5. Computing of O₂

5.1. Introduction

The calculation of ground spin state of O₂ has been an enduring topic in the early stages of the development of quantum chemistry. The ground state of O₂ is triplet ($^3\Sigma_g^{-}$) from experiments. One of the most well-known “failure” of VB theory is that the ground state of O₂ molecule is predicted as $^1\Delta_g$ with VB theory. Here we clarify again that the so-called “failure” comes from the naive application of hybridization followed by the simple Lewis pairing. The ab initio VB calculation is capable of correctly describing the ground state of O₂. In this section, both the triplet and singlet ground states of O₂ will be computed and compared.

5.2. Computations of the singlet state

The computations are proceeded at the equilibrium bond length of 1.227 Å, and the basis set is cc-pVDZ. For simplicity, O atoms are located in the Z axis. Totally 4 active oritals ($p_x$ and $p_y$ orbitals of each O atom) and the 6 active electrons on them are inclued in the calculation. (Readers can also include the two $p_z$ orbitals in the active space, but only the four mentioned $p_{\pi}$ orbitals are included in this case)

5.2.1. Input File

Here shows the XMVB input file for all the structures

singlet O2 VBSCF with all structures
$CTRL
STR=FULL NAO=4 NAE=6 IPRINT=3
ORBTYP=HAO FRGTYP=SAO
BASIS=CC-PVDZ NMUL=1
GUESS=READ
$END
$FRAG
1*6
SPZDXXDYYDZZ 1
SPZDXXDYYDZZ 2
PXDXZ 1
PXDXZ 2
PYDYZ 1
PYDYZ 2
$END
$ORB
1*4 2 1*4
1
2
1
2
1 2
3
4
5
6
$END
$GEO
O 0.0 0.0 0.0
O 0.0 0.0 1.227
$END
$GUS
     8     8     8     8    16     3     3     3     3
# ORBITAL          1  NAO =       8
-1.0001816769     1  -0.0219071559     2  -0.0208718204     3   0.0055675373     6
 0.0058885238     9   0.0012121651    10   0.0012098650    13   0.0014914363    15
# ORBITAL          2  NAO =       8
-1.0001807197    16  -0.0219222377    17  -0.0208875165    18  -0.0055748376    21
-0.0058896710    24   0.0012101317    25   0.0012115362    28   0.0014916466    30
# ORBITAL          3  NAO =       8
-0.0380292983     1   0.4956131485     2   0.5430800220     3  -0.1721288256     6
-0.1265645521     9   0.0040244173    10   0.0040243680    13  -0.0013604634    15
# ORBITAL          4  NAO =       8
-0.0380587693    16   0.4956124042    17   0.5430787036    18   0.1721299458    21
 0.1265653159    24   0.0040244421    25   0.0040243828    28  -0.0013600709    30
# ORBITAL          5  NAO =      16
-0.1873055284     1  -0.0320312577     2   0.1037620245     3  -0.7575342039     6
-0.1745797408     9   0.0069166781    10   0.0069169372    13  -0.0385887223    15
-0.1873136771    16  -0.0320328992    17   0.1037594786    18   0.7575341385    21
 0.1745797704    24   0.0069172963    25   0.0069171279    28  -0.0385889746    30
# ORBITAL          6  NAO =       3
 0.6907525874     4   0.4550551271     7   0.0256096827    12
# ORBITAL          7  NAO =       3
 0.6907525665    19   0.4550551519    22  -0.0256096585    27
# ORBITAL          8  NAO =       3
 0.6907525875     5   0.4550551269     8   0.0256096869    14
# ORBITAL          9  NAO =       3
 0.6907525683    20   0.4550551500    23  -0.0256096533    29
$END

The global keywords listed in $CTRL section are explained below:

STR=FULL XMVB generates all VB structures automatically according to a specific active space.
IPRINT=3 XMVB will print most information.
NAO=4 and NAE=6 Specify the active space with 4 active orbital and 6 active electrons respectively.
ORBTYP=HAO and FRGTYP=SAO The VB orbitals are described with fragments.
BASIS=CC-PVDZ The basis set is cc-pVDZ.
NMUL=1 The spin multiplicity (2S+1) is 1 (The default value).

$FRAG section describes the fragments used to construct VB orbitals. In this case, the first line 1*6 means that there are 6 fragments built and each fragment include only 1 atom. The details of fragments are described in the following lines. For example, SPZDXXDYYDZZ 1 means that the fragment includes s, p_z, d_xx, d_yy and d_zz basis functions on atom 1, which is the first F atom in this case.

$ORB section describes VB orbitals. The first line 1*4 2 1*4 means that there are 9 orbitals in this case, each of the first 4 orbitals consisting of only 1 fragment, the 5^th orbital consisting of 2 fragments, and each of the remaining 4 orbitals consisting of only 1 fragment. Each line below describes one orbital. For example, the first orbital includes only fragment 1, meaning that this orbital locates on the first O atom and belongs to the $\sigma$ space since the O-O lies on Z axis. Also, orbital 5 with fragment 1 2 shows that the orbital describes the O-O sigma bonding orbital.

$GEO section shows the geometry of O₂ molecule. Both Cartesian and internal coordinates are supported.

5.2.2. Computational Results

In this case, totally 10 VB structures are generated. The number of 0th ion structure is 6, and the number of 1th ion structures is 4. These structures can be expaned into 16 determinants.

Number of   0th ion  structures  is:          6  from          1 to          6
Number of   0th ion determinants is:         12  from          1 to         12
Number of   1th ion  structures  is:          4  from          7 to         10
Number of   1th ion determinants is:          4  from         13 to         16
Total number of  structures  is:         10
Total number of determinants is:         16

Number of Structures:          10
                   .
                   .
                   .
The following structures are used in calculation:

    1 *****  1:5    6   6   7   7   8   9
    2 *****  1:5    6   6   8   8   7   9
    3 *****  1:5    6   6   9   9   7   8
    4 *****  1:5    7   7   8   8   6   9
    5 *****  1:5    7   7   9   9   6   8
    6 *****  1:5    8   8   9   9   6   7
    7 *****  1:5    6   6   7   7   8   8
    8 *****  1:5    6   6   7   7   9   9
    9 *****  1:5    6   6   8   8   9   9
   10 *****  1:5    7   7   8   8   9   9

Note

In this (6,4) active space, there are at least 2 active orbitals are always doubly occupied. For instnce, in structure “1:5 6 6 7 7 8 9”, the active orbitals 6 and 7 are doubly occupied, and the electrons on orbitals 8 and 9 form a covalent bond. However, this structure belongs to “0th ion structure” instead of “2th ion structure”.

The user may find from the output file that the VBSCF converged after 43 iterations and the final energy is -149.59869269 hartree.

                       VBSCF converged in    43 iterations

Total Energy:    -149.59869269

First Excited:   -149.598693

The coefficients and weights shows the importance of each structure. Following are the coefficients and weights of generated VB structures. Both coefficients and weights show that structures 1 and 6 are dominant. Structure 1 means the covalent bodning between p_y orbitals of the two O atoms, while Structure 6 means the covalent bodning between p_x orbitals of the two O atoms

    ******  COEFFICIENTS OF STRUCTURES ******

 1       0.59132818  *****  1:5    6   6   7   7   8   9
 2      -0.00000000  *****  1:5    6   6   8   8   7   9
 3       0.00000000  *****  1:5    6   6   9   9   7   8
 4       0.00000000  *****  1:5    7   7   8   8   6   9
 5       0.00000000  *****  1:5    7   7   9   9   6   8
 6      -0.59163406  *****  1:5    8   8   9   9   6   7
 7       0.14679997  *****  1:5    6   6   7   7   8   8
 8       0.14680176  *****  1:5    6   6   7   7   9   9
 9      -0.14686415  *****  1:5    6   6   8   8   9   9
10      -0.14686585  *****  1:5    7   7   8   8   9   9
               .
               .
               .
    ******  WEIGHTS OF STRUCTURES ******

 1       0.40213218  *****  1:5    6   6   7   7   8   9
 2       0.00000000  *****  1:5    6   6   8   8   7   9
 3       0.00000000  *****  1:5    6   6   9   9   7   8
 4       0.00000000  *****  1:5    7   7   8   8   6   9
 5      -0.00000000  *****  1:5    7   7   9   9   6   8
 6       0.40253540  *****  1:5    8   8   9   9   6   7
 7       0.04881265  *****  1:5    6   6   7   7   8   8
 8       0.04881349  *****  1:5    6   6   7   7   9   9
 9       0.04885274  *****  1:5    6   6   8   8   9   9
10       0.04885354  *****  1:5    7   7   8   8   9   9

Finally, the bond order value (1.347) in atomic population analysis shows that O-O bond should be a double bond.

          ******  BOND ORDER  ******

ATOM 1        ATOM 2           DIST     BOND ORDER

 1 O           2 O            1.227       1.347

5.3. Computations of the triplet state

The geometry, basis set and active space are the same as the singlet calculation in the previuos section.

5.3.1. Input File

Here shows the XMVB input file for all the structures

triplet O2 VBSCF with all structures
$CTRL
STR=FULL NAO=4 NAE=6 IPRINT=3
ORBTYP=HAO FRGTYP=SAO
BASIS=CC-PVDZ NMUL=3
GUESS=READ
$END
$FRAG
1*6
SPZDXXDYYDZZ 1
SPZDXXDYYDZZ 2
PXDXZ 1
PXDXZ 2
PYDYZ 1
PYDYZ 2
$END
$ORB
1*4 2 1*4
1
2
1
2
1 2
3
4
5
6
$END
$GEO
O 0.0 0.0 0.0
O 0.0 0.0 1.227
$END
$GUS
     8     8     8     8    16     3     3     3     3
# ORBITAL          1  NAO =       8
-1.0001816769     1  -0.0219071559     2  -0.0208718204     3   0.0055675373     6
 0.0058885238     9   0.0012121651    10   0.0012098650    13   0.0014914363    15
# ORBITAL          2  NAO =       8
-1.0001807197    16  -0.0219222377    17  -0.0208875165    18  -0.0055748376    21
-0.0058896710    24   0.0012101317    25   0.0012115362    28   0.0014916466    30
# ORBITAL          3  NAO =       8
-0.0380292983     1   0.4956131485     2   0.5430800220     3  -0.1721288256     6
-0.1265645521     9   0.0040244173    10   0.0040243680    13  -0.0013604634    15
# ORBITAL          4  NAO =       8
-0.0380587693    16   0.4956124042    17   0.5430787036    18   0.1721299458    21
 0.1265653159    24   0.0040244421    25   0.0040243828    28  -0.0013600709    30
# ORBITAL          5  NAO =      16
-0.1873055284     1  -0.0320312577     2   0.1037620245     3  -0.7575342039     6
-0.1745797408     9   0.0069166781    10   0.0069169372    13  -0.0385887223    15
-0.1873136771    16  -0.0320328992    17   0.1037594786    18   0.7575341385    21
 0.1745797704    24   0.0069172963    25   0.0069171279    28  -0.0385889746    30
# ORBITAL          6  NAO =       3
 0.6907525874     4   0.4550551271     7   0.0256096827    12
# ORBITAL          7  NAO =       3
 0.6907525665    19   0.4550551519    22  -0.0256096585    27
# ORBITAL          8  NAO =       3
 0.6907525875     5   0.4550551269     8   0.0256096869    14
# ORBITAL          9  NAO =       3
 0.6907525683    20   0.4550551500    23  -0.0256096533    29
$END

The only difference of this input file from that in the previuos section is that the spin multiplicity is set to 3 for the triplet state.

NMUL=3 The spin multiplicity (2S+1) is 3.

5.3.2. Computational Results

In this case, totally 6 VB structures (or 6 VB determinants) are generated, and all of them are 0th ion structures.

Number of   0th ion  structures  is:          6  from          1 to          6
Number of   0th ion determinants is:          6  from          1 to          6
Total number of  structures  is:          6
Total number of determinants is:          6

Number of Structures:           6
                   .
                   .
                   .
The following structures are used in calculation:

    1 *****  1:5    6   6   7   7   8   9
    2 *****  1:5    6   6   8   8   7   9
    3 *****  1:5    6   6   9   9   7   8
    4 *****  1:5    7   7   8   8   6   9
    5 *****  1:5    7   7   9   9   6   8
    6 *****  1:5    8   8   9   9   6   7

Note

In this (6,4) active space, there are at least 2 active orbitals are always doubly occupied. Different from the calculation of singlet state, the remaining 2 electrons are in the same spin. Since two electrons with same spin cannot occupy on one spatial orbital, there is no “1th ion structures” in this case.

Note

For a structure with $2S+1\neq1$, the last $2S$ numbers means unpaired electrons with same spin instaed of covalent boinding in the singlet case. For instance, structure “1:5 6 6 7 7 8 9” in this triplet calculation means there are two unpaired electrons with same spin on orbitals 8 and 9.

The user may find from the output file that the VBSCF converged after 43 iterations and the final energy is -149.62744434 hartree.

                       VBSCF converged in    43 iterations

Total Energy:    -149.62744434

First Excited:   -149.440051

Note

Compared with the energy of singlet state (-149.59869269 hartree), we see the VB theroy correctly predict the ground state as triplet!

The coefficients and weights shows the importance of each structure. Following are the coefficients and weights of generated VB structures. Both coefficients and weights show that structures 3 and 4 are mainly dominant. Structure 3 means there are two unpaired electrons with same spin on p_x orbital of O2 and p_y orbital of O1. Structure 4 means there are two unpaired electrons with same spin on p_x orbital of O1 and p_y orbital of O2. The structures 2 and 5 also give nonneglegible contributions, but are less important than structures 3 and 4. This is because both structures 2 and 5 means there are two unpaired electrons with same spin on one O atom.

   ******  COEFFICIENTS OF STRUCTURES ******

1      -0.00000000  *****  1:5    6   6   7   7   8   9
2       0.24299537  *****  1:5    6   6   8   8   7   9
3      -0.55305281  *****  1:5    6   6   9   9   7   8
4      -0.55305265  *****  1:5    7   7   8   8   6   9
5       0.24299211  *****  1:5    7   7   9   9   6   8
6       0.00000000  *****  1:5    8   8   9   9   6   7
              .
              .
              .
   ******  WEIGHTS OF STRUCTURES ******

1       0.00000000  *****  1:5    6   6   7   7   8   9
2       0.12068238  *****  1:5    6   6   8   8   7   9
3       0.37931891  *****  1:5    6   6   9   9   7   8
4       0.37931871  *****  1:5    7   7   8   8   6   9
5       0.12068001  *****  1:5    7   7   9   9   6   8
6       0.00000000  *****  1:5    8   8   9   9   6   7

Finally, the bond order value (1.779) in atomic population analysis shows that the triplet O₂ is more stable than the singlet whose bond order is 1.347.

          ******  BOND ORDER  ******

ATOM 1        ATOM 2           DIST     BOND ORDER

 1 O           2 O            1.227       1.779

5.4. Computations with selected structures

In the previous calculations of O₂, the structures in the VB wave function are all automatically generated (STR=FULL). However, it’s possible to perform a VB calculation with selected structures. The structures can be selected by either chemical intuition or a previous VB calculation. Followings are input files of singlet and triplet O₂ with selected structures.

5.4.1. Input File

Here shows the XMVB input file with selected structures for both singlet and triplet O₂.

singlet O2 VBSCF with selected structures
$CTRL
NSTR=2 NAO=4 NAE=6 IPRINT=3
ORBTYP=HAO FRGTYP=SAO
BASIS=CC-PVDZ NMUL=1
GUESS=READ
$END
$FRAG
1*6
SPZDXXDYYDZZ 1
SPZDXXDYYDZZ 2
PXDXZ 1
PXDXZ 2
PYDYZ 1
PYDYZ 2
$END
$ORB
1*4 2 1*4
1
2
1
2
1 2
3
4
5
6
$END
$STR
1:5    6   6   7   7   8   9
1:5    8   8   9   9   6   7
$END
$GEO
O 0.0 0.0 0.0
O 0.0 0.0 1.227
$END
$GUS
     8     8     8     8    16     3     3     3     3
# ORBITAL          1  NAO =       8
-1.0001816769     1  -0.0219071559     2  -0.0208718204     3   0.0055675373     6
 0.0058885238     9   0.0012121651    10   0.0012098650    13   0.0014914363    15
# ORBITAL          2  NAO =       8
-1.0001807197    16  -0.0219222377    17  -0.0208875165    18  -0.0055748376    21
-0.0058896710    24   0.0012101317    25   0.0012115362    28   0.0014916466    30
# ORBITAL          3  NAO =       8
-0.0380292983     1   0.4956131485     2   0.5430800220     3  -0.1721288256     6
-0.1265645521     9   0.0040244173    10   0.0040243680    13  -0.0013604634    15
# ORBITAL          4  NAO =       8
-0.0380587693    16   0.4956124042    17   0.5430787036    18   0.1721299458    21
 0.1265653159    24   0.0040244421    25   0.0040243828    28  -0.0013600709    30
# ORBITAL          5  NAO =      16
-0.1873055284     1  -0.0320312577     2   0.1037620245     3  -0.7575342039     6
-0.1745797408     9   0.0069166781    10   0.0069169372    13  -0.0385887223    15
-0.1873136771    16  -0.0320328992    17   0.1037594786    18   0.7575341385    21
 0.1745797704    24   0.0069172963    25   0.0069171279    28  -0.0385889746    30
# ORBITAL          6  NAO =       3
 0.6907525874     4   0.4550551271     7   0.0256096827    12
# ORBITAL          7  NAO =       3
 0.6907525665    19   0.4550551519    22  -0.0256096585    27
# ORBITAL          8  NAO =       3
 0.6907525875     5   0.4550551269     8   0.0256096869    14
# ORBITAL          9  NAO =       3
 0.6907525683    20   0.4550551500    23  -0.0256096533    29
$END

“NSTR=2” means 2 structures are involved in the VB wave function. When “NSTR=value” is specified in the input, the “$STR” section, which describes the details of the selected structures, must be provided.

triplet O2 VBSCF with selected structures
$CTRL
NSTR=2 NAO=4 NAE=6 IPRINT=3
ORBTYP=HAO FRGTYP=SAO
BASIS=CC-PVDZ NMUL=3
GUESS=READ
$END
$FRAG
1*6
SPZDXXDYYDZZ 1
SPZDXXDYYDZZ 2
PXDXZ 1
PXDXZ 2
PYDYZ 1
PYDYZ 2
$END
$ORB
1*4 2 1*4
1
2
1
2
1 2
3
4
5
6
$END
$STR
1:5    6   6   9   9   7   8
1:5    7   7   8   8   6   9
$END
$GEO
O 0.0 0.0 0.0
O 0.0 0.0 1.227
$END
$GUS
     8     8     8     8    16     3     3     3     3
# ORBITAL          1  NAO =       8
-1.0001816769     1  -0.0219071559     2  -0.0208718204     3   0.0055675373     6
 0.0058885238     9   0.0012121651    10   0.0012098650    13   0.0014914363    15
# ORBITAL          2  NAO =       8
-1.0001807197    16  -0.0219222377    17  -0.0208875165    18  -0.0055748376    21
-0.0058896710    24   0.0012101317    25   0.0012115362    28   0.0014916466    30
# ORBITAL          3  NAO =       8
-0.0380292983     1   0.4956131485     2   0.5430800220     3  -0.1721288256     6
-0.1265645521     9   0.0040244173    10   0.0040243680    13  -0.0013604634    15
# ORBITAL          4  NAO =       8
-0.0380587693    16   0.4956124042    17   0.5430787036    18   0.1721299458    21
 0.1265653159    24   0.0040244421    25   0.0040243828    28  -0.0013600709    30
# ORBITAL          5  NAO =      16
-0.1873055284     1  -0.0320312577     2   0.1037620245     3  -0.7575342039     6
-0.1745797408     9   0.0069166781    10   0.0069169372    13  -0.0385887223    15
-0.1873136771    16  -0.0320328992    17   0.1037594786    18   0.7575341385    21
 0.1745797704    24   0.0069172963    25   0.0069171279    28  -0.0385889746    30
# ORBITAL          6  NAO =       3
 0.6907525874     4   0.4550551271     7   0.0256096827    12
# ORBITAL          7  NAO =       3
 0.6907525665    19   0.4550551519    22  -0.0256096585    27
# ORBITAL          8  NAO =       3
 0.6907525875     5   0.4550551269     8   0.0256096869    14
# ORBITAL          9  NAO =       3
 0.6907525683    20   0.4550551500    23  -0.0256096533    29
$END

5.4.2. Computational Results

The VBSCF calculations with selected structure for singlet and triplet O₂ are respectivlely -149.54466456 and -149.55699157 hartree. It’s not surprising the energies with selected structures are higher than those with all structures in the actve space. The calculations with selected structures again correctly predict the ground state of O₂ as triplet.

Readers can further check and compared other quantities in the output files.

5. Computing of O2

5.1. Introduction

5.2. Computations of the singlet state

5.2.1. Input File

5.2.2. Computational Results

5.3. Computations of the triplet state

5.3.1. Input File

5.3.2. Computational Results

5.4. Computations with selected structures

5.4.1. Input File

5.4.2. Computational Results

5. Computing of O₂