Quantum Mechanics Calculations

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Quantum Mechanics Calculations

• Based on the Schrödinger equation
• H? E?

2
The Schrödinger equation

• Schrödinger 's equation addresses the following
  questions
• Where are the electrons and nuclei of a molecule
  in space? configuration, conformation, size,
  shape, etc.
• Under a given set of conditions, what are their
  energies? heat of formation, conformational
  stability, chemical reactivity, spectral
  properties, etc.

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Schrödinger equation

• Schrödinger 's equation for molecular systems can
  only be solved approximately.
• The approximation methods can be categorized as
  either ab initio or semi-empirical.
• Semi-empirical methods use parameters that
  compensate for neglecting some of the time
  consuming mathematical terms in Schrödinger 's
  equation, whereas ab initio methods include all
  such terms.
• The parameters used by semi-empirical methods can
  be derived from experimental measurements or by
  performing ab initio calculations on model systems

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Ab Initio

• Limited from tens to 100 atoms.
• Can be applied to organics, organo-metallics, and
  molecular fragments (e.g., an enzyme catalytic
  site).
• Calculations are generally gas phase but
  continuum solvent can be simulated.
• Can be used to study ground states, transition
  states, and excited states.
• Specific implementations include GAMESS,
  GAUSSIAN, JAGUAR (PS-GVB), SPARTAN.

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Semi- Empirical

• Limited to hundreds of atoms, including small
  proteins (MOPAC 2000).
• Can be applied to organics, organo-metallics, and
  small oligomers (peptide, nucleotide,
  polysaccharide).
• Can be used to study ground states, transition
  states, and excited states.
• Specific implementations include SPARTAN, AMPAC,
  MOPAC, and ZINDO

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Assumptions

• Nuclei and electrons are distinguished from each
  other -gt electronic motion and nuclear motion are
  separable.
• Electron-electron interactions are usually
  averaged (SCF theory), but electron-nuclear
  interactions are explicit.
• Interactions are governed by nuclear and electron
  charges (potential energy) and electron motions
  (kinetic energy).
• Interactions determine the spatial distribution
  of nuclei and electrons and their energies.

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Standing Waves in a Clamped String
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Standing Waves in a Clamped String

• Waves can be generated by plucking (adding energy
  to) the string.
• Waves in a clamped string must adopt discrete
  wavelengths because the ends of the string cannot
  be displaced.

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Standing Waves in a Clamped String

• Such waves can be mathematically described by
  solving the wave equation shown below for ?.
• The one-dimensional form of the so-called wave
  equation is sufficient to describe the clamped
  string model

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Standing Waves in a Clamped String

• The solutions to the wave equation, ?, are called
  wavefunctions.
• ???is proportional to the energy density at any
  point along the wave (i.e. the amount of energy
  from the pluck that is stored in each part of the
  string).

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The Schrödinger equation for a one electron atom

• H? E?
• Where, H (-h2/8?2m) ???? V
• But V -Ze2/r
• So,
• (-h2/8?2m) ???-Ze2/r ? E?

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Properties of ?

• Each ??and its corresponding energy describe a
  single electron bound to a nucleus of charge Ze.
  
• They are called one-electron orbitals (often
  called hydrogen-like atomic orbitals).
• They are designated as 1s, 2s, 2p, etc.
• ?2 gives the probability density of finding an
  electron at a given point in space.
• Each ? can be contoured (i.e. graphed at a
  constant value) to give the familiar atomic
  orbitals.

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Molecular Orbital Theory

• The LCAO Approximation
• Let ?? represent a molecular wavefunction
• And ??represent an atomic wavefunction
• Then ??i ??cij ?j
• Plug into Schrödinger equation to get some pretty
  complicated equations - See NIH Guide to
  Molecular Modeling for additional details

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Semi-Empirical Methods

• MINDO
• MNDO
• AM1
• PM3
• Etc.

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Ab Initio Methods (Basis Sets)

• STO-3G
• Gaussian Orbital Basis Sets
• 3-21G()
• 6-31G
• 6-31G
• etc.

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Quantities Obtained from MO Calculations

• Molecular orbital energies, Ei, and coefficients
  cij.
• Total electronic energy, Eelec, calculated from
  the sum of the Coulomb integrals (Hij) and the
  molecular orbital energies Ei for all molecular
  orbitals in a molecule.
• Total nuclear repulsion energy Vnucleus-nucleus.
• Total energy, Etot, calculated from Eelec
  Vnucleus-nucleus.
• Heat of Formation, calculated from Etot -
  Eisolated atoms. The heat of formation is used
  for evaluating conformational energies.

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Quantities Obtained from MO Calculations

• Heat of Formation, calculated from Etot -
  Eisolated atoms. The heat of formation can used
  for evaluating conformational energies.
• Partial atomic charges, q, calculated from the
  molecular orbital coefficients using methods such
  as Mulliken population analysis, or using
  electrostatic potential fitting methods (Wendy
  Cornell, UCSF now at Novartis Pharmaceuticals).
• Electrostatic potential.
• Dipole moment.