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Orbital Symmetries
The symmetries of the orbitals and the electron states of an atom or ion in a symmetric environment can be deduced.
Atoms and atomic ions can be used to illustrate a chemically important use of symmetry and group theory.
Let us see how symmetry and group theory help us describe an atom or ion that is subjected to fields of various symmetries, such fields are exerted by surrounding ions or molecules. Studies of such effects on ions in crystals were made by Hans Bethe in 1929. The term crystal field, which stems from Bethe’s work, is now usually applied to an approach used in studies of transition metal ions surrounded by ligand molecules or ions. In the crystal field approach, the effect of the ligand is that of producing at the crystal ion a field with a symmetry that depends on the structural arrangement of the ligand about the ion.
Let us first describe the effect of a crystal field on orbitals, i.e. on the description we use for individual electrons. As in free atom systems we use these descriptions to develop atomic orbitals of a central atom. Such a field, with symmetry corresponding to the OH point group, occurs in many coordination compounds of the transition metals. The orbitals of a free atom are characterized, in part, by the angular properties of the wave function. We recognize, for a given value of n, one s orbital, three p orbitals and five d orbitals. These provide a basis for the construction or classification of orbitals that confirm to the symmetry of the imposed octahedral field.
An s orbital spherically symmetric and thus will transform according to the totally symmetric and thus transform according to the designation. Thus when a n atom is subjected to an octahedral field, the s orbitals can be given the symmetry designation a1g.
The d orbitals as shown in the squares and cross terms of the x, y, and z coordinates are octahedral. The location of these terms in the OH character table shows that in a field of OH symmetry the five d orbitals break up into two sets.
Symmetry properties of atomic orbitals, or atomic states, in an octahedral field, point group OH.
s
a1
p
t1
d
e = t2
f
a2 + t1 + t2
g
a1 + e + t1 + t2
Symmetry arguments alone do not indicate the relative energies of the sets of symmetry orbitals that stem from a given set of free atom orbitals. However, qualitative arguments can be made for fields of particular symmetries. Suppose, for example, that an ion is surrounded by a set of six electron-rich sites located along the Cartesian coordinates to form an octahedral field. An electron in an ep orbital will shows, be pointed at these electron repelling sites, whereas one in a t2g orbital will avoid these sites. We expect the eg orbitals to be at a higher than the t2g orbitals.
Now let us consider first the electron states that arise when a crystal field is applied to a free ion. We consider first an ion with closed shells and a simple d valence electron. The Ti3+ ion is an example. The electron configuration of the free ion is described as [inner shells]d1. The atomic state is described as 2D.
In a strong octahedral field, the electron will be in the lower energy t2g orbital, producing a 2T2g state, or a higher energy eg orbital, producing a higher energy 2Eg state. A schematic correlation diagram showing the development symmetry based states from the free atom states.
The energy details can be obtained only by quantum mechanical calculations or on an empirical basis from spectral studies. In aqueous solution the Ti3+ ion is present as the octahedral [Ti (H2O)6]3+ ion. It shows a weak absorption in the visible region with a maximum at about 20,400 cm-1. This absorption can be attributed to the transition from the 2T2g ground state to the excited 2Eg state. Thus, for the [Ti (H2O)6]3+ ion, the crystal field splitting factor, often designated Δ or 10Dq, has the value 20,400 cm-1, or 244 kJ mol-1.
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Atoms and atomic ions can be used to illustrate a chemically important use of symmetry and group theory.
Let us see how symmetry and group theory help us describe an atom or ion that is subjected to fields of various symmetries, such fields are exerted by surrounding ions or molecules. Studies of such effects on ions in crystals were made by Hans Bethe in 1929. The term crystal field, which stems from Bethe’s work, is now usually applied to an approach used in studies of transition metal ions surrounded by ligand molecules or ions. In the crystal field approach, the effect of the ligand is that of producing at the crystal ion a field with a symmetry that depends on the structural arrangement of the ligand about the ion.
| Operation | Character contribution | Operation | Character contribution |
| E | 3 | C4 | 1 |
| σ | 1 | S3 | -1 |
| I | -3 | S4 | 0 |
| C2 | -1 | S6 | 0 |
| C3 | 0 |
Let us first describe the effect of a crystal field on orbitals, i.e. on the description we use for individual electrons. As in free atom systems we use these descriptions to develop atomic orbitals of a central atom. Such a field, with symmetry corresponding to the OH point group, occurs in many coordination compounds of the transition metals. The orbitals of a free atom are characterized, in part, by the angular properties of the wave function. We recognize, for a given value of n, one s orbital, three p orbitals and five d orbitals. These provide a basis for the construction or classification of orbitals that confirm to the symmetry of the imposed octahedral field.
An s orbital spherically symmetric and thus will transform according to the totally symmetric and thus transform according to the designation. Thus when a n atom is subjected to an octahedral field, the s orbitals can be given the symmetry designation a1g.
The d orbitals as shown in the squares and cross terms of the x, y, and z coordinates are octahedral. The location of these terms in the OH character table shows that in a field of OH symmetry the five d orbitals break up into two sets.
Symmetry properties of atomic orbitals, or atomic states, in an octahedral field, point group OH.
s
p
d
f
g
Symmetry arguments alone do not indicate the relative energies of the sets of symmetry orbitals that stem from a given set of free atom orbitals. However, qualitative arguments can be made for fields of particular symmetries. Suppose, for example, that an ion is surrounded by a set of six electron-rich sites located along the Cartesian coordinates to form an octahedral field. An electron in an ep orbital will shows, be pointed at these electron repelling sites, whereas one in a t2g orbital will avoid these sites. We expect the eg orbitals to be at a higher than the t2g orbitals.
Now let us consider first the electron states that arise when a crystal field is applied to a free ion. We consider first an ion with closed shells and a simple d valence electron. The Ti3+ ion is an example. The electron configuration of the free ion is described as [inner shells]d1. The atomic state is described as 2D.
In a strong octahedral field, the electron will be in the lower energy t2g orbital, producing a 2T2g state, or a higher energy eg orbital, producing a higher energy 2Eg state. A schematic correlation diagram showing the development symmetry based states from the free atom states.
The energy details can be obtained only by quantum mechanical calculations or on an empirical basis from spectral studies. In aqueous solution the Ti3+ ion is present as the octahedral [Ti (H2O)6]3+ ion. It shows a weak absorption in the visible region with a maximum at about 20,400 cm-1. This absorption can be attributed to the transition from the 2T2g ground state to the excited 2Eg state. Thus, for the [Ti (H2O)6]3+ ion, the crystal field splitting factor, often designated Δ or 10Dq, has the value 20,400 cm-1, or 244 kJ mol-1.
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