P.A. Cox - Inorganic chemistry (793955), страница 17
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Overlapping atomic orbitals can givebonding and antibonding MOs. Electrons in bonding MOs have anincreased probability of being in the region between the nuclei.An MO diagram is a representation of the energies of bonding andantibonding MOs formed from atomic orbitals. Electrons areassigned to MOs in accordance with the exclusion principle, leadingto the same building-up procedure as for the periodic table of elements.The bond order is defined as half the number of net bondingelectrons.2p atomic orbitals can give rise to both σ and π MOs, the formeroverlapping along the molecular axis and the latter perpendicular toit. Multiple bonding in O2 and N2 arises from π as well as σ bonding.Trends in bond strengths and lengths follow predicted bond orders.The O2 molecule has two unpaired electrons, a fact not predicted bysimple electron-pair bonding models.Atomic orbitals (A2)Many-electron atoms (A3)Bonding and antibonding orbitalsJust as an atomic orbital (AO) is the wavefunction for an electron in an atom (Topic A2) so a molecular orbital (MO)is that for a molecule; an MO may extend over two or more atoms.
Exact MOs may be obtained by solution ofSchrödinger’s equation for the one-electron ionbut as in atoms the extension of the orbital approach to manyelectron systems involves approximations (see Topic A3). For MOs a further approximation is often made: rather thanusing the exact but very complicated wavefunction forit is convenient to express each MO as a linearcombination of AOs, the so-called LCAO approximation. This approximation needs to be refined considerablyfor quantitative calculations on computers, but it is adequate for qualitative purposes and provides very useful pictures ofhow chemical bonds form, and how they are related to the valence AOs of the atoms involved.Figure 1 shows how the LCAO method works for H2. The diagrams show how the value of a wavefunction variesalong the molecular axis.
Figure 1a shows wavefunctions and for the 1s valence AOs on the two H atoms. An MOwavefunction ψ is constructed by writing(1)C4—MOLECULAR ORBITALS: HOMONUCLEAR DIATOMICS71Fig. 1. Formation of a bonding and antibonding MO from the overlap of 1s AOs in H2 (see text)where c1 and c2 are numerical coefficients.
The square of ψ gives the electron probability distribution, and in ahomonuclear diatomic such as H2 (where each atom is the same) ψ2 must have the same value on each atom. Thuswe havefrom which either c1=c2 or c1=−c2. The former combination is the bonding MO, and the latter theantibonding MO, shown in Fig. 1b and c. respectively. Near each atomic nucleus both MOs resemble the 1s AO(apart from sign, which does not affect the probability distribution).
An electron close to one nucleus is hardly affectedby the presence of the other one. The important difference occurs in the region between the nuclei, where the bondingMO predicts an increase of electron density compared with the isolated atoms. This may be seen fromThe first two terms give the same electron density as found in the two 1s AOs, whereasrepresents an increase ofdensity in the region between the two atoms where the orbitals overlap. An electron in the bonding MO has lowerenergy than in the AO of an isolated atom, as it has enhanced probability of being close to both nuclei simultaneously.The electrostatic repulsion between the nuclei is effectively shielded, leading to a stable bond.
Conversely, in anantibonding MO an electron has higher energy, with a reduced probability of being in the internuclear region, andinternuclear repulsion being ‘deshielded’.In the ground state of H2 two electrons occupy the bonding MO, the antibonding one not being used. The simple MOmodel thus shows how bonding arises from an increase of electron density in the internuclear region, and how overlapof AOs is essential for this to happen.MO diagramsFigure 2 shows an MO diagram for H2. The AO energies of the two isolated atoms are displayed on the left- and righthand sides, and in the center are shown the bonding and antibonding MOs resulting from their overlap.
They are labeled1sσg and 1sσu, respectively, the σ designation referring to their symmetric nature about the molecular axis (see below)and g and u (from the German gerade and ungerade, respectively) to their even or odd behavior under inversion throughthe center of symmetry of molecule (the mid-point of the bond, see Topic C3).72SECTION C—STRUCTURE AND BONDING IN MOLECULESFig. 2. MO diagram for H2 showing the bonding MO occupied by two electrons with paired spins.Occupation of MOs is governed by the Pauli exclusion principle (see Topic A3).
In Fig. 2 two electrons areshown with opposite spin (one upward and one downward pointing arrow) corresponding to the electronconfiguration (1sσg)2. When more than two electrons are present some must occupy an orbital of higher energy.Thus He2 (four electrons) would have the electron configuration (1sσg)2(1sσu)2, which could be represented on a similarMO diagram.
Calculations show that the increase of energy (relative to isolated AOs) in forming the antibonding MO morethan compensates for the stabilization of the bonding MO. He2 is therefore an unstable molecule with higher energythan two individual He atoms. In MO theory the ‘repulsion’ between closed shells (the He core in this example) is seento be a consequence of the exclusion principle, which forces occupation of antibonding as well as bonding orbitals.The definition of bond order (BO) in MO theory recognizes that a ‘normal’ single bond is formed by twoelectrons (see Topic C1). We defineBO=(1/2)[(no. of electrons in bonding MOs)−(no. of electrons in antibonding MOs)]In the above examples, H2 and He2 have bond orders of one and zero, respectively.
Fractional values are possible, as inthe molecular ionsandwhich are both bonded but more weakly than H2, as indicated by the electronconfigurations and BO shown below:Second period diatomicsIn extending the MO theory to second period elements two additional principles need to be considered. First, onlyvalence-shell orbitals are shown in MO diagrams, as inner shells are too tightly bound to be involved in bonding,and do not overlap significantly. Second, both 2s and 2p valence orbitals need to be included (see Topics A2–A4).
Thethree AOs in a p shell differ only in their direction in space, and in an atom are degenerate (i.e. have the same energy).In diatomic molecules, however, they are distinguished by the way they overlap (see Fig. 3): pσ orbitals point along thedirection of the bond, and pπ orbitals (of which there are two equivalent and degenerate ones) are perpendicular to it.Each type of orbital can combine to form MOs in the same way as in H2. Two fundamental rules in the LCAO MOmodel are important:• the number of MOs formed is equal to the total number of starting AOs;• only AOs of the same symmetry type combine to make MOs.C4—MOLECULAR ORBITALS: HOMONUCLEAR DIATOMICS73Figure 3 shows that the pσ AOs, which point towards each other, overlap more strongly than do pπ MOs, so that boththe bonding stabilization and the antibonding destabilization is greater for pσ. The order of MO energies formed fromthe 2p AO on the two atoms is therefore:The symmetry designation of each MO is given and it should be noted that the g or u labeling for π MOs is the reverseof that for σ MOs.Figure 4 shows the MO diagram for dioxygen O2.
As in Fig. 2 the energies of 2s and 2p AOs of the separate atoms areshown and between them the MOs formed from overlap. Lowest in energy are 2sσg and 2sσu, respectively the bondingand antibonding combinations of 2s AOs. At higher energies are MOs resulting from 2p AOs. Figure 4 shows occupationof MOs in O2 (12 valence electrons). The BO calculated as above is two, as expected from the normal valence structure(see Topic C1). However, the degenerate 2pπg MOs are occupied by two electrons, and as in atoms, Hand’s firstrule shows that the ground state will be formed by putting one electron in each orbital with parallel spin (seeTopic A3). Thus O2 is correctly predicted to have two unpaired electrons and as a consequence is paramagnetic (seeTopic F7).One of the advantages of MO theory is that the same diagram can be used for molecules with different numbers ofelectrons.
The electron configurations and BOs for some molecules and ions, which can be derived using Fig. 4. areshown in Table 1 together with the experimental dissociation energies and bond lengths. Higher predicted BO valuescorrespond to stronger and shorter bonds.Fig. 3. Bonding MOs formed by σ and π overlap of 2p AOs. Negative regions of the wavefunction are shaded.Fig. 4.
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