For two atomic homonuclear molecules like H_{2}, O_{2}, N_{2} etc, the LCAO expansion represents an especially simple approach. The progress made with the ionic molecule H_{2}^{+} is as well valuable for other homonuclear species. We will treat theses molecules in the same way. Of course, we will face some additional points, especially for bonds established by more than one electron. Analogous to atomic orbitals, we introduce molecular orbitals of different energy. The molecule's electrons are distributed among them in a way that the orbitals with the lowest energies are occupied first and the Pauli principle is accomplished. For a first approximation, we regard these orbitals as linear combinations of atomic orbitals. Taking the two 1s orbitals of the hydrogen atom, we obtain the two MOs that already appeared in the previous chapter that dealt with H_{2}^{+}:
In order to have a concise presentation, the various coeffients for normalization are omitted in both equations. Moreover, greec minuscules σ (analogous to s for the atoms) are introduced to represent the MOs. A consideration of the indices follows below. Note that α_{A} and α_{B} in the secular equations are identical for homonuclear molecules: α_{A} = α_{B} = α. Therefore, both wave functions are introduced with the same weight but maybe with opposite signs.
Thus, in cases where the overlap integral S is, compared with 1, small, an originally degenerated pair of energy values α splits symmetrically into two values above and below the original energy level.
The level of reduced energy is called the bonding MO as it is below the energy level of the separated atoms. The level of increased energy is above the energy level of the separated atoms and is called antibonding MO. Horizontal lines in the following figure represent the two atomic and the two molecular orbitals and the respective energy of an electron within this orbital.
Now we allocate the MOs, beginning with the orbital lowest in energy and having the Pauli principle in mind. For H_{2} we obtain wave funcion σ_{g}(1) for one electron and σ_{g}(2) for the other. For the total wave function our approach yields:
The first two terms indicate that both electrons are located near different nuclei (for H: protons) whereas the last two terms describe two electrons located around one single nucleus (A resp. B). There is another approach by HeitlerLondon that just ignores the last two terms argueing that the repulsion between the two electrons will make this situation quite improbable (i.e. there is an interdependence between the electrons). But occasionally, this happens. Therefore, MO and HeitlerLonder's are two extreme approaches and a superior description is to be found between the two, e.g. in the socalled ionic approach with c as variation constant:
Ψ_{ges} = σ_{A}(1)Φ_{B}(2)+Φ_{A}(2)Φ_{B}(1) + c[Φ_{A}(1)Φ_{A}(2)+Φ_{B}(1)Φ_{B}(2)]
An overview to possible and impossible combinations of atomic orbitals of the types s, p and d is given in the following table:





















^{#}Possible combinations for p_{y} and d_{yz} are obtained by exchanging x and y in any of the two lines 
The figure below depicts the additive and subtractive combination of porbitals to give two MOs. By convention, direction z is in line with the axis between both nuclei. (Click here for further examples of MOs and for the specific cases of 1s+1s and 2s+2s. Subtracting one p_{z} orbital from another yields the bonding LCAO σ_{g}, whereas the linear combination obtained by adding the two p_{z} orbitals leads to the antibonding orbital σ_{u}. From combinations of p_{x} or p_{y} orbitals πorbitals emerge; thus, these orbitals are doubly degenerated. An additive combination of p_{x} or p_{y} AOs produces a marked overlap and therefore a bonding MO (π_{u}). In contrast, subtractive combination gives rise to antibonding MOs (π_{g}).
From combinations of two
2p_{z} atomic orbitals one bonding and one antibonding molecule orbital emerge, labelled as 
Combination of two 2p_{x}atomic orbitals gives as well rise to one bonding and one antibonding molecular orbital, labelled as 
The symbols for classification of molecular orbitals are similar to those used for atomic orbitals. The symbols s, p, d, ... for atoms correspond to σ, π and δ and refer to the angular momentum of the electron with respect to the axis between the two nuclei and to symmetry of the molecule orbital.
With the numbers in front of the symmetry symbol, molecular orbitals of one symmetry are classified with respect to their energy. The higher the number is, the higher is the energy of an electron within this state.
The sequence of MO energies implies structural properties of molecules in a similar way the energy levels of AOs are used to explain the properties of atoms. The currently available data support the following succession of molecular orbitals:

For atoms of the second row of the periodic table forming two atomic molecules, the σ_{g}MO is always below the π_{u}MO. Note that πMOs appear doubly degenerated. 1π_{u} is therefore used as an abbreviation for the pair of orbitals 1π_{xu } and 1π_{yu} that evidently come from the p_{x} and the p_{y} atomic orbital.
Now we are able to use the succession of MOs to describe homonuclear two atomic molecules. The electrons are distrubuted among the orbitals according energy and the Pauli principle. Due to degeneracy, up to four electrons are allocated to the π level. We already know that the two electrons of the hydrogen molecule H_{2} occupy the bonding MO, thus the molecules electronic configuration is (1σ_{g})^{2}. If we continue our consideration with He_{2}^{+}, we would expect the ground state configuration (1σ_{g})^{2}(1σ_{u}) and He_{2} would have (1σ_{g})^{2}(1σ_{u})^{2}. In fact, the molecule He_{2} is found to be instable in its ground state whereas it is stable in configurations with one electron in a molecular orbital of higher energy, e.g. in a configuration like (1σ_{g})^{2}(1σ_{u})(2σ_{g})).
The instability of He_{2} is one example of the general rule that occupation of both the bonding and the respective antibonding molecular orbital does not yield a chemical bond; in fact the antibond wins slightly over the bond because the splitting of the two involved atomic orbital energy levels is not perfectly symmetric.
The MO description tends to ignore the inner shell electrons and descriptions of electronic configurations just care for the former valence electrons. For example Li_{2} with two valence electrons is described with the inner shell electrons as
Li_{2} [ (1σ_{g})^{2}(1σ_{u})^{2}(2σ_{g})^{2}]
or, shortly by introducing capital letters for complete shells
Li_{2} [ KK(2σ_{g})^{2} ]
In a similar manner, we describe the electronic configuration of fluorine F_{2}
F_{2} [ KK(2σ_{g})^{2}(2σ_{u})^{2}(1π_{u})^{4}(3σ_{g})^{2}(1π_{g})^{4} ],
where (1π_{u})^{4} denotes the degenerated pair of (1π_{xu})^{2}(1π_{yu})^{2}. 2σ_{g} and 2σ_{u} as well as 1π_{u} and 1π_{g} are pairs of bonding and antibonding MOs. Therefore, of the 14 valence electrons only the pair of electrons in (3σ_{g})^{2} are regarded as bonding and this corresponds to the finding of a single bond of σtype in the molecule.
The total configuration (term symbols) is derived from the electronic configuration. Clicking will deliver details of the procedure and a summary for the term symbol notation. These term symbols consist of greek capital letters Σ, Π, Δ, ... (indicating the projection of the total angular momentum respective the axis between the two nuclei), the subscript g or u (for molecules with a centre of symmetry) and a preceding superscript (indicating the spinmultiplicity 2S+1 for the total spin S). The sign superscript for Σ states indicates the effect of reflecting the wave function on an arbitrary plane that contains both nuclei.
Examples:
^{2}Π_{u }total spin = 1/2; total angular momentum = 1; inversion changes the wave function's sign
^{3}Σ_{g}^{}_{ }
total spin = 1; total angular momentum = 0; inversion does not change the wave function's sign
Electronic configuration for two atomic homonuclear molecules in their ground state  
electronic configuration*  Number of
valence electrons
bond.  antibond. 
bonds 
Bonding
Energy in eV 
Term symbol  
H_{2}^{+}  1σ_{g}  1  0  ½  2.793  ^{2}Σ_{g}^{+} 
H_{2}  1σ_{g}^{2}  2  0  1  4.748  ^{1}Σ_{g}^{+} 
He_{2}^{+}  1σ_{g}^{2} 1σ_{u}  2  1  ½  2.470  ^{2}Σ_{u}^{+} 
He_{2}  1σ_{g}^{2} 1σ_{u}^{2} (=KK)  2  2  0  0.001  ^{1}Σ_{g}^{+} 
Li_{2}^{+}  KK 2σ_{g}  1  0  ½  1.46  ^{2}Σ_{g}^{+} 
Li_{2}  KK 2σ_{g}^{2}  2  0  1  1.068  ^{1}Σ_{g}^{+} 
Be_{2}  KK 2σ_{g}^{2} 2σ_{u}^{2}  2  2  0  0.096^{#}  ^{1}Σ_{g} 
B_{2}  KK 2σ_{g}^{2} 2σ_{u}^{2} 1π_{u}^{2}  4  2  1  3.08  ^{3}Σ_{g}^{−} 
C_{2}^{+}  KK 2σ_{g}^{2} 2σ_{u}^{2} 1π_{u}^{3}  5  2  1½  5.40  ^{2}Π_{u} 
C_{2}  KK 2σ_{g}^{2} 2σ_{u}^{2} 1π_{u}^{4}  6  2  2  6.32  ^{1}Σ_{g}^{+} 
N_{2}^{+}  KK 2σ_{g}^{2} 2σ_{u}^{2} 3σ_{g} 1π_{u}^{4}  7  2  2½  6.478  ^{2}Σ_{g}^{+} 
N_{2}  KK 2σ_{g}^{2} 2σ_{u}^{2} 3σ_{g}^{2} 1π_{u}^{4}  8  2  3  7.519  ^{1}Σ_{g}^{+} 
O_{2}^{+}  KK 2σ_{g}^{2} 2σ_{u}^{2} 3σ_{g}^{2} 1π_{u}^{4} 1π_{g}  8  3  2½  6.781  ^{2}Π_{g} 
O_{2}  KK 2σ_{g}^{2} 2σ_{u}^{2} 3σ_{g}^{2} 1π_{u}^{4} 1π_{g}^{2}  8  4  2  5.214  ^{3}Σ_{g}^{−} 
F_{2}^{+}  KK 2σ_{g}^{2} 2σ_{u}^{2} 3σ_{g}^{2} 1π_{u}^{4} 1π_{g}^{3}  8  5  1½  3.405  ^{2}Π_{g} 
F_{2}  KK 2σ_{g}^{2} 2σ_{u}^{2} 3σ_{g}^{2} 1π_{u}^{4} 1π_{g}^{4}  8  6  1  1.659  ^{1}Σ_{g}^{+} 
^{*} MOs are not always listed according their respective energy level
^{ # } calculated by R. Gdanitz (Dec. 1998) 
First: The notions of single, double and triple bond refer to the number of pairs of electrons with an effective contribution to bonding. σ and π denotes the type of bond. Typical electronic configurations are
single bond σ^{2}, double bond σ^{2}π^{2}, triple bond σ^{2}π^{4}.
For atoms of the first two rows, no other bonds exist. In contrast, for heavier atoms we face additional possibilities. E.g. there are strong bonds between metal atoms with the configuration σ^{2}π^{4}δ^{2} that would be regarded as quadrupole bond.
Second: The principle of maximal overlap predicts that π atomic orbitals with their lobes orthogonal to the axis between the nuclei will not overlap to the same extent as σ atomic orbitals where the distribution of charge is colinear with the axis. Therefore, π bonds are weaker than σ bonds. In analogy, the repulsion caused by antibonding σ orbitals exceed the attraction due to bonding π MOs.
Third: Hund's rule for the allocation of electrons in degenerated MOs (e.g. π_{x}, π_{y}) applies in the same way as with AOs in atoms. Thus, for molecular oxygen O_{2}, all levels up to 1π_{u} are occupied and electrons with parallel spin in the degenerated 1π_{g} levels cause the triplett ground state (S = 1).
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