Symmetry in Molecules: Classification, Symmetry Operations, and Vibrations, Exercises of Construction

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Molecular Symmetry

I. WHAT IS SYMMETRY AND WHY IT IS IMPORTANT?

Some object are ”more symmetrical” than others. A sphere is more symmetrical than a cube because it looks the same after rotation through any angle about the diameter. A cube looks the same only if it is rotated through certain angels about specific axes, such as 90 o, 180o, or 270o^ about an axis passing through the centers of any of its opposite faces, or by 120o^ or 240o^ about an axis passing through any of the opposite corners. Here are also examples of different molecules which remain the same after certain symme- try operations: N H 3 , H 2 O, C 6 H 6 , CBrClF. In general, an action which leaves the object looking the same after a transformation is called a symmetry operation. Typical symme- try operations include rotations, reflections, and inversions. There is a corresponding symmetry element for each symmetry operation, which is the point, line, or plane with respect to which the symmetry operation is performed. For instance, a rotation is carried out around an axis, a reflection is carried out in a plane, while an inversion is carried our in a point. We shall see that we can classify molecules that possess the same set of symmetry ele- ments, and grouping together molecules that possess the same set of symmetry elements. This classification is very important, because it allows to make some general conclusions about molecular properties without calculation. Particularly, we will be able to decide if a molecule has a dipole moment, or not and to know in advance the degeneracy of molecular states. We also will be able to identify overlap, or dipole moment integrals which necessary vanish and obtain selection rules for transitions in polyatomic molecules.

II. SYMMETRY OPERATIONS

The classification of objects according to symmetry elements corresponding to operations that leave at least one common point unchanged gives rise to the point groups. These are five kinds of symmetry operations and five kinds of symmetry elements of this kind. These symmetry operations are as follows.

• The identity, E, consists of doing nothing: the corresponding symmetry element is an entire object. In general, any object undergo this symmetry operation. The example of the molecule which has only the identity symmetry operation is C 3 H 6 O 3 , DN A,

III. THE SYMMETRY CLASSIFICATION OF MOLECULES

In order to classify molecules according to symmetry one can list their symmetry elements and collect together the molecules with the same list of elements. More precisely, we can collect together the molecules which belong to the same group.

A. Definition of the Group

According to the group theory, the symmetry operations are the members of a group if they satisfy the following group axioms:

1. The successive application of two operations is equivalent to the application of a mem- ber of the group. In other words, if the operations A and B belong to the same group then A · B = C, where C is also the operation from the same group. Note, that in general A · B 6 = B · A.
2. One of the operations in the group is the identity operation E. This means that A · E = E · A = A.
3. The reciprocal of each operation is a member of the group: if A belongs to a group, then A−^1 = B, where B is also the member of the group. Note, that A · A−^1 = A−^1 · A = E.
4. Multiplication of the operations is associative: A · B · C = (A · B) · C = A · (B · C).

B. Point Groups

Particularly we will consider the following point groups which molecules can belong to.

1. The groups C 1 , Ci, and Cs. A molecule belongs to the group C 1 if it has no elements other than identity E. Example: DNA. A molecule belongs to the group Ci, if it consist of two operations: the identity E and the inversion i. Example: meso-tartaric acid. A molecule belongs to the group Cs, if it consists of two elements: identity E and a mirror plane σ. Example: OHD.
2. The group Cn. A molecule belongs to the group Cn if it has a n-fold axis. Example: H 2 O 2 molecule belongs to the C 2 group as it has the elements E and C 2.

3. The group Cnv. A molecule belongs to the group Cnv if in addition to the identity E and a Cn axis, it has n vertical mirror planes σv. Examples: H 2 O molecule belongs to the C 2 v group as it has the symmetry elements E, C 2 , and two vertical mirror planes which are called σv and σ′ v. The N H 3 molecule belongs to the C 3 v group as it has the symmetry elements E, C 3 , and three σv planes. All heteroatomic diatomic molecules and OCS belong to the group C∞v because all rotations around the internuclear axis and all reflections across the axis are symmetry operations.
4. The group Cnh. A molecule belongs to the group Cnh if in addition to the identity E and a Cn axis, it has a horizontal mirror plane σh. Example: butadiene C 4 H 6 , which belongs to the C 2 h group, while B(OH) 3 molecule belongs to the C 3 h group. Note, that presence of C 2 and σh operations imply the presence of a center of inversion. Thus, the group C 2 h consists of a C 2 axis, a horizontal mirror plane σh, and the inversion i.
5. The group Dn. A molecule belongs to the group Dn if it has a n-fold principal axis Cn and n two-fold axes perpendicular to Cn. D 1 is of cause equivalent with C 2 and the molecules of this symmetry group are usually classified as C 2.
6. The group Dnd. A molecule belongs to the group Dnh if in addition to the Dn opera- tions it possess n dihedral mirror planes σd. Example: The twisted, 90o^ allene belongs to D 2 d group while the staggered confirmation of ethane belongs to D 3 d group.
7. The group Dnh. A molecule belongs to the group Dnh if in addition to the Dn opera- tions it possess a horizontal mirror plane σh. As a consequence, in the presence of these symmetry elements the molecule has also necessarily n vertical planes of symmetry σv at angles 360o/2n to one another. Examples: BF 3 has the elements E, C 3 , 3C 2 , and σh and thus belongs to the D 3 h group. C 6 H 6 has the elements E, C 6 , 3C 2 , 3C′ 2 and σh and thus belongs to the D 6 h group. All homonuclear diatomic molecules, such as O 2 , N 2 , and others belong to the D∞h group. Another examples are ethene C 2 H 4 (D 2 h), CO 2 (D∞h), C 2 H 2 (D∞h).
8. The group Sn. A molecule belongs to the group Sn if it possess one Sn axis. Example: tetraphenylmethane which belongs to the group S 4. Note, that the group S 2 is the same as Ci, so such molecules have been classified before as Ci.

TABLE I: Multiplication Table for the C 3 v Group

E C 3 + C 3 − σv σ v′ σ′′ v E E C 3 + C 3 − σv σ v′ σ′′ v C 3 + C 3 + C 3 − E σ v′ σ v′′ σv C 3 − C 3 − E C 3 + σ′′ v σv σ′ v σv σv σ v′′ σ′ v E C 3 − C 3 + σ′ v σ v′ σv σ′′ v C 3 + E C 3 − σ v′′ σ′′ v σ v′ σv C 3 − C 3 + E

Now we add to these four symmetry operations the identity operator E and show that all six symmetry operations joint a group. The ”multiplication table” presented below.

TABLE II: Multiplication Table for the C 2 v Group

E C 2 σv σ v′ E E C 2 σv σ v′ C 2 C 2 E σ′ v σv σv σv σ′ v E C 2 σ v′ σ′ v σv C 2 E

According to the Table II, the ”product” of each two symmetry transformations from six E, C 2 , σv, and σ′ v is equivalent to one of these transformations. It is clearly seen that the third and the fourth conditions of the group are also valid. Thus, these four operators build a group. This group is known as C 2 v group. The group order of C 2 v is 4. Each point group is characterized by each own multiplication table.

D. Some Consequences of Molecular Symmetry

As soon point the group of a molecule is identified, some statements about its properties can be done.

1. Polarity

As we have already discussed a polar molecule is one having a permanent electric dipole moment. For instance these are N aCl, O 3 , N H 3 , and many others. It is known that the rotational absorption transitions can occur only in polar molecules. The group theory give important instructions, how the molecular symmetry is related to the molecular polarity. For instance, if a molecule belongs to the group Cn, where n > 1, then it cannot have a component of the dipole moment perpendicular to the symmetry axis, because a dipole moment which exist in one direction perpendicular to the axis is cancelled by an opposing dipole. A dipole moment in these molecules can be only parallel to the molecular axis. The same is valid for any of the Cnv group molecule. The molecules which belongs to all other groups, but Cs, cannot have a permanent dipole moment, because they always have symmetry operations transforming one end of the molecule into another. Thus, only the molecules which belong to the Cn, Cnv, or Cs group can have a permanent dipole moment.

2. Chirality

A chiral molecule is that cannot be transformed to itself with any mirror transforma- tion. An achiral molecule can be transformed to itself with a mirror transformation. Chiral molecules are important because they are optically active in the sense that they can rotate the plane of polarized light passing through the molecular sample. A molecule may be chiral only if it does not have an axis of improper rotation Sn. Note, that the molecule with a center of inversion i belongs to S 2 group and thus cannot be chiral. Similarly, because S 1 = σ, any molecule with a mirror plane is achiral.

where kij , ki,j,k and ki,j,k,l are force constants. The lowest order terms in the expansion are quadratic and for small displacement only these terms can be preserved in eq. (3), while all other terms can be neglected. Corresponding expression for the potential V (^) N^0 is called the harmonic-oscillator approximation. In the harmonic-oscillator approximation the vibrational energy can be written as

E^0 vib =^12 ∑^3 N i=

mi u˙^2 i +^12 ∑^3 N i,j=

ki,j uiuj , (4)

where ki,j are harmonic force constants.

B. Normal Vibrational Modes

A standard result from classical mechanics is that the vibrational energy of a N -body harmonic oscillator (4) can be written in terms of 3N − 6 mass-weighted linear combinations of the uj which are called vibrational normal coordinates Qr:

E^0 vib =^12

3 N∑ − 6 r=

[ Q˙^2 r + λrQ^2 r ], (5)

where

m^1 i /^2 ui = ∑^3 N r=

lui,rQr. (6)

The quantum mechanical Hamiltonian of a vibrating polyatomic molecule can be obtained from eq.(5) by replacing the classical variables Qr and Q˙r by their quantum mechanical analogues. Great advantage of the vibrational energy expression in eq.(5) is that there is no cross terms in the potential energy. Therefore, the solution (wavefunction) of the corresponding Schr¨odinger equation is greatly simplified as can be presented as a product of the normal mode wavefunctions which are known solution of the harmonic oscillator problem:

Φvib = Φv 1 (Q 1 )Φv 2 (Q 2 )... Φv 3 N − 6 (Q 3 N − 6 ). (7)

The corresponding vibrational energy Evib is a sum of the each normal mode energy

Evib = Ev 1 + Ev 2 + · · · + Ev 3 N − 6 , (8)

where Evk = ωek(vk + 1/2).

Each of the 3N − 6 (3N − 5) vibrational normal coordinate Qr describes a collective normal mode of vibration. In general, any vibrational of the molecular system may be represented as a superposition of normal vibrations with suitable amplitudes. Within each of the normal mode k all nuclei move with the same frequency νk according to simple harmonic motion. Two, or more normal modes are degenerate if they all have the same frequency. As an example, consider vibration of a mass suspended by an elastic bar of rectangular cross section. If mass is displaced slightly from its equilibrium position in the x direction and then left, it will carry our simple harmonic in this direction with a frequency

νx = (^21) π

√ kx m ,^ (9)

where kx is a force constant in the x direction. If mass is displaced slightly from its equilibrium position in the y direction and then left, it will carry our simple harmonic in this direction with a frequency

νy = (^21) π

√ ky m ,^ (10)

where ky is a force constant in the y direction. If mass is displaced in a direction different from x and y, it will not carry out a simple harmonic oscillation, but more complicated type of motion, so named Lissajous motion. This is because the restoring force F whose components are Fx = −kxx and Fy = −kyy is not directed toward the origin since kx 6 = ky. However, this motion can be always presented as linear superposition of two simple harmonic motions of different frequency:

x = x 0 cos 2πνxt, y = y 0 cos 2πνyt, (11)

where x 0 , y 0 are coordinates of initial position of the mass (point A).

C. Symmetry of Normal Vibrations

We will now consider the effect of symmetry operations on the normal vibrations. Math- ematically there are two equivalent ways of carrying out a symmetry operation. We may either keep the coordinate frame fixed and transform the molecule (for instance, reflecting, or rotating the positions of nuclei), or we can keep the positions of nuclei and refer them to

Therefore, degenerate vibrations, in general, transformed under a symmetry oper- ation as a linear combination of each other. This result is valid for any number of the degenerate vibrations and any type of symmetry operations involved.

V. SYMMETRY OF VIBRATION AND ELECTRONIC WAVEFUNCTIONS

Since a vibrational eigenfunction is a function of the normal coordinates, its behavior with respect to symmetry operations depends on the symmetry behavior of the normal coordinates.

A. Molecules with Non-Degenerate Vibrations

The total vibrational wavefunction of a molecule can be always written as a product of the normal mode wavefunctions which are known solution of the harmonic oscillator problem, see eq. (7). The i-th harmonic oscillator wavefunction can be presented as

Φvi(Qi) = Nvie−^ α^2 i^ Q^2 i^ Hvi(√αiQi), (14)

where Hvi(√αiQi) is the Hermit polynomial of the vi-th degree and αi = ωi/h. If a non-degenerate vibration Qi is symmetric with respect to a symmetry operation A (that is Aˆ · Qi = Qi), the wavefunction Φvi(Qi) in eq. (14) is also symmetric for all values of the quantum number vi (that is, Aˆ · Φvi(Qi) = Φvi(Qi)). If a non-degenerate vibration Qi is antisymmetric with respect to this symmetry (that is Aˆ · Qi = −Qi), the wavefunction Φvi(Qi) behaves as Aˆ · Φvi(Qi) = Φvi(−Qi) = (−1)vi^ Φvi(Qi). Therefore, for antisymmetric vibration mode Qi the wavefunction Φvi(Qi) can be either symmetric, or antisymmetric depending of the value of the quantum number vi. In case if all normal vibrations are non-degenerate, the total vibrational eigenfunction Φ in eq. (7) will be symmetric with respect to a given symmetry operation when the number of component antisymmetric wave functions Φvi(Qi) is even. The total eigenfunction Φ will be antisymmetric when the number of component antisymmetric wave functions Φvi(Qi) is odd. Important result: Total vibrational eigenfunctions, corresponding to a non-degenerate vibration must be either symmetric, or antisymmetric with respect to the symmetry oper- ations of the group. The symmetric, or antisymmetric behavior of the total wavefunction

can be relatively easy obtained considering its explicit form which is a product of the eigen- functions of harmonic oscillators corresponding to different normal vibration modes.

B. Molecules with Degenerate Vibrations

If a molecule has a doubly degenerate vibrations they have the same frequencies ω 1 = ω 2 = ωi and the formula for the term values can be written as

G(v 1 , v 2 ) = ω 1 (v 1 +^12 ) + ω 2 (v 2 +^12 ) = ωi(vi + 1), (15)

where vi = v 1 + v 2 can be treated as a new vibrational quantum number. The corresponding total vibrational eigenfunction can be written as (see eq. (14))

Φi = Nvie−^ α^2 i^ (Q^21 +Q^22 )Hv 1 (√αiQ 1 )Hv 2 (√αiQ 2 ), (16)

where αi = ωi/h. If v 1 = v 2 = vi = 0, than H 0 (√αiQ) = constant and there is only one function Φi in eq. (16). Thus, the zero-point vibrational vi = 0 does not introduce a degeneracy. In this case, the same relations apply as in the previous section. If the degenerate vibration if excited by only one quantum, we have either v 1 = 1, v 2 = 0, or v 1 = 0, v 2 = 1 for which the wavefunctions Φi in eq. (16) are not the same. That is, there are two eigenfunctions for the state vi = v 1 + v 2 = 1 with the energy Gvi = 2ωi, see eq. (15). Therefore, the state vi is doubly degenerate. Note, that any linear combination of the two wavefunctions in eq. (16) is also an eigenfunction of the same energy level. If two quanta are excited (vi = 2), we may have either v 1 = 2, v 2 = 0, or v 1 = 1, v 2 = 1, or v 1 = 0, v 2 = 2, that is there is a triple degeneracy. In general, the degree of degeneracy if vi quanta of the double degenerate vibration are excited, is equal to vi + 1. Important result: Total vibrational eigenfunctions, corresponding to a degenerate vi- bration are neither symmetric, nor antisymmetric, but can in general be transformed under a symmetry operation as a linear combination of each other. However, there is only one zero-point v = v 1 = v 2 = 0 vibration wavefunction Φ 0 which must be either symmetric, or antisymmetric under a symmetry operation.

and antisymmetric behavior of the wavefunctions with respect to the corresponding sym- metry operation. Note, that in every normal vibration and eigenfunction there are species (irreducible representations) which are symmetric under all symmetry operations per- mitted within a group. These species are called totally symmetric and usually indicated by A, or A 1 , or A′. Particularly for the Cs group the totally symmetric species is indicated by A′^ and presented in the second line in Table III. It is seen, that the group Cs has two species, A′^ and A′′. The last column in the table indicate the group order, h = 2 and the simple functions of the coordinates x, y, z which belongs to a certain irreducible representation. These functions are very important, because they represent the symmetry of px, py, and px atomic orbitals which as we know are used for building the molecular orbitals. Therefore, these coordinates provide a simple way of understanding which species a normal mode, or wavefunction belongs to. For instance, consider the plane, but non-linear molecule of hydrazoic acid, N 3 H which belongs to the Cs group. It has, according to Table III normal vibrations which are sym- metric, or antisymmetric with respect to the molecular plane. During the former, all atoms remain in the plane, during the latter, they move in lines perpendicular to the plane. As another example consider the character table of the C 2 v symmetry group which is shown in Table IV

TABLE IV: Character Table for the C 2 v Group

C 2 v E C 2 σv σ v′ h = 4

A 1 +1 +1 +1 +1 z x^2 , y^2 , z^2

A 2 +1 +1 − 1 − 1 xy

B 1 +1 − 1 +1 − 1 x xz

B 2 +1 − 1 − 1 +1 y yz

As seen from Table IV, the C 2 v group has four species (irreducible representations). The totally symmetric species is called in this case A 1. Each of the other A 2 , B 1 and B 2 species are used to denote one-dimensional (non-degenerate) representations. A is used if the character under the principal rotation is +1, while B is used if the character is −1. If other higher dimensional representations are permitted, letter E denotes a two-dimensional irreducible representation and T denotes a three-dimensional representation. The symmetry species A 1 , A 2 , B 1 , and B 2 summarize the symmetry properties of the vibrational, or electronic molecular wavefunctions of a for polyatomic molecule. They are analogue to the symmetry labels Σ, Π, ∆ which are used for diatomic molecules. As an example we consider normal vibrations of the formaldehyde molecule H 2 CO which belongs to the group C 2 v. It is seen that the three normal vibrations ν 1 , ν 2 , and ν 3 are totally symmetric and thus belong to species A 1. The vibrations ν 4 and ν 5 belong to species B 1 (if we call the plane of the molecule the xz plane), and ν 6 belongs to species B 2. There is no normal vibration of species A 2 in this case. However, in more complicated molecules belonging to the same group there also can be normal vibrations belonging to species A 2. Let us now consider the symmetry of electronic orbitals. As we know, lowercase Greek letters σ, π, etc are used for denoting the symmetries of orbitals in diatomic molecules. Similarly, the lowercase Latin letters a 1 , a 2 , b 1 , and b 2 are used for denote the symmetry of orbitals in polyatomic molecules which belong to the A 1 , A 2 , B 1 , and B 2 irreducible representations, respectively. Alternatively, one says that the wavefunctions a 1 , a 2 , b 1 , and b 2 span the irreducible representations A 1 , A 2 , B 1 , and B 2. The functions in the 5-th and 6-th columns in Table IV represent the symmetry of different p and d atomic orbitals which span a certain irreducible representation. For instance, the symmetry of electronic wavefunctions in the H 2 O molecule are as follows. The atomic orbitals of the O atom are: O 2 px, O 2 py, and O 2 pz. Assuming that the molecular plane is Y Z we can see that the orbital O 2 px change sign under a 180^0 rotation, C 2 and under the reflection σ′ v, but remains the same under the reflection σv. Therefore, this orbital belongs to the B 1 irreducible representation. As we shall see, any molecular orbital built from this atomic orbital will be a b 1 orbital. It can also be seen in the similar way that O 2 py orbital changes sign under C 2 , but remain the same after σ′ v, thus it belongs to B 2 and can contribute to b 2 molecular orbital. Similarly, it can be shown that O 2 pz belongs to the A 1 irreducible representation.

Secondly, the symmetry species E in Table V is a double degenerate one. These species cannot be characterized simply by +1, or −1, as for non-degenerate case. As we know, the wavefunctions which belong to a degenerate vibration are neither symmetric, nor antisymmetric with respect to the symmetry operation of the group, but in general can be transformed as a linear combination of each other as

Φ′ v 1 = d 11 Φv 1 + d 12 Φv 2 + d 13 Φv 3 + · · · , Φ′ v 2 = d 21 Φv 1 + d 22 Φv 2 + d 23 Φv 3 + · · · , (18) Φ′ v 3 = d 31 Φv 1 + d 32 Φv 2 + d 33 Φv 3 + · · · , · · · = · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ,

where the primed wavefunctions in the lhs are ones after the symmetry operation while the non-primed wavefunctions in the rhs are the initial ones. In case of a double-degenerate state the number of the wavefunctions and the number of equations in eq. (18) is of cause equal to two. It can be shown, that for characterization of the behavior of the degenerate eigenfunctions under symmetry operations it is sufficient to label every symmetry operation with the value

χ = d 11 + d 22 + d 33 + · · · (19)

which is the sum of the diagonal expansion coefficients in the set of equations in eq. (18). The values χ in eq. (19) (as well as λ = ±1 symmetric indices for non-degenerate species) are called characters of the irreducible representation. These characters are given in the third line in Table V. As you can see the characters of the degenerate eigenfunctions are not limited by the values ±1, but can take other integer numbers including zero. Note, that the character of identity operator E is always equal to the degeneracy of the state. Therefore, for a C 3 v molecule any orbitals with a symmetry label a 1 and a 2 is non-degenerate, while a doubly degenerate pair of orbitals belong to e representation. Because there is not characters greater than 2 in Table V we can assume that no triply degenerate orbitals can occur in any C 3 v molecule. So far, we dealt with the symmetry classification of individual atomic orbitals. It is important to note that the same technique may be applied to the linear combinations of atomic orbitals which are used for building the molecular orbitals. This allows to

classify the molecular energy states and molecular orbitals with respect to the symmetry transformations of the molecule. As an example, we consider the linear combinations of electronic wavefunctions which belong to different representations in Table V. Particularly, for N H 3 case the combination

s 1 = sa + sb + sc, (20)

where sa, sb, and sc are s-orbitals of three hydrogen atoms, belongs to the species a 1. The combinations

s 2 = −sa +^12 (sb + sc) (21) s 3 = sb − sc

belongs to the doubly degenerate species e. For proving this statement let us consider the transformation of the combinations in eq. (21) under C 3 + and σv symmetry operations of the group Rotation C+ 3 :

s′ 2 = −sb +^12 (sc + sa) (22) s′ 3 = sc − sa

This can be easily proved from eqs. (21) and (22) that

s′ 2 = −^12 s 2 − 34 s 3 (23) s′ 3 = s 2 − 12 s 3

Reflection σv: (over the plane containing N − Ha bond)

s′ 2 = −sa +^12 (sb + sc) = s 2 (24) s′ 3 = sc − sb = −s 3

Similar expressions can be obtained for the symmetry operations C− 3 , σ′ v, and σ′′ v. It is seen that the wavefunctions s 2 and s 3 are transformed as a linear combination of each other and thus span the species E.