Molecular orbital
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In chemistry, a molecular orbital is a region in which an electron may be found in a molecule.[1] In general, atomic orbitals combine to form molecular orbitals. The probability of finding an electron in a given region can be obtained by solving the Schrödinger wave equation.
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[edit] Overview
In quantum chemistry, i.e. electronic structure theory, the molecular electronic states, which are the eigenstates of the electronic molecular Hamiltonian, are expanded (see configuration interaction expansion and basis) into linear combinations of anti-symmetrized products (Slater determinants) of one-electron functions. The spatial components of these one-electron functions are called molecular orbitals (MO). When one considers also their spin component, they are called molecular spin orbitals.
Most methods in computational chemistry today start 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 Pauli principle demands that they have opposite spin.
[edit] Qualitative discussion
For an imprecise, but qualitatively useful, discussion of the molecular structure, the molecular orbitals can be obtained from the "Linear combination of atomic orbitals molecular orbital method" ansatz (using eventually the concept of hybridized orbitals).
In this approach, the molecular orbitals are expressed as linear combinations of atomic orbitals, as if each atom were on its own.
The linear combination of atomic orbitals approximation for molecular orbitals was introduced in 1929 by Sir John Lennard-Jones. His ground-breaking paper showed how to derive the electronic structure of the fluorine and oxygen molecules from quantum principles. This qualitative approach to molecular orbital theory represents the dawn of modern quantum chemistry.
Some properties:
- The number of molecular orbitals is equal to the number the atomic orbitals included in the linear expansion,
- 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 is 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 level are closer.
[edit] Examples
[edit] H2
As a simple example consider the hydrogen molecule, H2, 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 (and hence more stable) than two free hydrogen atoms. This is called a covalent bond.
A Molecular Orbital Diagram of H2
[edit] He2
On the other hand, consider the hypothetical molecule of He2 with the atoms labelled He' and He. Again, the lowest-energy atomic orbitals, 1s' and 1s", do not transform according to the symmetries of the molecule, while 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 |
Similar to the molecule H2, 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. However, in its neutral ground state, each Helium atom contains two electrons in its 1s orbital, combining for a total of four electrons. Two electrons fill the lower energy bonding orbital, while the remaining two fill the higher energy antibonding orbital. Thus, the resulting electron density around the molecule does not support the formation of a bond between the two atoms (called a sigma bond), and the molecule does therefore not exist.
A Molecular Orbital Diagram of He2
[edit] Rare gases
Now let's move to larger atoms. Considering a hypothetical molecule of He2, since the basis set of atomic orbitals is the same as in the case of H2, we find that both the bonding and antibonding orbitals are filled, so there is no energy advantage to the pair. HeH would have a slight energy advantage, but not as much as H2 + 2 He, so the molecule exists only a short while. In general, we find that atoms such as He that have completely full energy shells rarely bond with other atoms. Except for short-lived Van der Waals complexes, there are very few noble gas compounds known.
[edit] Inner shells
Inner shell orbitals should not be included in the LCAO expansion. Molecular structure relies on the outermost (valence) electrons of the atoms, which are usually of comparable energy.
[edit] Ionic bonds
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 other's almost entirely to the antibonding orbitals. Thus, the situation is effectively that some electrons have been transferred from one atom to the other. This is called a (mostly) ionic bond.
[edit] More quantitative approach
To obtain quantitative values for the molecular energy levels, one needs to have molecular orbitals which are such that the configuration interaction (CI) expansion converges fast towards the full CI limit. The most common method to obtain such functions is the Hartree-Fock method which expresses the molecular orbitals as eigenfunctions of the Fock operator. One usually solves this problem by expanding the molecular orbitals as linear combinations of gaussian functions centered on the atomic nuclei (see linear combination of atomic orbitals and basis set (chemistry)). The equation for the coefficients of these linear combinations is a generalized eigenvalue equation known as the Roothaan equations which are in fact a particular representation of the Hartree-Fock equation.
Simple accounts often suggest that experimental molecular orbital energies can be obtained by the methods of ultra-violet photoelectron spectroscopy for valence orbitals and X-ray photoelectron spectroscopy for core orbitals. This however is incorrect as these experiments measure the ionization energy, the difference in energy between the molecule and one of the ions resulting from the removal of one electron. Ionization energies are linked approximately to orbital energies by Koopmans' theorem. While the agreement between these two values can be close for some molecules, it can be very poor in other cases.
[edit] See also
- Atomic orbital
- Computational chemistry
- Electron configuration
- HOMO/LUMO
- MO diagram
- Molecular radiation
- Quantum chemistry
[edit] References
- ^ Daintith, J. (2004). Oxford Dictionary of Chemistry. New York: Oxford University Press. ISBN 0-19-860918-3.
[edit] External links
- Visualizations of some atomic and molecular orbitals (Note: These visualisations run only on Apple Mac.)
- Java molecular orbital viewer shows orbitals of hydrogen molecular ion.
