Molecular Properties

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Molecular Properties
Computational Chemistry 5510 Spring 2006 Hai Lin
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Population Analysis

• What is an atom in a molecule?
• Classical View balls and sticks
• Quantum View positive nuclei surrounded by
  negative electron cloud (ambiguous definition of
  atoms)

• Population analysis
• Try to define an atom in a molecule and its
  associated electron population.
• ri (r) fi2(r)
• Methods
• Based on basis functions
• Fitting to target data
• Based on wave function properties

e-
e-
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Density Matrix

• Occupation number ni
• Number of electrons in the i-th molecule orbital
• Can be 0, 1, or 2 (single determinant wave
  function)
• Can be fractional numbers (correlated wave
  function)
• Density matrix D
• Electron density distribution in terms of atomic
  orbitals
• Summation of the product of MO coefficient and
  occupation number

• Overlap matrix S
• Overlap between atomic orbitals
• Sab ?cacb?

Total number of electrons
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Mulliken Löwdin Analyses

• Mulliken Analysis
• For a contribution involving two basis functions
• centered at the same nucleus A, attribute the
  contribution to the atom A.
• centered at two nuclei A and B, equally (5050)
  partition the contribution to atoms A and B.
• Löwdin Analysis
• Analysis of the density matrix in a orthogonal
  basis set
• Gross charge on atom A
• QA ZA - rA

Electron contribution
Nuclear charge
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Mulliken Löwdin Analyses (2)

• No solid theoretical justifications
• Can not be compared with experimental results.
  (Atomic charges are not experimentally measurable
  quantities!)
• Results depend heavily on basis sets. (Increasing
  the size of basis sets can bring even worse
  results!)
• Dipole, quadrupole etc moments are in general
  not conserved (E.g., the atomic charges produce
  different dipole moment from what the nuclei and
  electron wave functions do.)
• Some strange results (e.g., C atom have a
  charge of -2 e) can be obtained. In this regard,
  the Löwdin analysis is better than the Mulliken
  analysis.

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Electrostatic Potential Fit

• Find a set of atomic charges to reproduce the
  electrostatic potential (ESP) generated by the
  nuclei and electron wave function.

• Depends less on basis sets than Mulliken Löwdin
  analyses do.
• Depends on conformation (geometries).
• Charges for buried atoms are poorly determined
  (the ESP are mainly determined by atoms close to
  the surface).
• Accuracy of the charges are questionable to some
  extent (various sets of charges that are very
  different from each other can reproduce ESP
  nearly equally well.)

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Distributed Multipole Analysis

• The electrostatic potential arising from charge
  overlaps between basis functions can be expressed
  as a multipole expansion (charge, dipole,
  quadrupole etc.) around a point between two
  nuclei.
• In practice, locations of the multipoles can be
  distributed at atomic center or other locations
  (e.g., mid-point of a bond).

F
H
M
F
H

• More faithful representation of the charge
  distribution within a molecule that other methods
  using solely point charges.
• Computationally more expensive and less popular.

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Atoms in Molecules Analysis
r

• Based on the topology of electron density (the
  square of wave function) in spatial coordinates.
• Devide the electron density by surfaces where
  radcial gradients of all points are zero.
• Point charges derived do not necessarily
  reproduce the ESP.
• Sometimes gives unusual point charges.

R
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Example of Atomic Charges

• Carbon in CH4

• Oxygen in H2O

(Taken from textbook Page 233 All calculations
at the HF level of theory)
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Derivatives of Energy

• Molecular property measures the how the molecular
  energy changes w/r to perturbations.
• Internal perturbations geometric distortions
• E(R) E(R0) (?E/?R) (R - R0) ½ (?E2/?2R)
  (R - R0)2

• External perturbations electric fields, etc.
• E(F ) E(0) (?E/?F ) F ½ (?E2/?2F ) F 2

• Mixed perturbations vibrational intensities, etc.

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Analytic Calculations

• Some properties can be calculated analytically
  (fast!).
• Gradient Hessians
• Very important, allowing efficient gemoetry
  optimizations
• Not avaliable for all methods, esp. high level ab
  intio methods
• Dipole Moment
• Discribe the charge distributions of a system
• m ?yry?

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Numerical Calculations

• Finite Difference Procedures (expensive!)
• Gradient Hessians
• Slightly distort the geometric and see how energy
  changes.
• g E(r1) E(r0) / (r1 r0)
• Often the only way in high level ab intio
  calculations
• Dipole Moment
• Apply a small external electric field, see how
  energy changes.
• m E(F1) E(F0) / (F1 F0)
• Too large perturbation inaccurate results
• Too small perturbation subject to rounding
  errors

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Vibrational Frequencies

• Harmonic frequencies for normal-mode vibrations
  can be obtained from the Hessian at stationary
  points.
• Experimentally one measures the vibrational
  fundamental (or overtone) frequencies, which
  contain contributions due to anharmonicity.

E

• For a fair comparison, one should
• either derive the harmonic frequencies from
  experiments
• or calculate fundamental (or overtone)
  frequencies with inclusion of anharmonicity.

n1
n0
R
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Scaling the Frequencies

• Typically the calculated harmonic frequencies are
  systematically higher than experimental data.
• Apply scaling factors to calculated frequencies
  to bring them close to experimental data
• Select the proper scaling factors
• Different levels of theory HF, MP2, B3LYP,
  CCSD(T), etc.
• Different purpose harmonic frequencies,
  zero-point energies, etc.
• ZPE ½ ?wi
• Different modes stretches, bends, and torsions.
• Be careful! It is common that people scale the
  calculated harmonic frequencies to agree with
  experimental fundamental frequencies ? e.g., the
  widely used scaling factor of 0.8929 for the
  HF/6-31G(d) level of theory. See comments by A.
  P. Scott and L. Radom, J. Phys. Chem., 100 (41),
  16502 -16513.

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Frequencies Scaling Factors
Scale calculated harmonic frequencies to agree
with experimental fundamental frequencies or to
make zero-point energy agree with experiments.
Scaling factors in A. P. Scott L. Radom, J.
Phys. Chem., 100 (41), 16502 -16513.
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Very Anharmonic Cases

• Sometimes the harmonic approximation can be bad.
• Very flat potential energy surface
• (Large-amplitude motions of a protein backbone)
• Minima separated by low barriers
• (Torsions at the transition state inversion of
  NH3)
• With strong anharmonic coupling
• (CH stretch and bend in CHF3)

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IR Intensities

• Double Harmonic Approximation
• Potential energy surface is harmonic.
• E(Dr) k Dr2
• Dipole moment surface is linear.
• m(Dr) m0 a Dr
• Only fundamental transitions carry intensities.
• More realistic model will include both
  anharmonicities in potential energy and dipole
  moment.

E or m
Dr
E or m
Dr
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Summary

• Occupation Number, Density Matrix, and Overlap
  Matrix
• Population Analysis and Atomic Charges
• Mulliken Analysis
• Löwdin Analysis
• Electrostatic Potential Fit Analysis
• Distributed Multipole Analysis
• Atoms in Molecules Analysis
• Molecular Properties
• As Energy Derivatives
• Analytic vs. Numerical Calculations
• Vibrational Analysis
• Scaling Factor
• IR Intensities

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Your Homework

• Read the slides. If you have difficulty in
  understanding the math in the slides, find a
  reference.
• Read textbook (Take notes when you read.)
• 9.1, 9.2, 9.3, 9.7, and 9.8.
• 10.1.
• Questions
• What do an occupation number, a density matrix,
  and an overlap matrix describe?
• What are the problems in Mulliken, Löwdin, and
  ESP-fit analyses?
• How to calculate a dipole moment numerically?
• What should one do before comparing calculated IR
  spectra with experimental results?