Transcript Computational Spectroscopy

Computational Spectroscopy
Introduction and Context
Chemistry 713
(a) What is spectroscopy?
(b) Model Chemistries
What is Spectroscopy?
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A plot of light intensity or power as a function of
frequency, , or wavelength, 
Speed of light in a material with index of refraction
n is c/n=. (n=1 in a vacuum.)
Emission or absorption of light by atoms and
molecules depends on frequency.
For absorption and emission, the energy of a photon
of light E=h is equal to the difference in energy
between molecular energy levels: h = E2 - E1
For scattering, both energy and momentum (p=h/)
are conserved.
In certain circumstances, electrons or other particles
(=h/p) are used instead of photons.
What kinds of spectroscopy?
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Linear spectroscopy
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Spectral intensities are proportional to the intensity of the light at the
indicated frequency.
Nonlinear spectroscopies
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Spectral intensities are NOT proportional to the intensity of the light at the
indicated frequency, or
Spectral intensities depend on the light intensities at more than one
frequency. (e.g., 2-D and 3-D spectroscopies).
K=1->2
K=0->1
E1
E2
K<0
B
A
K<0
A linear absorption spectrum
of methylamine
1 n
(wavenumber is   
)
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K>0
E1
E1
K>0
B
B
E2
E1 E2
K>0
E1
K<0
A
A
K>0
K=2->3
K<0
c
2985
2990
2995
Wavenumber / cm
-1
3000
What kinds of molecular motions?
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Rotational spectra (gas phase)
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Vibrational spectra
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Determined by molecular geometry
Microwave and far infrared regions.
The electronic wavefunctions determine the chemical bonding, which in
turn determines the force constants between the atoms and hence the
vibrational frequencies.
Accompanied by rotational transitions (gas phase).
Absorption in the infrared region and Raman scattering.
Electronic Spectra
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Transitions between electronic states are accompanied by vibrational and
rotational transitions.
Visible and ultraviolet regions
Spectra from intrinsic spin
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Nuclear magnetic resonance (NMR) spectroscopy
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Electron spin resonance (ESR) spectroscopy
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Nuclear spins (I>0), when placed in a magnetic field, can have
only certain orientations relative to the magnetic field and these
orientations have different energies.
Transitions between these energy levels, typically in the radio
frequency region, give rise to NMR spectra.
Molecules with unpaired electron spins
Transitions between different orientations of electronic angular
momenta are in the microwave region for typical magnetic fields.
This course will focus on rotational, vibrational, and
electronic spectra, but projects on NMR spectra are
possible.
Diffraction of X-rays and electrons
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Measurement of scattered intensities vs scattering angle.
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Provides information about the relative positions of atoms in
condensed phases.
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X-ray diffractions of crystalline
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Relies on conservation of momentum rather than energy.
solids gives precise relative geometries of atoms (Youngs’ course
3150:645).
X-ray diffraction of liquids, polymers, and amorphous materials gives the
pair distribution function.
Low energy electron diffraction (LEED) gives the structure of ordered
surfaces.
Electron diffraction can give the structure of isolated gas molecules.
Diffraction per se is not covered in this course, but we will compute
molecular geometries.
Computation of Spectra
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First calculate the relevant molecular properties:
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Molecular geometry for rotational spectra
Force constants for vibrational spectra
Electronic energy states for electronic spectra.
Second, use this information to calculate transition
frequencies and intensities, that can be compared to
experiment.
Why bother to compute
molecular spectra?
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Spectroscopy is our most powerful means of gaining information about
the molecular world.
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Molecular structure and properties
Molecular dynamics
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Detection and quantification of molecular species
The desired information is not given directly, but is coded in the
spectra, often in a complicated way.
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Chemical reactions, energy transfer, protein folding, …
Computation of spectra helps to break the code.
Provide a deeper understanding than allowed by “rule of thumb”
interpretations of spectra.
Computational methods are now exceptionally powerful and are
increasingly accessible to every chemist.
Model Chemistries
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A computational model is defined by the concepts and
assumptions that go into it.
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Ideally defined in a general way that could be applied to any
chemical system, and reproducible by any user.
Specifically, the means used to calculate the molecular structure,
properties, and potential energy surface(s).
Molecular mechanics
Electronic structure methods
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Semi-empirical methods
Density functional theory
Ab initio methods
Foresman & Frisch, pp 3ff
Model Chemistries
Molecular Mechanics
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Based on an empirical potential energy hypersurface that
determines the forces exerted on the atoms in any
particular molecular geometry.1
Fxi x1, x 2 ,..., x 3N   
V
x i x ,x ,..., x
1
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2
3N
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V(x1,x2,…x3N) is given by an equation involving
the distances and angles between the N atoms
3
2
++
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 the atom types, e.g., H, C(sp ), C(sp ), Ca , etc.
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a set of parameters estimated from experimental data
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includes covalent, electrostatic, and van der Waals
interactions, and possible solvent effects
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defined in Sybyl, AMBER, MM3 2, CHARM 3, etc.
software packages
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From the front cover of “Energy
Landscapes with Applications to
Clusters, Biomolecules, and
Glasses”, by David J. Wales,
Cambridge University Press, 2003
1. There is a great description of the molecular mechanics method by John Wampler of the U of Georgia on the
web at www.bmb.uga.edu/wampler/399/lectures/mm1
2. Allinger, N. L., Yuh, Y. H., & Lii, J-H. (1989) Molecular Mechanics. The MM3 Force Field for Hydrocarbons. 1. J.
Am. Chem. Soc. 111, 8551-8565.
3. Brooks, B.R., Bruccoleri, R.E., Olafson, B.D., States, D.J., Swaminathan, S., Karplus, M. CHARMM: A program
for macromolecular energy, minimization, and dynamics calculations. J. Comp. Chem. (1983) 4, 187-217.
Model Chemistries
Molecular Mechanics
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Complicated molecules may have dozens,
thousands, or millions of minima.
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To find “the” molecular structure:
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Fig. 5.4, p 252 from Wales. On left are 1-D potential
energy surfaces, and on left are the corresponding
“disconnectivity” graphs.
How do we find “the real structure”?
The biologically active form of a protein may or
may not be the global minimum structure.
Often molecules are dynamic, sampling multiple
conformations during a process of interest.
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Start the molecule relatively hot, so that it can get
over the various saddle points.
Integrate the equations of motion of the atoms by
classical mechanics for a LONG time.
Assume that the molecules sample all of the relevant
geometries during that time. (ergodic assumption)
Gradually take away energy (anneal it) until the
molecule settles down into one or a few geometries.
Model Chemistries
Molecular Mechanics
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Advantages
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Very large systems can be treated because classical mechanics is
efficient and the computational effort for sufficiently systems
scales linearly with the system size.1
Pitfalls
 The empirical
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force field may not be calibrated for the system type of interest,
nor sufficiently accurate for the properties of interest.
The ergodic hypothesis may fail and consequently, the global
minimum or other important conformations may not be found.
1. C. J. Cramer, Essentials of Computational Chemistry - Theories and Models, Wiley 2004, p 14.
Model Chemistries
Electronic Structure Methods
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Semi-empirical methods
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Density Functional Theory (DFT)
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Some quantum mechanics + experimental data
Quantum mechanics + empirical exchange and correlation
functionals
Almost an ab initio method
Ab initio methods
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ab initio = “from the beginning”
Relies only on quantum mechanics + a small number of
fundamental constants:
h, c, and masses & charges of electrons and nuclei
Model Chemistries - Electronic Structure Methods
Semi-empirical Methods
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A very simplified quantum treatment of the bonding
The needed integrals are not evaluated explicitly, but are treated as parameters
to be calibrated from experimental data on reference systems.
Hückel theory treats only the  electrons and the strength of the interaction
between adjacent p orbital is taken as a parameter that can be varied to fit
experimental data.
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Gives a qualitative, and often semi-quantitative, treatment of aromaticity.
More sophisticate semi-empirical methods, e.g., AMI, MINDO/3, PM3, are
included in such packages as Gaussian, Spartan, and HyperChem.
Not extensively used any more because
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DFT is reasonably fast even for fairly large systems
Faster computers have made DFT and ab initio method tractable for larger systems
For very large systems empirical potentials are now better parametrized.
Ru(terpy)2 dimer
HOMO
Empirical
+ ab initio
MM2 geometry;
HF/3-21G Orbital
Model Chemistries - Electronic Structure Methods
Density Functional Theory (DFT)
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Strategy is to model electron correlation via general functionals of the electron density.
A functional is a function whose definition is also a function, that is, a function of a function.
Hohenberg-Kohn theorem (Phys. Rev. 136, B864 (1964)) says that the ground state energy is
equal to a functional of the electron density: E0=E0[0]
The problem is that it doesn’t tell us what the functional is, so we have to guess.
Strategy: calculate the energy without electron correlation first by ab initio means (HartreeFock (HF)).
Then use the resulting wavefunctions to get the electron density and come up with a
functional to treat only the contribution of electron correlation.
Some functionals depend only on the local density (Local Density Approximation (LDA));
others include the gradient of the density at each point. (gradient-corrected functionals).
In practice DFT calculations are not done just as a single step, but uses an iterative approach
analogous to the Hartree-Fock self-consistent field method.
The Becke 3 - Lee, Yang, Parr (B3LYP) functional is one of the most popular.
Many new functionals are reported in the literature.
The computer time required is only a little longer than Hartree-Fock - if it is coded
efficiently.
Model Chemistries - Electronic Structure Methods
ab initio Methods
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For a given molecular geometry (i.e., fixed nuclear coordinates, R), solve the
electronic Schroedinger equation:
H e e r;R  E R e r;R
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where He is the whole molecular Hamiltonian except the nuclear kinetic
energy and r represents the coordinates for all of the electrons, and e is the
electronic wave function. Repeat for all molecular geometries R of interest.
is called the Born-Oppenheimer approximation, and the relevant theory
This
can be found in Ira N. Levine, Quantum Chemistry, 5th ed, Prentice Hall
(2000).
The B.O. approx. is not perfect, but it is always used, at least as a starting
point, because we are helpless without it.
The nuclear motion can then be solved as a separate step. The electronic
energy E(R) is the potential energy in which the nuclei move.
The electronic Schroedinger equation is too difficult to be solve exactly, so
approximate methods must be used.
Model Chemistries - Electronic Structure Methods
ab initio Methods
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The repulsion between electrons is the term that causes the most
problem. This means that the motions of the electrons are intricately
correlated in a very complicated way.
As a practical matter, we still treat one electron at a time, moving in
the average field of the other electrons:
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Start by guessing the wavefunction of each electron, and use it to calculate
the average repulsive field of each electron.
One at a time, solve for the wavefunctions of each electron to get better
1-electron wavefunctions.
Recalculate the average repulsive field of each electron, and repeat the
process until a self-consistent field is obtained.
When the effects of the Pauli Principle (exchange interactions) are
included, this is call the Hartree-Fock Self-Consistent Field (HF) method.
The fact that electron motions are correlated is completely neglected by
the HF method.
Model Chemistries - Electronic Structure Methods
ab initio Methods
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More advanced methods treat electron correlation at some level of approximation.
DFT - quick, but we have to guess what density functional to use.
MP2 - Möller-Plesset Perturbation Theory
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MP4 - Möller-Plesset Perturbation Theory, 4th order
QCISD(T) - quadratic configuration interaction including single and double
excitations, and with triples treated perturbatively.
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Faster than QCISD(T)
Sometimes called “the gold standard” for single reference problems.
CASSCF - multi-reference method
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Multiple electron configurations can be included to account for electron correlation.
VERY time consuming.
CCSD(T) - coupled cluster theory including single and double excitations, and
with triples treated perturbatively.
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Electron correlation is treated by 2nd order perturbation theory
Good when multiple inequivalent resonance hybrids are important.
CASPT2 - a multi reference method with electron with a perturbative treatment of
electron correlation.
Model Chemistries - Electronic Structure Methods
ab initio Methods
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Even for the Hartree-Fock method, we can’t solve the electronic Schroedinger
equation exactly.
Express the wavefunction for each electron as a linear combination of atomiclike orbitals on each atom - called the “basis set”- then use variation method.
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The Hamiltonian becomes a MATRIX, that we will DIAGONALIZE to find
approximate eigenvalues (energies) and eigenvectors (wavefunctions).
Available basis sets include
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Just like vectors in 3-dimensions are expressed as linear combinations of the i, j, k
unit vectors
Wavefunctions are VECTORS in an infinite dimensional vector space.
Approximate with a finite number of basis vectors (= functions).
Split-valence basis sets: 3-21G, 6-31G*, 6-311++G**, etc
Dunning correlation-consistent basis sets: cc-pVDZ, aug-cc-pVTZ, etc
The larger the basis set, the more accurate the result and the more computer
time required.