Lecture 5. Many-Electron Atoms. Pt.3 Hartree-Fock Self-Consistent-Field Method

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Lecture 5. Many-Electron Atoms. Pt.3Hartree-Fock
Self-Consistent-Field Method
References

• Ratner Ch. 9, Engel Ch. 10.5, Pilar Ch. 10
• Modern Quantum Chemistry, Ostlund Szabo (1982)
  Ch. 2-3.4.5
• 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
• http//vergil.chemistry.gatech.edu/notes/hf-intro
  /hf-intro.html

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Hartree (single-particle) self-consistent-field
methodbased on Hartree products (D. R. Hartree,
1928)
Proc. Cambridge Phil. Soc. 24, 89
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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Constrained minimization with the Hartree product
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Hartree-Fock Self-Consistent-Field Methodbased
on Slater determinants (HartreePauli) (J. C.
Slater V. Fock, 1930) Z. Physik, 61, 126 Phys.
Rev. 35, 210
Restrict the search for the minimum E? to a
subset of ?, a Slater determinant.

• To build many-electron wave functions, assume
  that electrons are uncorrelated. (Hartree
  products of one-electron orbitals)
• To build many-electron wave functions, use Slater
  determinants, which is all antisymmetric products
  of N spin orbitals, to satisfy the Pauli
  principle.
• Use the variational principle to find the best
  Slater determinant (which yields the lowest
  energy) by varying the spatial orbitals ?i.

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Beyond Hartree the ground state of He (singlet)
? 1sgt ?
notation
1s
Slater determinant
? 1s2
Total spin quantum number S Ms 0 (singlet)
S2 ?(1,2) (s1 s2)2 ?(1,2) 0, Sz ?(1,2)
(sz1 sz2) ?(1,2) 0
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Energy of the Slater determinant of the He
atom the singlet ground state
spatial-symmetric
spin-antisymmetric
no spin in the Hamiltonian
Coulombic repulsion between two charge
distributions 1s(1)2 and 1s(2)2
Coulomb integral
lt1sh1sgt lt1sTVNe1sgtTssVs
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Coulombic repulsion between two charge
distributions 1s(1)2 and 1s(2)2
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Excited state of He (singlet and triplet states)
antisymmetric
spatial-symmetric
spin- symmetric
spatial-antisymmetric
spatial-symmetric
spatial-antisymmetric
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Energy of the Slater determinant of the He
atom a triplet first excited state
singlet
triplet
? (quiz)
Coulomb integral gt 0
includes in it wave function (final solution)!
where
Exchange integral (gt0)
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Energy of the Slater determinant of the He
atom a triplet first excited state
singlet
triplet
Coulomb integral gt 0
includes in it wave function (final solution)!
where
Exchange integral (gt0)
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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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Each term includes the wave function (the final
solution) in it!
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Hartree-Fock Self-Consistent-Field Methodbased
on Slater determinants (HartreePauli) (J. C.
Slater V. Fock, 1930)

• Each ? has variational parameters (to be changed
  to minimize E) including the effective nuclear
  charge ? (instead of the formal nuclear charge Z)
• Variational condition
• Variation with respect to the one-electron
  orbitals ?i, which are orthonormal

or its combination for lower E
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Constrained (due to the orthonormality of ?i)
minimization of EHF?SD leads to the HF equation.
Pilar Ch.10.1, Ostlund/Szabo Ch.1.3
vergil.chemistry.gatech.edu/notes/hf-intro/node7.h
tml
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Constrained minimization with the Slater
determinant
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After constrained minimization with the Slater
determinant
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Hartree-Fock equation (one-electron equation)
Fock operator effective one-electron operator
Two-electron repulsion operator (1/rij) is
replaced by one-electron operator VHF(i), which
takes it into account in an average way
Two-electron repulsion cannot be separated
exactly into one-electron terms. By imposing the
separability, the orbital approximation
inevitably involves an incorrect treatment of the
way in which the electrons interact with each
other.
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Hartree-Fock Self-Consistent Field (HF-SCF) Method

• Problem
• Fock operator (V) depends on the solution.
• The answer (solution) must be known in order to
  solve the problem!
• HF is not a regular eigenvalue problem that can
  be solved in a closed form.
• Solution (iterative approach)
• Start with a guessed set of orbitals.
• Solve the Hartree-Fock 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 to a
  self-consistent field).

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Hartree-Fock equation (One-electron equation)
spherically symmetric
Veff includes

• - Two-electron repulsion operator (1/rij) is
  replaced by one-electron operator VHF(i), which
  takes it into account in an average way.
• - Any one electron sees only the spatially
  averaged
• position of all other electrons.
• - VHF(i) is spherically symmetric.
• - (Instantaneous) electron correlation
• is ignored.
• Spherical harmonics (s, p, d, ) are valid
• angular-part eigenfunction (as for H-like
  atoms).
• - Radial-part eigenfunction of H-like atoms are
  not valid any more.

optimized
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Electron Correlation
Ref) F. Jensen, Introduction to Computational
Chemistry, 2nd ed., Ch. 4

• A single Slater determinant never corresponds to
  the exact wave function.
• 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), MCSCF, CAFSCF,

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Solution of HF-SCF equation gives
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Solution of HF-SCF equation Effective nuclear
charge (Z-? is a measure of shielding.)
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Aufbau (Building-up) principle