In chemistry, a molecular orbital (MO) is a mathematical function
describing the wave-like behavior of an electron in a molecule. This
function can be used to calculate chemical and physical properties
such as the probability of finding an electron in any specific region.
The term orbital was introduced by
Robert S. Mulliken
1 Overview 2 Formation of molecular orbitals 3 Qualitative discussion
3.1 Linear combinations of atomic orbitals (LCAO) 3.2 Bonding, antibonding, and nonbonding MOs 3.3 Sigma and pi labels for MOs
3.3.1 σ symmetry 3.3.2 π symmetry 3.3.3 δ symmetry 3.3.4 φ symmetry
3.4 Gerade and ungerade symmetry 3.5 MO diagrams 3.6 Bonding in molecular orbitals
3.6.1 Orbital degeneracy 3.6.2 Ionic bonds 3.6.3 Bond order 3.6.4 HOMO and LUMO
4.1 Homonuclear diatomics
4.1.1 H2 4.1.2 He2 4.1.3 Li2 4.1.4 Noble gases
4.2 Heteronuclear diatomics
5 Quantitative approach 6 References 7 External links
A molecular orbital (MO) can be used to represent the regions in a
molecule where an electron occupying that orbital is likely to be
found. Molecular orbitals are obtained from the combination of atomic
orbitals, which predict the location of an electron in an atom. A
molecular orbital can specify the electron configuration of a
molecule: the spatial distribution and energy of one (or one pair of)
electron(s). Most commonly a MO is represented as a linear combination
of atomic orbitals (the LCAO-MO method), especially in qualitative or
very approximate usage. They are invaluable in providing a simple
model of bonding in molecules, understood through molecular orbital
theory. Most present-day methods in computational chemistry begin by
calculating the MOs of the system. A molecular orbital describes the
behavior of one electron in the electric field generated by the nuclei
and some average distribution of the other electrons. In the case of
two electrons occupying the same orbital, the
displaystyle Psi =c_ a psi _ a +c_ b psi _ b
displaystyle Psi ^ * =c_ a psi _ a -c_ b psi _ b
displaystyle Psi ^ *
are the molecular wavefunctions for the bonding and antibonding molecular orbitals, respectively,
displaystyle psi _ a
displaystyle psi _ b
are the atomic wavefunctions from atoms a and b, respectively, and
displaystyle c_ a
displaystyle c_ b
are adjustable coefficients. These coefficients can be positive or negative, depending on the energies and symmetries of the individual atomic orbitals. As the two atoms become closer together, their atomic orbitals overlap to produce areas of high electron density, and, as a consequence, molecular orbitals are formed between the two atoms. The atoms are held together by the electrostatic attraction between the positively charged nuclei and the negatively charged electrons occupying bonding molecular orbitals. Bonding, antibonding, and nonbonding MOs When atomic orbitals interact, the resulting molecular orbital can be of three types: bonding, antibonding, or nonbonding. Bonding MOs:
Bonding interactions between atomic orbitals are constructive (in-phase) interactions. Bonding MOs are lower in energy than the atomic orbitals that combine to produce them.
Nonbonding MOs are the result of no interaction between atomic orbitals because of lack of compatible symmetries. Nonbonding MOs will have the same energy as the atomic orbitals of one of the atoms in the molecule.
Sigma and pi labels for MOs The type of interaction between atomic orbitals can be further categorized by the molecular-orbital symmetry labels σ (sigma), π (pi), δ (delta), φ (phi), γ (gamma) etc. These are the Greek letters corresponding to the atomic orbitals s, p, d, f and g respectively. The number of nodal planes containing the internuclear axis between the atoms concerned is zero for σ MOs, one for π, two for δ, three for φ and four for γ. σ symmetry Further information: Sigma bond A MO with σ symmetry results from the interaction of either two atomic s-orbitals or two atomic pz-orbitals. An MO will have σ-symmetry if the orbital is symmetric with respect to the axis joining the two nuclear centers, the internuclear axis. This means that rotation of the MO about the internuclear axis does not result in a phase change. A σ* orbital, sigma antibonding orbital, also maintains the same phase when rotated about the internuclear axis. The σ* orbital has a nodal plane that is between the nuclei and perpendicular to the internuclear axis. π symmetry Further information: Pi bond A MO with π symmetry results from the interaction of either two atomic px orbitals or py orbitals. An MO will have π symmetry if the orbital is asymmetric with respect to rotation about the internuclear axis. This means that rotation of the MO about the internuclear axis will result in a phase change. There is one nodal plane containing the internuclear axis, if real orbitals are considered. A π* orbital, pi antibonding orbital, will also produce a phase change when rotated about the internuclear axis. The π* orbital also has a second nodal plane between the nuclei. δ symmetry Further information: Delta bond A MO with δ symmetry results from the interaction of two atomic dxy or dx2-y2 orbitals. Because these molecular orbitals involve low-energy d atomic orbitals, they are seen in transition-metal complexes. A δ bonding orbital has two nodal planes containing the internuclear axis, and a δ* antibonding orbital also has a third nodal plane between the nuclei. φ symmetry Further information: Phi bond
Suitably aligned f atomic orbitals overlap to form phi molecular orbital (a phi bond)
Theoretical chemists have conjectured that higher-order bonds, such as phi bonds corresponding to overlap of f atomic orbitals, are possible. There is as of 2005 only one known example of a molecule purported to contain a phi bond (a U−U bond, in the molecule U2). Gerade and ungerade symmetry For molecules that possess a center of inversion (centrosymmetric molecules) there are additional labels of symmetry that can be applied to molecular orbitals. Centrosymmetric molecules include:
Homonuclear diatomics, X2 Octahedral, EX6 Square planar, EX4.
Non-centrosymmetric molecules include:
Heteronuclear diatomics, XY Tetrahedral, EX4.
If inversion through the center of symmetry in a molecule results in
the same phases for the molecular orbital, then the MO is said to have
gerade (g) symmetry, from the German word for even. If inversion
through the center of symmetry in a molecule results in a phase change
for the molecular orbital, then the MO is said to have ungerade (u)
symmetry, from the German word for odd. For a bonding MO with
σ-symmetry, the orbital is σg (s' + s'' is symmetric),
while an antibonding MO with σ-symmetry the orbital is σu, because
inversion of s' – s'' is antisymmetric. For a bonding MO
with π-symmetry the orbital is πu because inversion through the
center of symmetry for would produce a sign change (the two p atomic
orbitals are in phase with each other but the two lobes have opposite
signs), while an antibonding MO with π-symmetry is πg because
inversion through the center of symmetry for would not produce a sign
change (the two p orbitals are antisymmetric by phase).
A basis set of orbitals includes those atomic orbitals that are available for molecular orbital interactions, which may be bonding or antibonding The number of molecular orbitals is equal to the number of atomic orbitals included in the linear expansion or the basis set If the molecule has some symmetry, the degenerate atomic orbitals (with the same atomic energy) are grouped in linear combinations (called symmetry-adapted atomic orbitals (SO)), which belong to the representation of the symmetry group, so the wave functions that describe the group are known as symmetry-adapted linear combinations (SALC). The number of molecular orbitals belonging to one group representation is equal to the number of symmetry-adapted atomic orbitals belonging to this representation Within a particular representation, the symmetry-adapted atomic orbitals mix more if their atomic energy levels are closer.
The general procedure for constructing a molecular orbital diagram for a reasonably simple molecule can be summarized as follows: 1. Assign a point group to the molecule. 2. Look up the shapes of the SALCs. 3. Arrange the SALCs of each molecular fragment in increasing order of energy, first noting whether they stem from s, p, or d orbitals (and put them in the order s < p < d), and then their number of internuclear nodes. 4. Combine SALCs of the same symmetry type from the two fragments, and from N SALCs form N molecular orbitals. 5. Estimate the relative energies of the molecular orbitals from considerations of overlap and relative energies of the parent orbitals, and draw the levels on a molecular orbital energy level diagram (showing the origin of the orbitals). 6. Confirm, correct, and revise this qualitative order by carrying out a molecular orbital calculation by using commercial software. Bonding in molecular orbitals Orbital degeneracy Main article: Degenerate orbital Molecular orbitals are said to be degenerate if they have the same energy. For example, in the homonuclear diatomic molecules of the first ten elements, the molecular orbitals derived from the px and the py atomic orbitals result in two degenerate bonding orbitals (of low energy) and two degenerate antibonding orbitals (of high energy). Ionic bonds Main article: Ionic bond When the energy difference between the atomic orbitals of two atoms is quite large, one atom's orbitals contribute almost entirely to the bonding orbitals, and the others atom's orbitals contribute almost entirely to the antibonding orbitals. Thus, the situation is effectively that one or more electrons have been transferred from one atom to the other. This is called an (mostly) ionic bond. Bond order Main article: Bond order The bond order, or number of bonds, of a molecule can be determined by combining the number of electrons in bonding and antibonding molecular orbitals. A pair of electrons in a bonding orbital creates a bond, whereas a pair of electrons in an antibonding orbital negates a bond. For example, N2, with eight electrons in bonding orbitals and two electrons in antibonding orbitals, has a bond order of three, which constitutes a triple bond. Bond strength is proportional to bond order—a greater amount of bonding produces a more stable bond—and bond length is inversely proportional to it—a stronger bond is shorter. There are rare exceptions to the requirement of molecule having a positive bond order. Although Be2 has a bond order of 0 according to MO analysis, there is experimental evidence of a highly unstable Be2 molecule having a bond length of 245 pm and bond energy of 10 kJ/mol. HOMO and LUMO Main article: HOMO/LUMO The highest occupied molecular orbital and lowest unoccupied molecular orbital are often referred to as the HOMO and LUMO, respectively. The difference of the energies of the HOMO and LUMO, termed the band gap, can sometimes serve as a measure of the excitability of the molecule: The smaller the energy the more easily it will be excited. Examples Homonuclear diatomics Homonuclear diatomic MOs contain equal contributions from each atomic orbital in the basis set. This is shown in the homonuclear diatomic MO diagrams for H2, He2, and Li2, all of which containing symmetric orbitals. H2
As a simple MO example, consider the electrons in a hydrogen molecule, H2 (see molecular orbital diagram), with the two atoms labelled H' and H". The lowest-energy atomic orbitals, 1s' and 1s", do not transform according to the symmetries of the molecule. However, the following symmetry adapted atomic orbitals do:
1s' – 1s" Antisymmetric combination: negated by reflection, unchanged by other operations
1s' + 1s" Symmetric combination: unchanged by all symmetry operations
The symmetric combination (called a bonding orbital) is lower in
energy than the basis orbitals, and the antisymmetric combination
(called an antibonding orbital) is higher. Because the H2 molecule has
two electrons, they can both go in the bonding orbital, making the
system lower in energy (hence more stable) than two free hydrogen
atoms. This is called a covalent bond. The bond order is equal to the
number of bonding electrons minus the number of antibonding electrons,
divided by 2. In this example, there are 2 electrons in the bonding
orbital and none in the antibonding orbital; the bond order is 1, and
there is a single bond between the two hydrogen atoms.
On the other hand, consider the hypothetical molecule of He2 with the
atoms labeled He' and He". As with H2, the lowest energy atomic
orbitals are the 1s' and 1s", and do not transform according to the
symmetries of the molecule, while the symmetry adapted atomic orbitals
do. The symmetric combination—the bonding orbital—is lower in
energy than the basis orbitals, and the antisymmetric
combination—the antibonding orbital—is higher. Unlike H2, with two
valence electrons, He2 has four in its neutral ground state. Two
electrons fill the lower-energy bonding orbital, σg(1s), while the
remaining two fill the higher-energy antibonding orbital, σu*(1s).
Thus, the resulting electron density around the molecule does not
support the formation of a bond between the two atoms; without a
stable bond holding the atoms together, molecule would not be expected
to exist. Another way of looking at it is that there are two bonding
electrons and two antibonding electrons; therefore, the bond order is
0 and no bond exists (the molecule has one bound state supported by
the Van der Waals potential).
^ Mulliken, Robert S. (July 1932). "Electronic Structures of
Polyatomic Molecules and Valence. II. General Considerations".
Physical Review. 41 (1): 49–71. Bibcode:1932PhRv...41...49M.
^ Albright, T. A.; Burdett, J. K.; Whangbo, M.-H. (2013). Orbital
Interactions in Chemistry. Hoboken, N.J.: Wiley.
^ F. Hund, "Zur Deutung einiger Erscheinungen in den Molekelspektren"
[On the interpretation of some phenomena in molecular spectra]
Zeitschrift für Physik, vol. 36, pages 657-674 (1926).
^ F. Hund, "Zur Deutung der Molekelspektren", Zeitschrift für Physik,
Part I, vol. 40, pages 742-764 (1927); Part II, vol. 42, pages
93–120 (1927); Part III, vol. 43, pages 805-826 (1927); Part IV,
vol. 51, pages 759-795 (1928); Part V, vol. 63, pages 719-751 (1930).
^ R. S. Mulliken, "Electronic states. IV. Hund's theory; second
positive nitrogen and Swan bands; alternate intensities", Physical
Review, vol. 29, pages 637–649 (1927).
^ R. S. Mulliken, "The assignment of quantum numbers for electrons in
molecules", Physical Review, vol. 32, pages 186–222 (1928).
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