10 Electronic Spectra of Diatomic Molecules-I

Contents:

1. The Born-Oppenheimer Approximation

2. Classification of Electronic States

3. Vibrational Structure of Electronic Transitions

The students will be able to learn about the classification of electronic states, vibrational energy levels associated with two electronic states. The vibrational transitions accompanying an electronic transition called vibronic transitions that are divided into progressions and sequences.

1. The Born-Oppenheimer Approximation

Two or more atoms bind together to form a molecule in such a way that the total energy is lower than the sum of the energy of the constituents. The electrons of the inner shell of atoms forming molecules remain localized about each nucleus, though the valence electrons are distributed throughout the molecule and charge distribution of these electrons provide the binding force. These bonds are ionic or covalent nature and the force experienced by the electrons and nuclei is of comparable intensity. As the nuclei are about two thousand times heavier than the electrons, the motion of the nuclei is much slower than that of electrons. Hence, it is assumed that nuclei occupy fixed position in the atom.

The Hamiltonian for the molecule can be written as

H = T_n + T_e + V_ee + V_eN + V_nn…………(a)

Where Tn= kinetic energy operator of all the nuclei,

Te= kinetic energy operator of all the electrons

Vee= potential energy operator for coulumbic interaction between electron– electron

VeN = potential energy operator for coulombic attraction of all electrons and all nuclei

Vnn =coulombic repulsion between nucleus- nucleus

If ψ is an eigenfunction of H then

Hψ(r,R) = Eψ (r,R) ……(b)

Here r is electronic coordinate and R is internuclear distance The coordinates of the electrons and nuclei cannot be separated out because of term involved in Coulombic interaction between electrons and nuclei. Born and Oppenheimer were able to show that an approximate solution of Eq. (a) can be obtained by first solving the equation for the electrons alone with nuclear fixed positions. Therefore, the operator T_n is neglected assuming nuclear mass is infinite and hence the potential V_nn is constant. The Hamiltonian

H_E= T_e + V_ee + V_en

and Schrödinger wave equation is

H_Eψ_E(r,R) = E_E(R) ψ_E(r,R)

However, VeN depends on the position of the nuclei. Ee(R) can be obtained for different values of internuclear distances (from Eq. (b)). The potential energy can be obtained by adding Cnnto Ee(R), So

Here Z₁ and Z₂ are the atomic numbers of two nuclei 1 and 2, respectively. The value E_e decreases as the nuclei approaches each other because the attraction between electrons and nuclei increases and this is opposed by the repulsion between the two nuclei. There is an equilibrium bond length, that is, there will be a minimum distance at which these two balance. The energy rises sharply with the further decrease in the separation of nuclei. The electronic state is then a bound state. The curve representing the variation of E_e + V_nn is referred to as potential curve. The potential energy curve for each electronic state of the molecule has a different shape. The minima of potential curve falls at different equilibrium internuclear distances, in case of stable states. The electronic state is unstable if the potential curve has no minimum. In obtaining potential energy curve, nuclear kinetic energy term is ignored. Thus, V(R) does not depend on the mass of the nuclei. If V(R) is found for a given molecule, it applies to all length within Born – variation as V(R) is same for H2 , HD and D2 also have same bond length within Born- Oppenheimer approximation.

The variation of E_e as a function of R provides a part of the potential field in which nuclei move while V_nn provides the other part of this field. The Hamiltonian for the nuclear motion, is

H_n = T_n + V_nn + E_e (e)

and Schrödinger wave equation is

H_nΨ _n (R) = (T_n + V_nn + E_e) Ψ_n= E_nΨ_n(R) (f)

Here Ψ_n is a function of R only and represents the stationary state of the nuclei. The total wavefunction is

Ψ = Ψ_e (r,R) Ψ_n(R) (g)

The total energy E is given by

E = E_e+ E_n

The wavefunction Ψ_n, in a first approximation, can be expressed as

Ψn= ΨvΨr

whereΨv is the vibrational eigenfunction of a linear oscillator.

Thus to a first approximation

Ψ= Ψ_e Ψ_vΨ_r

The total energy E of the molecule apart from the translational energy is regarded as the sum of rotational energy E_r , vibrational energy E_v , and the electronic energy E_e, that is,

E = E_e+Ev+ E_r

Where E_e is defined as the energy of stable electronic state corresponding to the minimum of V. The minimum of lowest electronic state is chosen as zero point of the energy scale. This choice differs from atom to atom. The total energy in terms of wavenumbers is

As the electrons move faster than nuclei, the nuclei feel the potential energy of the averaged electronic distribution. This forms the basis of Born- Oppenheimer approximation. However, this approximation breaks down if electronic energy spacings are not large compared to vibrational spacings. This breakdown result in a situation where time scales of nuclear and electronic motions are not separable and it is not possible to separate nuclear and electronic energies.

2. Classification of Electronic States

In diatomic molecules, the electrons move in strong electrostatic field of nuclei that is considered to be cylindrically symmetric about the bond axis. The orbital angular momentum L precesses very rapidly about the direction of electrostatic field and only axial component of L is a constant of motion. The axial component field is characterized by the quantum number ML where ML = L, L-1, ……….., -L. As the internuclear field is of electrical nature, therefore energy is not changed by the exchange of ML↔ –ML. The absolute value of ML is designated by the symbol ᴧ and ᴧ =| ML|= 0, 1, 2, 3, ………..,L. The figure depicts the coupling of L about the electric field along the internuclear axis producing the axial component ᴧ

The symbols for the states are:

The quantized components MS of S, about the magnetic field direction, have value ħ MS. The quantum number MS is designated by the symbol ∑ that takes the values S, S – 1, S – 2, ……., – S. So, ∑ can take 2S + 1 values. There is no resultant magnetic field for the electronic state ∑ (⋀ = 0), and hence MS is not defined. These states have only one component, whatever be the multiplicity. Further, spin- orbit coupling can lift the degeneracy of 2S+1 ⋀ state. The symbol Ω is used to designate the quantum number of z component of total angular momentum(spin plus orbit) in diatomic molecules if coupling between L and S is weak. The quantum number Ω is written as a subscript to the term symbol. Thus the term symbol is 2S+1⋀Ω with Ω = ⋀+S, ⋀+S – 1 ,………., ⋀-S. As an example, a triplet ∆state is split by spin orbit coupling into three doubly degenerate levels.

The Vector addition of ⋀ + ∑ producing the total electronic angular momentum Ω is shown in the figure.

Symmetry properties of electronic functions need to be considered alongwith quantum numbers ∑, ⋀ and Ω. The reflection operator acing twice in succession on electronic wavefunction must give the original wavefunction, so,

The eigenvalue of 2 is 1. Thus the two eigenvalues of the operator are + 1 and -1. Then

Ψ e⁺and Ψ e^–represent the eigenfunction of with eigenvalues + 1 and -1, respectively,

The function Ψ ⁺remains unchanged while Ψ – changes sign under reflection operation. The electronic states which are doubly degenerate (⋀> 0) may be distinguished as + or -, for example, ∆+, ∆- etc. The electronic staste ∑is not degenerate but can still be classified as ∑+or ∑- .Homonuclear diatomic molecules also have inversion symmetry through the midpoint of the bond. With the midpoint taken as origin of the cartesian coordinate system and under inversion operation (xi, yi, zi) → (-xi, -yi, -zi) of electronic function twice in succession gives the same electronic function then

i² Ψ_e= (+1) Ψ_e

The eigenvalues of operator i are therefore +1 and -1. The wavefunction either remains unchanged or changed sign under inversion. The wavefunction is called even or g (gerade) if it remains unchanged upon inversion. On the other hand if wavefunction changes its sign upon inversion it is called odd or u (ungerade). The symbols g and u are written as subscripts to the term value, like, ∑_g, ∑_u.

Hund’s Rule

(i) The interaction between nuclear rotation and electronic angular momentum is assumed to be weak and also weak spin- orbit coupling. L and S precess independently about the bond axis in diatomic molecules. Along the bond direction the projections of L and S, that is, ⋀ and ∑ respectively add together to give Ω. So J = R +Ω

Hund’s Rule (i)

(ii) It is assumed that coupling of S and ⋀ is weak. S couples to the nuclear rotational axis. The magnetic field is also weak and S does not precess about the bond axis. S remains fixed in space. ⋀is added vectorally to I giving a resultant angular momentum N and N = R + ⋀ and then N combined with S to give total angular momentum J, So that

J = N + S

Hund’s Rule (ii)

Conclusevly, in case of diatomic molecules, S, ∑, ⋀ and Ω adequately represent the quantized electronic angular momentum. In an isolated molecule, however, the total angular momentum is due to vectorial coupling of nuclear angular momentum I, electronic spin angular momentum S and electronic orbital angular momentum L.

3. Vibrational Structure of Electronic Transitions

The total energy E and term value T of the molecule are given by

E = E_e+E_v+ E_r and

The wave number of spectral line for a transition from the upper electronic state T¹to lower electronic state T² is,

The spectral location corresponds to J = 0 →J = 0 rotational transitions. In an electronic transition, there are no strict selection rules on v and it may take any positive or negative integer. For v = 0 → v = 0 transition

The figure depicts a set of vibrational energy levels associated with two electronic states. The vibrational transitions accompanying an electronic transition are called vibronic transitions and are further divided into progressions and sequences.

A progression involves a series of vibronic transitions with a common lower or upper level.
As seen from the figure, for the v’’ progression members v’ = 3 level is common; and for v’ = 0 progression members all have v ‘’ = 0 level common with v’ = 0, 1, 2, 3, 4,….. .
A v’’ progression extends towards the lower wavenumber as v’’ increases and terminates in a continuum where the lower electronic state dissociates.
A v’ progression extends towards higher wavenumber as v’ increases and terminates in a continuum where the upper electronic state dissociates.
A group of transitions with the same value of v is referred to as a sequence. long sequences are observed mostly in emission due to population requirements.
The progressions and sequences are not mutually exclusive. Each member of a sequence is also a member of two progressions. The members of theprogressions are widely spaced with approximate separation of ?̅?′′ inemission and ?̅?’ in absorption. Members of the sequence are closelyspaced with approximate separation equal to ?̅?′′-?̅?’ as ?̅?′′ and ?̅?’ have slightly different values for the combining states.
The bands in each sequence are generally found to be grouped together and overlap each other partially in the spectrum.

Summary

As the electrons move faster than nuclei, the nuclei feel the potential energy of the averaged electronic distribution. This forms the basis of Born-Oppenheimer approximation. However, this approximation breaks down if electronic energy spacing are not large compared to vibrational spacing.
The vibrational transitions accompanying an electronic transition are called vibronic transitions and are further divided into progressions and sequences
The bands in each sequence are generally found to be grouped together and overlap each other partially in the spectrum.

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