Lecture: ab initio Methods Post Hartree-Fock Calculations

In the Hartree-Fock scheme, the many particle effects are approximated by the interaction of each electron with an average electric field produced by all the remaining electrons in the system. The error introduced by this SCF approximation is known as the correlation energy. The correlation energy is a small percentage of the total energy of the system, but is usually on the order of the strengths of chemical bonds or ionization energies. Thus, the correlation energy is not negligible. A number of different techniques are available to determine the correlation energy. The most widely used is the Configuration Interaction (CI) method. (Other methods are perturbation techniques or a cluster expansion of the system or direct solution of the integro-differential Hartree-Fock equations.)

Configuration Interaction

CI; the basic idea. For any system, there exists an infinite number of orbitals in addition to the HF orbitals found by the procedure outlined above. These higher energy orbitals can be used to construct other configurations with coefficients variationally determined. The CI wave function has the form,

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where tex2html_wrap_inline679, are an orthonormal set of N electron configurations.

Varying tex2html_wrap_inline681 leads to the CI equations,

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where I is the identity matrix.

The matrix elements of the Hamiltonian matrix are given by,

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If i and j are of different symmetry then, tex2html_wrap_inline683.

If M configurations are used in the CI expansion then one obtains M eigenenergies and the lowest energy is an upperbound to the true energy. This implies the total system need not be solved, only the lowest (or lowest two or three) eigenvalues need to be determined. In the above discussion, configuration means a symmtery adapted linear combination of Slater Determinants (denoted by D(i) ). Symmetry adapted means the configuration posseses all the symmetry of the molecular state being described. This is written as,

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The most common CI calculation is the Multi-Configuration Self Consistent Field (MCSCF) calculation. In the MCSCF, the configuration parameters tex2html_wrap_inline685 from Eq. (2.5), and the single particle expansion parameters Ci from Eq. (2.2) are varied simultaneously. This calculation is an iterative process. The tex2html_wrap_inline689 iteration involves forming the vector,

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One makes an initial guess to the tex2html_wrap_inline691 and tex2html_wrap_inline693. Then solve for the tex2html_wrap_inline679. Using the new tex2html_wrap_inline679, determine new parameters tex2html_wrap_inline691 and tex2html_wrap_inline693 and repeat the calculation until the parameters converge to a specified tolerance. The complication in the MCSCF is the determination of the matrix elements. Seighbahn uses a unitary group approach and a separation of the orbitals to the occupied and valance orbitals for rapid convergence.

To determine a molecular configuration, only the electrons outside of the closed shells contribute. Within a set of orbitals, matrix elements between determinants which differ by three or more spin orbitals are identically zero, hence single and double excitations of the reference HF configuration need to be considered. This is not really much of a limitation, there are many (infinitely many) excited configurations to consider.

These are the most common properties determined by ab initio calculations.

  1. Molecular Geometry
  2. Force Constants - Vibrational Spectrum
  3. Energy Barriers to Reactions - Internal, Rotational or Inversion
  4. Potential Surfaces
  5. Chemical Reactions - Transition State Theory
  6. Ionization Potentials
  7. Intermolecular Interaction Potential
  8. Solvation



Author: Ken Flurchick