Page 10 - 71 the abc of dft_opt

1.3. REINTERPRETING MOLECULAR ORBITALS 19 20 CHAPTER 1. INTRODUCTION
4
!
2 density
0
-2 Kohn-Sham potential
1s
1s
1s
-4 -2/r
!
-6
He atom
-8
0 0.2 0.4 r 0.6 0.8 1
Figure 1.2: The external and Kohn-Sham potentials for the He atom, thanks to Cyrus’ Umrigar’s very accurate density. Figure 1.3: Orbital diagram of H 2 bond formation.

electrons. If we can figure out some way to approximate this potential accurately, we have And a highly accurate approximate density functional calculation produces the full electronic
a much less demanding set of equations to solve than those of the true system. Thus we energy from these orbitals, resolving the paradox.
are always trying to improve a non-interacting calculation of a non-interacting wavefunction,
rather than that of the full physical system. In Fig. 1.1 there are also plotted the local
1.4 Sampler of applications
density approximation (LDA) and generalized gradient approximation (GGA) curves. LDA
is the simplest possible density functional approximation, and it already greatly improves on
HF, although it typically overbinds by about 1/20 of a Hartree (or 1 eV or 30 kcal/mol), In this section, we highlight a few recent applications of modern DFT, to give the reader a
feeling for what kinds of systems can be tackled.
which is too inaccurate for most quantum chemical purposes, but sufficiently reliable for many
solid-state calculations. More sophisticated GGA’s (and hybrids) reduce the typical error in Example from biochemistry, using ONIOM.
Example from solid-state, eg ferromagnetism.
LDA by about a factor of 5 (or more), making DFT a very useful tool in quantum chemistry.
Example using CP molecular dynamics, eg phase-diagram of C.
In section 1.5, we show how density functionals work with a simple example from elementary
quantum mechanics. Example from photochemistry, using TDDFT.

1.3 Reinterpreting molecular orbitals 1.5 Particle in a box

The process of bonding between molecules is shown in introductory chemistry textbooks as In this section, we take the simplest example from elementary quantum mechanics, and apply
linear combinations of atomic orbitals forming molecular orbitals of lower energy, as in Fig. density functional theory to it. This provides the underlying concepts behind what is going
1.3. on in the more previous sections of this chapter. Much of the rest of the book is spent
But later, in studying computational chemistry, we discover this is only the Hartree-Fock connecting these two.
picture, which, as stated above, is rarely accurate enough for quantum chemical calculations. In general, we write our Hamiltonian as
In this picture, we need a more accurate wavefunction, but then lose this simple picture of
ˆ
ˆ
H = T + V ˆ (1.7)
chemical bonding. This is a paradox, as chemical reactivity is usually thought of in terms of
frontier orbitals. ˆ ˆ
where T denotes the operator for the kinetic energy and V the potential energy. We begin
In the Kohn-Sham approach, the orbitals are exact and unique, i.e., there exists (at most) with the simplest possible case. The Hamiltonian for a 1-D 1-electron system can be written
one external potential that, when doubly occupied by two non-interacting electrons, yields as
the exact density of the H 2 molecule. So in this view, molecular orbital pictures retain their ˆ 1 d 2
significance, if they are the exact Kohn-Sham orbitals, rather than those of Hartree-Fock. H = − 2 dx 2 + V (x) (1.8)