In
molecular physics, the molecular term symbol is a shorthand expression of the
group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
and
angular momenta that characterize the state of a
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and b ...
, i.e. its electronic
quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
which is an
eigenstate
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
of the
electronic molecular Hamiltonian. It is the equivalent of the
term symbol In quantum mechanics, the term symbol is an abbreviated description of the (total) angular momentum quantum numbers in a multi-electron atom (however, even a single electron can be described by a term symbol). Each energy level of an atom with a giv ...
for the atomic case. However, the following presentation is restricted to the case of homonuclear
diatomic
Diatomic molecules () are molecules composed of only two atoms, of the same or different chemical elements. If a diatomic molecule consists of two atoms of the same element, such as hydrogen () or oxygen (), then it is said to be homonuclear. O ...
molecules, or other
symmetric molecules with an inversion centre. For heteronuclear diatomic molecules, the ''u/g'' symbol does not correspond to any exact symmetry of the
electronic molecular Hamiltonian. In the case of less symmetric molecules the molecular term symbol contains the symbol of the
group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
to which the molecular electronic state belongs.
It has the general form:
where
*
is the total
spin quantum number
In atomic physics, the spin quantum number is a quantum number (designated ) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. The phrase was originally used to describe t ...
*
is the projection of the orbital angular momentum along the internuclear axis
*
is the projection of the total angular momentum along the internuclear axis
*
indicates the symmetry or ''parity'' with respect to
inversion
Inversion or inversions may refer to:
Arts
* , a French gay magazine (1924/1925)
* ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas
* Inversion (music), a term with various meanings in music theory and musical set theory
* ...
(
) through a centre of symmetry
*
is the reflection symmetry along an arbitrary plane containing the internuclear axis
Λ quantum number
For atoms, we use ''S'', ''L'', ''J'' and ''M
J'' to characterize a given
state
State may refer to:
Arts, entertainment, and media Literature
* ''State Magazine'', a monthly magazine published by the U.S. Department of State
* ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States
* ''Our S ...
. In linear molecules, however, the lack of spherical symmetry destroys the relationship
, so ''L'' ceases to be a
good quantum number In quantum mechanics, given a particular Hamiltonian H and an operator O with corresponding eigenvalues and eigenvectors given by O, q_j\rangle=q_j, q_j\rangle, the q_j are said to be good quantum numbers if every eigenvector , q_j\rangle remai ...
. A new set of
operators have to be used instead:
, where the ''z''-axis is defined along the internuclear axis of the molecule. Since these
operators commute with each other and with the
Hamiltonian on the limit of negligible spin-orbit coupling, their
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s may be used to describe a molecule state through the quantum numbers ''S'', ''M
S'', ''M
L'' and ''M
J''.
The cylindrical symmetry of a linear molecule ensures that positive and negative values of a given
for an
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have n ...
in a
molecular orbital
In chemistry, a molecular orbital is a mathematical function describing the location and 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 find ...
will be
degenerate in the absence of spin-orbit coupling. Different molecular orbitals are classified with a new quantum number, λ, defined as
:
Following the spectroscopic notation pattern, molecular orbitals are designated by a lower case Greek letter: for λ = 0, 1, 2, 3,... orbitals are called σ, π, δ, φ... respectively, analogous to the Latin letters s, p, d, f used for atomic orbitals.
Now, the total ''z''-projection of ''L'' can be defined as
:
As states with positive and negative values of ''M
L'' are degenerate, we define
:Λ = , ''M
L'', ,
and a capital Greek letter is used to refer to each value: Λ = 0, 1, 2, 3... are coded as Σ, Π, Δ, Φ... respectively (analogous to S, P, D, F for atomic states).
The molecular term symbol is then defined as
:
2''S''+1Λ
and the number of electron degenerate states (under the absence of spin-orbit coupling) corresponding to this term symbol is given by:
* (2''S''+1)×2 if Λ is not 0
* (2''S''+1) if Λ is 0.
Ω and spin–orbit coupling
Spin–orbit coupling lifts the degeneracy of the electronic states. This is because the ''z''-component of spin interacts with the ''z''-component of the orbital angular momentum, generating a total electronic angular momentum along the molecule axis J
z. This is characterized by the ''M
J'' quantum number, where
:''M
J'' = ''M
S'' + ''M
L''.
Again, positive and negative values of ''M
J'' are degenerate, so the pairs (''M
L'', ''M
S'') and (−''M
L'', −''M
S'') are degenerate: , and represent two different degenerate states. These pairs are grouped together with the quantum number Ω, which is defined as the sum of the pair of values (''M
L'', ''M
S'') for which ''M
L'' is positive. Sometimes the equation
:Ω = Λ + ''M
S''
is used (often Σ is used instead of ''M
S''). Note that although this gives correct values for Ω it could be misleading, as obtained values do not correspond to states indicated by a given pair of values (''M
L'', ''M
S''). For example, a state with (−1, −1/2) would give an Ω value of Ω = , −1, + (−1/2) = 1/2, which is wrong. Choosing the pair of values with ''M
L'' positive will give a Ω = 3/2 for that state.
With this, a level is given by
:
Note that Ω can have negative values and subscripts ''r'' and ''i'' represent regular (normal) and inverted multiplets, respectively. For a
4Π term there are four degenerate (''M
L'', ''M
S'') pairs: , , , . These correspond to Ω values of 5/2, 3/2, 1/2 and −1/2, respectively.
Approximating the spin–orbit Hamiltonian to first order
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
, the energy level is given by
:''E'' = ''A'' ''M
L'' ''M
S''
where ''A'' is the spin–orbit constant. For
4Π the Ω values 5/2, 3/2, 1/2 and −1/2 correspond to energies of 3''A''/2, ''A''/2, −''A''/2 and −3''A''/2. Despite of having the same magnitude, levels of Ω = ±1/2 have different energies associated, so they are not degenerate. With that convention, states with different energies are given different Ω values. For states with positive values of ''A'' (which are said to be ''regular''), increasing values of Ω correspond to increasing values of energies; on the other hand, with ''A'' negative (said to be ''inverted'') the energy order is reversed. Including higher-order effects can lead to a spin-orbital levels or energy that do not even follow the increasing value of Ω.
When Λ = 0 there is no spin–orbit splitting to first order in perturbation theory, as the associated energy is zero. So for a given ''S'', all of its ''M
S'' values are degenerate. This degeneracy is lifted when spin–orbit interaction is treated to higher order in perturbation theory, but still states with same , ''M
S'', are degenerate in a non-rotating molecule. We can speak of a
5Σ
2 substate, a
5Σ
1 substate or a
5Σ
0
substate. Except for the case Ω = 0, these substates have a degeneracy of 2.
Reflection through a plane containing the internuclear axis
There are an infinite number of planes containing the internuclear axis and hence there are an infinite number of possible reflections. For any of these planes, molecular terms with Λ > 0 always have a state which is symmetric with respect to this reflection and one state that is antisymmetric. Rather than labelling those situations as, e.g.,
2Π
±, the ± is omitted.
For the Σ states, however, this two-fold degeneracy disappears, and all Σ states are either symmetric under any plane containing the internuclear axis, or antisymmetric. These two situations are labeled as Σ
+ or Σ
−.
Reflection through an inversion center: u and g symmetry
Taking the molecular center of mass as origin of coordinates, consider the change of all electrons' position from (''x
i'', ''y
i'', ''z
i'') to (−''x
i'', −''y
i'', −''z
i''). If the resulting wave function is unchanged, it is said to be ''gerade'' (German for even) or have even
parity
Parity may refer to:
* Parity (computing)
** Parity bit in computing, sets the parity of data for the purpose of error detection
** Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the ...
; if the wave function changes sign then it is said to be ''ungerade'' (odd) or have odd parity. For a molecule with a center of inversion, all orbitals will be symmetric or antisymmetric. The resulting wavefunction for the whole multielectron system will be ''gerade'' if an even number of electrons are in ''ungerade'' orbitals, and ''ungerade'' if there are an odd number of electrons in ''ungerade'' orbitals, regardless of the number of electrons in ''gerade'' orbitals.
An alternative method for determining the symmetry of an
MO is to rotate the orbital about the axis joining the two nuclei and then rotate the orbital about a line perpendicular to the axis. If the sign of the lobes remains the same, the orbital is ''gerade'', and if the sign changes, the orbital is ''ungerade''.
Wigner-Witmer correlation rules
In 1928
Eugene Wigner
Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his co ...
and E.E. Witmer proposed rules to determine the possible term symbols for diatomic molecular states formed by the combination of a pair of atomic states with given atomic
term symbol In quantum mechanics, the term symbol is an abbreviated description of the (total) angular momentum quantum numbers in a multi-electron atom (however, even a single electron can be described by a term symbol). Each energy level of an atom with a giv ...
s.
[ Reprint 2nd ed. with corrections (1989): Krieger Publishing Company. ] For example, two like atoms in identical
3S states can form a diatomic molecule in
1Σ
g+,
3Σ
u+, or
5Σ
g+ states. For one like atom in a
1S
g state and one in a
1P
u state, the possible diatomic states are
1Σ
g+,
1Σ
u+,
1Π
g and
1Π
u.
[ The parity of an atomic term is ''g'' if the sum of the individual angular momentum is even, and ''u'' if the sum is odd.
]
Alternative empirical notation
Electronic states are also often identified by an empirical single-letter label. The ground state is labelled X, excited states of the same multiplicity (i.e., having the same spin quantum number) are labelled in ascending order of energy with capital letters A, B, C...; excited states having different multiplicity than the ground state are labelled with lower-case letters a, b, c...
In polyatomic molecules (but not in diatomic) it is customary to add a tilde (e.g. , ) to these empirical labels to prevent possible confusion with symmetry labels based on group representations.
See also
*Molecular orbital theory
In chemistry, molecular orbital theory (MO theory or MOT) is a method for describing the electronic structure of molecules using quantum mechanics. It was proposed early in the 20th century.
In molecular orbital theory, electrons in a molec ...
References
{{reflist
Molecular physics
Quantum chemistry
Atomic physics
Spectroscopy