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Ch121a Atomic Level Simulations of Materials and MoleculesPowerPoint Presentation

Ch121a Atomic Level Simulations of Materials and Molecules

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### Ch121a Atomic Level Simulations of Materials and Molecules

BI 115

Hours: 2:30-3:30 Monday and Wednesday

Lecture or Lab: Friday 2-3pm (+3-4pm)

Lecture 7, April 15, 2011

Molecular Dynamics – 3: vibrations

William A. Goddard III, wag@wag.caltech.edu

Charles and Mary Ferkel Professor of Chemistry, Materials Science, and Applied Physics, California Institute of Technology

Teaching Assistants Wei-Guang Liu, Fan Lu, Jose Mendoza, Andrea Kirkpatrick

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

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

Vibration in one dimension – Harmonic Oscillator

No friction

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, w, 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=d = A cos(wt) then

–Mw2 Acos(wt) = -k A cos(wt)

Hence –Mw2 = -k or w = Sqrt(k/M).

Stiffer force constant k higher w and

higher M lower w

E= ½ k d2

Reduced Mass

M2

M1

Put M1 at R1 andM2 at R2

CM = Center of mass

Fix Rcm = (M1R1 + M2R2)/(M1+ M2) = 0

Relative coordinate R=(R2-R1)

Then Pcm = (M1+ M2)*Vcm = 0

And P2 = - P1

Thus KE = ½ P12/M1 + ½ P22/M2 = ½ P12/m

Where 1/m = (1/M1 + 1/M2) or m = 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, m

For molecules the energy is harmonic near equilibrium but for large distortions the bond can break.

(n + ½)2

Successive vibrational levels are closer by

(n + ½)3

(n + ½)2

Exact solution

Real potentials are more complex; in general:

(Philip Morse a professor at MIT, do not manufacture cigarettes)

The simplest case is the Morse Potential:

Now on to multiple atoms for large distortions the bond can break.

Eigenvalue problem

or

N atoms => 3N degrees of freedom

However, 3 degrees for translation, get l = 0

3 degrees for rotation is non-linear molecule, get l = 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

Vibration for a molecule with N particles for large distortions the bond can break.

There are 3N degrees of freedom (dof) which we collect together into the 3N vector, Rk where k=1,2..3N

The 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, (dR)m = Rnew –R0

Expanding the force in a Taylor’s series leads to

Fk = -(∂E(Rnew)/∂Rk) = -(∂E/∂Rk)0 - Sm (∂2E/∂Rk∂Rm) (dR)m

Where we have neglected terms of order d2.

Writing the Hessian as Hkm = (∂2E/∂Rk∂Rm) with (∂E/∂Rk)0 = 0,

we get

Fk = - Sm Hkm (dR)m = Mk (∂2Rk/∂t2)

To find the normal modes we write (dR)m = Am cos wt leading to

Mk(∂2Rk/∂t2) = Mkw2 (Ak cos wt) = Sm Hkm (Amcos wt)

Here the coefficient of cos wt must be {Mkw2 Ak - Sm Hkm Am}=0

Solving for the Vibrational modes for large distortions the bond can break.

The normal modes satisfy

{Mkw2 Ak - Sm Hkm Am}=0

To solve this we mass weight the coordinates as Bk = sqrt(Mk)Ak leading to

Sqrt(Mk) w2 Bk - Sm Hkm [1/sqrt(Mm)]Bm}=0 leading to

Sm Gkm Bm = w2 Bk where Gkm = Hkm/sqrt(MkMm)

G is referred to as the reduced Hessian

For M degrees of freedom this has M eigenstates

Sm Gkm Bmp = dkp Bk (w2)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

Saddle points for large distortions the bond can break.

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

For practical simulations for large distortions the bond can break.

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

Normal Modes of Vibration H for large distortions the bond can break. 2O

H2O

D2O

Sym. stretch

3657 cm-1

2671 cm-1

Ratio: 0.730

1595 cm-1

1178 cm-1

Bend

Ratio: 0.735

Antisym. stretch

3756 cm-1

2788 cm-1

Ratio: 0.742

Isotope effect: n ~ sqrt(k/M):

Simple nD/nH ~ 1/sqrt(2) = 0.707:

More accurately, reduced masses

mOH = MHMO/(MH+MO)

mOD = MDMO/(MD+MO)

Ratio = sqrt[MD(MH+MO)/MH(MD+MO)]

~ sqrt(2*17/1*18) = 0.728

Most accurately

MH=1.007825

MD=2.0141

MO=15.99492

Ratio = 0.728

The Infrared (IR) Spectrum for large distortions the bond can break.

Characteristic vibrational modes

- EM energy absorbed by interatomic bonds in organic compounds
- frequencies between 4000 and 400 cm-1 (wavenumbers)
- Useful for resolving molecular vibrations

13

http://webbook.nist.gov/chemistry/

http://wwwchem.csustan.edu/Tutorials/INFRARED.HTM

Normal Modes of Vibration CH for large distortions the bond can break. 4

1

3

2

3

Sym. stretch

Anti. stretch

Sym. bend

Sym. bend

A1

T2

E

T2

3019 cm-1

CH4

2917 cm-1

1534 cm-1

1306 cm-1

CD4

2259 cm-1

1092 cm-1

996 cm-1

1178 cm-1

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)

MC: Morse bond stretch and cosine angle bend

MCX: include 1 center cross terms

H2CO

The QM Harmonic Oscillator constants

The Schrödinger equation H = e

for harmonic oscillator

energy

wavefunctions

reference http://hyperphysics.phy-astr.gsu.edu/hbase/shm.html#c1

Raman and IR spectroscopy constants

- 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

IR and Raman selection rules for vibrations constants

The intensity is proportional to dm/dR averaged over the vibrational state

The polarizability is responsible for Raman

where e is the external electric field at frequency n

- For both, we consider transition matrix elements of the form

The electrical dipole moment is responsible for IR

IR selection rules, continued constants

- 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

For IR, we expand dipole moment

Raman selection rules constants

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

For Raman, we expand polarizability

Translation and Rotation Modes constants

Both K and V are constant l=0

- center of mass translation
- Dxa= DxDya=0 Dza=0
- Dxa=0 Dya=Dy Dza=0
- Dxa=0 Dya=0 Dza=Dz
- center of mass rotation (nonlinear molecules)
- Dxa=0 Dya=-caDqxDza=baDqx
- Dxa= caDqyDya=0 Dza=-aaDqy
- Dxa= -baDqzDya=aaDqxDza=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 K and V are constant l=0

21

Classical Rotations constants

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

The moment of inertia about an axis q is defined as

Quantum Rotations constants

For symmetric rotors, two of the moments of inertia are equivalent, combine:

Eigenfunctions are spherical harmonic functions YJ,K or Zlm with eigenvalues

The rotational Hamiltonian has no associated potential energy

Transition rules for rotations constants

- 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

Raman scattering constants

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

Raman selection rules constants

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

For Raman, we expand polarizability

Another simple way of looking at Raman constants

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 substract

The peak of the incident light is called the Rayleigh line

The Sorption lineshape - 1 constants

- 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

c: speed of light

n: index of refraction of the medium

28

The Sorption lineshape - II constants

- Absorption cross section a(w)
- Define absorption linshape I(w) as
- It is more convenient to express I(w) in the time domain

Beer-Lambert law Log(P/P0)=abc

I(w) is just the Fourier transform of the autocorrelation function of the dipole moment

ensemble average

29

Non idealities and surprising behavior constants

- 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

Electromagnetic Spectrum constants

How does a Molecule response to an oscillating external electric field (of frequency w)? Absorption of radiation via exciting to a higher energy state ħw ~ (Ef- Ei)

Figure taken from Streitwiser & Heathcock, Introduction to Organic Chemistry, Chapter 14, 1976

31

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