Quantum Chemistry Methods: Ab Initio and DFT Approaches

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). .

2. . 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

3. 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

6. 1) Ab initiomethods .

16. 2) Methods based on Density Functional Theory (DFT)

18. 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

19. 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

20. 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

21. 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

22. 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 meta-GGAs 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

28. 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.

29. 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

30. 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

31. Hybrid functionals mix exact (i.e. Hartree–Fock) and DFT exchange and describe correlation at the DFT level. All functionals exist in a spin-degenerate 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

32. 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 ground-state 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

33. Usual DFT calculations actually significantly underestimate the highest occupied molecular orbital lowest unoccupied molecular orbital HOMO-LUMO 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.