Introduction to Quantum mechanics and Molecular Spectra

1
Introduction toQuantum mechanics and Molecular
Spectra

• Ka-Lok Ng
• Asia University

2
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

3
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.
4
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
5
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
6
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
7
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
8
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
9
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.
10
Eigenvalues of QM operator

• Eigenvalues of QM operator must be real !
• Example

The two values for l are real
11
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.
12
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
13
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 !

14
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
15
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.
16
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

17
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
18
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
19
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
20
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

21
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
22
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.

23
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
24
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
25
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
26
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
27
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
28
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
29
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.
30
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
31
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.