Heteronuclear Molecules AB

As well for a treatment of heteronuclear molecules, the qualitative aspects of the LCAO method apply. To get a successful combination of the kind Ψ = cAΦA + cBΦB

The possible combinations of s-, p- and d-orbitals have already been presented in a table. For atomic orbitals ΦA and ΦB of different symmetry, e.g. respectively a reflection on a mirror plane, the wave functions are said to be orthogonal and the integral ∫ΦAΦBdτ becomes zero.

Let us apply the conditions for an effective linear combination of atomic orbitals to the hydrogene chloride molecule. The first condition, for which mathematical evidence will be delivered below, forbids combinations between a hydrogene AO and 1s-, 2s- and 2p-orbitals of of chlorine for these AOs (the 3s-orbital might be classified as a borderline case) are of too low energy. In contrast, the atomic orbitals 3px, 3py and 3pz remain as candidates for combinations with the 1s-AO which is the orbital of lowest energy of the hydrogene atom. Taking symmetry into account, the 3pz-AO of chlorine is the orbital that has to be introduced in an LCAO approach that shall describe HCl. As we will learn in one of the following chapters (Hybridization), the 3s-AO of Cl is to some degree involved in this combination, but this does not call the description presented here into question. From these considerations and from the building-up principle, the ground state of the hydrogene chloride molecule is:

(1s)2 (2s)2 (2pz)2 (2px)2 (2py)2 (3s)2 (3px)2 (3py)2 {cAH(1s) + cBCl(3pz)}2

Where the chlorine atomic orbitale are simply denoted as 1s, 2s, ... 3py. Though these orbitals do not contribute to bonding, they appear like MOs of related symmetry, i.e. the s- and pz-orbitals as σ- and the px and py π-molecular orbitals. A σ-state represents the orbital emerging from the {cAH(1s) + cBCl(3pz)}. The energy level of this MO is below the π-orbitals occupied by the (3px)and (3py) electrons of chlorine. With numbers added, the electronic configuration of the molecules is described as:

(1σ)2 (2σ)2(3σ)2 (1π)4 (4σ)2 (5σ)2(2π)41Σ

The relation between an orbital of the molecule AB in the LCAO approach and the atomic orbitals, i.e. the components for this approach (ΦA and ΦB) are worth being considered, especially in the case where this molecular orbital is occupied by electrons that can be regarded as valence electrons of the unbound atoms A and B. Assuming a possibility to separate the atoms that form molecule AB, we expect such electrons to appear again in the two atomic orbitals ΦA and ΦB. (We already encountered this aspect when treating the hydrogene cation). Thus, it is allowed to regard these electrons as originating from a ΦA- or ΦB-state, now sharing a state Ψ = cAΦA + cBΦB. Reversely, we are able to imagine the two atoms approaching each other without interaction and then pairing both electrons.When the molecule AB is formed, there is no possibility to discriminate between electron A and B, i.e. both belong to that AB molecule. In fact, this impossibility is the reason for a chemical bond.

Let us have a look on the mathematical treatment. For all relevant solutions of the secular equation for a heteronuclear molecule AB

cAA−E) + cB(β − E·S) = 0
cA(β − E·S) + cBB−E) = 0

the determinant becomes zero, i.e. the values for energy E accomplish equation

A-E)(αB-E) - (β-ES)2 = 0

To avoid the effort of deriving the exact solution of this equation quadratic in E, we assume that the energies of the bonding MO and the lowest of the two AOs differ only slightly. In case that ΦA is this AO, E ≈ αA−ΔE reflects this assumption without further restrictions to the general applicability. The ΔE represents the shift in energy that corresponds with the bonding effect. Replacing E in the quadratic equation yields

A - αA+ ΔE)(αB - αA + ΔE) = (β - (αA - ΔE)S)2

As ΔE << αA, we can write

ΔE·(αB - αA) = (β - αAS)2

and the following expression for the bonding energy is obtained

ΔE = [ β - αAS ]2 / [ αBA ].

For the energy level for the bonding MO we get:
Ebonding =  αA- [ β - aAS / [ αB - αA ]

In an analogous way, we derive the energy of the antibonding MO, starting with E ≈ αB-ΔE.

A - αB+ ΔE)(αB - αB +ΔE) = (β - (αB - ΔE) S)2

As ΔE << αB, we can write

A - αB)·ΔE = (β - αBS)2

and the following expression for repulsion is obtained

ΔE = − [ β - αBS ]2 / [ αBA ]

For the energy level for the antibonding MO we get:
Eantibonding =  αB+ [ β - aBS / [ αB - αA ]

Fig. 1: In a heteronuclear molecule the bonding MO is closer to the more electronegative atom (A in the case shown here ). In the respective wave function Ψbonding, the atomic orbital ΦA is dominant. In contrast, the antibonding MO the original atomic orbital of the atom B (of lower electronegativity) are of similar energy and ΦB is dominant in Ψantibonding.

The larger the difference between αA and αB, the larger is the denominator in the fraction and, in consequence, the smaller is the splitting of the MO energy levels. These levels are shown in Fig. 1. With almost identical values of αA and αB, the approximation above is not applicable anymore, but recalling the previous chapter homonuclear molecules, we are well prepared to treat this case where we are approach the situation αA = αB.

To obtain the coefficients cA and cB that represent the weight of the single atomic orbitals, we introduce the obtained expression for Ebonding into equation (αA - E) cA + (β - ES) cB = 0. Using again the approximation E ≈ aA- ΔE or ΔE<<aA we find:

cA/cB  =  (β - aAS)/ΔE = - [αB - aA]/[β - αAS]

ψbonding = cA [ ΦA + AS - β]/B - αA]·ΦB]

Coefficient cA emerges when Ψbonding is normalized. In analogy, for the antibonding MO we recieve

cB/cA =  (β - aBS)/BA]

ψantibonding = cA [ ΦA - BS - β]/B - αA]·ΦB]

As a threedimensional plot, Fig. 2 illustrates a linear combination with differently weighted atomic wave functions. ΦA is an 2pz-, ΦB is an 1s-orbital. The ratio cB/cA is 3/7.

Fig .2: Example for the wave function that emerges from a linear, differently weighted combination of an 2pz and an 1s atomic orbital.

Fig. 3 shows a correlation diagram for a heteronuclear diatomic molecule (comp. homonuclear case). Just σ and π appear as the g-u-classification is only relevant for homonuclear diatomics. Fig. 4 qualitatively shows the distribution of the wave function's amplitude for the single atoms and the orbitals of an hypothetical united atom as the two extremes in this approach. The intermediate case between shall deliver an impression of the conditions in a molecule AB.

Fig.3: Correlation diagram for a heteronuclear diatomic molekule (compare Fig. 1 Energy Levels in homonuclear molecules). Only σ and π molecular orbitals are of interest. The vertical dashed line represents the sequence of MO energies found in the Molecule NO (according to R.S. Mulliken, Rev. mod. Phys. 4, 1 (1932)).

1s <=> <=> 1sA
2s <=> <=> 1sB
2px <=> <=> 2pxA
3px <=> <=> 2pxB
  united atom   molecule   single atoms  
Fig. 4: Examples of the correlation between orbitals of a united atom and the single atoms for a heteronuclear diatomic molecule. The atomic orbitals of the separate atoms on the right yield in one bonding and one antibonding combination one molecular orbital (in the middle) and, in the hypothetical case of an internuclear distance zero, atomic orbital (left). Pay attention to the case of the antibonding combination depicted in the last row. Here, an additional node appears and an AO with an increased quantum number (3px) is introduced.


Electron configuration and term symbols of the lowest states of XH
XH molecule ground state first excited state
LiH, BeH+
NaH, MgH+
CuH, ZnH+
K(2sσ)2  1Σ+
KL(3sσ)2  1Σ+
KLM(4sσ)2  1Σ+
2sσ2pσ: 1Σ+ [3Σ+]
3sσ3pσ: 1Σ+ [3Σ+]
4sσ4pσ: 1Σ+ [3Σ+]
MgH, AlH+
KL(3sσ)2 3pσ:   2Σ+
KLMsp(4sσ)2 3dσ:   2Σ+
KLMNOspd(6sσ)2 6pσ:   2Σ+
(3sσ)2 3pπ: 2Πr
(4sσ)2 3dπ: 2Πr
(6sσ)2 6pπ: 2Πr
BH, CH+ K(2sσ)2 (2pσ)21Σ+ (2sσ)2 2pσ2pπ: 1Π, 3Π
CH K(2sσ)2 (2pσ)2 2pπ: 2Πr (2sσ)2 2pσ (2pπ)2:  [4Σ-], 2Δ, 2Σ+, 2Σ-
NH, OH+ K(2sσ)2 (2pσ)2 (2pπ)2   3Σ-, 1Δ, 1Σ+ (2sσ)2 2pσ (2pπ)3: 3Π, 1Π
OH K(2sσ)2 (2pσ)2 (2pπ)3: 2Πi (2sσ)2 2pσ (2pπ)4 : 2Σ+
K(2sσ)2 (2pσ)2 (2pπ)4: 1Σ+
KL(3sσ)2 (3pσ)2 (3pπ)4 : 1Σ+
(2sσ)2 (2pσ)2 (2pπ)3 3sσ:  [3Π], [1Π]
(3sσ)2 (3pσ)2 (3pπ)3 4sσ: 3Π, 1Π
NiH KLMsp(3dσ)2 (3dπ)4 (3dδ)3 (4sσ)2  2Δ (3dσ)2 (3dπ)4 (3dδ)3 4sσ4pσ: 2Δ, 2Δ, 4Δ
For the excited electron configurations the closed atomic shells have not been repeated. The states in brackets have not been observed. Msp means that in the M shell only the groups 3s and 3p are closed, and similarly in other case.

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