Lecture 11. Quantum Mechanics. Hartree-Fock Self-Consistent-Field Theory

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Lecture 11. Quantum Mechanics. Hartree-Fock
Self-Consistent-Field Theory

• Outline of todays lecture
• Postulates in quantum mechanics
• Schrödinger equation
• Simplify Schrödinger equation
• Atomic units, Born-Oppenheimer approximation
• Solve Schrödinger equation with approximations
• Variation principle, Slater determinant, Hartree
  approximation,
• Hartree-Fock, Self-Consistent-Field, LCAO-MO,
  Basis set

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References

• Molecular Quantum Mechanics, Atkins Friedman
  (4th ed. 2005), Ch. 1 8
• Essentials of Computational Chemistry. Theories
  and Models, C. J. Cramer,
• (2nd Ed. Wiley, 2004) Chapter 4
• Molecular Modeling, A. R. Leach (2nd ed.
  Prentice Hall, 2001) Chapter 2
• Introduction to Computational Chemistry, F.
  Jensen (1999) Chapter 3
• A Brief Review of Elementary Quantum Chemistry
• http//vergil.chemistry.gatech.edu/notes/quantrev
  /quantrev.html
• Molecular Electronic Structure Lecture
• http//www.chm.bris.ac.uk/pt/harvey/elstruct/intr
  oduction.html
• Wikipedia (http//en.wikipedia.org) Search for
  Schrödinger equation, etc.

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Potential Energy Surface Quantum Mechanics

• How do we obtain the potential energy E?
• MM Evaluate analytic functions (FF)
• QM Solve Schrödinger equation

3N (or 3N-6 or 3N-5) Dimension PES for N-atom
system
Continuum Modeling
Atomistic Modeling
Geometry
Quantum Modeling
Charge Force Field
For geometry optimization, evaluate E, E (
E) at the input structure X (x1,y1,z1,,xi,yi,zi
,,xN,yN,zN) or li,?i,?i.
Length
1?m
1cm
0.1 nm
1 nm
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The Schrödinger Equation
The ultimate goal of most quantum chemistry
approach is the solution of the time-independent
Schrödinger equation.
?
(1-dim)
Hamiltonian operator ? wavefunction (solving a
partial differential equation)
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Postulate 1 of Quantum Mechanics

• The state of a quantum mechanical system is
  completely specified by the wavefunction or state
  function that depends on the coordinates
  of the particle(s) and on time.
• The probability to find the particle in the
  volume element located at r at time
  t is given by . (Born
  interpretation)
• The wavefunction must be single-valued,
  continuous, finite, and normalized (the
  probability of find it somewhere is 1).
• lt??gt

Probability density
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Born Interpretation of the Wavefunction
Probability Density
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Probability Density Examples
B 0
A 0
A B
nodes
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Postulate 2 of Quantum Mechanics

• Once is known, all properties of the
  system can be obtained
• by applying the corresponding operators to the
  wavefunction.
• Observed in measurements are only the
  eigenvalues a which satisfy
• the eigenvalue equation

eigenvalue
eigenfunction
(Operator)(function) (constant number)?(the
same function)
(Operator corresponding to observable)? (value
of observable)??
Schrödinger equation Hamiltonian operator ?
energy
with
(Hamiltonian operator)
(e.g. with )
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Physical Observables Their Corresponding
Operators
with
(Hamiltonian operator)
(e.g. with )
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Observables, Operators Solving Eigenvalue
Equations an Example
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The Uncertainty Principle
When momentum is known precisely, the position
cannot be predicted precisely, and vice versa.
When the position is known precisely,
Location becomes precise at the expense of
uncertainty in the momentum
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Postulate 3 of Quantum Mechanics

• Although measurements must always yield an
  eigenvalue,
• the state does not have to be an eigenstate.
• An arbitrary state can be expanded in the
  complete set of
• eigenvectors ( as
  where n ? ?.
• We know that the measurement will yield one of
  the values ai, but
• we don't know which one. However,
• we do know the probability that eigenvalue ai
  will occur ( ).
• For a system in a state described by a
  normalized wavefunction ,
• the average value of the observable
  corresponding to is given by
• 
  lt?A?gt

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The Schrödinger Equation for Atoms Molecules
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Atomic Units (a.u.)

• Simplifies the Schrödinger equation (drops all
  the constants)
• (energy) 1 a.u. 1 hartree 27.211 eV 627.51
  kcal/mol,
• (length) 1 a.u. 1 bohr 0.52918 Å,
• (mass) 1 a.u. electron rest mass,
• (charge) 1 a.u. elementary charge, etc.

(before)
(after)
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Born-Oppenheimer Approximation

• Simplifies further the Schrödinger equation
  (separation of variables)
• Nuclei are much heavier and slower than
  electrons.
• Electrons can be treated as moving in the field
  of fixed nuclei.
• A full Schrödinger equation can be separated into
  two
• Motion of electron around the nucleus
• Atom as a whole through the space
• Focus on the electronic Schrödinger equation

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Born-Oppenheimer Approximation
(before)
(electronic)
(nuclear)
E
(after)
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Electronic Schrödinger Equation in Atomic Unit
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Variational Principle

• Nuclei positions/charges number of electrons in
  the molecule
• Set up the Hamiltonian operator
• Solve the Schrödinger equation for wavefunction
  ?, but how?
• Once ? is known, properties are obtained by
  applying operators
• No exact solution of the Schrödinger eq for
  atoms/molecules (gtH)
• Any guessed ?trial is an upper bound to the true
  ground state E.
• Minimize the functional E? by searching through
  all acceptable
• N-electron wavefunctions

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Hartree Approximation (1928)Single-Particle
Approach
Nobel lecture (Walter Kohn 1998) Electronic
structure of matter

• Impossible to search through
• all acceptable N-electron
• wavefunctions.
• Lets define a suitable subset.
• N-electron wavefunction
• is approximated by
• a product of N one-electron
• wavefunctions. (Hartree product)

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Antisymmetry of Electrons and Paulis Exclusion
Principle

• Electrons are indistinguishable. ? Probability
  doesnt change.
• Electrons are fermion (spin ½). ? antisymmetric
  wavefunction
• No two electrons can occupy the same state (space
  spin).

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Slater determinants

• A determinant changes sign when two rows (or
  columns) are exchanged.
• ? Exchanging two electrons leads to a change in
  sign of the wavefunction.
• A determinant with two identical rows (or
  columns) is equal to zero.
• ? No two electrons can occupy the same state.
  Paulis exclusion principle

antisymmetric
0
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N-Electron Wavefunction Slater Determinant

• N-electron wavefunction aprroximated by a product
  of N one-electron
• wavefunctions (hartree product).
• It should be antisymmetrized (
  ).

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Hartree-Fock (HF) Approximation

• Restrict the search for the minimum E? to a
  subset of ?, which
• is all antisymmetric products of N spin
  orbitals (Slater determinant)
• Use the variational principle to find the best
  Slater determinant
• (which yields the lowest energy) by varying
  spin orbitals

(orthonormal)
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Hartree-Fock (HF) Energy
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Hartree-Fock (HF) Energy Evaluation
finite basis set
Molecular Orbitals as linear combinations of
Atomic Orbitals (LCAO-MO)
where
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Hartree-Fock (HF) Equation Evaluation

• No-electron contribution (nucleus-nucleus
  repulsion just a constant)
• One-electron operator h (depends only on the
  coordinates of one electron)
• Two-electron contribution (depends on the
  coordinates of two electrons)

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1. Potential energy due to nuclear-nuclear Coulombic
   repulsion (VNN)

In some textbooks ESD doesnt include VNN, which
will be added later (Vtot ESD VNN).

1. Electronic kinetic energy (Te)
2. Potential energy due to nuclear-electronic
   Coulombic attraction (VNe)

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• Potential energy due to two-electron interactions
  (Vee)
• Coulomb integral Jij (local)
• Coulombic repulsion between electron 1 in
  orbital i and electron 2 in orbital j
• Exchange integral Kij (non-local) only for
  electrons of like spins
• No immediate classical interpretation entirely
  due to antisymmetry of fermions

gt 0, i.e., a destabilization
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(No Transcript)
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Hartree-Fock (HF) Energy Integrals
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Self-Interaction

• Coulomb term J when i j (Coulomb interaction
  with oneself)
• Beautifully cancelled by exchange term K in HF
  scheme

? 0
0
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Constrained Minimization of EHF?SD
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Hartree-Fock (HF) Equation

• Fock operator effective one-electron operator

• two-electron repulsion operator (1/rij) replaced
  by one-electron operator VHF(i)
• by taking it into account in average way

Two-electron repulsion cannot be separated
exactly into one-electron terms. By imposing the
separability, the Molecular Orbital Approximation
inevitably involves an incorrect treatment of
the way in which the electrons interact with each
other.
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Self-Consistent Field (HF-SCF) Method

• Fock operator depends on the solution.
• HF is not a regular eigenvalue problem that can
  be solved in a closed form.
• Start with a guessed set of orbitals
• Solve HF equation
• Use the resulting new set of orbitals
• in the next iteration and so on
• Until the input and output orbitals
• differ by less than a preset threshold
• (i.e. converged).

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Koopmans Theorem

• As well as the total energy, one also obtains a
  set of orbital energies.

• Remove an electron from occupied orbital a.

Orbital energy Approximate ionization energy
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Koopmans Theorem Examples
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Restricted vs Unrestricted HF
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Accuracy of Molecular Orbital (MO) Theory
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Electron Correlation

• A single Slater determinant never corresponds to
  the exact wavefunction.
• EHF gt E0 (the exact ground state energy)
• Correlation energy a measure of error introduced
  through the HF scheme
• EC E0 - EHF (lt 0)
• Dynamical correlation
• Non-dynamical (static) correlation
• Post-Hartree-Fock method
• Møller-Plesset perturbation MP2, MP4
• Configuration interaction CISD, QCISD, CCSD,
  QCISD(T)

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Summary