Lecture 9. Many-Electron Atoms

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Lecture 9. Many-Electron Atoms
References

• Engel, Ch. 10
• Ratner Schatz, Ch. 7-9
• Molecular Quantum Mechanics, Atkins Friedman
  (4th ed. 2005), Ch.7
• Computational Chemistry, Lewars (2003), Ch.4
• A Brief Review of Elementary Quantum Chemistry
• http//vergil.chemistry.gatech.edu/notes/quantrev
  /quantrev.html

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Helium First (1 nucleus 2 electrons)

• Electron-electron repulsion
• Indistinguishability

newly introduced
Electron-electron repulsion
H atom electron at r1
H atom electron at r2
Correlated
r12 term removes spherical symmetry in He.
Cannot solve the Schrödinger equation
analytically (not separable/independent any more)
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Many-electron (many-body) wave function
To first approximation electrons are treated
independently.
H atom orbital
An N-electron wave function is approximated by a
product of N one-electron wave functions
(orbitals). (Hartree product)
Orbital Approximation or Hartree
ApproximationSingle-particle approach or
One-body approach
Does not mean that electrons do not sense each
other.(Well see later.)
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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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Electron has intrinsic spin angular momentum,
which has nothing to do with orbital angular
momentum in an atom.
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spin
space
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Electrons are indistinguishable.
? Probability doesnt change.
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Antisymmetry of electrons (fermions)
Electrons are fermion (spin ½). ? antisymmetric
wavefunction
Quantum postulate 6 Wave functions describing a
many-electron system must change sign (be
antisymmetric) under the exchange of any two
electrons.
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Ground state of Helium
Slate determinants provide convenient way to
antisymmetrize many-electron wave functions.
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Excited state of Helium
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Slater determinant and Pauli exclusion principle

• A determinant changes sign when two rows (or
  columns) are exchanged.
• ? Exchanging two electrons leads to a change in
  sign of the wave function.
• 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
4 quantum numbers (space and spin)
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N-electron wave function Slater determinant

• N-electron wave function is approximated by
• a product of N one-electron wave functions
  (hartree product).
• It should be antisymmetrized.
  

but not antisymmetric!
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Ground state of Lithium
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Variational theorem and Variational method
If you know the exact (true) total energy
eigenfunction
True ground state energy
For any approximate (trial) ground state wave
function
Better trial function
Lower E (closer to E0)
Minimize E? by changing ?!
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Example Particle in a box ground state
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Approximation to solve the Schrödinger
equation using the variational principle

• Nuclei positions/charges number of electrons in
  the molecule
• Set up the Hamiltonian operator
• Solve the Schrödinger equation for wave function
  ?, 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 wave functions

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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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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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• Assume that electrons are uncorrelated.
• Use Slater determinant for many-electron wave
  function
• Each ? has variational parameters (to change to
  minimize E)
• including effective nuclear charge ?

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Constrained Minimization of EHF?SD
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Hartree-Fock (HF) Equation (one-electron 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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HF equation (one-electron equation)
Any one electron sees only the spatially averaged
position of all other electrons.(Electron
correlation ignored)
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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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Solution of HF-SCF equation gives
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Hartree-Fock (HF) Energy
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Hartree-Fock (HF) Energy Integrals
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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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Aufbau (Building-up) Principle
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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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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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The lowest energy term for p 2 and d 6
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Restricted vs. Unrestricted HF