Computing of O\ :sub:`2` =============================== Introduction -------------- The calculation of ground spin state of O\ :sub:`2` has been an enduring topic in the early stages of the development of quantum chemistry. The ground state of O\ :sub:`2` is triplet (:math:`^3\Sigma_g^{-}`\) from experiments. One of the most well-known "failure" of VB theory is that the ground state of O\ :sub:`2` molecule is predicted as :math:`^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 O\ :sub:`2`. In this section, both the triplet and singlet ground states of O\ :sub:`2` will be computed and compared. 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 (:math:`p_x`\ and :math:`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 :math:`p_z`\ orbitals in the active space, but only the four mentioned :math:`p_{\pi}`\ orbitals are included in this case) Input File ^^^^^^^^^^ Here shows the XMVB input file for all the structures .. code-block:: text 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). ``$FRAG`` section describes the fragments used to construct VB orbitals. In this case, the first line ``1*6`` means that there are 6 fragments built and each fragment include only 1 atom. The details of fragments are described in the following lines. For example, ``SPZDXXDYYDZZ 1`` means that the fragment includes *s*, *p*\ :sub:`z`\ , *d*\ :sub:`xx`\ , *d*\ :sub:`yy`\ and *d*\ :sub:`zz`\ basis functions on atom 1, which is the first F atom in this case. ``$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\ :sup:`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 \ :math:`\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 O\ :sub:`2`\ molecule. Both Cartesian and internal coordinates are supported. 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. .. parsed-literal:: 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. .. parsed-literal:: 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*\ :sub:`y`\ orbitals of the two O atoms, while Structure 6 means the covalent bodning between *p*\ :sub:`x`\ orbitals of the two O atoms .. parsed-literal:: \*\*\*\*\*\* 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. .. parsed-literal:: \*\*\*\*\*\* BOND ORDER \*\*\*\*\*\* ATOM 1 ATOM 2 DIST BOND ORDER 1 O 2 O 1.227 1.347 Computations of the triplet state ---------------------------------- The geometry, basis set and active space are the same as the singlet calculation in the previuos section. Input File ^^^^^^^^^^ Here shows the XMVB input file for all the structures .. code-block:: text 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. Computational Results ^^^^^^^^^^^^^^^^^^^^^^^ In this case, totally ``6`` VB structures (or ``6`` VB determinants) are generated, and all of them are 0th ion structures. .. parsed-literal:: 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 :math:`2S+1\neq1`\, the last :math:`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. .. parsed-literal:: 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*\ :sub:`x`\ orbital of O2 and *p*\ :sub:`y`\ orbital of O1. Structure 4 means there are two unpaired electrons with same spin on *p*\ :sub:`x`\ orbital of O1 and *p*\ :sub:`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. .. parsed-literal:: \*\*\*\*\*\* 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 O\ :sub:`2` is more stable than the singlet whose bond order is 1.347. .. parsed-literal:: \*\*\*\*\*\* BOND ORDER \*\*\*\*\*\* ATOM 1 ATOM 2 DIST BOND ORDER 1 O 2 O 1.227 1.779 Computations with selected structures --------------------------------------- In the previous calculations of O\ :sub:`2`, 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 O\ :sub:`2` with selected structures. Input File ^^^^^^^^^^ Here shows the XMVB input file with selected structures for both singlet and triplet O\ :sub:`2`. .. code-block:: text 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. .. code-block:: text 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 Computational Results ^^^^^^^^^^^^^^^^^^^^^^^ The VBSCF calculations with selected structure for singlet and triplet O\ :sub:`2` 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 O\ :sub:`2` as triplet. Readers can further check and compared other quantities in the output files.