Stability of HartreeFock Solutions

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Stability of Hartree-Fock Solutions

• The Symmetry Dilemma

Lara Thompson August 15, 2001
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Talk Outline

• Schrödingers equation for many-electron problem
• some concepts in orbital theory
• Hartree-Fock approximation
• LCAO approximation
• p electron approximation
• PPP approximation
• Stability of HF solutions
• cyclic polyenes
• rules for stable symmetric solutions
• conclusions

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Many-Electron Problem
The time independent Schrödinger equation for a
fully interacting many-electron system is
This is neglecting the inter-nuclear repulsion.
where ? is the N-electron wavefunction,
ri are the electron positions, da
are the positions of the ions and Za
are the ionic charges. This equation is
impossible to solve directly so approximate
solutions must be sought.
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Orbital Theory
atomic orbital (AO) the wavefunction of an
electron in orbit about an atoms nucleus
molecular orbital (MO) the wavefunction of an
electron associated to a molecule, dependent only
on the coordinates of that electron
molecular spin-orbital (MSO) the wavefunction of
an electron in a molecule which in addition to
space coordinates also contains the spin
coordinates of that electron
Total N-particle wavefunctions are approximated
as a single anti-symmetrized product (AP) of
MSOs (Slater determinant)
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More Orbital Theory

• electron shell a set of MSOs in which
• every MO occurs twice, namely, once with either
  spin
• if there is degeneracy due to molecular symmetry,
  the MOs in the shell form a complete degenerate
  set
• closed electron shell an AP which is made up of
  complete electron shells

An example of the outer shells of ground state
benzene. Each level shows an electron shell. Note
all MSOs in a shell have the same energy.
In orbitals of identical energy, electrons remain
unpaired if possible in order to minimize
electron-electron repulsion (Hunds Rule).
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Hartree-Fock Approximation
The Independent Particle Model, IPM, and the
Variation Principle are evoked

• the total N-electron wavefunction is approximated
  by a single anti-symmetric product function
• the potential affecting each electron is averaged

According to the Variational Principle, the best
solution is a set of MOs that minimize the energy
expectation value. Since the Hamiltonian is spin
independent the spin components can be integrated
out.
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Hartree-Fock Equation
Taking the first variation and setting dE 0
yields the Hartree-Fock equations
where is the Fock operator, fi are
the MOs and ei are the Lagrange
multipliers introduced to enforce
orthonormality of the MOs.

• the Fock operator is Hermitian so that the
  eigenvalues ei are real and the eigenvectors fi
  are orthogonal
• the HF equation is a pseudo-eigenvalue problem

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Self-Consistent Field Method
The Fock operator
Hamiltonian due to lone electron in nuclear frame
Coulomb term interaction with other electrons
Exchange term arises from Pauli principle
Since the Fock operator depends on the solution f
itself, the HF equation must be solved
iteratively. This process is called the
self-consistent field method.
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Orbital Energies ei
The N functions fi with the lowest eigenvalues ei
are called the ground state orbitals. The
eigenvalues ei are called the Hartree-Fock
orbital energies. This is in accordance with the
Aufbau Principle electrons are successively
assigned to the available orbital of lowest
energy. A degenerate eigenvalue represents
multiple electron orbitals with the same energy,
and hence the corresponding set of eigenvectors f
lie in the same shell.
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LCAO Approximation
For atoms, the problem of solving HF equations is
greatly simplified by the central symmetry.
For molecules, this loss of symmetry requires
further approximations. The MOs of a molecule
are represented by a Linear Combination of Atomic
Orbitals.
The solution is the best LCAO MOs for a closed
state, considering the AOs, ?p, as given. The
coefficients Cip then become variational
parameters to be determined.
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p Electron Approximation
Hydrocarbon molecules with conjugated double
bonds (single and double bonds alternate) are of
great interest to chemists. In these planar
systems, the bonding electrons of carbon take
part in s and sp2 hybrid orbitals leaving one
electron on each carbon to take part in a pz. The
first set are called s electrons and the latter
is a p electron.
In the p electron approximation, only the p
electrons are considered explicitly the s
electrons are lumped into the core Hamiltonian.
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PPP Approximation
For the Pariser, Parr and Pople (PPP)
approximation,

• the s system is treated as a non-polarizable core
• the zero-differential overlap (ZDO) approximation
  is employed
• the overlap Spq is neglected unless pq
• the two-electron integrals
• the Mataga-Nishimoto approximation for ?ij

The effective Hamiltonian in AO representation
where P is the density matrix
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Stability of HF Solutions
So far, iteratively finding a HF solution ensures
only that dE 0. This isnt necessarily a
minimum! Stability of HF solutions must be
assured by checking that d2Egt0 that is, the
solution presents a local minimum in
energy. There is no way of assuring that the
solution is a global minimum without exploring
the entire solution space. Thouless was the first
to explore d2E. Paldus and Cizek first
systematically categorised instabilities.
If the eigenvalues of the Hermitian matrix are
all positive, the solution is stable. The
eigenvalues are categorised as singlet and
triplet instabilities. The latter break spin
symmetry and will not be considered.
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The Symmetry Dilemma
If the Hamiltonian has the symmetry ?k (that is
they commute), then there exist a set of
simultaneous eigenvectors ?i.
But when applying the variational theorem, the
symmetry requirement becomes a constraint.
Theorem If there exists a Kekulé structure
possessing the molecules symmetry, then the HF
solution having that symmetry is stable. However,
that doesnt guarantee that there isnt a
solution with broken symmetry with lower energy.
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Cyclic Polyenes
CNHN

• for non-degenerate ground state
• N4n2

• fis given as Bloch orbitals

• explore stability

• instability implies broken symmetry solution and
  bond length alternation
• explore the role of approximate symmetry

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N6 Benzene
We can model the stability phenomenon by varying
the coupling constant, b.
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Losing Symmetry
-3.08
-3.05
0
-2.98
-0.46
-0.43
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Summary

• overviewed approximations leading to
  realistically soluble PPP equations
• IPM and Variation Principle
• LCAO approxn
• p electron approxn
• PPP semi-empirical approxn
• examined stability issues
• cyclic polyenes
• approximate symmetry

Conclusions

• conjugated hydrocarbons really want to alternate
• solutions will pseudo-alternate while retaining
  symmetry but eventually lose symmetry to
  alternate fully
• this pseudo-alternation is in support of the
  existence of domain walls