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Introduction toQuantum mechanics and Molecular

Spectra

- Ka-Lok Ng
- Asia University

Contents

- The postulates of quantum mechanics (QM)The wave

equation Schrodinger equation - Quantum mechanical operators
- Eigenvalues of QM operators
- Wave functions
- The particle in a 1D box
- Physical methods of determining the 3D structure

of proteins - References
- House J.E. Fundamentals of quantum chemistry, 2nd

ed. Elsevier 2004 - Whitford D. Proteins structure and function. J.

Wiley 2005. - http//www.spaceandmotion.com/Physics-Erwin-Schrod

inger.htm - Molecular spectra, see http//spiff.rit.edu/classe

s/phys315/lectures/lect_14/lect_14.html - http//cref.if.ufrgs.br/hiperfisica/hbase/molecul

e/molec.htmlc2

The postulates of quantum mechanics (QM)

- Postulate I
- For any possible state of a system, there is a

function y of the coordinates of the parts of the

system and time that completely describes the

system.

Y Is called a wave function. For two particles

system,

The wave function square Y2 is proportional to

probability. Since Y may be complex, we are

interested in YY, where Y is the complex

conjugate (i ? -i) of Y. The quantity YYdt is

proportional to the probability of finding the

particles of the system in the volume element, dt

dxdydz.

that is the probability of finding the particle

in the universe is 1 ? normalization condition.

The postulates of quantum mechanics (QM)

- Orthogonal of two wave functions

Example sinq and cosq are orthogonal functions.

Fourier series expansion sin(nq) and cos(nq)

orthogonal functions

The Wave Equation

- In 1924 de Brogile shown that a moving particle

has a wave character. This idea was demonstrated

in 1927 by Davisson and Germer when an electron

beam was diffracted by a nickel crystal. - According to the de Brogile relationship, there

is a wavelength associate with a moving particle

which is given by

where l, h, m and v denote the wavelength,

Plancks constant, mass and velocity. Erwin

Schrodinger adapted the wave model to the problem

of the hydrogen atom and propose the Schrodinger

equation. The model needs to describe a

three-dimensional wave. Classical physics the

flooded planet problem the waveforms that would

result form a disturbance of a sphere that is

covered with water The classical 3D wave equation

is

where f is the amplitude function and v is the

phase velocity of the wave.

Schrodinger equation

The Wave Equation

- The Schrodinger wave equation

The 1D wave equation solution http//www-solar.mcs

.st-and.ac.uk/alan/MT2003/PDE/node12.html

The 2D wave equation solution http//www.math.harv

ard.edu/archive/21b_fall_03/waves/index.html

Operators

- Postulate II
- With every physical observable q there is

associated an operator Q, which when operating

upon the wavefunction associated with a definite

value of that observable will yield that value

times the wavefunction F, i.e. QF qF.

H

http//hyperphysics.phy-astr.gsu.edu/hbase/quantum

/qmoper.html

Operators

- (1) The operators are linear, which means that
- O(Y1 Y2) OY1 OY2
- The linear character of the operator is related

to the superposition of states and waves

reinforcing each other in the process - (2) The second property of the operators is that

they are Hermitian (the 19th century French

mathematician Charles Hermite). - Hermitian matrix is defined as the transpose of

the complex conjugate () of a matrix is equal to

itself, i.e. (M)T M

In QM, the operator O is Hermitian if

C. Hermite

http//commons.wikimedia.org/wiki/ImageCharles_He

rmite_circa_1887.jpg

Eigenvalues of QM operator

- Postulate III
- The permissible values that a dynamical variable

may have are those given by OF aF, where F is

the eigenfunction of the QM operator (Hermitian)

O that corresponds to the observable whose

permissible real values are a. - The is postulate can be stated in the form of an

equation as

O F a F

operator wave function eignevalue

wave function

Example Let F e2x and Od/dx ? dF/dx

d(e2x)/dx 2 e2x ? F is an eigenfunction of the

operator d/dx with an eigenvalue of 2.

Eigenvalues of QM operator

- Eigenvalues of QM operator must be real !
- Example

The two values for l are real

Wave functions

- Postulate IV
- The state function Y is given as a solution of
- where is the total energy operator, that is

the Hamiltonian operator. - The hamiltonian function is the total energy,

TV, where T is the kinetic energy and V is the

potential energy. In operator form

Schrodinger equation

where is the operator for kinetic energy and

is the operator for potential energy. In

differential operator form, the time dependent

Schrodinger equation is

where qi is the generalized coordinates, m is the

mass of the particle.

The particle in a one-dimensional box

- We treat the behavior of a particle that is

confined to motion in a box - The coordinate system for this problem is show at

the right - The Hamiltonian operator, H, is H T V p2/2m

V - where p is the momentum, mass is the mass of the

particle, and V is the potential energy - Outside the box V 8, so the Schrodinger equation

For the equation to be valid, y must be 0 ?

Boundary condition the probability of finding

the particle outside the box is zero Inside the

box, V 0, so the Schrodinger equation becomes

http//www.everyscience.com/Chemistry/Physical/Qua

ntum_Mechanics/a.1128.php

The particle in a one-dimensional box

- Where k2 8p2mE/h2. This is a linear

differential equation with constant coefficients,

which have a solution fo the form Y A cos(kx)

B sin(kx). - The constant A and B must be evaluated using the

boundary conditions. Boundary conditions are

those requirments that must be met becase of the

the physical limits fo the system. - For the probability of finding the particle to

vanish at the walls of the box, that is Y 0

both at x?0 and x?L. - At x?0
- Y 0 A B(0) ? A 0
- At x?L
- Y 0 B sin kL
- Since B ? 0, otherwise the complete wavefunction

0 ! - ? sin kL 0 that is kL np ? quantization

condition !

The particle in a one-dimensional box

- quantization condition ? kL np
- Recall k2 8p2mE/h2
- k2 L2 8p2mE/h2 L2 n2p2, where n 1, 2 . is

the quantum number

- Zero-point energy
- One quantum number arise from a 1D system

E n2 E 1/L2 E m To determine the

wavefunction Y, one uses the normalization

condition

n 1, 2, 3

http//www.everyscience.com/Chemistry/Physical/Qua

ntum_Mechanics/a.1128.php

The particle in a one-dimensional box

- Consider a carbon chain like
- CC-CC-C
- as an arrangement where the p electrons can move

along the chain. If we take an average bond

length of 1.40 Angstrom, the entire chain would

be 5.60 Angstrom length, Therefore, the energy

difference between the n1 and n2 state would be

The energy corresponds to a wavelength of 344 nm,

and the maximum in the absorption spectrum of

1,3-pentadiene is found at 224 nm. Although this

not close agreement, the simple model does

predict absorption in the UV region of the

spectrum.

Molecular Spectra

- Three types of energy levels in molecules
- electronic large energy separations (200-400

kJ/mol) ? optical or UV - vibrational medium energy separations (10-40

kJ/mol) ? Infrared - rotational small energy separations (10-40

J/mol) ? microwave - All the energy levels are quantized

Molecular Spectra

- For a spectral line of 6000 Angstrom, which is in

the visible light region, the corresponding

energy is E hc/l 3.3x10-12 erg - ? a molar quantity multiply by Avogadros number

? E 200 kJ/mol - Diatmoic molecule can be viewed as if they are

held together by bonds that have some stretching

and bending (vibrational) capability, and the

whole molecule can rotate as a unit.

http//cref.if.ufrgs.br/hiperfisica/hbase/molecule

/molec.htmlc2

Molecular Spectra

- Normal mode of vibration for the CO2 molecule
- behaves much like a simple harmonic oscillator
- The vibrational energies can therefore be

described by the relation (?1/2)h?, where ?,

the vibrational quantum number 0,1,2,3. and

?the classical frequency - the symmetric stretch mode
- the asymmetric stretch mode
- the bending mode

http//www.phy.davidson.edu/StuHome/shmeidt/Junior

Lab/CO2Laser/Theory.htm

Vibrational and Rotational energy levels

transition spectra

- The CO2 molecule is free to rotate. The energies

of the rotational modes (E h2/(8p2I), where I

is the moment of inertia)) are smaller than for

vibrational modes. Hence, the energy levels for

two vibrational states with the rotational

divisions looks like

Vibrational and Rotational energy levels

transition spectra for HCl

http//universe-review.ca/F12-molecule.htm

Applications of the Vibrational Energy Levels

- determination of bond lengths
- determination of bond force constants
- determination of bond dissociation energy
- qualitative and quantitative chemical analysis

Selection rules for energy level transitions

- Selection rules are divided into high probability

or allowed transitions - and Forbidden transitions of much lower

probability - Forbidden transition symmetry-forbidden and

spin-forbidden transitions - Spin-forbidden transitions involve a change in

spin multiplicity defined as 2S1 where S is the

electron spin number. Spin multiplicity reflects

electron pairing (see Table). For a favourable

transition there is no change in multiplicity

(DS0)

Number of unpair electrons Electron spin S 2S1 Multiplicity

0 0 1 Singlet

1 ½ 2 Doublet

2 1 3 Triplet

3 3/2 4 quartet

Selection rules for energy level transitions

- Symmetry-forbidden transitions reflect

redistributin of charge during transitions in a

quantity called the transition dipole moment. - Differences in dipole moment arise from the

different electron distributions of ground and

excited states - For an allowing transition it requires a change

in dipole moment - EM radiation can induce Rotational transitions

only in molecule with a permanent dipole moment. - Not all molecules have dipole moments!
- (1) only polar molecules can absorb and emit

electromagnetic photons - (2) non-polar molecules H2 ,CO2 ,CH4
- (3) energy transfer can take place during

collisions - The intensity of the signals in a rotational

spectrum increase with the molecular dipole

moment.

Fluorescence Spectroscopy

- Fluorescence excited molecules decay to the

ground state via the emission of a photon with DS

0 ( no change in spin multiplicity, S1 ? S0) - Emission is occurs at longer wavelengths than the

corresponding absorbance band - Quantum yield of Fluorescence emission
- photons emitted/number of photons absorbed
- maximum value of quantum yield is 1
- Photophysical properties of a fluorophore can be

used to obtain information on its immediate

molecular environment. Relaxation of a

fluorophore from its excited state can be

accelerated by fluorescence resonance energy

transfer (FRET). - FRETcan be used to characterize protein-protein

interactions as observed in signaling complexes

of ion channel proteins.

http//www.physiologie.uni-freiburg.de/fluorscence

.html

Raman Spectroscopy

- Chandrasekhara Venkata Raman (1888-1970) who

discovered in 1928 that light interacts with

molecules vai absorbance, transmission or

scattering - the first Indian Nobel Laureate in physics
- Raman made many major scientific discoveries in

acoustics, ultrasonic, optics, magnetism and

crystal physics - Scattering can occur at the same wavelength when

it is known as Rayleigh scattering (elastic, n0)

or it can occur at altered frequency (change in

the colour of the scattered light) when it is the

Raman effect

Figure. See http//www.search.com/reference/Raman_

spectroscopy

Figure. See http//www.inphotonics.com/raman.htm

http//www.vigyanprasar.gov.in/dream/feb2002/artic

le1.htm

Raman Spectroscopy

- some weaker bands of shifted frequency are

detected. Moreover, while most of the shifted

bands are of lower frequency n0 - ni, there are

some at higher frequency, n0 ni. By analogy to

fluorescence spectrometry, the former are called

Stokes bands and the latter anti-Stokes bands.

The Stokes and anti-Stokes bands are equally

displaced about the Rayleigh band however, the

intensity of the anti-Stokes bands is much weaker

than the Stokes bands and they are seldom

observed.

http//www.gfz-potsdam.de/pb4/pg2/equipment/raman/

raman.html

Raman Spectroscopy Application

- commonly used in chemistry
- provides a fingerprint by which the molecule can

be identified. The fingerprint region of organic

molecules is in the range 500-2000 cm-1. - to study changes in chemical bonding, e.g. when a

substrate is added to an enzyme. - Raman gas analyzers have many practical

applications, for instance they are used in

medicine for real-time monitoring of anaesthetic

(???) and respiratory gas mixtures during

surgery. - In solid state physics, spontaneous Raman

spectroscopy is used to, among other things,

characterize materials, measure temperature, and

find the crystallographic orientation of a

sample.

http//www.search.com/reference/Raman_spectroscopy

Nuclear Magnetic Resonance Spectroscopy

- In 1945, the NMR phenomenon was given by F. Bloch

and E. M. Purcell (both share the 1952 Nobel

Prize) - NMR spectra are observed upon the pulse

absorption of a photon (radio frequency) of

energy and the transition of nuclear spins from

ground to excited states

Bloch F (1905-1983)

Purcell E.M. (1912-1997)

http//nobelprize.org/nobel_prizes/physics/laureat

es/1952/ http//cancer.stanford.edu/research/miles

tones/ http//www.pulseblaster.com/gallery/1.html

Nuclear Magnetic Resonance Spectroscopy

- For 1H there are two orientations. In one

orientation the protons are aligned with the

external magnetic field (north pole of the

nucleus aligned with the south pole of the magnet

and south pole of the nucleus with the north pole

of the magnet) and in the other where the nuclei

are aligned against the field (north with north,

south with south)

A spinning nucleus is equivalent to a magnet

http//www.brynmawr.edu/Acads/Chem/mnerzsto/The_Ba

sics_Nuclear_Magnetic_Resonance20_Spectroscopy_2.

htm http//vam.anest.ufl.edu/simulations/nuclearma

gneticresonance.php

Nuclear Magnetic Resonance Spectroscopy

- Nuclei possessing angular moment (also called

spin) have an associated magnetic moment (current

generate magnetic field). Certain atomic nuclei,

such as 1H, 13C, 15N and 31P have spin S½ and

2H, 14N have spin S1, 18O has S5/2). - For nuclei such as 12C is the most common isotope

is NMR silent, that is not magnetic. If a nucleus

is not magnetic, it can't be studied by nuclear

magnetic resonance spectroscopy. For the

purposes, biomolecular NMR spectroscopy requires

proteins enriched with 1H, 13C or 15N or ideally

all nuclei. - Nuclear transitions differed in frequency from

one nucleus to another but also showed subtle

differences according to the nature of the

chemical group (chemical shift effect). - Methyl protons resonating at a frequency ?

amide proton ? a-carbon proton ? b-carbon proton

? The chemical environment of such nuclei are

different - Probe by NMR and this technique can be exploited

to give information on the distances between

atoms in a molecules. These distances can then

be used to derive a 3D model of the molecule.

The frequency range needed to excite protons is

relatively high. It ranges from 300 MHz to 900

MHz.

Nuclear Magnetic Resonance Spectroscopy

- Why do we see peaks ?
- When the excited nuclei in the beta orientation

start to relax back down to the alpha

orientation, a fluctuating magnetic field is

created. This fluctuating field generates a

current in a receiver coil that is around the

sample. The current is electronically converted

into a peak. - Why do we see peaks at different positions?
- because nuclei that are not in identical

structural situations do not experience the

external magnetic field to the same extent. The

nuclei are shielded or deshielded due to small

local fields generated by circulating s, p and

lone pair electrons.

NMR spectra

Solid-state 900 MHz (21.1 tesla) NMR spectrometer

at the Canadian National Ultrahigh-field NMR

Facility for Solids.

http//www.answers.com/topic/solid-state-nuclear-m

agnetic-resonance http//www.scielo.br/scielo.php?

pidS0100-41582002000500017scriptsci_arttext

Nuclear Magnetic Resonance Spectroscopy

- Limitations for NMR methods
- 1. For small proteins with size lt 100 kD
- 2. Require highly concentrated protein solutions

on the order of 1-2 mM. - 3. pH of solution lt 6.