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CH2422 – Quantum Chemistry

Although classical physics is often applied within chemistry to explain interactions on a visible scale,

the same approximations are not as applicable when describing phenomena on a much smaller scale

i.e. particles at the molecular scale and during very small energy transfers.

Classical physics assumes that;

The precise position and momentum of a particle can be precisely determined

simultaneously

The motion of an object can assume an arbitrary energy

Waves and particles are distinct

Quantum mechanical theories are necessary, therefore, to make up for the shortcomings of these

assumptions and to develop an understanding of several phenomena, e.g. molecular and atomic

structures, chemical bonding, spectroscopy.

Quantum Effects in Chemistry

Spectroscopy and the Quantisation of Energy Levels

Spectroscopy as a technique depends upon quantum effects to provide information about chemical

species, for example different molecules will provide different and characteristic UV spectra. These

spectra show absorptions and emissions at very particular frequencies, thus disproving the notion

that a system may possess an arbitrary energy.

Instead, atoms and molecules are said to exist within quantum states, these states have a defined

energy and so energy levels are quantised and only certain transitions are permitted.

There are exceptions, e.g. ionisation, here the combined ion-electron system can take any energy

above the ionisation limit, this energy level is quantised, however, beyond this limit the kinetic

energy of the free electron is continuously varied and is therefore not subject to quantisation.

The Particle Nature of Light

The evidence for the particle nature of light is best described by the photo-electric effect. Upon

irradiation with UV light a metal surface will only eject electrons once a characteristic frequency

threshold has been surpassed. The kinetic energy of the emitted electrons increases as the

frequency increases while increasing the intensity of the light will have no effect. This shows that

electromagnetic radiation is transferring energy in strict quanta and that light itself must therefore

be able to behave as a particle, these ‘light particles’ are called photons and can be considered a

discrete ‘packet of information’.

This phenomenon, in conjunction with the quantisation of molecular energy levels, is the basis of

absorption and emission spectroscopy.

The Wave Nature of Matter

While studying the scattering of electrons from the surface of a nickel sample C.J. Davisson and L.

Germer, having accidentally melted their sample and reformed it as a single crystal, were able toconfirm a theory proposed by de Broglie two years prior – that matter is able to possess wave-like

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characteristics, completing the quantum picture of wave-matter duality. The electrons scattering

from the nickel surface displayed a clear interference pattern, the arranged crystal lattice was able

to act as a diffraction grating thus proving that electrons, classically considered a form of matter,

were able to behave as waves, similar diffraction patterns are also displayed for objects of much

higher masses – including protons and neutrons.

Molecular theory, in particular our understanding of bonding and anti-bonding interactions are now

very heavily reliant on this quantum understanding of the electron, destructive and constructive

interference led to the current molecular orbital theory that has provided explanations for real-

world phenomena for decades.

Wave-Particle Duality

Therefore on a microscopic scale it has been shown that particles may take on the characteristics of

waves, and that waves may take on the characteristics of particles.

This duality was first theorised by de Broglie in the form of the de Broglie relationship (Equation 1).

The presence of Planck’s constant sets the scale on which wave-particle duality becomes a

significant effect as anything with a macroscopic momentum will have an overwhelmingly large

denominator, thus leading to a negligible wavelength. This explains why classical physics seemed to

provide an accurate fit before the discovery of the constituents of the nucleus, electrons and other

incredibly small particles.

Equation 1 - the de Broglie relationship1

Wavefunctions and the Schrödinger Equation

In quantum mechanics the wave nature of a system is expressed quantitatively through the

wavefunction, Ψ. The wavefunction is defined everywhere in 3N dimensional space as a single-

valued and finite number, the physical properties of a system are dependent on it.

The wavefunction is determined by the Schrödinger equation (Equation 2) set within boundary

conditions both in space and time. The Hamiltonian operator, Ĥ, acts on and transforms the

mathematical function it prefaces and allows for multiple answers to exist for any one form of the

equation. It is the boundary conditions that determine which mathematical solutions are realisticallypossible. It is often expressed as the total energy of a system and can, for many systems, be

considered a combination of the kinetic energy operator, T, and the potential energy operator, V.

Equation 2 - The time-dependent Schrödinger Equation2

Using the Schrödinger equation in a practical context it is often easier to consider stationary states;

these are time-independent systems, i.e. they can be considered to provide a consistent average

1 λ=wavelength, h=Planck’s constant (6.626 x 10-34

J.s), ρ=momentum (ρ=mv=mass x velocity)2 Ĥ=the Hamiltonian operator, Ψ=the wavefunction, i=√-1, ħ=h/2π, t=time

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wavefunction called a standing-wave, atomic and molecular orbitals are examples of standing-

waves.

Deriving the Time-Independent Schrödinger Equation

In one dimension the Schrödinger equation takes the form;

Equation 3 - One dimensional time-dependent Schrödinger equation3

Equations of this form can be solved by separation of variables, a trial solution (Equation 4) is

formed which can then be substituted to obtain Equation 5.

Equation 4 - The trial solution used in separating the variables within the Schrödinger equation4

Equation 5 - The result of substituting the trial solution into the Schrödinger equation

By dividing both sides of Equation 5 by θΨ Equation 6 can be obtained.

Equation 6

Since only the right-hand side of Equation 6 is a function of x only the left side will change upon

altering the value of x, since both sides are equal and the left-hand side is unaffected by variation of

x, the right-hand side must be equal to a constant. This constant is given the symbol, E, as the

dimensions are those of an energy (the same as those of V). The time-dependent equation (Equation

3) can therefore be split into the two differential equations, Equation 7 and Equation 8.

Equation 7 - The differential equation formed by the right-hand side of Equation 65

Equation 8 - The differential equation formed by the left-hand side of Equation 6

In principle any value of E is allowed, however with the introduction of typical boundary conditions

there are only certain values of E that give a non-zero value of ψ, this is the mathematical origin of

quantisation. Quantisation can therefore be rationalised as the requirement to fit an integral

3

m=the mass of the particle, V=the potential energy operator4 θ and ψ are fractions of the total wavefunction

5E=the total, conserved energy of the system

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number of wavelengths with wavefunctions sufficing the Schrödinger equation and within the

potential energy boundaries of a system as any non-integral sum of waveforms would lead to

destructive interference.

Equation 8 has the solution shown in Equation 9 and so the complete wavefunction (Ψ=ψθ) has the

form given in Equation 10. This wavefunction satisfies Equation 7 and may be written as shown in

Equation 11.

Equation 9 - The solution to Equation 8

⁄

Equation 10 - The form of the complete wavefunction6

⁄

Equation 11 - The time-independent Schrödinger equation

Since Equation 7 takes the form of a standing-wave equation it is legitimate, when only considering

the spatial dependence of the wavefunction, to consider Equation 11 a wave equation. For as long as

the potential energy is independent of time and system is in a state of energy, E, it is possible to

construct the time-dependent wavefunction from the time-independent wavefunction by

multiplying the latter by e-iEt/ ħ

.

Construction of the Hamiltonian operator, Ĥ

When constructing a Hamiltonian operator there are two steps;

1) Consider the expression that defines the energy as if it were obeying classical mechanics by

forming an equation that includes information on the coordinates (x,y and z in Cartesian

format) and momentum, px, for each particle

2) Replace each value of px with –i ħ d/dx. This comes from the following proof;

Firstly, consider the constituents of the Hamiltonian (Equation 12).

Equation 12 - The Hamiltonian shown as the sum of potential and kinetic energies7

Substituting in for T with the classical equation for kinetic energy (Equation 13) and further

substituting for the quantum expression for momentum (Equation 14) will give an

expression for T (Equation 15) as defined in step 2 above.

Equation 13 - The classical mechanics expression for kinetic energy8

6

The constant of proportionality in Equation 9 is absorbed into the normalisation constant for ψ 7T=the kinetic energy operator

8v=velocity

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Equation 14 - The quantum mechanical expression for momentum9

Equation 15 - The final form of T according to quantum mechanics

The Born Interpretation and Normalisation

The wavefunction contains all information for a system it is not directly measurable, nor does it have

any physical meaning. The Born interpretation states that Ψ Ψ’ (often simply Ψ2) will provide a

probability density. The probability of finding an object between x and dx in one dimension is

therefore equal to Ψ2(x)dx, in 3 dimensions the probability becomes Ψ2dV

(10).

If this is to be true then it must also be true that the overall probability of finding the particle, for

example in one dimension beteen the points x=0 and x=L (Equation 16), must be equal to 1.

Equation 16 - A mathematical representation of the Born interpretation for the wavefunction of a particle in one

dimension in a box of length, L

∫

Since the wavefunction term appears on both sides of the Schrödinger equation (Equation 11) it is

possible to multiply it by any value without any effect on the overall solution. A value is therefore

chosen that ensures the probability of finding the particle within the entire system is equal to 1, this

function is then considered normalised.

9 =Laplace operator, a differential operator given by the divergence of a function on Euclidean space

10V here is not the potential energy operator but simply denotes volume

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Constructing and Normalising the Hamiltonian Operator, Ĥ, for Conceptual Systems

A Particle in a 1D Box

Imagine a particle confined within two walls to a region of space of length, L. This system is called a

one-dimensional square well, or a particle in a box, and can be represented graphically (Figure 1).

The two walls have a potential energy that rises abruptly to infinity, thus providing boundary

conditions for the permissible wavefunctions and ensuring that all acceptable energies are

subsequently quantised.

The particle in this box will have a potential energy of 0, and so the first step is to construct a

Hamiltonian in which the only term is that of kinetic energy (Equation 17).

Equation 17 - The Hamiltonian for a particle in a box11

This equation is the same as the Hamiltonian for free translational motion. Again, it is due to the

imposed boundary conditions that the number of acceptable wavefunctions for this system is so

many fewer than there are for free translational motion.

Wavefunctions must be everywhere continuous, and so the biggest implication of the boundary

conditions is that at x=0 and x=L, the wavefunction must be equal to 0. From here it is possible to

form a general solution (Equation 18) into which the two boundary conditions can be tested

(Equation 19 and Equation 20).

Equation 18 - The general solution for a particle in a box

⁄

Equation 19 - The wavefunction at x=012

11 The substitutions described in ‘Construction of the Hamiltonian’ (page 4) are shown here stepwise

12cos 0=1 and sin 0=0

L0 Position, x P o t e n t i a l e n e r g y ,

V ( x )

∞

Figure 1 - A graphical representation of a particle in a box

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Since Equation 19 shows that the wavefunction is equal to C at x=0, it is sensible to define C as being

equal to 0, eliminating the cos term from the equation and beginning the process of fitting the

wavefunction into the boundary conditions.

Equation 20 - The wavefunction at x=L

It would be possible, using Equation 20, to define D as being equal to 0 as well. This would, however,

mean that the wavefunction is 0 everywhere and so an alternative approach must be considered in

order to make the sin function in the equation disappear at x=L. This is achieved if kL is equal to an

integer multiple of π13 (Equation 21).

Equation 21 - The allowed values for k

The energy of the system can be determined (Equation 22) and is found to also be dependent on n, n

is therefore a quantum number and may be used to label the state of a system and is responsible for

the quantisation of energy.

Equation 22 - Stepwise calculation of the formula for the allowed energy values

⁄

The difference in energy levels for this system can therefore be given by Equation 23.

Equation 23 - Calculating the difference between energy levels

Equation 23 also shows that as L increases, ΔE decreases and so at a large enough distance (beyond

the scale of Planck’s constant, present in the numerator) the change between energy levels can be

treated as done in classical terms. ΔE also decreases as the mass, m, of the object increases and so

again, at larger masses the quantisation of energy is unnecessary.

The only thing yet to be calculated is D, this constant plays the role of normalising the wavefunction

(Equation 24) so as to satisfy the Born interpretation.

13Sin(π)=0 in radians, since sin nx = n sin x it follows that any multiple of π will give a result of 0, as required by

the boundary conditions

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The Hamiltonian for Other Systems

In the following equations the terms for kinetic energy are in blue, and the terms for potential

energies in green.

A Particle in a 3D Box

Note that while the Hamiltonian is additively separable, the wavefunction is multiplicatively

separable, i.e. ψlmn=Ψl(x) Ψm(y) Ψn(z), this gives nodal structure in 3 dimensions.

The energy is the sum of energies in each degree of freedom;

Harmonic Oscillator

Equation 25 - Hamiltonian for a harmonic oscillator15

The Schrödinger equation arising from Equation 25 is shown in Equation 26.

Equation 26 - The Schrödinger equation for a harmonic oscillator

The solution for this Schrödinger equation gives an equation for the energy values shown in

Equation 27 - The allowed energy levels of the harmonic oscillator16

( )

According to classical physics the motion of an oscillator ought to be restricted by ‘turning points’(points at the edge of the oscillating motion on which the particle momentarily stops before

changing direction) given by x= . In quantum mechanics, however, Ψ2is non-zero (Figure 2)

beyond these classical barriers. This has the implication that the potential energy is in fact greater

than the total energy, in turn implying that the kinetic energy must be negative. This is the basis for

‘quantum tunnelling’, a phenomena utilised by scanning electron microscopy (SEM).

15

k= the force constant, m=mass16

ν=the vibrational quantum number

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Figure 2 - The forms of the first 7 wavefunctions found upon application of the equations for a harmonic oscillator

It is worth noting in Figure 2 that as the energy level increases, the wave takes a shape more in

alignment with classical predictions, i.e. the probability of finding the particles at the classical turning

points is the most noticeable cf. the probability of finding the particles at the classical turning points

is the most noticeable cf. Ψ0 where the probability density is almost entirely ‘within’ the parabola

and not on the line of the curve.

Hydrogenic Atom

Equation 28 - The Hamiltonian for a hydrogenic atom

The second term of Equation 28 is the motion of the nucleus, this is not entirely necessary as the

electron can be considered to move around the nucleus while the nucleus remains stationary. By

replacing the terms for nuclear and electronic motion with those of centre-of-mass coordinates and

the relative electronic position it is possible to reach Equation 29. The solution to the Schrödinger

equation is then shown by Equation 30.

Equation 29 - The Hamiltonian of a hydrogenic molecule in which the nuclei is considered a stationary centre of mass17

Equation 30 - Solution to the Schrödinger equation for a hydrogenic atom

[

(

)

]

The allowed energy levels are therefore;

17The reduced mass, μ=memn/me+mn

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Equation 31 - The energies allowed by the Schrödinger equation (Equation 30)18

The energies given by Equation 31 have negative values; this implies that the electron has a lower

energy than it would were it free, as would be predicted.

Hydrogen Molecule Ion (H2+)

19

Equation 32 - The Hamiltonian for a hydrogen molecule ion

Hydrogen Molecule20

Equation 33 - The Hamiltonian for a hydrogen molecule

Particle on a Ring

The particle on a ring is a particularly useful description as it can apply to both particles moving in a

circular motion as well as the gyration of a particle around its own centre of mass.

The Hamiltonian for a particle on a ring (Equation 34) can be simplified if polar coordinates areadopted and x and y are replaced by (r cos φ) and (r sin φ) respectively, where φ varies from 0 to 2π

(Equation 35).

Equation 34 - The Hamiltonian for a particle on a ring, considered to have no potential energy and motion on the xy

plane

Equation 35 - The Hamiltonian after substitution of polar coordinates

If r is constant the derivatives with respect to r can be disregarded, giving Equation 36.

Equation 36 - The final Hamiltonian for a particle on a ring21

18RH is the Rydberg constant

19

A and B denote the two hydrogen nuclei20Where A and B are defined above, 1 and 2 denote the two electrons

21Inertia = I = mr

2

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Equation 37 - The energy levels of a particle on a ring22

Equation 37 allows for the calculation of energy levels for a particle on a ring. m l is a dimensionless

number with integral values (both positive and negative). This expression defines four characteristic

features of the energy levels for this scenario;

The separation of neighbouring levels increases as ml increases (the ml term is squared and

so the relationship is non-linear)

The separation of energy levels is small for systems with large moments of inertia, i.e. those

beyond a quantum scale

There is no zero-point energy (at E0 the energy is 0)

All energy levels other than the ground state are double degenerate i.e. ml=1 will have the

same energy as ml=-1, ml essentially represents the direction of travel around the ring

22ml=0, ±1, ±2…

( )

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Penetration of Potential Barriers

Imagine, again, a barrier of infinite width that has a potential energy labelled V(x) (Figure 3).

Figure 3 - A barrier of infinite width

The potential of a particle inside the barrier is equal to 0 at x<0 and V at x≥0 and the two

Hamiltonians are shown below (Equation 38 and Equation 39).

Equation 38 - Hamiltonian for the particle where x<0

Equation 39 - Hamiltonian for the particle where x≥0

The general solutions to these equations are shown by Equation 40 and Equation 41. By consideringonly the case in which E<V, where classically it would be impossible for the particle to be found in

the barrier, and applying this to Equation 41 it is implied that k’ is imaginary. k’ can therefore be

redefined as iκ where kappa is real, giving Equation 42.

Equation 40 - Solution to the Hamiltonian for x<0

( )

Equation 41 - Solution to the Hamiltonian for x≥0

( )

Equation 42 - Modification to the solution for the Hamiltonian at x≥0

( )

This wavefunction is therefore a mixture of decaying and increasing exponentials, i.e. it does not

oscillate when E<V. The increasing exponential must be dismissed as, across an infinite barrier, this

implies an infinite amplitude. The wavefunction is therefore an exponentially decaying function of

the form exp(-κx). Most importantly, at x≥0 the wavefunction is not equal to zero, the particle can

exist within the classically forbidden region, this is penetration. Kappa is a measure of how quickly

the exponential decays to zero and the penetration depth, 1/κ, decreases with increasing particle

t

t i

l

r

,

(

)

0 Position, x

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mass and increasing height of the barrier (i.e. V-E). Again this effect is limited to the smaller

particles, protons and electrons are able to penetrate ‘forbidden zones’ to an appreciable extent.

Crossing a Barrier of Finite Width - Quantum Mechanical Tunnelling

Imagine a barrier, of potential energy V(x), similar to that imagined for the particle in a box but with

a defined width,0<x< L (Figure 4).

At x<0 and x≥L , V(x)=0 and at 0≤x<L, V(x)=V.

The general solutions to the time-independent Schrödinger equations can easily be written

(Equation 43,Equation 44 and Equation 45).

Equation 43 - The wavefunction at x<0 for a finite width barrier

( )

Equation 44 - The wavefunction at 0≤x<L for a finite width barrier

( )

Equation 45 - The wavefunction at L≤x for a finite width barrier

( )

A and B represent ‘incoming’ and ‘outgoing’ waves, an incoming wave contributes to the total

wavefunction with a component of linear momentum towards the target whereas an outgoing wavehas a contribution with a component of linear momentum away from the target. It can be seen that

either side of the barrier the wave acts as before, it is able to move in both directions and oscillates

as a function of both a positive and negative exponential. For the same reasons as discussed above,

the B’ term is the only of significance (as the A’ term would imply an infinite amplitude) and as a

result the wave decays exponentially in this region. As a result, the emerging wave will be of a lower

amplitude than the original wave on the left.

t

t i

l

r

,

(

)

0 L Position, x

Figure 4 - A barrier of finite width

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To consider a simple, workable example, imagine the wave is formed from a cos function23

.

In this case;

Equation 46 - The form of the wavefunction on the left of the barrier when representing the wave as a cos function

√

Equation 47 - The form of the wavefunction in the barrier when representing the wave as a cos function

Equation 48 - The form of the wavefunction on the right of the barrier when representing the wave as a cos function

√

The transmission coefficient is given the label T and accounts for the decay across the barrier and isgiven by;

Equation 49 - The value of the transmission coefficient, T

Expectation Values

23

A sin wave would also work but if this were to be an expansion of the particle in a box (a usefulapproximation in worked examples) the sin wave would be 0 at the barrier and so tunnelling could not occur, a

cos wave will have an amplitude of 1 at the barrier and so could decay across the barrier as required

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