Beyond Density-functional Theory

Without being too harsh on density-functional theory, there are clear weaknesses, especially if effects of strong electronic correlation come into play. When the Coulomb interactions between the electrons are becoming large. 

As has been alluded to already (see Section 2.12.1), some insulating transition-metal oxides (also called Mott insulators) are falsely predicted as being metals when their electronic structures are calculated on the basis of the LDA because the full amount of the local Coulomb repulsion experienced by the electrons within the d orbitals is underestimated, and the inclusion of some extra repulsion U is needed to theoretically change them into insulators. This is how the LDA+U method is typically justified [174]. The method works well for transition-metal oxides and may also be used for correlated metals, in particular, the / elements. One may well ask whether a method that goes by the noble name of first principles or ab initio should be augmented by the introduction of a somewhat arbitrary U energy parameter, but that is matter of taste. The calculation of U requires additional approximations (the so-called constrained LDA method may deal with the problem [175]) and the spatial extent of the basis orbitals effectively determines the size of U, as expected by any quantum chemist. 

25) Here we use the jargon of the theoretical physics community which often classifies LDA-challenging materials as correlated . 

A remarkable DMFT test case is given by the correlated 5/ element plutonium where the LDA fails both for reproducing the equilibrium volume and the magnetic properties [178], in contrast to DMFT. An important psychological advantage of DMFT is that it has increased the mutual understanding of two very different theoretical communities.  

The question for a more systematic inclusion of electronic correlation brings us back to the realm of molecular quantum chemistry [51,182]. Recall that (see Section 2.11.3) the exact solution (configuration interaction. Cl) is found on the basis of the self-consistent Hartree-Fock wave function, namely by the excitation of the electrons into the virtual, unoccupied molecular orbitals. Unfortunately, the ultimate goal oi full Cl is obtainable for very small systems only, and restricted Cl is size-inconsistent the amount of electron correlation depends on the size of the system (Section 2.11.3). Thus, size-consistent but perturbative approaches (Moller-Plesset theory) are often used, and the simplest practical procedure (of second order, thus dubbed MP2 [129]) already scales with the fifth order of the system s size N, in contrast to Hartree-Fock theory ( N ). The accuracy of these methods may be systematically improved by going up to higher orders but this makes the calculations even more expensive and slow (MP3 N, MP4 N ). Fortunately, restricted Cl can be mathematically rephrased in the form of the so-called coupled clus- 

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Finally, can we dare to ask what is the future of first-principle MD It would be hard to be highly predictive. However we would like to quote the following directions of research QM/MM methods to treat quantum systems in an environment [92-94,225,226,269-272], Gaussian basis sets [23,30,38, 63,110,172] or Gaussian augmented plane waves methods [168] in search for order N methods [273,274] etc. Also, in order to go beyond Density Functional Theory, Quantum-Monte Carlo techniques are very attractive [119]. Some of these topics are already well-advanced and are discussed here in this book. 

In the following we will concentrate on the quality of results obtained for these quantities from density functional theory. A more general discussion of polarizabilities, hyperpolarizabilities etc., is beyond the scope of the present book, but can be found in many textbooks on physical or theoretical chemistry, such as Atkins and Friedman, 1997. 

The general theory of the quantum mechanical treatment of magnetic properties is far beyond the scope of this book. For details of the fundamental theory as well as on many technical aspects regarding the calculation of NMR parameters in the context of various quantum chemical techniques we refer the interested reader to the clear and competent discussion in the recent review by Helgaker, Jaszunski, and Ruud, 1999. These authors focus mainly on the Hartree-Fock and related correlated methods but briefly touch also on density functional theory. A more introductory exposition of the general aspects can be found in standard text books such as McWeeny, 1992, or Atkins and Friedman, 1997. As mentioned above we will in the following provide just a very general overview of this... 

Gorling, A., 1999, Density-Functional Theory Beyond the Hohenberg-Kohn Theorem , Phys. Rev. A, 59, 3359. 

There are also important features of current implementations of density functional theory which leave some biological processes beyond the scope of the DFT studies ... 

Density-functional theory, developed 25 years ago (Hohenberg and Kohn, 1964 Kohn and Sham, 1965) has proven very successful for the study of a wide variety of problems in solid state physics (for a review, see Martin, 1985). Interactions (beyond the Hartree potential) between electrons are described with an exchange and correlation potential, which is expressed as a functional of the charge density. For practical purposes, this functional needs to be approximated. The local-density approximation (LDA), in which the exchange and correlation potential at a particular point is only a function of the charge density at that same point, has been extensively tested and found to provide a reliable description of a wide variety of solid-state properties. Choices of numerical cutoff parameters or integration schemes that have to be made at various points in the density-functional calculations are all amenable to explicit covergence tests. 

Recent calculations of hyperfine parameters using pseudopotential-density-functional theory, when combined with the ability to generate accurate total-energy surfaces, establish this technique as a powerful tool for the study of defects in semiconductors. One area in which theory is not yet able to make accurate predictions is for positions of defect levels in the band structure. Methods that go beyond the one-particle description are available but presently too computationally demanding. Increasing computer power and/or the development of simplified schemes will hopefully... 

The quantum-mechanical energy curve was calculated at the B3LYP/6-311++G level of hybrid density-functional theory, as described in Appendix A. However, due to B3LYP convergence failures beyond Ji 3A, the quantities shown in Figs. 2.4—2.8 were calculated at HF/6-311++G" level. 

The study of behavior of many-electron systems such as atoms, molecules, and solids under the action of time-dependent (TD) external fields, which includes interaction with radiation, has been an important area of research. In the linear response regime, where one considers the external held to cause a small perturbation to the initial ground state of the system, one can obtain many important physical quantities such as polarizabilities, dielectric functions, excitation energies, photoabsorption spectra, van der Waals coefficients, etc. In many situations, for example, in the case of interaction of many-electron systems with strong laser held, however, it is necessary to go beyond linear response for investigation of the properties. Since a full theoretical description based on accurate solution of TD Schrodinger equation is not yet within the reach of computational capabilities, new methods which can efficiently handle the TD many-electron correlations need to be explored, and time-dependent density functional theory (TDDFT) is one such valuable approach. 

The time-dependent density functional theory [38] for electronic systems is usually implemented at adiabatic local density approximation (ALDA) when density and single-particle potential are supposed to vary slowly both in time and space. Last years, the current-dependent Kohn-Sham functionals with a current density as a basic variable were introduced to treat the collective motion beyond ALDA (see e.g. [13]). These functionals are robust for a time-dependent linear response problem where the ordinary density functionals become strongly nonlocal. The theory is reformulated in terms of a vector potential for exchange and correlations, depending on the induced current density. So, T-odd variables appear in electronic functionals as well. 

We will of course be rather more focused here. We shall be concerned with the generic computational strategies needed to address the problems of phase behavior. The physical context we shall explore will not extend beyond the structural organization of the elementary phases (liquid, vapor, crystalline) of matter, although the strategies are much more widely applicable than this. We shall have nothing to say about a wide spectrum of techniques (density functional theory [1], integral equation theories [2], anharmonic perturbation... 

Density functional theory has become the standard tool to compute the geometry, energy, and vibrational frequencies of adsorbed species. Computational chemistry has developed into an almost routine ingredient of research in catalysis and surface science. Thus, the reader is referred to textbooks [21, 22] and reviews [23, 24] relating to this fascinating subject which - unfortunately - is beyond the scope of this book. 

In the implementation of density functional methods for atoms and molecules, accurate wave functions can be utilized to test or construct the energy density functional[l]. Except for one-electron systems, however, these wave functions are necessarily approximate. Beyond hydrogen, the most precise wave functions available are obtained by the use of variational methods. For the simplest, few-electron systems, such calculations are capable of producing energies and wave functions of very high accuracy, more than sufficient for the present requirements of density functional theory. In this article we review the development and present state of accurate, variational calculations on simple atomic and molecular systems. In order to facilitate comparison of various alternative... 

A mode coupling theory is recently developed [135] which goes beyond the time-dependent density functional theory method. In this theory a projection operator formalism is used to derive an expression for the coupling vertex projecting the fluctuating transition frequency onto the subspace spanned by the product of the solvent self-density and solvent collective density modes. The theory has been applied to the case of nonpolar solvation dynamics of dense Lennard-Jones fluid. Also it has been extended to the case of solvation dynamics of the LJ fluid in the supercritical state [135],... 

Currently, research in our laboratory continues on real-space models of exchange and correlation hole functions in inhomogeneous systems. We anticipate that this work will ultimately generate completely non-empirical parameter-free beyond-LDA density functional theories. The quality of molecular dissociation energies and related properties obtainable with existing semi-empirical gradient-corrected DFTs approaches chemical accuracy, and we hope these future theoretical developments will continue this trend. 

To date, most applications of TDDFT fall in the regime of linear response. The linear response limit of time-dependent density functional theory will be discussed in Sect. 5.1. After that, in Sect. 5.2, we shall describe the density-functional calculation of higher orders of the density response. For practical applications, approximations of the time-dependent xc potential are needed. In Sect. 6 we shall describe in detail the construction of such approximate functionals. Some exact constraints, which serve as guidelines in the construction, will also be derived in this section. Finally, in Sects. 7 and 8, we will discuss applications of TDDFT within and beyond the perturbative regime. Apart from linear response calculations of the photoabsorbtion spectrum (Sect. 7.1) which, by now, is a mature and widely applied subject, we also describe some very recent developments such as the density functional calculation of excitation energies (Sect. 7.2), van der Waals forces (Sect. 7.3) and atoms in superintense laser pulses (Sect. 8). 

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