The Density Functional Method

Section 19.4 Excited States of the Helium Atom. Degenerate Perturbation Theory 

This method attempts to calculate the molecular energy and other properties from the electron probability density, which was introduced in Section 18.3 for a two-electron system. For a system with N electrons in a state corresponding to the wave function (ri, F2,. ..,tn), the electron probability density is obtained by integrating over the spin coordinates of all particles and over the space coordinates of all of the electrons except one  

Walter Kohn, 1923-, an Austrian-American physicist, shared the 1998 Nobel Prize in chemistry with John A. Pople for his work on density functional theory 

Since the electron probability density depends on only three space coordinates (represented by ri), it contains much less information than the wave function, which depends on the 3N space coordinates of N electrons as well as their spins. However, Hohenberg and Kohn proved a remarkable theorem For the ground electronic state of a molecule in the Born-Oppenheimer approximation, there is only one wave function and only one energy that correspond to a given electron probability density. This is surprising, since electron correlation involves effects of the mutual repulsion of pairs of electrons. It would seem that a function of the six coordinates of two electrons would be required to describe the energy of the molecule. However, their theorem implies that the effects of electron correlation are uniquely expressed in a function of only three coordinates. 

Since the electronic wave function is uniquely determined by the electron probability density and since the energy in the Born-Oppenheimer approximation is determined by the electronic wave function, the ground-state energy is a. functional of the ground-state electron probability density. A functional is a rule that associates a numerical value of a dependent variable with an independent function in much the same way that an ordinary function associates a numerical value of a dependent variable with a numerical value 

The density functional (DF) method of electronic structure calculations is based on the feet that the ground state of a many-electron system is completely determined by its electron density. When compared with correlated ab initio methods, the density functional method can be applied to much larger systems than those approachable by traditional ab initio methods. The density functional method has long been employed in studying electronic structures of solid-state systems and has become a widely applicable, advanced computational method in chemistry. In this section, we briefly comment on the concepts of DF theory and compare them with those of HF theory. 

The essential aspects of DF and HF theories can be summarized as follows [20] The total energy of a system is determined by its wavefunction in HF theory, and by its total electron density p in DF theory. 

The total energy is an expectation value of the total Hamiltonian H in HF theory, while it is decomposed into three density-dependent terms in DF theory (i.e., a kinetic energy term T(p), a Coulomb energy term U(p), and a many-body term xc(p) which contains all exchange and electron correlation effects). 

The state wavefunction is described by a Slater determinant using one-electron functions Vf, (/= I, 2,. . . , n) in HF theory, and the total density p is decomposed into single-electron densities, which originate from one-electron functions in DF theory. 

The total energy assumes a minimum upon variation of the wavefunction in HF theory, and upon the variation of the total electron density in DF theory. 

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To overcome these limitations, the hybrid QM-MM potential can employ ad initio or density function methods in the quantum region. Both of these methods can ensure a higher quantitative accuracy, and the density function methods offer a computaitonally less expensive procedure for including electron correlation [5]. Several groups have reported the development of QM-MM programs that employ ab initio [8,10,13,16] or density functional methods [10,41-43]. 

Sec. Ill is concerned with the description of models with directional associative forces, introduced by Wertheim. Singlet and pair theories for these models are presented. However, the main part of this section describes the density functional methodology and shows its application in the studies of adsorption of associating fluids on partially permeable walls. In addition, the application of the density functional method in investigations of wettability of associating fluids on solid surfaces and of capillary condensation in slit-like pores is presented. 

An important conceptual, or even philosophical, difference between the orbital/wavefunction methods and the density functional methods is that, at least in principle, the density functional methods do not appeal to orbitals. In the former case the theoretical entities are completely unobservable whereas electron density invoked by density functional theories is a genuine observable. Experiments to observe electron densities have been routinely conducted since the development of X-ray and other diffraction techniques (Coppens, 2001).18... 

More detailed information can be obtained by calculating the charge contours by a procedure called the density-functional method. We will not explain how it works... 

The replacement of Equation (15) corresponds to the density functional method. But the exchange-correlation energy is generally unknown. Therefore, the unknown... 

In these equations, (24)-(26), orthonormal orbits are denoted by indices Vs. Equation (26) means that the orbiting electron interacting with itself, that is self-interaction, exists. This is unphysical. In order to remove this unphysical term, the SIC is taken into account by the following procedure. The SIC for the LDA in the density functional method has been treated for free atoms and insulators [16], and found an important role in determining the energy levels of electrons. However, no established formula is known to take into account the SIC for semiconductors and metals. As a way of trial, in the present calculation, the atomic SIC potential is introduced for each angular momentum in a way similar to the SIC potential for atoms [17] as follows ... 

Chapter 1 gives a short description of ab initio methods, Hartree-Fock and post-Hartree-Fock, focusing on the Gaussian computer programs. Chapter 2 describes semi-empirical calculations and their applications to biological systems. Chapter 3 addresses itself to electrostatic properties of molecules, as determined by quantum-chemical methods. The density functional method is discussed in chapter 4. Chapter 5 compares theoretically obtained parameters to experimental data. 

Equilibrium structures of the 5//-bcnzotriazolo[l,2-tf]bcnzotriazol-6-ium and 6//-benzotriazolo[2,l -zdbcn/otriazol-5-ium inner salts 17 and 18, respectively, and their pyrido azaanalogs 19-22 have been studied by the density functional method B3LYP/6-31 lG(d,p) and found to have a fully conjugated planar geometry <2000CPH(254)375>. 

Although the resonant level model successfully explains a few general aspects of chemisorption, it has nevertheless many shortcomings. The model gives no information on the electronic structure of the chemisorption bond it does not tell where the electrons are. Such information is obtained from a more refined model, called the density functional method. We will not explain how it works but merely give the results for the adsorption of Cl and Li on jellium, reported by Lang and Williams [20]. 

The scheme analyzed so far is, in a way, a simplification of the Hartree-Fock scheme. As such, it is only a model approximation. The most serious drawback is the replacement of a fundamentally quantum mechanical term, whose very nature is to be non local, by a local approximation. Of course, when the system is in an electronic degenerate state, or when the BO approximation is no longer valid, the density functional method cannot be applied. For a discussion of this and other limitations the reader is referred to the paper by Bersuker [117],... 

The preparation and reactions of metal cluster ions containing three or more different elements is an area with a paucity of results. The metal cyanides of Zn, Cd (258), Cu, and Ag (259) have been subjected to a LA-FT-ICR study and the Cu and Ag complex ions reacted with various reagents (2,256). The [M (CN) ]+ and [M (CN) +11 ions of copper, where n = 1-5, were calculated to be linear using the density functional method. The silver ions were assumed to have similar structures. The anions [M (CN) +1 of both copper and silver were unreactive to a variety of donor molecules but the cations M (CN) H + reacted with various donor molecules. In each case, where reactions took place, the maximum number of ligands added to the cation was three and this only occurred for the reactions of ammonia with [Cu2(CN)]+, [Cu3(CN)2]+, [Ag3(CN)2]+, and [ Ag4(CN)3]+. Most of the ions reacted sequentially with two molecules of the donor with the order of reactivity being Cu > Ag and NH3 > H2S > CO. 

A. Nagy, in Recent Advances in Computational Chemistry, vol. 1 Recent Advances in the Density Functional Methods, Eds. V. Barone, A. Bencini, and P. Fantucci (World Scientific, Singapore, 2002), Part III, p. 247. 

The density functional methods assessed in this study (B3LYP, BLYP, and LDA) all perform much worse for the enthalpies of formation of the larger molecules in the G3/99 set. This is due to a cumulative effect in the errors for the larger molecules in this test set. The errors are found to be approximately proportional to the number of pairs of electrons in the molecules but the methods are not improved significantly when a higher-level correction such as that used in G2 or G3 theory is added the DFT methods. Further correction schemes may be necessary to improve the performance of density functional methods for large molecules. 

Let us now turn to the density functional methods. All of them correctly predict the para-ortho ordering, but considering that this is the case even for a Hartree-Fock treatment this is a somewhat hollow victory. Without exception, all functionals wrongly predict p-protonation. Arguably, this small energy difference falls within the error margin of any type of calibration for (semi-) empirical DFT functionals. 

Static deformation density maps can be compared directly with theoretical deformation densities. For tetrafluoroterephthalonitrile (l,4-dicyano-2,3,5,6-tetra-fluorobenzene) (Fig. 5.13), a comparison has been made between the results of a density-functional calculation (see chapter 9 for a discussion of the density-functional method), and a model density based on 98 K data with a resolution of (sin 0//)max = 1.15 A -1 (Hirshfeld 1992). The only significant discrepancy is in the region of the lone pairs of the fluorine and nitrogen atoms, where the model functions are clearly inadequate to represent the very sharp features of the density distribution. 

A commonly used quantity to present the information obtained from a first-principles calculation based on the density-functional method is the local density of states (LDOS) at every energy value below the Fermi level at zero absolute temperature. Because every state has an energy eigenvalue, the information with both spatial and energetic distributions is important for many experiments involving energy information. The LDOS p(r, ) at a point r and at an energy level E is defined as... 

The density functional methods were, in the views of many, legitimized by the introduction of the first Hohenberg-Kohn theorem [10]. The consequence of this celebrated theorem is that for a non-degenerate ground state and a given external potential, v(r), the electronic ground state energy can be expressed as... 

Because of its computational efficiency and good results for molecular properties, notably structural parameters, the MP2 level of theory is one of the most popular methods to include correlation effects on computed molecular properties. The other widely applied method is the density functional method, which will be introduced later. 

Molecular Orientation in Fluids near Solid-Liquid Interface as Studied by the Density Functional Method... 

The density functional method as applied by Tarazona to deal with classical fluids has been used to calculate the orientation of triatomic molecular fluids near the solid-liquid interface. The results give valuable suggestions about the effect of molecular shape on the orientation of real molecules such as liquid crystals near the solid-fluid interface. 

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