Homonuclear Molecules A2

For two atomic homonuclear molecules like H2, O2, N2 etc, the LCAO expansion represents an especially simple approach. The progress made with the ionic molecule H2+ 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 H2+:

ΦA(1s) + ΦB(1s) =  σg
ΦA(1s) − ΦB(1s) =  σu

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 αA and αB in the secular equations are identical for homonuclear molecules: αA = αB = 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 H2 we obtain wave funcion σg(1) for one electron and σg(2) for the other. For the total wave function our approach yields:

Ψges = σg(1)·σg(2) = [ΦA(1) + ΦB(1)]·[ΦA(2) + ΦB(2)]
= ΦA(1)ΦB(2)+ΦA(2)ΦB(1) + ΦA(1)ΦA(2)+ΦB(1)ΦB(2)

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 Heitler-London 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 Heitler-Londer's are two extreme approaches and a superior description is to be found between the two, e.g. in the so-called ionic approach with c as variation constant:

Ψges = σA(1)ΦB(2)+ΦA(2)ΦB(1) + c[ΦA(1)ΦA(2)+ΦB(1)ΦB(2)]

In advance to a detailed treatment of homonuclear molecules, we consider the fundamental criteria that a LCAO like Ψ = cAΦA + cBΦB for arbitrary atoms A and B. The resulting combination can be regarded as successful, i.e. possibly yields one wave function for a bonding MO, in cases where ...
1. The energy of ΦA and ΦB for the single atoms A and B ought to be similar. Therefore, the valence orbitals of two atoms are able to combine whereas for electrons from the valence shell and electrons from inner shells, no relevant combination is found. The latter are of very low energy.
2. The product ΦA·ΦB ought to be large, a point which is often referred to as "principle of maximal overlap".
3. The functions ΦA and ΦB ought to have the same symmetry with respect to the molecule's axis.
When dealing with heteronuclear molecules, we shall discuss the first criterion. The second criterion leads sometimes to the conclusion that S reaches a peak. But simply considering the case of H2+ will prove that this is wrong. Here, S would reach a maximum for R=0. (Hence, one precondition for nuclear fusion was simply obsolete). The last criterion is a consequence of the fact that there is no combination of ΦA and ΦB in cases where β equals zero for some reason, i.e. an overlap between the two orbitals is not possible or specific properties in the symmetry of ΦA and ΦB produce identical but oppositely signed contributions to the integral β. In such cases, it is said that ΦA and ΦB will not combine for symmetry reasons and therefore, a molecular orbital containing both functions does not exist. (For details, click here).

An overview to possible and impossible combinations of atomic orbitals of the types s, p and d is given in the following table:

 ΦA combines withΦB but does not combine with s s, pz, dz² px, py, dx²-y², dxy, dyz, dxz pz s, pz, dz² px, py, dx²-y², dxy, dyz, dxz px # px, dxz s, py, pz, dx²-y², dz², dxy, dyz dxz # px, dxz s, py, dx²-y², dz², dxy, dyz dx²-y² dx²-y² s, px, py, pz, dz², dxy, dyz, dxz dz² s, pz, dz² px, py, dx²-y², dxy, dyz, dxz #Possible combinations for py and dyz are obtained by exchanging x and y in any of the two lines

The figure below depicts the additive and subtractive combination of p-orbitals 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 pz orbital from another yields the bonding LCAO σg, whereas the linear combination obtained by adding the two pz orbitals leads to the antibonding orbital σu. From combinations of px or py orbitals π-orbitals emerge; thus, these orbitals are doubly degenerated. An additive combination of px or py AOs produces a marked overlap and therefore a bonding MO (πu). In contrast, subtractive combination gives rise to antibonding MOs (πg).

 From combinations of two 2pz atomic orbitals one bonding and one antibonding molecule orbital emerge, labelled as 3σg and 3σu. Axis z points to the right. Note the signs of the lobes and the way they combine. Dashed lines indicate nodal planes. Combination of two 2pxatomic orbitals gives as well rise to one bonding and one antibonding molecular orbital, labelled as 1πu and 1πg. The direction denoted as x is orthogonal to the axis between the nuclei. Note the signs of the lobes. In contrast to the previous combination of two pz atomic orbitals now addition of the AOs yields a bonding MO. Dashed lines indicate nodal planes. πu and πg are doubly degenerated as analogous linear combinations are possible with 2py atomic orbitals.

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.

• The symmetry operation of rotation does not affect a σ-molecular orbital (e.g the MOs of H2+). The orbital's distinctive feature is axial symmetry;
• A π molecular orbital consists of one positive and one negative lobe on the two sides of a nodal plane. Rotation by 180° changes but the sign of the wave function's amplitude;
• A δ molecular orbital emerging from dxy atomic orbitals with nuclei A and B as centres shows two nodal planes. These planes divide the orbital in four lobes. Therefore, rotation of the orbital by 90° changes the sign the wave function. Respective greek letters denote as well the higher MOs and the analogy with latin letters for AOs is kept.
One physical property related to MOs is an angular momentum of electrons. With respect to the axis between the nuclei, which is denoted as direction z, multiples of Planck's constant are found for a projection of this momentum on this axis (values 0, h, 2h, 3h, ... for σ, π, δ). In analogy with the s, p, d, ... - atomic orbitals, the rotational motion around an axis is associated with an angular momentum. The more azimuthal nodes appear, the higher is the angular momentum.
• for a σ-MO (λ = 0) the projection of the angular momentum is 0, thus h;
• for a π-MO (λ = 1) the projection is 1h;
• for a δ-MO (λ = 2) the projection is 2h.
The indices g and u (for german even/gerade and odd/ungerade) refer to symmetry as well, i.e. distinctive properties of the molecule with respect to a centre of symmetry. Molecules with a centre of symmetry will appear unchanged when submitted to Inversion. This symmetry operation connects any point on one side of the centre of symmetry with another beyond this centre. In case the wave function's amplitude for this point is not affected by inversion, it is said to be even and the MO's symbol appears with index g. In contrast, where inversion introduces a change in sign of the wave function, these wave functions are classified as odd and the MO's symbol appear with index u.

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:

 1σg < 1σu < 2σg < 2σu < 1πu < 3σg < 1πg < 3σu < 4σg < 4σu < 2πu < 5σg < 2πg < 5σu

For atoms of the second row of the periodic table forming two atomic molecules, the σg-MO is always below the πu-MO. Note that π-MOs appear doubly degenerated. 1πu is therefore used as an abbreviation for the pair of orbitals 1πxu and 1πyu that evidently come from the px- and the py 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 H2 occupy the bonding MO, thus the molecules electronic configuration is (1σg)2. If we continue our consideration with He2+, we would expect the ground state configuration (1σg)2(1σu) and He2 would have (1σg)2(1σu)2. In fact, the molecule He2 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 (1σg)2(1σu)(2σg)).

The instability of He2 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 Li2 with two valence electrons is described with the inner shell electrons as

Li2 [ (1σg)2(1σu)2(2σg)2]

or, shortly by introducing capital letters for complete shells

Li2 [ KK(2σg)2 ]

In a similar manner, we describe the electronic configuration of fluorine F2

F2 [ KK(2σg)2(2σu)2(1πu)4(3σg)2(1πg)4 ],

where (1πu)4 denotes the degenerated pair of (1πxu)2(1πyu)2. 2σg and 2σu as well as 1πu and 1πg are pairs of bonding and antibonding MOs. Therefore, of the 14 valence electrons only the pair of electrons in (3σg)2 are regarded as bonding and this corresponds to the finding of a single bond of σ-type 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:
2Πu   total spin = 1/2; total angular momentum = 1; inversion changes the wave function's sign
3Σg-  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. (bond-antib)/2= bonds Bonding Energy in eV Term symbol H2+ 1σg 1 0 ½ 2.793 2Σg+ H2 1σg2 2 0 1 4.748 1Σg+ He2+ 1σg2 1σu 2 1 ½ 2.470 2Σu+ He2 1σg2 1σu2 (=KK) 2 2 0 0.001 1Σg+ Li2+ KK 2σg 1 0 ½ 1.46 2Σg+ Li2 KK 2σg2 2 0 1 1.068 1Σg+ Be2 KK 2σg2 2σu2 2 2 0 0.096# 1Σg B2 KK 2σg2 2σu2 1πu2 4 2 1 3.08 3Σg− C2+ KK 2σg2 2σu2 1πu3 5 2 1½ 5.40 2Πu C2 KK 2σg2 2σu2 1πu4 6 2 2 6.32 1Σg+ N2+ KK 2σg2 2σu2 3σg 1πu4 7 2 2½ 6.478 2Σg+ N2 KK 2σg2 2σu2 3σg2 1πu4 8 2 3 7.519 1Σg+ O2+ KK 2σg2 2σu2 3σg2 1πu4 1πg 8 3 2½ 6.781 2Πg O2 KK 2σg2 2σu2 3σg2 1πu4 1πg2 8 4 2 5.214 3Σg− F2+ KK 2σg2 2σu2 3σg2 1πu4 1πg3 8 5 1½ 3.405 2Πg F2 KK 2σg2 2σu2 3σg2 1πu4 1πg4 8 6 1 1.659 1Σg+ * MOs are not always listed according their respective energy level  #  calculated by R. Gdanitz (Dec. 1998)

Summary

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 σ2, double bond σ2π2, triple bond σ2π4.

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 σ2π4δ2 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. πx, πy) applies in the same way as with AOs in atoms. Thus, for molecular oxygen O2, all levels up to 1πu are occupied and electrons with parallel spin in the degenerated 1πg levels cause the triplett ground state (S = 1).

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