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1© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
Ch121a Atomic Level Simulations of Materials and Molecules
William A. Goddard III, [email protected] and Mary Ferkel Professor of Chemistry,
Materials Science, and Applied Physics, California Institute of Technology
BI 115Hours: 2:30-3:30 Monday and Wednesday
Lecture or Lab: Friday 2-3pm (+3-4pm)
Teaching Assistants Wei-Guang Liu, Fan Lu, Jose Mendoza, Andrea Kirkpatrick
Lecture 7, April 15, 2011Molecular Dynamics – 3: vibrations
2© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
Outline of today’s lecture
• Vibration of molecules– Classical and quantum harmonic oscillators– Internal vibrations and normal modes– Rotations and selection rules
• Experimentally probing the vibrations– Dipoles and polarizabilities– IR and Raman spectra– Selection rules
• Thermodynamics of molecules– Definition of functions– Relationship to normal modes– Deviations from ideal classical behavior
3© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
Simple vibrations
• Starting with an atom inside a molecule at equilibrium, we can expand its potential energy as a power series. The second order term gives the local spring constant
• We conceptualize molecular vibrations as coupled quantum mechanical harmonic oscillators (which have constant differences between energy levels)
• Including Anharmonicity in the interactions, the energy levels become closer with higher energy
• Some (but not all) of the vibrational modes of molecules interact with or emit photons This provides a spectroscopic fingerprint to characterize the molecule
4© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
Vibration in one dimension – Harmonic Oscillator
Consider a one dimensional spring with equilibrium length xe which is fixed at one end with a mass M at the other. If we extend the spring to some new distance x and let go, it will oscillate with some frequency, which is related to the M and spring constant k.To determine the relation we solve Newton’s equation M (d2x/dt2) = F = -k (x-xe)Assume x-x0= = A cos(t) then –Mcos(t) = -k A cos(t) Hence –M= -k or Sqrt(k/M). Stiffer force constant k higher and higher M lower
No friction
E= ½ k 2
5© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
Reduced Mass
M1M2
Put M1 at R1 and M2 at R2
CM = Center of mass Fix Rcm = (M1R1 + M2R2)/(M1+ M2) = 0Relative coordinate R=(R2-R1)Then Pcm = (M1+ M2)*Vcm = 0 And P2 = - P1
Thus KE = ½ P12/M1 + ½ P2
2/M2 = ½ P12/
Where 1/ = (1/M1 + 1/M2) or = M1M2/(M1+ M2) Is the reduced mass.Thus we can treat the diatomic molecule as a simple mass on a spring but with a reduced mass,
6© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
2
)2/1()2/1(2
2
a
vnE
e
en
(n + ½)2
For molecules the energy is harmonic near equilibrium but for large distortions the bond can break.
The simplest case is the Morse Potential:
2/1
2
)2(
)1()(
e
axe
hcD
ka
ehcDxV
Exact solution
Real potentials are more complex; in general: (n + ½)2 (n + ½)3
Successive vibrational levels are closer by
(Philip Morse a professor at MIT, do not manufacture cigarettes)
7© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
Now on to multiple atoms
• N atoms => 3N degrees of freedom• However, 3 degrees for translation, get = 0• 3 degrees for rotation is non-linear molecule, get = 0• 2 degrees if linear (but really a restriction only for diatomic• The remaining (3N-6) are vibrational modes (just 1 for diatomic)• Derive a basis set for describing the vibrational modes by
solving the eigensystem of the Hessian matrix
Eigenvalue problem
or
8© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
Vibration for a molecule with N particles
Fk = -(∂E(Rnew)/∂Rk) = -(∂E/∂Rk)0 - m (∂2E/∂Rk∂Rm) (R)m
Where we have neglected terms of order 2. Writing the Hessian as Hkm = (∂2E/∂Rk∂Rm) with (∂E/∂Rk)0 = 0, we get
Fk = - m Hkm (R)m = Mk (∂2Rk/∂t2) To find the normal modes we write (R)m = Am cos t leading to
Mk(∂2Rk/∂t2) = Mk 2 (Ak cos t) = m Hkm (Amcos t)
Here the coefficient of cos t must be {Mk 2 Ak - m Hkm Am}=0
There are 3N degrees of freedom (dof) which we collect together into the 3N vector, Rk where k=1,2..3NThe interactions then lead to 3N net forces, Fk = -(∂E(Rnew)/∂Rk) all of which are zero at equilibrium, R0
Now consider that every particle is moved a small amount leading to a 3N distortion vector, (R)m = Rnew – R0
Expanding the force in a Taylor’s series leads to
9© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
Solving for the Vibrational modes
The normal modes satisfy
{Mk 2 Ak - m Hkm Am}=0To solve this we mass weight the coordinates as Bk = sqrt(MkAk
leading to
Sqrt(Mk) 2 Bk - m Hkm [1/sqrt(Mm)]Bm}=0 leading to
m Gkm Bm = 2 Bk where Gkm = Hkm/sqrt(MkMm) G is referred to as the reduced HessianFor M degrees of freedom this has M eigenstates
m Gkm Bmp = kp Bk (2)p where the eigenvalues are the squares of the vibrational energies.If the Hessian includes the 6 translation and rotation modes then there will be 6 zero frequency modes
10© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
Saddle points
If the point of interest were a saddle point rather than a minimum, G would have one negative eigenvalue. This leads to an imaginary frequency
11© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
For practical simulations
• We can obtain reasonably accurate vibrational modes from just the classical harmonic oscillators
• N atoms => 3N degrees of freedom• However, there are 3 degrees for translation, n = 0 • 3 degrees for rotation for non-linear molecules, n = 0 • 2 degrees if linear• The rest are vibrational modes
12© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
Normal Modes of Vibration H2O
1595 cm-1
3657 cm-1
3756 cm-1
H2O D2O
1178 cm-1
2671 cm-1
2788 cm-1
Sym. stretch
Antisym. stretch
Bend
Isotope effect: ~ sqrt(k/M):Simple D/H ~ 1/sqrt(2) = 0.707:
Ratio: 0.730
Ratio: 0.735
Ratio: 0.742
More accurately, reduced massesH = MHMO/(MH+MO)D = MDMO/(MD+MO)Ratio = sqrt[MD(MH+MO)/MH(MD+MO)] ~ sqrt(2*17/1*18) = 0.728
Most accuratelyMH=1.007825MD=2.0141MO=15.99492Ratio = 0.728
13© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
13
•EM energy absorbed by interatomic bonds in organic compounds
•frequencies between 4000 and 400 cm-1 (wavenumbers)
•Useful for resolving molecular vibrations
http://webbook.nist.gov/chemistry/
http://wwwchem.csustan.edu/Tutorials/INFRARED.HTM
The Infrared (IR) SpectrumCharacteristic vibrational modes
14© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
Normal Modes of Vibration CH4
2917 cm-1 3019 cm-1 1534 cm-1CH4
CD4 1178 cm-1 2259 cm-1 1092 cm-1
Sym. stretch
1
Anti. stretch
3
Sym. bend
2
Sym. bend
3
A1 T2 E T2
1306 cm-1
996 cm-1
15© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
Fitting force fields to Vibrational frequencies and force constants
Hessian-Biased Force Fields from Combining Theory and Experiment; S. Dasgupta and W. A. Goddard III; J. Chem. Phys. 90, 7207 (1989)
H2CO
MC: Morse bond stretch and cosine angle bendMCX: include 1 center cross terms
16© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
The Schrödinger equation H =
for harmonic oscillator22
22
2
1
2kx
xm
The QM Harmonic Oscillator
energy
wavefunctions
reference http://hyperphysics.phy-astr.gsu.edu/hbase/shm.html#c1
17© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
Raman and IR spectroscopy
• IR– Vibrations at same frequency as radiation– To be observable, there must be a finite dipole derivative– Thus homonuclear diatomic molecule (O2 , N2 ,etc.) does not
lead to IR absorption or emission.
• Raman spectroscopy is complimentary to IR spectroscopy.– radiation at some frequency, n, is scattered by the molecule to
frequency, n’, shifted observed frequency shifts are related to vibrational modes in the molecule
• IR and Raman have symmetry based selection rules that specify active or inactive modes
18© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
IR and Raman selection rules for vibrations
The electrical dipole moment is responsible for IR
rrr 3),()( dtt
The polarizability is responsible for Raman
)()()( ttt For both, we consider transition matrix elements of
the form
iini Q
t
)(
|)(|'
,
The intensity is proportional to d/dR averaged over the vibrational state
where is the external electric field at frequency
19© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
IR selection rules, continued
• For IR, we expand dipole moment
....)( 00
ii i
We see that the transition elements are
The dipole changes during the vibration
Can show that n can only change 1 level at a time
iiii
nQnQ
||')( 0
20© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
Raman selection rules
• For Raman, we expand polarizability
....)( 00
ii i
substitute the dipole expression for the induced dipole
Same rules except now it’s the polarizability that has to change
For both Raman and IR, our expansion of the dipole and alpha shows higher order effects possible
=
21© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
21
•center of mass translation x= x y=0 z=0
x=0 y=y z=0
x=0 y=0 z=z•center of mass rotation (nonlinear molecules) x=0 y=-cx z=bx
x= cy y=0 z=-ay
x= -bz y=ax z=0•linear molecules have only 2 rotational degrees of freedom
•The translational and rotational degrees of freedom can be removed beforehand by using internal coordinates or by transforming to a new coordinate system in which these 6 modes are separated out
Both and V are constant =0
Translation and Rotation Modes
Both K and V are constant =0
22© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
Classical Rotations
• The moment of inertia about an axis q is defined as
)(2 qxmIk
kkqq xk(q) is the perpendicular distance to the axis q
Can also define a moment of inertia tensor where (just replace the mass density with point masses and the integral with a summation. Diagonalization of this matrix gives the principle moments of inertia!
the rotational energy has the form
qqqq
q qq
q
qqqqrot
IJ
I
JqIE
2)(
2
12
2
kk
kkk
m
mrR0
23© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
Quantum Rotations
The rotational Hamiltonian has no associated potential energy
zz
z
yy
y
xx
x
I
J
I
J
I
JH
222
222
JJJM
M
JJJK
J
KIII
JJMKJE
J
J
JJrot
,...,1,
,...,1,
,...2,1,0
)2
1
2
1(
2
)1(),,( 22
2
zJIII
JH )
2
1
2
1(
2
2
For symmetric rotors, two of the moments of inertia are equivalent, combine:
Eigenfunctions are spherical harmonic functions YJ,K or Zlm with eigenvalues
24© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
Transition rules for rotations
• For rotations– Wavefunctions are spherical harmonics– Project the dipole and polarizability due to rotation
• It can be shown that for IR– Delta J changes by +/- 1– Delta MJ changes by 0 or +/-1– Delta K does not change
• For Raman– Delta J could be 1 or 2– Delta K = 0– But for K=0, delta J cannot be +/- 1
25© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
Raman scattering
• Phonons are the normal modes of lattice vibrations (thermal + zero point energy)
• When a photon absorbs/emits a single phonon, momentum and energy conservation the photon gains/loses the energy and the crystal momentum of the phonon. – q ~ q` => K = 0– The process is called anti-Stokes for absorption and
Stokes for emission.– Alternatively, one could look at the process as a
Doppler shift in the incident photon caused by a first order Bragg reflection off the phonon with group velocity v = (ω/ k)*k
26© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
Raman selection rules
• For Raman, we expand polarizability
....)( 00
ii i
substitute the dipole expression for the induced dipole
Same rules except now it’s the polarizability that has to change
For both Raman and IR, our expansion of the dipole and alpha shows higher order effects possible
=
27© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
Another simple way of looking at Raman
)cos()cos(2
1)cos(2)(
)cos()cos(2
12)(
)cos()(2)()()(
intint00
0int0
0
tttttt
ttt
ttttt
Take our earlier expression for the time dependent dipole and expose it to an ideal monochromatic light (electric field)
We get the Stokes lines when we add the frequency and the anti-Stokes when we substractThe peak of the incident light is called the Rayleigh line
28© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
28
•The external EM field is monochromatic
•Dipole moment of the system
•Interaction between the field and the molecules
•Probability for a transition from the state i to the
state f (the Golden Rule)
•Rate of energy loss from the radiation to the
system
•The flux of the incident radiation
)ωcos(ε)( 0 tEtE
)(μ)( tEt
)]ωω(δ)ωω(δ[|με|2
π)ω(
220 fififi if
EP
fffi ωωω
)ω(ωρ)ω( fifii f
irad PE
20π8
Ecn
S c: speed of lightn: index of refraction of the medium
The Sorption lineshape - 1
29© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
29
•Absorption cross section ()
•Define absorption linshape I() as
•It is more convenient to express I() in the time domain
S
Erad )ω()ω(α
)ωω(δ|με|ρ3)1(ω4π
)ω(α3)ω(
2
ωβ2
fi
i fi if
e
cnI
dtet
dteiffiI
dte
ti
tEE
i
i fi
ti
if
ω
ω]-[
ω
)(μ)0(μ2π
1
|με||με|ρ2π
3)ω(
2π
1ω)(δ
I() is just the Fourier transform of the autocorrelation function of the dipole moment
ensemble average
Beer-Lambert law Log(P/P0)=bc
The Sorption lineshape - II
30© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
Non idealities and surprising behavior
• Anharmonicity – bonds do eventually dissociate
• Coriolis forces– Interaction between vibration
and rotation
• Inversion doubling• Identical atoms on rotation
– need to obey the Pauli Principle– Total wavefunction symmetric
for Boson and antisymmetric for Fermion
),()1(),( ,,
JJ MJJ
MJ
NVEN
YY
31© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07
31
Fig
ure
taken fro
m S
treitw
iser &
Heath
cock,
Intro
ductio
n to
Org
anic C
hem
istry, C
hapte
r 1
4, 1
97
6Electromagnetic Spectrum
How does a Molecule response to an oscillating external electric field (of frequency )? Absorption of radiation via exciting to a higher energy state ħ ~ (Ef - Ei)