1 PHY583 – Notes

Molecular Structure

Intro

Elements (except inert gases) generally combine to form chemical compounds whose basic unit is molecules

Physical & chemical properties of molecules derived from the arrangement, interaction & electronic structure of their constituent atoms

Properties of molecules can be determined from their spectra

Just like atoms, molecules emit & absorb photons with accompanying electronic transition among the allowed energy levels

Emission & absorption spectrum is different for each molecule & acts as fingerprint to its electronic structure

Unlike atoms, molecules can rotate & vibrate

The rotational & vibrational energy are quantized & give rise to its own unique spectra

Vibration-rotation spectrum inform us the arrangement & strength of interaction between individual atoms that form the molecule

Bonding Mechanism

Combination of two atoms are due to net attractive force between them

Etot bound molecule < Etot separated atoms

Etot separated atoms - Etot bound molecule = E binding of molecules

Bonding mechanisms in molecules are primarily due to electrostatic forces

At close distance, attractive & repulsive forces come into play

At very large distance attractive force is dominant while at close distance repulsive force dominates

Potential energy of 2 atoms can be +ve or –ve depending on separation distance

Total potential energy U of two atoms:

푈 = − +

푟 = 푖푛푡푒푟푛푢푐푙푒푎푟 푠푒푝푎푟푎푡푖표푛 푑푖푠푡푎푛푐푒 푏푒푡 푡푤표 푎푡표푚푠

A & B constants associated with attractive & repulsive force

m & n small integer

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Fig. 11.1 (p373) Total particle energy as a function of the internuclear separation for a system of two atoms.

Pot energy at large separation is negative corresponding to net attractive force.

At equilibrium separation, attractive & repulsive force just balance and the pot energy is minimum.

Potential Energy Diagram (U vs r) for Molecules

Simplest case: U of two point charges q1 & q2

푈 =

푟 = 푑푖푠푡푎푛푐푒 푏푒푡푤푒푒푛 푐ℎ푎푟푔푒푠; = 9.0 × 10 푁푚 /퐶

When q1 & q2 are the same sign: U is +ve for all values of r

The force is repulsive, U increases as r decreases. Since work is done to bring the charges together, their U increases.

Fig. 40-7(a) p1074 Giancoli Physics for Scientists & Engineers 4th Ed.

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When q1 & q2 are opposite sign: U is ve for all values of r since the product q1q2 is ve.

The force is attractive, U becomes more ve as r

decreases. U −

Fig. 40-7b (Giancoli)

Potential Energy Diagram for the formation of covalent bond e.g H2 molecules.

U vs r of one H atom in the presence of another is given below.

Fig. 40-8

Starting at large r, U decreases as the atoms approach, because the electrons concentrate between the two nuclei, attraction occurs.

At very short distances, the electrons would be “squeezed out” – there is no room for them between the two nuclei.

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Without electrons between them, each nucleus would feel a repulsive force due to the other, so the curve rises as r decreases further.

At optimum separation distance ro, the potential energy is lowest.

Here, greatest stability for H2 molecules.

ro is the average separation of atoms for H2 molecules.

Depth of well = binding energy. i.e. how much energy must be put to the system to separate the two atoms to infinity where U = 0.

For H2 molecules Ebinding 4.5 eV & ro = 0.074 nm

In molecules made of larger atoms, eg O2 or N2, repulsion also occurs at short distances because the closed inner electron shells begin to overlap & the exclusion principle forbids them coming too close. A reasonable approximation to the potential energy, at least in the vicinity of ro is given as:

푈 = −퐴푟

+퐵푟

A & B constants associated with attractive & repulsive force.

m & n small integer.

For many bonds, the U-r curve has the shape of Fig. 40-9.

Fig. 40-9 Potential-Energy diagram for a bond requiring an activation energy.

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The atoms do not interact spontaneously, but some additional energy must be supplied to the system to get it over the “hump” (or barrier) in the potential energy diagram.

This required energy is called the activation energy.

The curve 40-9 is more common than 40-8.

Eg. To make water from O2 & H2.

The O2 & H2 molecules must first be broken into O & H atoms by supplying energy which is represented by the activation energy.

Then H & O atoms combine to form H2O molecules with the release of a lot more energy than was put in initially. This explosion of energy when H2O molecules were formed from H & O atoms provides further activation energy where additional H2 & O2 are broken up & recombined to form H2O.

Ionic Bonds

- two atoms combine such that one/more electrons transferred from one to the other

- Fundamentally caused by Coulomb attraction between oppositely charged ions

e.g NaCl

Na with configuration 1s22s22p63s gives up its 3s valence electron to form Na+ ion.

Energy needed to ionise Na to Na+ is 5.1 eV.

Cl with configuration 1s22s22p5 is one electron short of the closed-shell structure of argon.

Since closed-shell configuration is energetically more favourable, Cl is more stable than neutral Cl atom.

Energy released when an atom takes on an electron is the electron affinity of the atom.

Electron affinity for Cl is 3.7 eV.

Energy required to form Na+ and Cl from isolated atoms is 5.1 – 3.7 = 1.4 eV.

The 1.4 eV is called the activation energy of the molecule.

As ions are brought closer together, their mutual energy decreases due to electrostatic attraction.

At sufficiently small separation, the energy of formation becomes negative, indicating that the ion pair is preferred over neutral Na & Cl atoms.

Fig. 11.2 (p375) Total energy versus the internuclear separation for Na+ & Cl ions.

At very large separation distance, the energy of the system of ions is 1.4 eV.

The total energy is minimum at 4.2 eV at the equilibrium separation 0.24 nm.

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The energy required to break the Na+-Cl bond and form neutral Na & Cl atoms, called dissociation energy, is 4.2 eV.

When the two ions are closer than 0.24 eV, the electrons in closed shell begins to overlap, that results in repulsion between the closed shells.

This repulsion is partly electrostatic and partly a result of the identity of electrons.

Because they must obey the exclusion principle (Chp. 9) some electrons in overlapping shells are forced into higher states and the system energy increases, as if there were a repulsive force between the ions.

Fig. 40-10 (Giancoli) Potential energy diagram for NaCl bond.

U = 0 for free Na & Cl neutral atoms. Activation energy to form Na+Cl from neutral Na & Cl

atom = 5.14 3.61 = 1.53 eV (Giancoli). Binding energy (energy to separate NaCl into Na & Cl atoms) = 4.2 eV.

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Covalent bonds

A covalent bond between two atoms is one in which electrons supplied by either one or both atoms are shared by the two atoms.

Eg. Diatomic molecules H2, F2 & CO

Classical orbit model for covalent bond between two 1s electrons of H2 molecules is given in Fig. 11.3 (p375)

Eg. of more complex stable molecules formed by covalent bonds are: H2O, CO2 & CH4.

Classical orbit model for four covalent bonds in CH4 molecule is given Fig. 11.4a (p376)

Van der Waals Bonds

Ionic & covalent are bonds within molecules.

Van der Waals & hydrogen bonds occur between molecules.

Atoms that do not form ionic or covalent bonds are attracted to each other by van der Waals forces.

van der Waals force arises when an electrically neutral molecule has centers of positive and negative charge that do not coincide which results in the molecule forming an electric dipole

the interaction between electric dipoles causes two molecules to attract each other.

3 types of van der Waals forces 1. Dipole-dipole force: interaction between 2 molecules, each having a

permanent electric dipole moment. E.g. HCL & H2O. In effect, one molecule interacts with the electric field produced by another molecule.

2. Dipole-induced force: results when a polar molecule having a permanent electric dipole moment induces a dipole moment in a nonpolar molecule. The electric field of the polar molecule creates the dipole moment in a nonpolar molecule, which then results ii an attractive force between the molecules.

3. Dispersion force: an attractive force that occurs between two nonpolar molecules. Although the average dipole moment of a nonpolar molecule is zero, but charge fluctuations can cause two nonpolar molecules near each other to have dipole moments that are correlated in time so as to produce an attractive van der Waals force.

Hydrogen Bonds

The type of bonding between a hydrogen atom in certain molecule.

Attraction of two negative ions by an intermediate hydrogen atom.

E.g. hydrogen difluoride ion (HF2) in Fig 11.5 (p377). Two negative fluorine ions are bound by the positively charged protons.

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Bonds within a water molecule are covalent, but the bonds between water molecules in ice are hydrogen bonds.

Hydrogen bond is relatively weak compared with other chemical bonds. It can be

broken with input energy 0.1 eV.

MOLECULAR ROTATION & VIBRATIONS

The energy of an individual molecule in the gaseous phase of a substance can be divided into 4 categories: 1. Electronic energy: due to interactions between the molecule’s electrons & nuclei. 2. Translational energy: due to motion of the molecule’s centre of mass through space. 3. Rotational energy: due to rotation of the molecule about its centre of mass. 4. Vibrational energy: due to the vibration of the molecule’s constituent atoms.

E = Eel + Etrans + Erot + Evib

Eel is very complex because it involves the interactions of many charged particles.

Etrans is not important in interpreting molecular spectra since it not related to internal structure.

Molecular Rotation

Rotation of molecules about its centre of mass.

Confine to diatomic molecules even though can be extended to polyatomic molecules.

Diatomic molecules: 2 rotational degrees of freedom

The energy of a rigid rotating molecule is all kinetic.

Let m1 & m2 be the atomic masses with speed v1 & v2 respectively.

v1 = r1 & v2 = r2

r1 & r2 are distances of m1 & m2 to the axis of rotation.

L = m1v1r1 + m2v2r2 = (m1r12 + m2r2

2) = I ..........(1)

Erot = ½ m1v12 + ½ m2v2

2 = ½ I2 ............(2)

Rotational energy & angular momentum: Solving (1) & (2), we have

퐸 = ..................(3)

I depends on the position of the axis of rotation. For the axis passing through the centre of mass,

m1r1 = m2r2

In terms of the atomic separation Ro = r1 + r2

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푅 =푚푚

+ 1 푟 = 1 +푚푚

푟

Then,퐼 = 푅 = 휇푅

is the reduced mass of the molecule.

Unlike the moment of inertia I, angular momentum L is a dynamical variable. In the transition to quantum mechanics, L2 becomes quantized.

The quantization rule: L2 = ℓ(ℓ + 1)ℏ , ℓ = 0, 1, 2, …

Allowed energies for rotation: 퐸 = ℏ ℓ(ℓ + 1) … . . (5) ℓ = 0, 1, 2, …

ℓ is the rotational quantum number.

Rotational energy of molecules is quantized & depends on the moment of inertia of the molecule.

The allowed rotational energy level of a diatomic molecule is given in Fig. 11.6 p.378.

10 The spacing between the adjacent rotational level (ℓ values) are not equal and given by:

Δ퐸 = 퐸ℓ − 퐸ℓ = ℏ ℓ ............(6) where ℓ is the quantum number of the higher energy state.

In going from one energy state to the next, the molecule loses (gains) energy, E.

The loss (gain) is typically accompanied by photon emission (or absorption) at the frequency 휔 =ℏ

.

Thus photon should be observed at frequency: 휔 = ℏ , 2휔 , 3휔 , …in excellent agreement with

experiment.

The energy required to excite a molecule into rotation is quite small 104 eV.

From the data of wavelength or frequency of absorption lines (Table 11.1p380) the moment of inertia (I) & the bond length of molecules can be determined.

11 Molecular Vibration

A molecule is a flexible structure, whose atom are bonded together by what can be considered “effective spring”.

If disturbed, the molecule can vibrate, taking on vibrational energy.

Atomic displacement in the direction of the molecular axis give rise to oscillations along the line joining the atoms.

For these longitudinal vibrations, the system is effectively one-dimensional, with the coordinate of each atom measured along the molecular axis.

The elastic energy of the two-atom pair when the effective spring is stretch by the net amount 휉 − 휉 is given by:

U = ½ K (12)2 where K is the force constant.

1 & 2 are the displacement of m1 & m2 from equilibrium.

The allowed energies of vibrations:

퐸 = 휐 + ℏ휔 , 휐 = 0,1,2, … .....................(7)

휐 is the vibrational quantum number, is the classical frequency of vibration.

The relationship between K & is given by:

K = 2 , 휇 = = the reduced mass of the molecule

12 Allowed vibrational energy of a diatomic molecule is given in Fig. 11.8 p.382.

When 휐 = 0 , Evib = ℏ휔 called zero-point energy. The vibration of zero-point motion, is present

even when the molecule is not excited

From equation (7), the energy difference between any two successive vibrational levels is the equal

given by Evib = ℏ휔.

A typical value of Evib 0.3 eV.

Normally most molecules, are in the lowest energy state because the thermal energy at ordinary

temperatures ( 0.025 eV) is insufficient to excite the molecule to the next available vibrational state.

However, electromagnetic radiation can stimulate transitions to the first excited level. Such transition would be accompanied by the absorption of a photon to conserve energy.

Once excited, the molecule can return to the lower vibrational state by emitting a photon of the same energy.

13 For molecular vibrations, photon absorption and emission occur in the infrared region of the spectrum.

Since larger force constant K describes stiffer springs, K indicates the strength of the molecular bond.

MOLECULAR SPECTRA

In general, molecules rotates & vibrates simultaneously. To a first approximation, these motions are independent of each other & a total rotational & vibrational energy of the molecules are given by adding equation (5) & (7):

퐸 =ℏ

2퐼ℓ(ℓ + 1) + 휐 +

12ℏ휔… … … … … … . (8)

Equation 8 constitute the simplest approximation to the rotation-vibration spectrum of any molecule.

14 Each level is indexed by two quantum numbers, ℓ 푎푛푑 휐, specifying the state of rotation & vibration respectively.

For each allowed value of the vibrational quantum number 휐, there is a complete set of rotational levels, corresponding to ℓ = 0,1,2, …

The rotation-vibration levels of a typical diatomic molecule is given in Fig. 11.10p.385.

Normally, the molecule will take on the configuration with lowest energy, ℓ = 0 푎푛푑 휐 = 0.

External influences such as temperature or the presence of electromagnetic radiation, can change the molecular condition, resulting in a transition from one rotation-vibration level to another.

The change in molecular energy must be compensated by absorption or emission of energy in some other form.

When electromagnetic radiation is involved, the transitions – referred to as optical transitions – are subject to other conservation laws as well, since photons carry both momentum & energy.

15 Any optical transition between molecular levels with energy E1 & E2, must be accompanied by photon emission or absorption at frequency:

푓 =|퐸 − 퐸 |

ℎ 표푟 Δ퐸 = ±ℎ푓 … … … … … … (11.11)

Because hf is the photon energy, this is energy conservation for the system of molecule plus photon.

Equation 11.11 indicates a kind resonance between the molecule & photon. Unless photons of the correct frequency (energy) are available, no transition is possible.

Similar restrictions apply to other quantities that we know must be conserved in the process of transition.

In particular, the initial & final states for an optical transition also must differ by exactly one angular momentum unit:

|ℓ − ℓ | = 1 표푟 Δℓ = ±1 … … … … … … . (11.12)

The inference from Eqn 11.12 is that the photon carries angular momentum in the amount of ℏ, i.e. photon is a spin 1 particle with spin quantum number s = 1

Eqn. 11.12 then expresses angular momentum conservation for the system molecule plus photon.

Eqn 11.11 & 11.12 are called selection rules for optical transition.

For lower vibrational levels, there is also restriction on the vibrational quantum number 휐:

|휐 − 휐 | = 1 표푟 Δ휐 = ±1 … … … … … … … … … … . . (11.13)

Eqn. 11.13 reflects the harmonic character of the interatomic forces rather than any photon property, for higher vibrational energies, the concept of an effective spring joining the atoms is inaccurate, and eqn. 11.13 ceases to be valid.

Selection rules greatly restrict the number of photon frequencies or wavelengths observed in molecular spectra since transitions are prohibited unless all rules are obeyed simultaneously.

For instance, a pure rotational transition would not normally be observed, since this requires Δ휐 = 0 in violation of eqn 11.13.

In the same way, a pure vibrational transition (Δℓ = 0) is forbidden, and we conclude that optical transitions usually involved both molecular vibration & rotation.

The spectrum of a particular molecule can be predicted by considering a collection of such molecules initially undisturbed. At ordinary temperatures there is insufficient energy to excite any but the 휐 = 0 vibrational mode, although the molecules will be in various states of rotation.

Since a pure rotational transition is forbidden, optical absorption must result from transition in which 휐 increases by one unit but ℓ either increases or decreases, also by one unit (Fig. 11.11a).

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Therefore, the molecular absorption spectrum consists of two sequences of lines, represented by Δℓ = ±1, with Δ휐 = +1 for both cases.

The energies of the absorbed photons are readily calculated from eqn. 11.14:

The ℓ in eqn. 11.14 is rotational quantum number of the initial state.

17 The first of eqn. 11.14, generates a series of equally spaced lines at frequencies above the

characteristic vibration frequency .

The second of eqn. 11.14 generates a series below this frequency.

Adjacent lines are separated in (angular) frequency by the fundamental unit ℏ

.

ℓ cannot be zero if the transition is one for which ℓ decreases (Δℓ = −1).

Fig. 11.11b shows the expected frequencies in the absorption spectrum for the molecule; these same frequencies appear in the emission spectrum.

The absorption spectrum of the HCL molecule in Fig. 11.12 follows this pattern very well and reinforces our model.

Except, with one perculiarity: each line in the HCL spectrum is split into a doublet. This doubling

occurs because the sample is a mixture of two chlorine isotopes 퐶푙 and 퐶푙 whose different

masses give rise to two distinct values for ICM.

Not all the spectral lines appear with the same intensity, because even the allowed transitions occur at different rates (number of photons absorbed per second). Transition rates are mainly governed by the population of the initial and final states, and these depends on the degeneracy of the levels as well the temperature of the system.

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Molecules, like atoms, often simply scatter radiation without having first to absorb and later re-emit it. In fact, photons of any energy can be scattered, so no resonance is involved.

In Rayleigh scattering, the photon energy is unchanged by the collision. Rayleigh scattering is stronger at shorter photon wavelength, and this selectivity in the scattering that accounts for the blue colour of the daytime sky.

Raman scattering can also occur, with the photon losing (or gaining) energy in the collision. Because the photon energy changes, the Raman effect is an example of an inelastic process.

For such processes, energy is still conserved overall, with the photon energy loss or gain compensated by a suitable change in the rotational and/or vibrational state of the molecule.

Where rotation is involved, the energy exchange in Raman scattering is consistent with the selection rule ∆ℓ = ±2 .

Fig. 11.13 shows a typical Raman process where an incoming photon with energy E is scattered and emerges with a reduced energy E’, the difference being expended to excite a higher rotational state of the molecule.

The excitation energy E can be found from Eqn. 11.10 using ℓ, 휐 for the quantum numbers of the initial state and ℓ + 2, 휐 for the final state (Δℓ = 2,Δ휐 = 0).

The scattered photon has lower frequency f’ compared to the original; the Raman shift f – f’ is just

the excitation energy or

푓 − 푓 = ℏ (2ℓ + 3) .......................(11.15) Raman shift

19 Measurement of the Raman Shift can be used to determine the moment of inertia of the molecules, which gives important clues about the molecular structure.

Raman spectra provide a kind of “fingerprint” for molecules, and have been used successfully to identify minerals in lunar soil samples.

Raman scattering is relatively weak & can be observed only if the incident radiation is sufficiently intense.

With the advent of powerful monochromatic laser sources, Raman spectroscopy has found application in the remote monitoring of pollutants.

Example: the scattering produced by a laser beam directed on the plume from an industrial smokestack can be used to monitor the effluent for level of those molecules that produce recognizable Raman spectral lines.

In the Raman process the electronic state of the molecule is unchanged.

Molecular spectra for which changes occur in the electronic state as well as in the vibrational and/or rotational states of the molecule are called electronic spectra.

Because the electronic energy levels of a molecule are separated by much larger energies (~ 1푒푉) than vibrational (or rotational) levels, electronic transitions give rise to spectral lines that lie in the visible or ultraviolet range.

For the same reason, a complete set of vibrational levels may be associated with each electronic level, just as a complete rotational spectrum accompanies each vibrational level.