Numerical methods vs. basis sets in quantum chemistry

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Numerical methods vs. basis sets in quantum
chemistry

• M. Defranceschi
• CEA-Saclay

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Various molecular ab initio models

• Minimization problem
• where

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For sake of simplicity

• non relativistic equations,
• time-independent model,
• nuclei are points at fixed known positions,
• real-valued functions,
• spin not explicitly considered,
• simplifications to make it more convenient

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Vector space considered

• Physical functions ? are square integrable
  functions (three dimensional measure in the
  Lebesgue sense)
• Hilbert space
• Reduced to a subspace

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Overwhelming numerical difficulties

• Problem too difficult to be solved numerically
• Vector space too large
• Non linear terms

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Two classes of simplification

• Rigourous energy/approximated wavefunction
• Hartree-Fock approx.
• Restriction to a set of functions

• Rigourous density/approximated energy
• Density functional approx.

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Hartree-Fock settings
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Mathematical fundation

• Define the energy functional E(?) on a set X of
  functions ? (the set of all the possible states
  of the molecule).
• Then find a function (the
  ground-state) satisfying some given constraint
  (i.e. constant number of
  electrons) and minimize the energy E on the
  convenient set of states

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Notion of physical space

• What are the variables of ?
• Physical notion coordinates can be either
  position or momentum (or both) or any other
  quantity.

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First ideas in position space

• Analytical solutions
• Numerical solutions
• Radial function of
  in a one-center approximation
• Spheroidal cooordinates for diatomic molecules
• Complete numerical integration for diatomic
  molecules
• In the case of atoms numerical integration are
  reliable

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The quantum chemist procedure

• Molecules are not considered as a whole but as
  constructed from atoms.
• Use of atomic basis sets
• Slater type
• Gaussian type
• Any functions which contain the correct physical
  information.
• The procedure most widely used consists in
  writing the molecular orbitals as LCAO which
  belong to a given complete set of the Sobolev
  space

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Manageable approximate solutions

• Infinite basis sets are impracticable
• Truncated basis sets
• Large expansions but often to small
• Tendency towards linear dependence
• Inherent deficiencies for GTO
• cusp problem
• wrong asymptotic decay

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Some attempts of numerical solutions

• Integration over a numerical grid
• Finite element method
• Momentum space direct numerical integration
• Numerical solution using a wavelet basis

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Finite element method

• Very accurate results for even time-independent
  problems for simple systems.
• Large storage requirement for the FE matrices for
  extended three-dimensional systems
• Removal of the singularities inherent in the
  nuclear potentials.

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Momentum space approach(1)

• In position space HF equations are
  integro-differential
• FT of operators and not of functions

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Momentum space approach(2)

• In momentum space HF equations are first order
  integral equations

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Momentum space approach(3)

• The solutions for bound states ( ) can
  be obtained by an iterative procedure starting
  with a LCAO in momentum space ( a modified
  Lanczos procedure).
• Enables to recover basis functions, and then
  basis sets not limited in size
• Enables to recover the asymptotic behavior.
• Removal of the singularities inherent in the
  nuclear and interelectronic potentials.

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Momentum space approach(4)
trial function first iterate Slater function
1.162569 1.247735 1.248098
0.724396 0.705793 0.750513
1.358986 1.727544 2.322622
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Momentum space approach(5)

• Disadvantage of a FT of a function is that all
  information about its support or its
  singularities is lost.
• A function with high variations of momenta is
  hardly interpretable
• A compactly supported function requires a lot of
  sinusoidal functions

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A wavelet approach(1)

• The idea is to realize a decomposition with
  vanishing functions which leads to a momentum
  representation involving a position parameter
• Functions depending on two variables linked to
  momentum and position are used

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A wavelet approach(2)

• A representation is obtained by means of a
  decomposition of the Schrödinger operator onto an
  orthonormal wavelet basis.
• scaling function
• wavelet mother

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A wavelet approach(3)

• The approach is related to multiresolution
  analysis, which is a decomposition of the Hilbert
  space into a chain of closed subspaces.
• The family defined by the scaling function
  constitutes an o.n. basis set for Vj. Let Wj be
  the space containing nthe difference in
  information between Vj-1 and Vj. It allows to
  decompose

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A wavelet approach (4)

• The two part of a Fock operator has to be treated
  in two differents ways
• The NS form of the Laplacian operator is solved
  iteratively
• The NS form of the potential term is obtained by
  a quadrature formula

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A wavelet approach(5)

• The matrix representation of an operator applied
  to a vector may be depicted

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A wavelet approach (6)
STO-1G STO-2G STO-3G Slater
0.4244132 0.4244099 7.78 10-6 0.4857612 0.4857904 6.01 10-5 0.4967535 0.4968063 1.06 10-4 0.5000000 0.5002572 5.14 10-4
-0.8488264 -0.8486610 1.95 10-4 -0.9715744 -0.9711148 4.73 10-4 -0.9937322 -0.9929722 7.65 10-4 -1.000000 -0.9985268 1.47 10-3
-0.4244132 -0.4242511 3.82 10-4 -0.4858132 -0.4853243 1.01 10-3 -0.4969787 -0.4961660 1.64 10-3 -0.500000 -0.4982696 3.46 10-3
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Conclusion

• The numerical development is far from the state
  of the art of the current quantum chemistry
  practice based on the use of atomic basis sets.