General Performance Overview of Basic NDDO Models

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General Performance Overview of Basic NDDO Models
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Performance of NDDO Models

• Energetics Primary energetic observable is heat
  of formation
• NDDO Neglect of diatomic differential overlap
• MNDO Modified neglect of differential overlap
• AM1 and PM3

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Performance of NDDO Models

• Table 5.2 unsigned errors (kcal/mol)

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Performance of NDDO Models

• AM1 and PM3 have grater accuracies
• PM3 appears to be the best model
• Greatest advantage Hypervalent molecules, e.g.
  IF7, PBr5.

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Performance of NDDO Models

• Analysis of the errors show, that they are random
  
• They reflect the noise introduced to the Schr.
  Eq. by the NDDO approximations
• Problem Comparing energies between isomers
  there is no guarantee that the errors will cancel
• Problem The extra electron of an anion is
  occupying the same STO (valence orbital) as an
  uncharged molecule -gt anomalously high energy is
  computed
• Problem Radicals are calculated too stable -gt
  bond dissociation energies are too low

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Performance of NDDO Models

• Another energetic quantity of interest is
  ionization potential (IP)
• Koopmans Theorem The negative IP is the energy
  of the highest occupied molecular orbital
• This enables a reasonable prediction of IP for
  organic molecules
• Errors are in the range of 0.5-0.7 eV
• For inorganic molecules PM3 is still applicable,
  while MNDO and AM1 have increased errors

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Performance of NDDO Models

• Energetics associated with conformational changes
  and reactions
• MNDO have some shortcomings
• Steric crowding is too strongly disfavoured -gt
  unrealistically high heats of formation for
  sterically packed molecules and reactions with
  crowded TS structures
• Small ring structures are predicted too stable
• Semiempirical methods are in general unreliable
  when calculating weak interactions, like
  dispersion forces (London forces) or hydrogen
  bonding
• Dispersion forces are electron-correlation forces
  -gt HF methods are based on neglecting
  correlation!
• Hydrogen bonding was the primary reason for
  expanding MNDO to AM1 and PM3 -gt Interactions in
  the water dimer. However, in other systems the
  methods still fail!
• Energetic barriers to rotate around bonds having
  partial double bond character tend to be
  significantly too low 15 kcal/mol
  underestimation

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Performance of NDDO Models

• Geometries
• Correct molecular structures are dependent on the
  proper location of minima in the energy landscape
  -gt energetics of conformation
• Details are modeled with a reasonable degree of
  accuracy -gt bond lengths and angels can have
  errors

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Performance of NDDO Models

• Organic molecules H, C, N, O, F, Cl, Br, I
• Bondlength errors
• AM1 0.027 Å
• pM3 0.022 Å
• Angle Errors
• AM1 2.3
• PM3 2.8

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Performance of NDDO Models

• Other molecules Al, Si, P, S
• MNDO 0.054 Å 4.3 (up to 9)
• AM1 0.050Å 3.3
• PM3 0.036Å 3.9
• Dihedral angels are particularly difficult
• Errors 21.6, 12.5, 14.9

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Ongoing Developments in Semiempirical MO Theory

• Semiempirical methods are still used not
  because of their accuracy, but because they
  compete effectively in terms of computational
  time
• Is used if you have an enourmously large
  molecule, or want to compare a large number of
  small molecules
• The developments aim at improving the size
  horizon, so that larger systems can be simulated

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Ongoing Developments in Semiempirical MO Theory

• Structure-Activity Relationship (SAR)
• Used to understand how the features of a
  biologically active molecule contribute to the
  specific activity
• SARs typically take the form of linear equations
  that quantify the activity as a function of
  variables associated with the molecule
• Variables Molecular weight, dipole moment,
  hydrophobic surface area, vapor pressure,
  geometry...

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Ongoing Developments in Semiempirical MO Theory

• Once a SAR is developed it can be used to
  prioritize further research by focusing on the
  molecules having the highest activity, as
  predicted by SAR.
• If you have synthesized and characterized a large
  amount of molecules you can create a SAR for a
  some particular bio-target
• Can it be used to predict new molecules with
  certain properties?

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Ongoing Developments in Semiempirical MO Theory

• One effecient method is to create SARs not with
  experimental molecular properties, but with
  predicted ones
• Combined with experimental databases new
  compounds can be examined in a purely
  computational fashion -gt new targets for synthesis

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Ongoing Developments in Semiempirical MO Theory

• Implementing d-orbitals
• So far d-orbitals have not been included in NDDO
  models
• D-orbitals are necessary to accurately model
  non-metals from the third row and lower,
  especially in hypervalent situations
• Increase the flexibility with which the wave
  function may be described
• -gt Geometry optimization
• -gt MNDO/d

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Ongoing Developments in Semiempirical MO Theory

• MNDO/d
• For H, He and other first row atoms the original
  MNDO parameters are unchanged
• For heavier atoms, d orbitals are included as a
  part of the basis set
• MNDO/d represent an enourmous improvement over
  AM1 and PM3 in its ability to handle hypervalent
  molecules the error is reduced by more than
  half!
• However, it still performs poorly with respect to
  intermolecular interactions and hydrogen bonds
• PM3(tm) tm transition metals
• SAM1D semi-ab inito model 1

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Ongoing Developments in Semiempirical MO Theory

• Specific Reaction Parameters (SPR) models
• An SPR model is one where the initial parameters
  have been manually adjusted to a particular
  problem or class of problem

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Ongoing Developments in Semiempirical MO Theory

• Linear Scaling
• The motivation is to permit QM calculations to ba
  carried out on biomolecules, e.g. Proteins or
  polynucleic acids
• -gt Methods that scale linearly with system size
• They permit calculation of charge distribution,
  charge-charge interactions, and polarization
• -gt Greater sensitivity in group-group
  interactions

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The HF Limit

• HF Limit definition Solution of the HF equations
  with an infinite basis set
• Fig. 6.4

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The HF Limit

• This gives rise to an extrapolation equation -gt
  Going from one point to the other with reduced
  computation
• However, there are issues
• If the property is sensitive to geometry should
  the geometry be optimized at each point or should
  a single geometry be chosen to allow
  extrapolation?
• What form does the extrapolation curve have? Is
  any curve fitting method applicable?
• -gt Must be addressed manually in each case

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Effective Core Potentials

• When dealing with atoms with many electrons there
  is a problem
• A large number of basis functions is needed to
  describe them
• However, many of these electrons are core
  electrons -gt only few valence electrons
• Solution Replace electrons with analytical
  functions that represent the combined
  nuclear-electronic core to the remaining electrons

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Effective Core Potentials

• Essentially the ECP is a point charge with
  reduced magnitude by the number of core electrons
• In ab initio theory there is however more to the
  problem
• ECPs must represent both Coulomb repulsion
  effects but also Pauli principle...

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Effective Core Potentials

• The core electrons of very heavy elements reach
  velocities close to the speed of light
• -gt Relativistic effects, which can be significant
  for chemical properties
• ECPs can be modified to include relativistic
  effects in contrast to treating each electron
  with a non-relativistic Hamiltonian operator
• -gt Removes the need to search for suitable wave
  functions

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Effective Core Potentials

• Key issue
• How many electrons should be part of the core?
• Large core All electrons, but valence
• Small core Scale back to the next lower shell -gt
  Polarization can be included

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Sources

• Environmental Molecular Sciences Laboratory
  Gaussian Basis Set Order Form
• Website with many pre-defined basis-sets

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SCF Convergence

• SCF Self consistent field
• There is no guarantee that SCF processes will
  converge to a stable solution
• -gt SCF oscillation The solution oscillates
  between two discrete values
• -gt Convergence schemes
• Start with a simple basis set -gt The result is
  used in a augmented basis -gt This result is used
  again in a larger basis etc.

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SCF Convergence

• Another problem arises when the separation
  between the highest and lowest occupied MO is
  very low
• This can mean that occupation of either orbital
  leads to HF eigenfunctions with similar energy
• -gt Geometry optimization at a low level of theory

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Symmetry

• The presence of symmetry in a molecule can be
  used with advantage in electronic structure
  calculations -gt computational efficiency
• -gt Removes degrees of molecular freedom
• Example Benzene
• Calculating without symmetry 30 dimensions
• With symmetry 2 dimensions (C-C, C-H bond length)

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Symmetry

• Pitfall
• The minima in the symmetry energy may not be
  minima in the full-dimension energy
• Fig. 6.8

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Symmetry

• Pitfall
• Unpaired electrons, that can be in to different
  orbitals, that are fundamentally different
• Fig. 6.9

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Symmetry

• The initial guess defines the end result
• -gt The electronic state symmetry can not change
• Problem The two states both exist, but one is
  ground state and the other is excited state
• It is crucial that calculations are carried out
  for the right wavefunction

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Symmetry

• Solutions
• Start out with strict symmetry constraints, and
  the allowing them to relax later
• Consider electron switches from one orbital to
  another -gt if the energy drops it was an excited
  state

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Open Shell Systems

• Restricted Open shell HF theory
• Allows wavefunctions to be eigenfunctions to S2
  (spin) operator
• -gt Spin magnetic moments for unpaired electrons
• -gt But no spin polarization / spin coupling

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Open Shell Systems

• To allow two spins to occupy different regions of
  space (coupling) it is necessary to treat the two
  electrons individually
• -gt Unrestricted HF theory (UHF)
• Fig. 6.10

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Open Shell Systems

• UHF includes spin polarization
• But, the wavefunctions are not eigenfunctions of
  S2
• Thus, the UHF and ROHF models require more
  attention due to unphysical behaviour