.00944  -0.01004  -0.00358   0.00000   0.00727
           

Molecular orbital 8 (the HOMO) is now described by as much as 48 MO coefficients, each of the two hydrogen atoms contributing 6, and carbon and oxygen contributing 18 basis functions. With this increased flexibility of the basis set, the ease of interpretation of the single MO coefficients is almost completely lost. In this situation, analysis of the orbital structure using one of the population analysis schemes (Mulliken population analysis, Natural Population Analysis, AIM . . ) or simly plotting the molecular orbitals in 2D- or 3D-diagrams is clearly preferrable.

last changes: 09.01.2005, HZ
questions & comments to: zipse@cup.uni-muenchen.de