HartreeFock Theory

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Hartree-Fock Theory
Computational Chemistry 5510 Spring 2006 Hai Lin
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Limitations of Molecular Mechanics
The bond-breaking and bond-forming cannot be
described.
Energy
MM Description Based on the Bonding Pattern of
Product
MM Description Based on the Bonding Pattern of
Reactant
A
A
B
C
B
C
Reactant
Product
Correct description
Progress of Reaction
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Quantum Mechanics
Macroscopic
Microscopic
Quantum mechanics is the law governing the
behavior of nuclei and electrons.
Correct Description for Bond-breaking and
Bond-forming
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Basis of Quantum Chemistry
Schrödinger equation Hy Ey
Erwin Schrödinger Paul A. M. Dirac
Dirac (1929) The underlying physical laws
necessary for the mathematical theory of a large
part of physics and the whole of chemistry are
thus completely known.
Nobel Prize in Physics 1933 "for the discovery
of new productive forms of atomic theory" 
However, it can be solved exactly only for
one-electron systems (e.g., a hydrogen atom) and
numerically for any a system having more
electrons.
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Accurate Quantum Mechanical Methods
Accurate quantum mechanical computation is a
powerful tool in study of chemistry.
Nobel Prize in Chemistry 1998
John A. Pople
Walter Kohn
for his development of the density-functional
theory
for his development of computational methods in
quantum chemistry
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Schrödinger Equation
H Tn Te Vnn Vee Vne
e1
e2
Kinetic energy of nuclei Kinetic energy of
electrons Coulombic energy between
nuclei Coulombic energy between
electrons Coulombic energy between nuclei and
electrons
n1
n2
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Approximations

• To solve the Schrödinger equation approximately,
  assumptions are made to simplify the equation
• Born-Oppenheimer approximation allows separate
  treatment of nuclei and electrons. (ma gtgt me)
• Hartree-Fock independent electron approximation
  allows each electron to be considered as being
  affected by the sum (field) of all other
  electrons.
• LCAO Approximation represents molecular orbitals
  as linear combinations of atomic orbitals (basis
  functions).

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Born-Oppenheimer Approximation

• Nuclei are much heavier than electrons (ma / me ?
  1836) and move much slower.
• Effectively, electrons adjust themselves
  instantaneously to nuclear configurations.
• Electron and nuclear motions are uncoupled, thus
  the energies of the two are separable.

• For a given nuclear configuration, one calculates
  electronic energy.
• As nuclei move continuously, the points of
  electronic energy joint to form a potential
  energy surface on which nuclei move.

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Many-electron Wave function
Hartree product All electrons are independent,
each in its own orbital.
YHP(x1, x2, , xN )
fi(x1)
fj(x2)
fk(xN)
Pauli principle Two electrons can not have all
quantum number equal. This requires that the
total (many-electron) wave function is
anti-symmetric whenever one exchanges two
electrons coordinates.
e1
e2
ei

Y(x1, x2, , xN )
- Y(x2, x1, , xN )
eN
Slater determinant satisfies the Pauli principle.
fj(x1)
fk(x1)
fi(x1)
fj(x2)
fk(x2)
fi(x2)
Y(x1, x2, , xN ) (1/N!)½

fj(xN)
fk(xN)
fi(xN)
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Many-electron Wave function (2)
Example A two-electron system.
Hartree product Both electrons are independent.
YHP(x1, x2) fi(x1) fj(x2)
Slater determinant satisfies the Pauli principle.
e1
fj(x1)
fi(x1)
Y(x1, x2) (1/2)½
e2
fj(x2)
fi(x2)
Y(x1, x2) (1/2)½ fi(x1) fj(x2) - fi(x2) fj(x1)
Y(x2, x1) (1/2)½ fi(x2) fj(x1) - fi(x1)
fj(x2) - Y(x1, x2)
The total (many-electron) wavefuntion is
anti-symmetric when one exchanges two electrons
coordinates x1 and x2.
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Hartree-Fock Approximation

• Each electron feels all the other electrons as
  a whole (field of charge), .i.e., an electron
  moves in a mean-field generated by all the other
  electrons.

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The Fock Operator
Exchange operator
Coulomb operator
Core-Hamiltonian operator
Exchange energy due to another electron (A pure
quantum mechanical term due to the Pauli
principle, no classical interpretation)
Coulombic energy term for the given electron due
to another electron
Kinetic energy term and nuclear attraction for
the given electron
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Self-consistency

• Each electron feels all the other electrons as
  a whole (field of charge), .i.e., an electron
  moves in a mean-field generated by all the other
  electrons.

• The Fock equation for an electron in the i-th
  orbital contains information of all the other
  electrons (in an averaged fashion), i.e., the
  Fock equations for all electrons are coupled with
  each other.

ek
ej
ei

• All equations must be solved together
  (iteratively until self-consistency is obtained).
  
• Self-consistent field (SCF) method.

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Refresh Your Mind Eigenvalue Eigenvector
Generally, one can construct a matrix for an
operator, e.g., the Hamiltonian H, using a set of
basis functions fi.
f1 f2 ... fn
f1 f2 ... fn
H11 H12 ... H1n H21 H22 ... H2n Hn1
Hn2 ... Hnn
Where Hij ?fi H fj? ?fi(x) H(x) fj (x) dx
After diagonalization, one obtains eigenvalues
(energy levels) and eigenvectors (wavefunctions).
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Linear Combination of Atomic Orbitals

• Each one-electron molecular orbital is
  approximated by a linear combination of atomic
  orbitals (basis functions).
• f c1 c1 c2 c2 c3 c3
• where f is the molecular orbital wavefunction,
  ci represents atomic orbital wavefunction, and ci
  is the corresponding expansion coefficients.
• The resulting Fock equations are called
  Roothaan-Hall equations.
• This reduces the problem of finding the best
  functional form for the molecular orbitals to the
  much simpler one of optimizing a set of
  coefficients (cn) in a linear equation.

y
x
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Variational Principle

• Based on the LCAO approximation, each
  one-electron molecular orbital is approximated as
  a linear combination of atomic orbitals.
• f c1 c1 c2 c2 c3 c3
• The energy calculated from any approximated wave
  function is higher than the true energy.
• The better the wave function, the lower the
  energy.
• One can vary the coefficients to minimize the
  calculated energy.
• At the energy minimum, dE 0, and one has the
  best approximation of the true energy that is
  what we want.

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Summary

• The need of quantum mechanical treatment for
  chemical problems
• Schrödinger equation
• Born-Oppenheimer approximation
• Hartree-Fock approximation
• Self-consistency
• Many-electron wavefunction
• Linear combination atomic orbitals
• Variational principle

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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 1.1, 3.1, 3.2, 3.5 (page 65).
  Take notes when you read.
• Questions
• When does the Born-Oppenheimer approximation
  break down? (P56)
• The Born-Oppenheimer approximation leads to the
  electronic Schrödinger equation (Eq. 3.17 in the
  textbook). Which terms are also neglected in such
  an equation? (P57)
• Why the Hartree-Fock approximation is also called
  a mean-field approximation?
• What is meaning of LCAO approximation? Do you
  think that it is a good idea? Why?