Subject: DFT and spectroscopy

From: Gustavo Mercier <[email-masked]>
Date: Sat, 11 Dec 1993 15:55:26 -0500 (EST)

Hi, Netters!

I recently posted a message on the application of ADF/DFT to the d-d
transitions of hexaquao first row transition metals. The results of my
computations were rather puzzling and I requested some help from the NET.

A little reading has shed some light into the problem. Given the
interest of some subscribers in this topic I post a reply to my own
message : - )).

The problem is at a very FUNDAMENTAL level.

The application of DFT to spectroscopy is not so simple as to just
compute the ground state and excited state densities and evaluate
the energy differences as I did!!

Oliveira has an article in v. 21 of Adv. in Quantum Chem. (1990, p 135,
ed. S. Trickey, APress) titled "Density Functional Treatment of
Excited States." In it he states:

  LIKE THE GROUND - STATE FORMALISM, FOR PRACTICAL APPLICATIONS
  THE ENSEMBLE FORMALISM [FOR EXCITED STATES] REQUIRES AN APPROXIMATION
  FOR THE EXCHANGE - CORRELATION ENERGY FUNCTIONAL. HERE, UNFORTUNATELY,
  THE lda IS inadequate. FOR, SINCE THE MTH-EXCITED- AND THE GROUND-
  STATE ENERGIES OF A HOMOGENEOUS ELECTRON GAS DIFFER INFINITESIMALLY,
  IN THE LDA THE ENSEMBLE - AVERAGED AND THE GROUND-STATE EXCHANGE-
  CORRELATION ENERGY ARE NECESSARILY IDENTICAL. BY CONTRAST, IN
  A SMALL SYSTEM (E.G. AN ATOM), THE GROUND STATE DIFFERING MARKEDLY
  FROM THE EXCITED STATES, ONE EXPECTS THOSE TWO FUNCTIONALS TO BE
  VERY DIFFERENT AND THE LDA TO BE UNSATISFACTORY.

[] my own.

Apparently, the LDA is not good for excited state computations and he
goes on to describe solutions to this problem within a rigorous theory
of DFT for excited states. The key point is that a new exchange - correlation
potential is need for a computational method that wishes to treat
BOTH ground and excited states!

My interpretation of the above is as follows:

Practical molecular applications of DFT to the ground state, such as 
those implemented in ADF, use the K-S equations within the LDA applying
an xc functional most commonly derived from simulations on an homogeneous
electron gas. Different from the electron gas, the density in a molecule
varies drastically in space and you may expect the LDA to fail.
Nevertheless, given the reference state used in K-S, the contribution
from xc is small and as a whole the errors are small.

To extend an xc functional derived from an electron gas simulation
to computations of molecular excited state densities
is inadequate because the difference in the spatial variation of the density
between the excited state and the ground state are also very large. Therefore,
you introduce error at two levels: the spatial variation is significantly
different from that of an electron gas, but the difference of this variation
between the g.s. and the e.s. are also large. The latter is not recovered
by the xc functinal derived from an electron gas because the excited and
g.s. densities of the gas only have an INFINITESIMAL difference!

So in conclusion my approach to computing d-d transitions is fundamentaly
INCORRECT! Unfortunately, the application of DFT to spectroscopy is more
complex than I expected.

I would appreciated comments expanding this topic or correcting my mistakes!

Results will be summarized to the net.

good luck
Gus Mercier
[email-masked]