This notion was first suggested by Fermi (1930) and later proven to be exact by W. Kohn
in 1964. Professor Kohn was given the Nobel prize for chemistry in 1998 as the founder of density functional theory (DFT)
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Index
I.
Short introduction to density functional theory
1. The total energy as a function of electron density
2. Electron correlation
1. Local DFT
2. Non-local DFT
3.Ongoing Search for More Accurate Functional
III. Performance of Approximate DFT in Transition Metal Chemistry
1. Calculated Geometries
2. Calculated bond energies
3. Inorganic Spectroscopy
3.1 Electronic Excitations
3.2 Vibrational Spectroscopy
I. Short introduction to
density functional theory
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1. The total energy as a function
of electron density
Density functional theory is based on the notion that
the total energy, E, of an
electronic system is determined by the electron density
r, or E(r).
This notion was first suggested
by Fermi (1930) and later proven to be exact by W. Kohn
in 1964. Professor Kohn was given the Nobel
prize for chemistry in 1998 as the founder of density functional
theory (DFT)
In the DFT theory of Kohn ( see e.g. Ziegler, T. "Approximate Density Functional Theory as a Practical Tool in Molecular Energetics and Dynamics" Chem. Rev., 1991, 91,651. ) the total energy is written in terms of the energy of n non-interacting electrons and a term Eex that takes into account the complicated correlated motion of the electrons.
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2. Electron correlation
As the electrons move they try to avoid the repulsive interaction from each other.
by creating a hole (or no-fly zone) into which other
electrons will not penetrate. This is called the exchange-correlation hole
and gives rise to the term Eex.
(See Ziegler, T. "Approximate Density Functional
Theory as a Practical Tool in Molecular Energetics and Dynamics" Chem.
Rev., 1991, 91,651. Also E.J.Baerends and O.V. Gritsenko Phys.
Chem. A 1997,101,5383 )
In electron theory the difficult
part is to describe this hole or Eex. It follows from the work
of Kohn that the shape of the hole or Eex in principle can be
expressed as a function of the electron density. However, we do not know
the exact form of this relationship.
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II.Approximate DFT
1. Local DFT
The first generation of approximate DFT took the shape of the exchange-correlation hole function or Eex from the electron gas where the hole-function is known as a function of the electron density. This approximation became known as local DFT and it has been used extensively
2. Non-local DFT
Axel Becke, John Perdew and others have improved on the simple local electron gas model where the hole function correspond to an electron density that is assumed to be constant throughout. In their non-local theory the hole function or Eex includes corrections from the fact that the electron density in atoms, molecules and solids changes with position.
3.Ongoing Search for More Accurate Functionals
Attempts to developed better functionals in which the relation between the hole function ( or Eex) and the electron density is expressed more accurately are still ongoing :
1. A. D. Becke, "A new mixing of Hartree-Fock and local density-functional theories," J. Chem. Phys. 98, 1372-1377 (1993).
2. . K. Burke, J. P. Perdew and Y. Wang, in Electronic Density Functional
Theory: Recent Progress and New Directions, Ed. J. F.
Dobson, G. Vignale and M. P. Das (Plenum, 1998).
3. J. P. Perdew, K. Burke and Y. Wang, Phys. Rev. B 54, 16533 (1996).
4. C. Adamo and V. Barone, Chem. Phys. Lett. 274, 242 (1997).
5. P. M. W. Gill, Mol. Phys. 89, 433 (1996).
6. A. D. Becke, J. Chem. Phys. 104, 1040 (1996).
7. A.D. Daniels, J. M. Milliam and G.E. Scuseria, J. Chem. Phys. 107, 425 (1997).
III. Performance of Approximate
DFT in Transition Metal Chemistry
For a general introduction see: Ziegler, Tom. "The
1994 Alcan Award lecture: Density Functional Theory as a Practical Tool
in Studies of Organometallic Energetics and Kinetics. Beating the Heavy
metal Blues with DFT." Can.J.Chem 1995; 73, 743.
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In order to describe a chemical
reaction from the reactant over the transition state to the product well
, DFT would have to be able to determine structures accurately.
Geometries calculated by the local DFT method (LDA) and the non-local method LDA/NL are in general in goodagreement with experiment. Bond distances are usually calculated with an accuracy of 0.02 angstrom or better with the LDA/NL method. However, exceptions exist
Here is a full geometry optimization of Fe2CO9 compared to experiment
A DFT description of a chemical reaction also requires accurate (relative) energies of the species involved. It would be useless if DFT predicted (green curve) the reaction to be exothermic with a small barrier that experimentally (red) was observed to be endothermic with a large barrier.
Here are a number of calculated
M-CO bond disociation energies (kJ/mol). Local DFT (LDA) affords bond dissociation
energies that are much too high- it overestimates bond energies. With non-local
corrections LDA/NL the bond energies are in good agreement with experiment.
Non-local corrections are in general required to obtain good estimates
of (relative) energies. The error margin for LDA/NL is around 5 Kcal/mol
or 20 kJ/mol. (See :Ziegler,
Tom. "The 1994 Alcan Award lecture: Density Functional Theory as a Practical
Tool in Studies of Organometallic Energetics and Kinetics. Beating the
Heavy metal Blues with DFT." Can.J.Chem 1995; 73, 743.
)
Sigma bonds are also estimated with high accuracy
by LDA/NL
We can conclude that both energies and geometries are well represented by LDA/NL methods - to the point where LDA/NL can be used to study the potential energy surface of transition metal complexes.
3. Inorganic Spectroscopy
Molecules in transition metal
chemistry are characterized extensively by spectroscopic methods in which
a molecule is subjected to light as well as constant electric or
magnetic fields. The response from this perturbation (the spectra) is used
to deduce information about the composition or structure of the investigated
species.
DFT can be used to simulate and interprete experimental spectropic observations. Typical fields where this is possible are (a) ionizations; (b) electronic excitations; (c) molecular vibrations; (d) NMR and ESR spectroscopy
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3.1 Electronic Excitations
Above the ground state potential energy surface are the potential surfaces of the excited states. They are important for photo-chemistry and other related fields. DFT is able to treat excited states on the same footing as the ground state, either by traditional methods (Ziegler, Rauk and Baerends TCA ,1977, 43, 261) or by schemes based on time-dependent perturbation theory (C. A. Ullrich and E. K. U. Gross, Phys. Rev. Lett. 74, 872 (1995)).
The electronic
spectrum of the tetrahedral complex MnO4-
(permagnate) has been studied extensively and used as a testing ground
for new theoretical methods . Below is a complete assignment of the
electronic spectrum of that molecule. The excitations are from oxygen lone-pairs
to empty d-orbitals of the formally d0 metal center. This study
also examined the structure of the excited states and their distortion
from tetrahedral geometry.
(see: Dickson, Ross M., and Tom Ziegler. "A Density
Functional Study of the Electronic Spectrum of Permanganate." Int.J.Quantum
Chem.1996,58,681. )
DFT on the LDA and LDA/NL levels is able to calculate
vibrational frequencies and normal modes of transition metal complexes
with high accuracy. Calculated frequencies can be used to identify
intermediates (spectroscopic finger printing ). It is also required for
evaluating the entropy of activation in elementary reaction steps. Finally,
frequency calculations can be used to generate molecular mechanics force
fields for transition metal complexes
Force constants are the second derivatives of
the molecular energy with respect to two nuclear displacements. They can
be calculated directly from analytically derived expressions.
(See: Berces,A.; Ziegler,T. Application of Density
Functional Theory to the Calculation of Force Fields and Vibrational Frequencies
of Transition metal Complexes. Top.in Cur. Chem. 1996,182,42-85 )
Here is the full assignment for the vibrational
spectrum of ferrocene
