Lecture: ab initio Methods Density Functional Theory for Molecules


The molecular Hamiltonian for M atoms with N electrons, can be written in Atomic Units. (See the appendix: Units) as,

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where

and

tex2html_wrap_inline313 is the distance between the atomic centers,


tex2html_wrap_inline315 is the distance between the electrons,

and

tex2html_wrap_inline317 is the distance between an electron and an atomic center.


The total energy, which is the expectation value of the Hamiltonian, is written as

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In matrix form, the Hartree-Fock-Roothan equations are,

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where C is the expansion coefficient matrix, tex2html_wrap_inline323 and tex2html_wrap_inline325 is the energy.

Thomas - Fermi Theory

Thomas (1927) and Fermi (1928)

Using a cubic division of space, the kinetic energy of the electrons can be expressed as

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where tex2html_wrap_inline329

This is a relation between the total kinetic energy and the electron density. This is the Thomas-Fermi kinetic energy functional.

This is based on statistical mechanics and Thomas and Fermi applied this to electrons in atoms and molecules!

Local Density Approximation LDA

Electronic properties are determined as a functional of the electron density by applying locally relations appropriate for a homogeneous electronic system.

The usefulness of this approach is that long range computations are removed. The accuracy of this approach is remarkable. (No more surprising than the accuracy of Hartree-Fock schemes)

Density Functional Theory

The fundamental quantity is the electron density, N, given by,

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TO begin, consider extending the Thomas-Fermi kinetic energy functional to cover the total energy of the system. Recall the object is to obtain an expression for the energy of the system. Initially neglecting exchange-correlation terms, using terms first order in the density matrix and classical electrostatic interactions,

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with the constraint of conservation of the number of electrons;

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For the ground state, then

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which gives;

eqnarray101

where tex2html_wrap_inline337 is the electrostatic potential at point tex2html_wrap_inline339 due to the nuclei and the entire electron density.

Hohenberg and Kohn (1964) and Kohn and Sham (1965) Theorems.

Theorem 1: The electron density tex2html_wrap_inline341 , through normalization, determines the electron number N and the external potential V(|vecr). And with Schrodinger's equation, all the molecular properties.

Theorem 2: The energy functional tex2html_wrap_inline345 is a minimum for the true electron density. This requires electron number conservation.

Thus, minimize the energy functional

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with respect to variations in tex2html_wrap_inline347 (not variations in the wave functions as in Hartree-Fock) with the constraint of electron number conservation

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gives;

eqnarray135

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where tex2html_wrap_inline353 is the basis function and tex2html_wrap_inline355 are the expansion coefficients.

These are the Kohn-Sham orbital equations.

Approximations for the exchange correlation term.

This looks a lot like the equations arising from the variational approach for Hartree-Fock theory. However there are crucial differences;

  1. The fundamental quantity is the electron density
  2. The explicit use of the local density approximation for this expression.
  3. The results obtained are the electron densities, not wave functions.
  4. The integrals involved are not over atomic or molecular orbitals.
  5. The form of the exchange-correlation term is crucial for accuracy and tractability.


Author: Ken Flurchick