Determination of Molecular Properties

1
Property Determination

• Weve seen how to evaluate the energy and
  approximate wave function for a given molecular
  geometry
• Molecular properties you can obtain directly from
  the wave function
• Atomic point charges
• Leading order term in an electrostatic potential
• Use in molecular simulations
• Try to explain anomolous chemistry
• General idea is to use known electron density
  function to assign charge to each atom
• Procedures vary by method

2
Property Determination

• Mulliken Population Analysis
• Widely-used, widely-criticized
• Begin with constraint that the wave function be
  minimized
• Easiest way is to force each fi to be normalized
• If the basis functions are normalized, we may
  write

3
Property Determination

• The previous expression may also be written as
• Let there be ni electrons in fi
• Define the number of electrons in cr due to
  orbital fi as nr,i ni ari2
• The number of electrons in the overlap region
  between cs and cr is given by
• Calculate the net population in cr (function on a
  single atom), and in overlap region

4
Property Determination

• If we want charges on atoms, what do we do about
  the overlap populations?
• Mulliken assumed that the overlap populations
  should split evenly
• Can now calculate the gross population in cr
• Charge on atom X is then given by
• Results are usually intuitive, but highly basis
  set dependent

5
Property Determination

• Calculated charges on H
• Meaningful comparisons between charges may only
  be made between calculations using the same model
  chemistry and basis set

6
Property Determination

• Other charge-determination algorithms
• Charges from Electrostatic Potentials
• CHELPG
• Merz, Singh, Kollman (MK,ESP,MSK)
• All have the same idea
• Calculate the electrostatic potential at a grid
  of points around the molecule
• Use a least-squares fit to find point charges
  which best mimic the calculated electrostatic
  potential
• Methods differ in number of grid points, and how
  to select them
• Less arbitrary than Mulliken, and more widely
  accepted
• Again, only meaningful to compare calculations
  using the same model chemistry and basis set

7
Property Determination

• Comparison of Mulliken and CHELPG charges
• Also shows basis set effects form dC,dO

Mulliken
CHELPG
8
Property Determination

• While the electrostatic potential and the
  electron density are properties of the wave
  function, atomic point charges are not
• Ultimately a futile task, as you are trying to
  match a continuous distribution of electron
  density (charge) with a series of d-functions
• However, as the results are often used in
  empirical functions or qualitatively, they can be
  quite useful

9
Property Determination

• A second property obtainable from the approximate
  wave function is the ionization potential
• Recall the definition of ei
• We can call ei an orbital energy
• Via Koopmans Theorem, the energy required to
  remove an electron from an orbital is -ei
  (technically not correct)

Energies in a.u.
10
Property Determination

• Multipole moments and polarizabilities
• Apply a uniform external electric field to the
  molecule and ask for the effect on the energy of
  the system
• Use perturbation theory to obtain
• Q is the strength of the electric field, and the
  operator d is the electric dipole-moment operator
• The answer we want is

11
Property Determination

• We already have y0, so this is a quick
  calculation
• m is the first-order correction to the system
  energy due to the perturbation of the electric
  field
• The second-order correction yields the
  polarizability
• Requires calculating the perturbation to the wave
  function
• Formula not so straight forward
• Comparison of dipole moments for various basis
  sets
• Results are general In the limit of infinite
  basis set, H-F overestimates dipole moments

12
Property Determination

• Electrostatics
• Mulliken charges nearly free with SCF energy
  calculation
• Electrostatic-potential based charges have a
  minimal cost
• Dipole moments are nearly free with SCF energy
  calculation
• Unlike charge distributions, dipole moment is a
  property of the wave function
• Quadrupole and higher moments are also nearly
  free
• Same routine as with dipole moments, different
  operator
• Polarizabilities are nearly free with a harmonic
  frequency calculation

13
Property Determination

• All the previous calculations assumed the nuclear
  configuration was known
• Calculations took place at a fixed molecular
  geometry
• Where does one obtain geometry?
• Experimental/crystallographic
• Find the minimum of your model chemisty energy
  surface
• Full molecular energy, not just the electronic
  energy
• Nuclear motion re-introduced to the problem

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Property Determination

• Nuclear motion is only introduced formally No
  need to try to uncouple correlation of nuclear
  motion
• Instead, look at energy equation
• Energy minimization corresponds to solving
• subject to the 2nd derivative matrix being
  positive definite

15
Property Determination

• This will turn out to involve derivatives of the
  (rstu) and (ruts) integrals, along with spatial
  derivatives of VNN
• For gaussian-type basis functions, these
  derivatives are analytical, and many algorithms
  are available to efficiently find a minimum
• Procedure
• Solve SCF equations at a given point
• Calculate derivatives with respect to the nuclear
  coordinates
• Step to new point
• Repeat until derivative convergence criteria is
  met
• From this calculation, we can determine the
  optimum bond lengths, bond angles, and dihedral
  angles in a molecule

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Property Determination

• Performance for a couple of small molecules
• Water
• Ammonia

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Property Determination

• General Results
• A-H bonds mean absolute errors (Hehre, Radom,
  v.R. Schleyer, and Pople, 1987)
• A-B bonds

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Property Determination

• The extension of this method to transition states
  is mathematically simple, but algorithmically
  difficult
• Energy minimizations converge readily from most
  reasonable starting points
• Transition states require starting points in much
  closer proximity to desired end point to achieve
  convergence

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Property Determination

• Evaluation of molecular energies
• Usually not terribly useful by themselves, but
  differences between molecular energies are
• reaction energies (thermodynamic)
• barrier heights (kinetic)
• conformational equilibria
• Basis set effects on molecular energies
• Water
• 1 Hartree627.5 kcal/mol
• Difference is the correlation
• energy

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Property Determination

• Formaldehyde
• Hartree-Fock methods do not
• accurately reproduce molecular
• energies
• However, errors are consistent across many
  molecules
• All singly-bonded carbons have a particular
  error, all doubly-bonded oxygens have a
  particular error, etc.
• Differences between molecular energies can be
  accurate

21
Property Determination

• Isodesmic reactions
• Isodesmicsame and type of bonds on both sides
  of reaction
• Reaction energies (kcal/mol) are corrected for
  zero-point vibration
• The quantity compared is DE at 0 K, and in a
  vacuum
• Using vibrational frequencies, rotational
  temperatures, and translational constants, we
  could compare results to enthalpy changes at some
  finite temperature

22
Property Determination

• A second type of quantity where the number and
  types of bonds do not change are calculated
  barriers to rotation and inversion
• H-F predicted rotational barriers (staggered vs.
  eclipsed)
• Energies found by optimizing geometry subject to
  a constraint on the angle of relative rotation
  between the two halves of the molecule

23
Property Determination

• Barrier to inversion trigonal pyramid vs.
  planar configurations, tetrahedral vs. planar
• Optimize geometry with constraints
• General conclusion H-F performs well for
  problems where the number and types of bonds do
  not change during reaction

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Property Determination

• But, when the number or types of bonds do change
  ...
• Bond Dissociation Reactions A Failing of H-F
  Theory

• Inaccuracy stems from poor description of pairs
  of bonding
• electrons
• Underestimate the correlation, underestimate
  the amount of
• energy needed to split bond
• One would also expect poor description of
  transition states

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Property Determination

• In Summary For H-F using at least a 6-31g
  basis set
• Bond lengths and angles are quite accurate (1)
• Vibrational frequencies systematically high by
  10-12
• Isodesmic reaction energies are accurate to 2-4
  kcal/mol
• Zero-point vibrational energies accurate to 1
  kcal/mol (post-scaling)
• Protonation/Deprotonation reactions accurate to
  10 kcal/mol DE O(100 kcal/mol)
• Atomization and homolytic bond-breaking have
  large errors (25-40 kcal/mol)
• Reaction barriers have large errors (Energies and
  geometries)

26
Property Determination

• Harmonic frequencies
• Used to characterize stationary points
• all positive minimum
• one negative transition state
• Used with statistical mechanics formulas for
  thermochemistry
• all calculations are performed at 0 K and in a
  vacuum
• use frequencies to correct for zero-point
  vibration, finite temperature effects, entropies
• Approximate molecular energy curve in the
  neighborhood of a stationary point as a parabola
• Working equations resemble those for the harmonic
  oscillator
• Solution requires second derivatives of the
  (rstu) and (ruts) integrals, along with the
  second derivatives of VNN

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Property Determination

• Harmonic frequencies
• For H-F theory, the second derivatives are
  analytic
• Generally, most computationally intensive part of
  a study
• In practice, scales as N3.5 where N is the number
  of basis functions
• Typical Results
• Water
• It has been found that H-F frequencies are
  generally 10-12 too high
• Standard suggested scale factor is 0.8929

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