Transcript Chapter 1.

Chapter 1. Introduction, perspectives, and aims. On the science
.
of simulation and modelling. Modelling at bulk, meso, and nano
scale. (2 hours).
Chapter 2. Experimental Techniques in Nanotechnology. Theory
and Experiment: “Two faces of the same coin” (2 hours).
Chapter 3. Introduction to Methods of the Classic and Quantum
Mechanics.
Force
Fields,
Semiempirical,
Plane-Wave
pseudpotential calculations. (2 hours)
Chapter 4. Introduction to Methods and Techniques of Quantum
Chemistry, Ab initio methods, and Methods based on Density
Functional Theory (DFT). (4 hours)
Chapter 5. Visualization codes, algorithms and programs.
GAUSSIAN; CRYSTAL, and VASP. (6 hours).
.
Chapter 6. Calculation of physical and chemical properties of
nanomaterials. (2 hours).
Chapter 7. Calculation of optical properties. Photoluminescence.
(3 hours).
Chapter
8.
Modelization
of
the
growth
mechanism
of
nanomaterials. Surface Energy and Wullf architecture (3 hours)
Chapter
9. Heterostructures Modeling. Simple and complex
metal oxides. (2 hours)
Chapter 10. Modelization of chemical reaction at surfaces.
Heterogeneous catalysis. Towards an undertanding of the
Nanocatalysis. (4 hours)º
Chapter 4. Introduction to Methods and
Techniques of Quantum Chemistry, Ab initio
methods, and Methods based on Density
Functional Theory (DFT). (4 hours)
Juan Andrés y Lourdes Gracia
Departamento de Química-Física y Analítica
Universitat Jaume I
Spain
&
CMDCM, Sao Carlos
Brazil
Sao Carlos, Novembro 2010
1) Ab initio methods
.
2) Methods based on Density
Functional Theory (DFT)
Density functional theory (DFT) is nowadays one of the most (if
not the most) used methods of performing electronic structure
calculations in the ground state of atoms, molecules, and solids. Its
success stems from the simplification of the Schrödinger equation
through the Hohenberg-Kohn (HK) theorems and also from the
practical implementation of the self-consistent Kohn-Sham (KS)
equations.
These look like (and scale like) the Hartree-Fock equations,
where the several terms that make up the ground-state energy
(which HK proved to be a functional of the electron density, E0[F])
can be calculated exactly except one, corresponding to a small
fraction of the total energy and named exchange correlation (xc)
energy functional, Exc[F], which is unknown.
L. P. Viegas, A. Branco and A. J. C. Varandas, J. Chem. Theory Comput. 2010.
In press
KS-DFT is therefore exact in principle, while in practice Exc[F] must be
approximated, thus being the main source of error in the theory. Such
approximate functionals are often constructed4 by constraint satisfaction
(nonempirical functionals) or by fitting them to experimental and/or ab initio
data (semiempirical functionals). It is believed that increasing the number
of satisfied constraints is a step toward the exact and universal functional,
but while this approach seems theoretically attractive, its progress has
shown to be somewhat slow.
On the other hand, semiempirical functionals rapidly achieved
widespread success, particularly through the popular B3LYP
functional.The semiempirical approach has the advantage of making
accurate predictions for systems which belong (or are similar) to the
training set, but it carries two main problems: one is the possible failure for
systems outside the training set, and the other is that the functionals
sometimes do not respect some of the known exact constraints.
L. P. Viegas, A. Branco and A. J. C. Varandas, J. Chem. Theory Comput. 2010.
In press
However, their low computational cost together with a
huge predictive character for systems that cannot be
correctly described by nonempirical functionals explains the
great success of the semiempirical approach.
While in ab initio theory one knows exactly how to
proceed to improve the quality of the results, in KS-DFT this
is not so obvious and straightforward. One way to hierarchize
and develop improved exchange-correlation functionals is by
adding to them increasingly complex ingredients, therefore
creating the possibility of satisfying more constraints.
L. P. Viegas, A. Branco and A. J. C. Varandas, J. Chem. Theory Comput. 2010.
In press
This is the basic philosophy behind the “Jacob’s ladder” of density
functional approximations to the exchange-correlation energy, where
functionals (nonempirical or semiempirical) are assigned to different
rungs of the ladder, according to the complexity of their ingredients. As
one naturally expects, on going up the ladder, accuracy and
computational cost will generally increase.
The first rung is the local spin density approximation (LSDA), often
referred to as the “mother of all approximations”. It uses as ingredients
the spin densities FR(r) and F(r) and is by construction exact for uniform
densities or densities that vary very slowly over space. Many electronic
systems do not respect these conditions (e.g., atoms and molecules),
making LSDA more useful in solids. However, despite overestimating
atomization and binding energies, LSDA gives good results in predicting
properties like molecular geometries and vibrational frequencies, being a
useful structural tool except for thermochemistry.
L. P. Viegas, A. Branco and A. J. C. Varandas, J. Chem. Theory Comput. 2010.
In press
The second rung is the generalized gradient
approximation (GGA) which introduces the density gradients
∇FR(r) and ∇F(r) as additional ingredients. GGAs show a
good improvement for thermochemistry relative to LSDA.
The third rung is the metageneralized gradient
approximation (meta-GGA), with ∇2FR(r) and ∇2F(r) being
additional ingredients or, more commonly, the Kohn-Sham
orbital kinetic energy densities τR(r) and τ(r). The metaGGAs mainly improve atomization energies while keeping
computational cost similar to the previous rungs
L. P. Viegas, A. Branco and A. J. C. Varandas, J. Chem. Theory Comput. 2010.
In press
Most of DFT’s problems originate from trying to
accomodate the fact that the energy is known to be a
functional of the density into a computational construct,
which introduces errors in the kinetic and exchange
energies, in addition to the correlation that it is suppose to
include, and manifests itself in the above list of failures.
Most of these problems originate with the potential, VXC,
instead of the functionals, E[ρ(1)], per se, as an energy
functional is relatively insensitive to small changes in the
density and can often provide good energies once a
realistic density, ρ(1), is inserted.
Hierarchy of exchange–correlation functionals
During the last two decades many approximations to the
exchange–correlation functional that would produce, within
DFT, exactly the same ground-state as the solution of the
many body Schrödinger equation, have been proposed.
Perdew refers to the hierarchy of approximate functionals
as the “Jacob’s ladder” of DFT. The lowest rung on this ladder
is the Local density approximation (LDA). At this level, the
local exchange–correlation energy, Exc[n(r)], is taken to be
the same as in a homogeneous electron gas of the same
density, as derived from quantum Monte-Carlo simulations.
J. Hafner, Computer Physics Communications 177 (2007) 6–13
The Generalized Gradient Approximation (GGA) introduces a
dependence of Exc on the local gradient of the electron density,
|∇n(r)|. Many different forms of the GGA have been proposed in
the literature. For materials simulations it is good practice to adopt
parameter-free functionals derived from known expansion
coefficients and sum-rules of many-body theory [10,11] and to
avoid empirical parameterizations popular in molecular quantum
chemistry.
Meta-GGA functionals introduce the kinetic energy density [or
the Laplacian n(r) of the electron density] as an additional variable
[12].
Hyper-GGA uses the one-electron orbitals (instead of the
many-electron wavefunctions) to evaluate the Hartree–Fock
exchange energy; this is often referred to as “exact exchange”.
J. Hafner, Computer Physics Communications 177 (2007) 6–13
Hybrid functionals mix exact (i.e. Hartree–Fock) and DFT exchange
and describe correlation at the DFT level. All functionals exist in a spindegenerate and spin-polarized (for magnetic calculations) version. LDA
and GGA are by far the most commonly used functionals.
The GGA corrects the over-binding tendency of the LDA (albeit with a
certain trend to over-correct for heavy elements) and yields a correct
answer in some cases where the LDA fails quite spectacularly [such as
the prediction of the correct ground-state of Fe (ferromagnetic) and Cr
(antiferromagnetic) which are both predicted to be nonmagnetic in the
LDA] and the location of the energy barrier for the dissociation of small
molecules over metallic surfaces.
Climbing the DFT ladder further to the meta-GGA or hyper-GGA does
not lead to a systematic improvement over the GGA.
J. Hafner, Computer Physics Communications 177 (2007) 6–13
Hybrid functionals are enormously popular in molecular chemistry, but
their application to solid-state and materials problems is still in a
pioneering stage.
Preliminary studies
show a great promise for insulating and
semisemiconducting systems, although serious difficulties are evident if
these functionals are applied to metals. Novel range-separated functionals
may offer a viable solution also for metals.
All levels of DFT have been implemented in VASP, and at the end of
this paper I shall comment specifically on the latest results for groundstate properties obtained with hybrid functionals and for the description of
excited states based on many-body perturbation theory.
J. Hafner, Computer Physics Communications 177 (2007) 6–13
Usual DFT calculations actually significantly underestimate the highest
occupied molecular orbital lowest unoccupied molecular orbital HOMOLUMO energy gaps, i.e., band gaps, except for those of metal crystals.
Especially in solid-state physics, many studies have been reported that
identify why orbital energy gaps are usually underestimated. Perdew et al
proved that poor orbital energy gaps are attributed to the discontinuity of
exchange-correlation potentials.
J. P. Perdew, R. G. Parr, M. Levy, and J. L. J. Balduz, Phys. Rev. Lett. 49,
1691 1982.
J. P. Perdew and M. Levy, Phys. Rev. Lett. 51, 1884 1983.
L. J. Sham and M. Schlüter, Phys. Rev. Lett. 51, 1888 1983.