Hartree-Fock Theory

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Hartree-Fock Theory
Erin Dahlke Department of Chemistry University of
Minnesota VLab Tutorial May 25, 2006
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Elementary Quantum Mechanics

• The Hamiltonian for a many-electron system can
  be written as

Potential energy
Kinetic energy
where the first two terms represent the kinetic
energy of the electrons and nuclei, respectively,
the third term is the nuclear-electron
attraction, the fourth term is the
electron-electron repulsion, and the fifth term
is the nuclear-nuclear repulsion.
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Elementary Quantum Mechanics
Atomic units

• The Hamiltonian for a many-electron system can
  be written in atomic units as

Kinetic energy
Potential energy
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The Born-Oppenheimer Approximation
To a good approximation one can assume that the
electrons move in a field of fixed nuclei
Kinetic energy
Potential energy
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What is a Wave Function?
For every system there is a mathematical function
of the coordinates of the system ? the wave
function (?) This function contains within it
all of the information of the system.
In general,for a given molecular system, there
are many different wave functions that are
eigenfunctions of the Hamiltonian operator, each
with its own eigenvalue, E.
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Properties of a Wave Function

1. The wave function must vanish at the boundaries
   of the system
2. The wave function must be single-valued.
3. The wave function must be continuous.

• A description of the spatial coordinates of an
  electron is not enough. We must also take into
  account spin (?). (Spin is a consequence of
  relativity.)
• spin up
• ? spin down
• A many-electron wave function must be
  antisymmetric with respect to the interchange of
  coordinate (both space and spin) of any two
  electrons. ? Pauli exclusion principle

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Spin Orbitals and Spatial Orbitals
orbital - a wave function for a single
electron spatial orbital - a wave function that
depends on the position of the electron and
describes its spatial distribution
spin orbital - a wave function that depends on
both the position and spin of the electron
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Hartree Products
Consider a system of N electrons
The simplest approach in solving the electronic
Schrödinger equation is to ignore the
electron-electron correlation. Without this term
the remaining terms are completely separable.
Consider a system of N non-interacting electrons
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Hartree Products
Each of the one-electron Hamiltonians will
satisfy a one-electron Schrödinger equation.
Because the Hamiltonian is separable the wave
function for this system can be written as a
product of the one-electron wave functions.
This would result in a solution to the
Schrödinger equation
in which the total energy is simply a sum of the
one-electron orbital energies
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Hartree Method
Instead of completely ignoring the
electron-electron interactions we consider each
electron to be moving in a field created by all
the other electrons
From variation calculus you can show
that where

• Self Consistent Field (SCF) procedure
• 1. Guess the wave function for all the occupied
  orbitals
• Construct the one-electron operators
• Solve the Schrödinger equation to get a new guess
  at the wave function.

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Hartree Method
The one-electron Hamiltonian h(i) includes the
repulsion of electron i with electron j, but so
does h(j). The electron-electron repulsion is
being double counted.
Sum of the one-electron orbital eigenvalues
One half the electron-electron repulsion energy
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Hartree Products
What are the short comings of the Hartree Product
wave function?
Electron motion is uncorrelated - the motion of
any one electron is completely independent of the
motion of the other N-1 electrons.
Electrons are not indistinguishable
Wave function is not antisymmetric with respect
to the interchange of two particles
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Slater Determinants
Consider a 2 electron system with two spin
orbitals. There are two ways to put two
electrons into two spin orbitals.
Neither of these two wave functions are
antisymmetric with respect to interchange of two
particles, nor do they account for the fact that
electrons are indistinguishable Try taking a
linear combination of these two possibilities
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Slater Determinants
Try the linear combination with the addition.
This wave function is not antisymmetric!
What about the linear combination with the
subtraction.
This wave function is antisymmetric!
An alternative way to write this wave function
would be
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Slater Determinants
For an N electron system
rows correspond to electrons, columns correspond
to orbitals
Slater Determinant
?
Electrons are not indistinguishable
Wave function is not antisymmetric with respect
to the interchange of two particles
?
?
Electron motion is uncorrelated
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Slater Determinants
Is the electron motion correlated? Consider a
Helium atom in the singlet state
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Slater Determinants
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Slater Determinants
What happens to the spin terms?
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Slater Determinants
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Slater Determinants
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Slater Determinants
After integrating out the spin coordinates, were
left with only two terms in the integral
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Coulomb Repulsion
Classically the Coulomb repulsion between two
point charges is given by where q1 is the
charge on particle one, q2 is the charge on
particle 2, and r12 is the distance between the
two point charges. An electron is not a point
charge. Its position is delocalized, and
described by its wave function. So to describe
its position we need to use the wave function for
the particle, and integrate its square modulus.
(in atomic units the charge of an electron (e) is
set equal to one.
This is the expression for the Coulomb repulsion
between an electron in orbital i with an electron
in orbital j. Since is always positive, as
is the Coulomb energy is always a positive
term and causes destabilization.
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Slater Determinants
Consider a Helium atom in the triplet state
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Slater Determinants
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Slater Determinants
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Slater Determinants
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Slater Determinants
exchange integral
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Exchange Integral
The exchange integral has no classical
analog. The exchange integral gets its name
from the fact that the two electrons exchange
their positions as you go from the left to the
right in the integrand. The exchange integrals
are there to correct the Coulomb integrals, so
that they take into account the
antisymmetrization of the wave function. Electron
s of like spin tend to avoid each other more than
electrons of different spin, so the
destabilization predicted by the Coulomb
integrals is too high. The exchange integrals,
which are always positive, lower the overall
repulsion of the electron (25).
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Slater Determinants
So no two electrons with the same spin can occupy
the same point in space ?? the Pauli Exclusion
principle
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HartreeFock Theory
What if we apply Hartree theory to a Slater
determinant wave function?
one-electron Fock operator
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HartreeFock Theory
Coulomb operator
Exchange operator
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Self Interaction

• The sums in these equations run over all values
  of j, including j i.
• Each of these terms contains a self-interaction
  term
• - a Coulomb integral between an electron and
  itself
• - an exchange integral between an electron and
  itself
• Since both J and K contain the self-interaction
  term, and since were subtracting them from each
  other, the self-interaction cancels.

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What is a Wave Function?
For every system there is a mathematical function
of the coordinates of the system ? the wave
function (?) This function contains within it
all of the information of the system.
In general,for a given molecular system, there
are many different wave functions that are
eigenfunctions of the Hamiltonian operator, each
with its own eigenvalue, E.
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HartreeFock Theory
For a molecular system we dont know what the
true wave function is. In general in order to
approximate it we make the assumption that the
true wave function is a linear combination of
one-electron orbitals.
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HartreeFock Theory
What happens to our Coulomb and exchange
operators??
We call Pmn the density matrix
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HartreeFock Theory
One can write the one-electron Schrödinger
equation as
Where we can define the following matrix elements
Fock matrix
overlap matrix
We can then rearrange the one-electron
Schrödinger equation to get
Where we will have one such equation for each
electron is our system.
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HartreeFock Theory
In order find a non-trivial solution to this set
of equations one can set up and solve the secular
determinant
Solution of the secular determinant determines
the coefficients cyi which can, in turn be used
to solve for the one-electron energy eigenvalues,
?i.
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Flow chart of the implementation of HartreeFock
Theory
no
yes
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Limitations of HartreeFock Theory - Energetics

• HartreeFock theory ignores electron correlation
• - cannot be used (accurately) in any process in
  which the
• total number of paired electrons
  changes.
• Even if the total number of paired electrons
  stays the same, if the nature of the bonds
  changes drastically HF theory can have serious
  problems. (i.e., isomerization reactions)
• Does well for protonation/deprotonation
  reactions.
• Can be used to compute ionization potentials and
  electron affinities.
• Will not do well for describing systems in which
  there are dispersion interactions, as they are
  completely due to electron correlation effects,
  except by cancellation of errors.
• HF charge distributions tend to be over polarized
  which give electrostatic interactions which are
  too large.

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Limitations of HartreeFock Theory - Geometries

• HF theory tends to predict bonds to be too short,
  especially as you increase the basis set size.
• Bad for transition state structures due to the
  large correlation associated with the making and
  breaking of partial bonds.
• Nonbonded complexes tend to be far too loose, as
  HF theory does not account for dispersion
  interactions.

Limitations of HartreeFock Theory - Charge
Distributions

• The magnitude of dipole moments is typically
  overestimated by 1025 for medium sized basis
  sets.
• Results are erratic with smaller basis sets.

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Summary

• HartreeFock theory is an approximate solution
  to the electronic Schrödinger equation which
  assumes that each individual electron i, moves
  in a field created by all the other electrons.
• Introduces the concept of exchange energy
  through the use of a Slater determinant wave
  function.
• Ignores all other electron correlation.
• Contains a self-interaction term which cancels
  itself out.
• Often underbinds complexes.
• Predicts bond lengths which are too short.

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References
Szabo, A. Ostlund, N. S. Modern Quantum
Chemistry. An Introduction to Advanced Electronic
Structure Theory. Dover Publications Mineola,
NY 1996. Pilar, F. L. Elementary Quantum
Mechanics, 2nd Ed. Dover Publications Mineola,
NY 2001. Cramer, C. J. Essentials of
Computational Chemistry. Wiley Chichester 2002.