Physical Chemistry 2nd Edition

1
Physical Chemistry 2nd Edition
Chapter 26 Computational Chemistry

• Thomas Engel, Philip Reid

2
Objectives

• Discover the usage of numerical methods.
• Discussion is the Hartree-Fock molecular orbital
  model.

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Outline

1. The Promise of Computational Chemistry
2. Potential Energy Surfaces
3. Hartree-Fock Molecular Orbital Theory A
   Direct Descendant of the Schrödinger
   Equation
4. Properties of Limiting Hartree-Fock Models
5. Theoretical Models and Theoretical Model
   Chemistry

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Outline

1. Moving Beyond Hartree- Fock Theory
2. Gaussian Basis Sets
3. Selection of a Theoretical Model
4. Graphical Models
5. Conclusion

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26.1 The Promise of Computational Chemistry

• Sufficient accuracy can be obtained from
  computational chemistry.
• Approximations need to be made to realize
  equations that can be solved.
• No one method of calculation is likely to be
  ideal for all application.
• Hartree-Fock theory leads to ways to improve on
  it and to a range of practical quantum chemical
  models.

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26.2.1 Potential Energy Surfaces and Geometry

• Energy minima give the equilibrium structures of
  the reactants and products.
• Energy maximum defines the transition state.
• Reactants, products, and transition states are
  all stationary points on the potential energy
  diagram.

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26.2.1 Potential Energy Surfaces and Geometry

• In the one-dimensional case, 1st derivative of
  the potential energy with respect to the reaction
  coordinate is zero
• For many-dimensional case, each independent
  coordinate, Ri, gives rise to 3N-6 second
  derivatives

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26.2.1 Potential Energy Surfaces and Geometry

• Stationary points where all second derivatives
  are positive are energy minima
• where ?i normal coordinates
• Stationary points where all but one are positive
  are saddle points
• where ?i reaction coordinate

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26.2.2 Potential Energy Surfaces and Vibrational
Spectra

• The vibrational frequency for a diatomic molecule
  A-B is
• k is the force constant which is defined as
• And µ is the reduced mass.

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26.2.3 Potential Energy Surfaces and
Thermodynamics

• The energy difference between the reactants and
  products determines the thermodynamics of a
  reaction.
• The ratio is as follow,

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26.2.3 Potential Energy Surfaces and
Thermodynamics

• The energy difference between the reactants and
  transition state determines the rate of a
  reaction.
• The rate constant is given by the Arrhenius
  equation and depends on the temperature

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26.3 Hartree-Fock Molecular Orbital Theory A
Direct Descendant of the Schrödinger
Equation

• 3 approximations need to realize a practical
  quantum mechanical theory for multielectron
  Schrödinger equation
• Born-Oppenheimer approximation
• Hartree-Fock approximation
• Linear combination of atomic orbitals (LCAO)
  approximation

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MATHEMATICAL FORMULATION OF THE HARTREE-FOCK
METHOD

• The Hartree-Fock and LCAO approximations, taken
  together and applied to the electronic
  Schrödinger equation, lead to a set of matrix
  equations now known as the Roothaan-Hall
  equations
• where c unknown molecular orbital coefficients
  e orbital energies S
  overlap matrix F Fock matrix

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MATHEMATICAL FORMULATION OF THE HARTREE-FOCK
METHOD

• For Fock matrix,
• where Hcore core Hamiltonian
• Coulomb and exchange elements are given by

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MATHEMATICAL FORMULATION OF THE HARTREE-FOCK
METHOD

• P is called the density matrix
• The cost of a calculation rises rapidly with the
  size of the basis set

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26.4 Properties of Limiting Hartree-Fock Models

• For computation, it is expected to have errors
  in
• Relative energies
• Geometries
• Vibrational frequencies
• Properties such as dipole moments

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26.4.1 Reaction Energies

• Hartree-Fock models is compare with homolytic
  bond dissociation energies.
• For example in methanol,

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26.4.1 Reaction Energies

• The poor results seen for homolytic bond
  dissociation reactions do not necessarily carry
  over into other types of reactions as long as the
  total number of electron pairs is maintained.

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26.4.2 Equilibrium Geometries

• Systematic discrepancies are also noted in
  comparisons involving limiting Hartree-Fock and
  experimental.
• They are geometries and bond distances.
• The reason is that limiting Hartree-Fock bond
  distances is shorter than experimental values.

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26.4.3 Vibrational Frequencies

• The error in bond distances for limiting
  Hartree-Fock models calculated frequencies are
  larger than experimental frequencies.
• The reason is that the Hartree-Fock model does
  not dissociate to the proper limit of two
  radicals as a bond is stretched.

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26.4.4 Dipole Moments

• Electric dipole moments are compared, the
  calculated values are larger than experimental
  values.

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26.5 Theoretical Models and Theoretical Model
Chemistry

• Limiting Hartree-Fock models do not provide
  results that are identical to experimental
  results.
• Theoretical model chemistry is a detailed theory
  starting from the electronic Schrödinger
  equation and ending with a useful scheme.

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26.6 Moving Beyond Hartree-Fock Theory

• Improvements will increase the cost of a
  calculation.
• 2 approaches to improve Hartree-Fock theory
• Increases the flexibility by combining it with
  wave functions corresponding to various excited
  states.
• Introduces an explicit term in the Hamiltonian to
  account for the interdependence of electron
  motions.

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26.6.1 Configuration Interaction Models

• Improvements will increase the cost of a
  calculation.
• 2 approaches to improve Hartree-Fock theory
• Increases the flexibility by combining it with
  wave functions corresponding to various excited
  states.
• Introduces an explicit term in the Hamiltonian to
  account for the interdependence of electron
  motions.

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26.6.2 Møller-Plesset Models

• Møller-Plesset models are based on Hartree-Fock
  wave function and ground-state energy E0 as exact
  solutions.
• where small perturbation ?
  dimensionless parameter

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MATHEMATICAL FORMULATION OF MØLLER-PLESSET MODELS

• Substituting the expansions into the Schrödinger
• equation and gathering terms in ?n yields

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MATHEMATICAL FORMULATION OF MØLLER-PLESSET MODELS

• Multiplying each by ?0 and integrating over all
  space yields the following expression for the
  nth-order (MPn) energy

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MATHEMATICAL FORMULATION OF MØLLER-PLESSET MODELS

• In this framework, the Hartree-Fock energy is the
  sum of the zero- and firstorder Møller-Plesset
  energies
• The first correction, E(2) can be written as
  follows

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MATHEMATICAL FORMULATION OF MØLLER-PLESSET MODELS

• The integrals (ij ab) over filled (i and j)
  and empty (a and b) molecular orbitals account
  for changes in electronelectron interactions as
  a result of electron promotion,
• in which the integrals (ij ab) and (ib ja)
  involve molecular orbitals rather than basis
  functions.
• The two integrals are related by a simple
  transformation,

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26.6.3 Density Functional Models

• Density functional theory is based on the
  availability of an exact solution for an
  idealized many-electron problem.
• The Hartree-Fock energy may be written as
• where ET kinetic energy EV the
  electronnuclear potential energy EJ
  Coulomb EK interaction energy

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26.6.3 Density Functional Models

• For idealized electron gas problem
• where EXC exchange/correlation energy
• Except for ET, all components depend on the total
  electron density, p(r)

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MATHEMATICAL FORMULATION OF DENSITY FUNCTIONAL
THEORY

• Within a finite basis set (analogous to the LCAO
  approximation for Hartree Fock models), the
  components of the density functional energy,
  EDFT, can be written as follows

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MATHEMATICAL FORMULATION OF DENSITY FUNCTIONAL
THEORY

• Better models result from also fitting the
  gradient of the density. Minimizing EDFT with
  respect to the unknown orbital coefficients
  yields a set of matrix equations, the Kohn-Sham
  equations, analogous to the Roothaan-Hall
  equations
• Here the elements of the Fock matrix are given by

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MATHEMATICAL FORMULATION OF DENSITY FUNCTIONAL
THEORY

• FXC is the exchange/correlation part, the form of
  which depends on the particular
  exchange/correlation functional used. Note that
  substitution of the Hartree-Fock exchange, K, for
  FXC yields the Roothaan-Hall equations.

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26.6.4 Overview of Quantum Chemical Models

• An overview of quantum chemical models.

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26.7 Gaussian Basis Sets

• LCAO approximation requires the use of a finite
  number of well-defined functions centered on each
  atom.
• Early numerical calculations use nodeless
  Slater-type orbitals (STOs),
• If the AOs are expanded in terms of Gaussian
  functions,

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26.7.1 Minimal Basis Sets

• The minimum number is the number of functions
  required to hold all the electrons of the atom
  while still maintaining its overall spherical
  nature.
• This simplest representation or minimal basis set
  involves a single (1s) function for hydrogen and
  helium.
• In STO-3G basis set, basis functions is expanded
  in terms of three Gaussian functions.

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26.7.2 Split-Valence Basis Sets

• Minimal basis set is bias toward atoms with
  spherical environments.
• A split-valence basis set represents core atomic
  orbitals by one set of functions and valence
  atomic orbitals by two sets of functions
• for lithium to neon
• for sodium to argon

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26.7.3 Polarization Basis Sets

• Minimal (or split-valence) basis set functions
  are centered on atoms rather than between atoms.
• The inclusion of polarization functions can be
  thought about either in terms of hybrid orbitals.

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26.7.4 Basis Sets Incorporating Diffuse Functions

• Calculations involving anions can pose problems
  as highest energy electrons may only be loosely
  associated with specific atoms (or pairs of
  atoms).
• In these situations, basis sets may need to be
  supplemented by diffuse functions.

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26.8 Selection of a Theoretical Model

• Hartree-Fock models have proven to be successful
  in large number of situations and remain a
  mainstay of computational chemistry.
• Correlated models can be divided into 2
  categories
• Density functional models
• Møller-Plesset models
• Transitionstate geometry optimizations are more
  time-consuming than equilibrium geometry
  optimizations, due primarily to guess of geometry.

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26.8.1 Equilibrium Bond Distances

• Hartree-Fock double bond lengths are shorter than
  experimental distances.
• Treatment of electron correlation involves the
  promotion of electrons from occupied molecular
  orbitals to unoccupied molecular orbitals.

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26.8.2 Finding Equilibrium Geometries

• An equilibrium structure corresponds to the
  bottom of a well on the overall potential energy
  surface.
• Equilibrium structures that cannot be detected
  are referred to as reactive intermediates.
• Geometry optimization does not guarantee that the
  final geometry will have a lower energy than any
  other geometry of the same molecular formula.

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26.8.3 Reaction Energies

• Reaction energy comparisons are divided into
  three parts
• Bond dissociation energies
• Energies of reactions relating structural isomers
• Relative proton affinities.

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26.8.4 Energies, Enthalpies, and Gibbs Energies

• Quantum chemical calculations account for
  thermochemistry by combining the energies of
  reactant and product molecules at 0 K.
• Residual energy of vibration is ignored.
• We would need 3 corrections
• Correction of the internal energy for finite
  temperature.
• Correction for zero point vibrational energy.
• Corrections of entropy.

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26.8.5 Conformational Energy Differences

• Hartree-Fock models overestimate differences by
  large amounts.
• Correlated models also typically overestimate
  energy differences but magnitudes of the errors
  are much smaller than those seen for Hartree-Fock
  models.

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26.8.6 Determining Molecular Shape

• The problem of identifying the lowest energy
  conformer in simple molecules is when the number
  of conformational degrees of freedom increases.
• Sampling techniques will need to replace
  systematic procedures for complex molecules, thus
  Monte Carlo methods is used.

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26.8.7 Alternatives to Bond Rotation

• Single-bond rotation is the most common mechanism
  for conformer interconversion.
• 2 other processes are known
• Inversion is associated with pyramidal nitrogen
  or phosphorus and involves a planar transition
  state.
• Pseudorotation is associated with trigonal
  bipyramidal phosphorus and involves a
  square-based-pyramidal transition state.

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26.8.8 Dipole Moments

• Dipole moments from the two Hartree-Fock models
  are larger than experimental values due to
  behavior of the limiting Hartree-Fock model.
• Recognize that electron promotion from occupied
  to unoccupied molecular orbitals takes electrons
  from where they are to where they are not.

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26.8.9 Atomic Charges Real or Make Believe?

• Charge distributions assess overall molecular
  structure and stability.
• Mulliken population analysis can be used to
  formulate atomic charges.

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MATHEMATICAL DESCRIPTION OF THE MULLIKEN
POPULATION ANALYSIS

• The Mulliken population analysis starts from the
  definition of the electron density, ?(r), in the
  framework of the Hartree-Fock model
• Summing over basis functions and integrating over
  all space leads to an expression for the
• total number of electrons, n

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MATHEMATICAL DESCRIPTION OF THE MULLIKEN
POPULATION ANALYSIS

• where Sµv are elements of the overlap matrix
• It is possible to equate the total number of
  electrons in a molecule to a sum of products of
  density matrix and overlap matrix elements as
  follows

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MATHEMATICAL DESCRIPTION OF THE MULLIKEN
POPULATION ANALYSIS

• According to Mullikens scheme, the gross
  electron population for basis function is given
  by
• Atomic electron populations, qA, and atomic
  charges, QA, follow, where ZA is the atomic
  number of atom A

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26.8.10 Transition-State Geometries and
Activation Energies

• Transition-state theory states that all reactants
  have the same energy, or that none has energy in
  excess of that needed to reach the transition
  state.
• Hartree-Fock models overestimate the activation
  energies by large amounts.

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26.8.11 Finding a Transition State

• There is less effort (energy) by passing through
  a valley between two mountains (pathway B).
• Saddle point referred to a maximum and minimum in
  the transition state.

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26.9 Graphical Models

• Molecular orbitals, electron density and
  electrostatic potential can be defined a isovalue
  surface or isosurface
• Most common graphical models are on electron
  density surfaces and electrostatic potential.

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26.9.1 Molecular Orbitals

• Molecular orbitals, ?, are written as
• Highest energy occupied molecular orbital (HOMO)
  holds the highest energy electrons and is attack
  by electrophiles, while lowest energy unoccupied
  molecular orbital (the LUMO) provides the lowest
  energy space for additional electrons and attack
  by nucleophiles.

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26.9.2 Orbital Symmetry Control of Chemical
Reactions

• HOMO and LUMO (frontier molecular orbitals) could
  be used to rationalize why some chemical
  reactions proceed easily whereas others do not.

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26.9.3 Electron Density

• Electron density ?(r) is written in terms of
• Depending on the value, isodensity surfaces can
  either serve to locate atoms to delineate
  chemical bonds or to indicate overall molecular
  size and shap.

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26.9.4 Where Are the Bonds in a Molecule?

• An electron density surface can be used to know
  the location of bonds in a molecule.
• Electron density surfaces is also use as the
  description of the bonding in transition states

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26.9.5 How Big Is a Molecule?

• The size of a molecule can be defined according
  to the amount of space that it takes up in a
  liquid or solid.
• The electron density provides an alternate
  measure of how much space molecules actually take
  up.

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26.9.6 Electrostatic Potential

• The electrostatic potential,ep, is defined as
• Note that electrostatic potential represents a
  balance between repulsion of the point charge by
  the nuclei and attraction of the point charge by
  the electrons.

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26.9.7 Visualizing Lone Pairs

• The octet rule dictates that each main-group atom
  in a molecule will be surrounded by eight valence
  electrons.
• A comparison between electrostatic potential
  surfaces for ammonia in both the observed
  pyramidal and unstable trigonal planar geometries.

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26.9.8 Electrostatic Potential Maps

• Most commonly used property map is the
  electrostatic potential map.
• It gives the value of the electrostatic potential
  at locations on a particular surface.

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26.9.8 Electrostatic Potential Maps

• Electrostatic potential maps are used to
  distinguish between molecules in which charge is
  localized from those where it is delocalized.

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26.9.10 Conclusions

• Inability of the calculations to deal highly
  reactive molecules that
• difficult to synthesize
• with reaction transition states
• Limitations of quantum chemical calculations are
• Practical and numerical results not match
• Important quantities cannot be yield
• Calculations apply strictly to isolated molecules
  (gas phase)

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Example 27.1

• Are the three mirror planes for the NF3 molecule
  in the same or in different classes?
• b. Are the two mirror planes for H2O in the same
  or in different classes?

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Solution

• a. NF3 belongs to the C3v group, which contains
• the rotation operators
  and the
• vertical mirror planes
  . These
• operations and elements are illustrated by this
• figure