Theory density-functional

Density-functional formalism is based on two theorems by Hohenberg and Kohn 

Firstly, Hohenberg and Kohn showed that the external potential is a unique functional of the electron density n(r), and hence that the ground state, and the energy functionals 

Hoheriberg and Kohn showed secondly that the energy functional (7.7) assumes its minimum value, the ground-state energy, for the correct ground-state density. Hence, if the universal functional F[n] = T + U were known it would be relatively simple to use this variational principle to determine the ground-state energy and density for any specified external potential. Unfortunately, the functional is not known, and the full complexity of the many-electron problem is associated with its determination. 

In this situation it is useful to note that the theorems described above apply equally well to the case of non-interacting electrons, i.e. to a system with the Hamiltonian 

The ground state s of this single-particle problem is simply a Slater determinant obtained by populating the lowest-lying one-electron orbitals defined by the Schrodinger equation 

The Density Functional theorem states that the total ground state energy is a unique functional of the electron density, p [40]. This simple but enormously powerful result means that it is possible, in principle, to provide an exact description of all electron correlation effects within a one-electron (i.e. orbital-based) scheme. Khon and Sham (KS) [41] have derived a set of equations which embody this result. They have an identical form to the one-electron Hartree Fock equations. The difference is that the exchange-correlation term, Vxc, is not the same. 

LDA-based methods seem to be very promising ab initio approaches for routine application to TM systems. They scale much less severely than HF-based methods (formally N3) and are therefore capable of treating much larger systems for the same computational cost. Moreover, since the underlying DFT encompasses an intrinsically better treatment of electron correlation, they are inherently more accurate that the single determinant HF approximation. The problems that plague the HF description of [Mn04], for example, are absent in the DFT treatment [33], 

Approximate DFT approaches have been available for many decades. The most well known method to inorganic chemists is probably the Xa method first introduced by Slater and Johnson [15]. The Xa scheme utilises only the exchange part of the Local Density Approximation (9). 

Slater and Johnson s original method [15,43], which employed the mathematics of multiple scattering (or scattered waves) [44], adjusted the a scaling factors to yield atomic energies equal to those from accurate Hartree-Fock 

Density functional theory (DFT), developed within solid state physics, is based on the theorem of Hohenberg and Kohn that the ground state energy of a system depends on the electron density. It can be applied to calculations performed either with localised basis sets or by combination of plane waves. Both approaches have been applied to microporous solids, although the plane wave methods have been used more commonly. The SIESTA code, for example,permits DFT calculations using localised basis sets as does GAUSSIAN. 

DFT methods have become increasingly useful as their implementation within computer codes has benefited from improvements in computational techniques. These include the important steps introduced by Car and Parrin-ello and other more recent developments. Such changes have enabled the method to utilise fully increases in computing speed available through the implementation of parallel computing. As a result, DFT calculations have become an attractive method for the study of fully periodic microporous solids 

It should be noted, however, that DFT simulations tend to simulate dispersive interactions poorly, so that in many cases interactions of non-polar adsorbates within the pores of microporous solids are better treated using well-parametrised forcelield methods that use established atom-atom pan-potentials. 

Density Functional Theory aims to find the ground state electron density rather than the wavefunction. 

The density depends only on three spatial coordinates instead of 3N, reducing the complexity of the task enormously. The Hohenberg-Kohn principles prove that the electron density is the most central quantity determining the electronic interactions and forms the basis of an exact expression of the electronic ground state. 

The basic lemma of Hohenberg and Kohn [4] states that the ground state electron density of a system of interacting electrons in an arbitrary external potential determines this potential uniquely. The proof is given by the variational principle. If we consider a Hamiltonian Hi of an external potential Vi as 

Considering another potential V2, which cannot be obtained as Vi + constant, with a ground state wavefunction ij/2, which generates the same electron density, the ground state energy is 

This is the indirect proof that no two different external potentials can generate the 

Olivier [104] developed a method, based on the above theory, for looking at all the pores from the smallest to the largest. Traditionally the Kelvin or BJH theory is used for large pores and die t plot, Dubinin approach, MP method or the Horvath Kavazoe method for micropores. All but the last are based on mechanistic models the Horvath Kavazoe is based on a quasi-thermodynamic approach. 

The theory was applied by Seaton et al. [9] who forced the pore size distribution to fit a functional form. Olivier and Conklin [105] extract the pore size distribution from experimental data so that the final results more accurately reflect the structure of the sample. 

The basis for Density Functional Theory (DFT) is the proof by Hohenberg and Kohn that the ground-state electronic energy is determined completely by the electron density 

A note on semantics a function is a prescription for producing a number from a set of variables (coordinates). A functional is similarly a prescription for producing a number from a function, which in turn depends on variables. A wave function and the electron density are thus functions, while an energy depending on a wave function or an electron density is a functional. We will denote a function depending on a set of variables with parentheses, /(x), while a functional depending on a function is denoted with brackets, 

Early attempts at deducing functionals for the kinetic and exchange energies considered a non-interacting umform electron gas. For such a system it may be shown that T[p] and K[p] are given as 

The foundation for the use of DFT methods in computational chemistry was the introduction of orbitals by Kohn and Sham. The main problem in Thomas-FermL models is that the kinetic energy is represented poorly. The basic idea in the Kohn and Sham (KS) formalism is splitting the kinetic energy functional into two parts, one of which can be calculated exactly, and a small correction term. 

Assume for the moment a Hamilton operator of the following form with 0 A 1.  

The local density approximation (LDA) is the oldest and simplest of the functional types stiU in use. It is based on the idea of a imiform electron gas, a homogeneous 

DFT calculations offer a good compromise between speed and accuracy. They are well suited for problem molecules such as transition metal complexes. This feature has revolutionized computational inorganic chemistry. DFT often underestimates activation energies and many functionals reproduce hydrogen bonds poorly. Weak van der Waals interactions (dispersion) are not reproduced by DFT a weakness that is shared with current semi-empirical MO techniques. 

As mentioned earlier, density functional theory (DFT) does not yield the wave-function directly. Instead it first determines the probability density p and calculates the energy of the system in terms of p. Why is it called density functional theory and what is a functional anyway We can define functional by means of an example. The variational integral E( p) = f pH pdt / / (pipit is a functional of the trial wavefunction ip and it yields a number (with energy unit) for a given p. In other words, a functional is a function of a function. So, in the DFT theory, the energy of the system is a functional of electron density p, which itself is a function of electronic coordinates. 

One important reason for DFT s ever-increasing popularity is that even the most elementary calculation includes correlation effects to a certain extent but 

Electronic wavefunctions symbolized in this text as I e(ri, S], ra, S2. r , s ) depend on the spatial (r) and spin (s) variables of all the m electrons. The electron density on the other hand depends only on the coordinates of a single electron. I discussed the electron density in Chapter 5, and showed how it was related to the wavefunction. The argument proceeds as follows. The chance of finding electron 1 in the differential space element dti and spin element ds] with the other electrons anywhere is given by 

The integration is over all the space and spin coordinates of electrons 2, 3. m. Many of the operators that represent physical properties do not depend on spin, and so we often average-out over the spin variable when dealing with such properties. The chance of finding electron 1 in the differential space element dt] with either spin, and the remaining electrons anywhere and with either spin is 

the integration is now over the spin variable for electron 1, as well as the space and spin variables for all the remaining electrons. 

Finally, many of the common molecular electronic properties depend only on the chance of finding any electron in dx and this is obviously m times the above quantities. We focus attention on points in space (written r) and interest ourselves in the electron probability density 

The actual electron density is —eP(r), but authors speak about / (r), which is strictly the electron number density, as if it were the same thing. I will follow this sloppy (but common) usage from time to time. 

Parr and al. gave in 1988 a theoretical support to the absolute hardness. In the density ftmc-tional theory two basic parameters were introduced. Any chemical system can be characterized by its chemical potential, p, and its absolute hardness, T. The chemical potential measures the escaping tendency of an electronic cloud, while absolute hardness determines the resistance of the species to lose electrons. The exact definitions of these quantities are  

However, according to frontier orbital method, the relationship between t] and the HOMO and LUMO energies is reduced to  

Of course the absolute softness is the reciprocal of the absolute hardness. The apparent success of the density-functional theory is to provide two parameters from which we can calculate the number of electrons transferred, resulting mainly from the charge transfer between two molecules, i.e., fix)m electrons flow until chemical potential reaches an equilibrium. As a first approximation, the number of electron transferred is given by  

This number varies from 0 to 1 and it is in most cases a fractional number. As an example for the interaction between CI2 and substituted aromatic compounds, N ns varies 

AH enthalpy of acid-base adduct formation E ability of the acid to participate in electrostatic bonding 

According to Hohenberg and Kohn (1964), the electron density p(r) uniquely determines the wave function, p(r) V(r) v /, and the energy is a functional of p(r)  

The proof of this theorem is relatively simple but will not be reproduced here (the interested reader may consult ref. 9). It is based on the variational principle and uses that from the integral of the electron density one knows the total number of electrons and accordingly the kinetic-energy and the electron-electron-interaction parts of the total Hamilton operator. Only the external potential (which above was only the Coulomb potential of the nuclei, but which may contain other parts, too) is unspecified, but assuming that this can be written as a sum of identical single-particle terms, Hohenberg and Kohn proved that also this is uniquely determined within an additive constant. 

This first theorem states that any ground-state property is a unique functional of the density. But most often it is used exclusively as stating that Ee is a functional of p r), although the theorem does not give any expression for this functional. Assuming, however, that we know its precise form, the second theorem states that 

This second theorem gives accordingly a variational principle for the density functionals. Eqs. (16) and (17) can then be combined into 

The main problem related to practical applications of the theorems of Hohenberg and Kohn is obvious the theorems provide proofs for the existence of certain relations but do not give any specific forms for these. In order to obtain a practical scheme for electronic-structure calculations, Kohn and Sham2 reformulated the problem of calculating Ee from the electron density p(r). They observed that parts of Ee directly can be written as functionals of p, 

ifri are the orbitals of the model system and the total density of this is given as 

The basis for DFT was given by Hohenberg and Kohn17 who showed that the ground-state energy, E, of a system of N interacting electrons in an external field (e.g. the field from the nuclei) is determined uniquely by the electron density, p(r), i.e, the ground-state 

The ground-state energy and charge density are obtained by minimizing the energy functional with the constraint that the number of particles is conserved. This minimization results in the following Euler equation18  

If the same minimization procedure is applied to a system of non-interacting particles moving in an effective potential, Ve[, the corresponding Euler equation would be  

If we identify the last three terms of eq. (2.26) with Veff, then clearly the same Euler equations will result since T0 was defined as the kinetic energy of a system of non-interacting particles. For non-interacting particles, the Schrodinger equation can be written 

Kohn and Sham 8 suggested a local density approximation (LDA) for the second integral  

In a series of seminal papers, Hohenberg, Kohn and Sham developed a different way of looking at the problem, which has been called Density Functional 

Theory (DFT). The basic ideas of Density Functional Theory are contained in the two original papers of Hohenberg, Kohn and Sham, [22, 23] and are referred to as the Hohenberg-Kohn-Sham theorem. This theory has had a tremendous impact on realistic calculations of the properties of molecules and solids, and its applications to different problems continue to expand. A measure of its importance and success is that its main developer, W. Kohn (a theoretical physicist) shared the 1998 Nobel prize for Chemistry with J.A. Pople (a computational chemist). We will review here the essential ideas behind Density Functional Theory. 

The basic concept is that instead of dealing with the many-body Schrodinger equation, Fq. (2.1), which involves the many-body wavefunction P ( r ), one deals with a formulation of the problem that involves the total density of electrons (r). This is a huge simplification, since the many-body wavefunction need never be explicitly specified, as was done in the Hartree and Hartree-Fock approximations. Thus, instead of starting with a drastic approximation for the behavior of the system (which is what the Hartree and Hartree-Fock wavefunctions represent), one can develop the appropriate single-particle equations in an exact manner, and then introduce approximations as needed. 

In the following discussion we will make use of the density n(r) and the one-particle and two-particle density matrices, denoted by y(r, r ), F(r, r lr, r ), respectively, as expressed through the many-body wavefunction  

These quantities are defined in detail in Appendix B, Eqs. (B.13)-(B.15), where their physical meaning is also discussed. 

Currently the most popular approach to carrying out electronic structure calculations is density functional theory (DFT). The central concept is the electron probability density, p r). This density is a function of the three spatial coordinates only, instead of the 3 N coordinates that are necessary to describe the Schrodinger wave function. The potential and kinetic energies of the molecule are expressed in terms of the electron density such that the total energy contains an unknown, but universal functional of p r). Note that the original wave function possesses all the coordinates for eveiy electron in the system. In practice it is convenient to retain the wavefunction of individual electrons to calculate the electron density. The theoretical basis of DFT can be found in [36,37]. It is necessary to make some approximation to the functional 

For non-periodic systems it is standard practice to use localised basis fiinctions to describe the individual electron wavefunctions. In the case of periodic systems, the use of plane waves is the most favoured method, because of the simplicity of the Fourier analysis of plane waves. There are many excellent accounts of the principles and applications of DFT and we refer the reader to these [38]. 

As we have seen before, the vibrational spectra of molecules require the calculation of the second derivatives of the energy with respect to the atom coordinates about their equilibrium positions. Many DFT program codes can perform the calculation of the dynamical matrix either using analytical second derivatives or numerical differences. As a result their output includes a set of eigenvectors and vibrational frequencies which are the input to programs that calculate INS spectra ( 5.3). 

The robust and convincing performance of a cheap DFT level such as B3LYP/6-31G(d) is emphasised by the data in Table 7.4, which presents some notoriously problematical cases. Considering the time and effort (i.e. cost , cf. Table 7.5), DFT does unbelievably well This, however, does not imply that this level of theory treats every problem well careful validation is always required, and this is particularly true of DFT methods. 

In a nutshell (DFT methods) Electron density description of chemical phenomena, straightforward interpretation Large molecular systems can be treated Variable accuracy, validation necessary Little basis set dependence No systematic improvements. 

Recently, there have been several successful attempts at the calculation of spin-spin coupling constants at the density functional theory (DFT) level. In view of the success of the DFT methodology, it would in particular be interesting to see how well DFT performs with respect to the calculation of spin-spin coupling constants. In particular, recent results demonstrated that DFT does not suffer from the triplet-instability problems that have plagued the application of Hartree-Fock theory to the calculation of spin-spin 

Sychrovsky et performed, for the first time, a complete implementation of coupled perturbed density functional theory (CPDFT) for the calculation of spin-spin coupling constants with pure and hybrid DFT. They analyzed the dependence of DFT with respect to the calculation of coupling constants on the exchange-correlation (XC) functionals used. They demonstrated the importance of electron correlation effects and showed that the hybrid functional leads to the best accuracy of calculated spin-spin 

is the one-electron perturbation Hamiltonian operator. Equation (4.19) has to be solved simultaneously with eqs. (4.20) in a self-consistent fashion by CPDFT. This self-consistent procedure is avoided in SOS-DFPT, where the approximation Fg w Hg is invoked. The spin-spin coupling constants for benzene calculated by Sychrovsky et al. with the B3LYP functional are shown in Table 4.3. A good agreement between the calculated and measured spin-spin coupling constants is obtained for /(C,C), /(C,H), and /(H,H). 

Helgaker et alP presented a fully analytical implementation of spin-spin coupling constants at the DFT level. They used the standard procedure for linear response theory to evaluate second-order properties of PSO, FC and SD mechanisms. Their calculation involves all four contributions of the nonrelativistic Ramsey theory. They tested three different XC functionals -LDA (local density approximation), BLYP (Becke-Lee-Yang-Parr), and B3LYP (hybrid BLYP). All three levels of theory represent a 

Kamienska-Trela et al performed DFT calculations of /(C,C) coupling constants for variously mono- and disubstituted pyridine N-oxides and mono-substituted pyridines. The DFT data reproduced very well the experimental coupling values and revealed that the FC contribution is the dominating factor which governs the magnitude of the CC coupling across one bond. 

The Vext operator is equal to Vne for A = 1, for intermediate A values, however, it is assumed that the external potential Vext(A) is adjusted so that the same density is obtained for both A = 1 (the real system) and A = 0 (a hypothetical system with noninteracting electrons). For the A = 0 case the exact solution to the Schrddinger equation is given as a Slater determinant composed of (molecular) orbitals, for which the 

The function/(r) is usually dependent upon other weU-defined functions. A simple example of a functional would be the area under a curve, which takes a function/(r) defining the curve between two points and returns a number (the area, in this case). In the case of DFT the function depends upon the electron density, which would make Q a functional of p(r), in the simplest case/(r) would be equivalent to the density (i.e. /(r) = p(r)). If the function /(r) were to depend in some way upon the gradients (or higher derivatives) of p(r) then the functional is referred to as being non-locaT, or gradient-corrected. By contrast, a local functional would only have a simple dependence upon p(r). In DFT the energy functional is written as a sum of two terms  

In order to minimise the energy we introduce this constraint as a Lagrangian multiplier (-p), leading to  

Equation (3.40) is the DFT equivalent of the Schrodinger equation. The subscript Vgxt indicates that this is under conditions of constant external potential (i.e. fixed nuclear positions) It is interesting to note that the Lagrange multiplier, p, can be identified with the chemical potential of an electron cloud for its nuclei, which in turn is related to the 

The second landmark paper in the development of density functional theory was by Kohn and Sham who suggested a practical way to solve the Hohnberg-Kohn theorem for a set of interacting electrons [Kohn and Sham 1965J. The difficulty with Equation (3.37) is that we do not know what the function f [p(r)] is. Kohn and Sham suggested that F[p(r)] should be approximated as the sum of three terms  

The second term, Eh(p), is also known as the Hartree electrostatic energy. The Hartree approach to solving the Schrodinger equation was introduced briefly in Section 2.3.3 and almost immediately dismissed because it fails to recognise that electronic motions are correlated. In the Hartree approach this electrostatic energy arises from the classical interaction between two charge densities, which, when summed over all possible pairwise interactions, gives  

Although the theorem of DFT is in terms of the total electron density, in practical calculations the density is described in terms of orbitals, which are solved self-consistently as in HF theory. The wild popularity of DFT is due to its low computational expense (comparable to HF) and generally good accuracy, even for difficult systems such as radicals that contain transition metal atoms. For high precision, however, DFT is inappropriate, since convergent series are unavailable. Current functionals do not include the dispersion interaction, which is necessary for describing weakly bound molecular complexes [94]. Thus, DFT methods should be avoided wherever van der Waals interactions are important. 

Most molecular DFT software employs basis sets, as in MOT software. However, additional numerical integration is required, which is typically done on a spatial grid. The use of a grid is an additional numerical approximation. As expected, a finer grid produces a more precise result but increases the computational cost. 

Many choices of functional are available and their number is increasing. Some are intended to excel for both thermochemistry and reaction barrier heights [95,96]. In applied work, it may be acceptable to test the performance of several functionals on a related, benchmark system, and then adopting the most successful functional for the prediction of interest. However, the best functional may be different for a different project. 

To improve the accuracy by including the interaction between electrons without running into excessive computational problems many approximate solutions of the wave equation have been proposed. Today the most commonly used approach is called Density Functional Theory (DFT). 

DFT uses the density of the electron distribution as a fundamental variable. The many body wave function depends on 37Vspatial variables (3 per electron), the electron density only on 3 variables (the space coordinates). 

In a many body case the nuclei interact among themselves through Coulomb forces and generate a static external potential Lthat determines the spatial distribution of the moving electrons. The stationary state of the electrons is characterized by the wave function W r) and is described in DFT by  

Once the configuration of the transition state has been identified its energy E is available. The partition functions, entropies and enthalpies of the activated complex can then be calculated and substituted together with those of the reactants into the expression for the rate coefficient k. 

The application of equations (1.7.1-11) and (1.7.3-1) will be illustrated in Section 2.6 of Chapter 2 dealing with the modeling of the rate of catal)dic reactions. 

In Kohn-Sham DFT based approaches, expressions that are of similar structure as Eqs. (9a) and (9b) are obtained, but in the form of contributions from all occupied Kohn-Sham MOs The excited-state wavefunctions are at the same time formally replaced by the unoccupied MOs, and the many-electron perturbation operators /T(M41, etc. by their one-electron counterparts //(M-41, etc. Orbital energies e and ea formally substitute the total energies of the states (see later). Thus, similar interpretations of NMR parameters can be worked out in which the highest occupied MO-lowest unoccupied MO gap (HLG) plays a highly important role. It must be emphasized, though, that there is no one-to-one correspondence between the excited states of the SOS equations and the unoccupied orbitals which enter the DFT expressions, nor between excitation energies and orbital energy differences, i.e., there are no one-determinantal wavefunctions in Kohn-Sham DFT perturbation theory which approximate the reference and excited states. 

As already mentioned, the MOs in Eqs. (10a) and (10b) are assumed to be exact, i.e., represented at the complete basis set limit. In practice, the Kohn-Sham equations are converted into the respective self-consistent-field matrix equations in the basis, and the perturbation treatment is carried out from there. Any dependence of the basis set (GIAOs) on the perturbation (Bext in the case of shielding tensors) is in this way naturally covered. 

A completely different route to the A-electron problem is provided by DPT. On an operational level it can be thought of as an attempt to improve on the HE method by including correlation effects into the self-consistent field procedure. 

The existence of the first HK theorem is quite surprising since electron-electron repulsion is a two-electron phenomenon and the electron density depends only on one set of electronic coordinates. Unfortunately, the universal functional is unknown and a plethora of different forms have been suggested that have been inspired by model systems such as the uniform or weakly inhomogeneous electron gas, the helium atom, or simply in an ad hoc way. A recent review describes the major classes of presently used density functionals [10]. 

The connection to HF theory has been accomplished in a rather ingenious way by Kohn and Sham (KS) by referring to a fictitious reference system of noninteracting electrons. Such a system is evidently exactly described by a single Slater determinant but, in the KS method, is constrained to share the same electron density with the real interacting system. It is then straightforward to show that the orbitals of the fictitious system fulfil equations that very much resemble the HF equations  

The first term is the familiar one-electron operator, the second term represents the Coulomb potential, and the third term is called exchange-correlation potential. HF and DFT differ only in this last term. In HF theory there is only a nonlocal exchange term, while in DFT the term is local and supposed to cover both exchange and correlation. It arises as a functional derivative with respect to the density  

There are several things known about the exact behavior of Vxc(r) and it should be noted that the presently used functionals violate many, if not most, of these conditions. Two of the most dramatic failures are (a) in HF theory, the exchange terms exactly cancel the self-interaction of electrons contained in the Coulomb term. In exact DFT, this must also be so, but in approximate DFT, there is a sizeable self-repulsion error (b) the correct KS potential must decay as 1/r for long distances but in approximate DFT it does not, and it decays much too quickly. As a consequence, weak interactions are not well described by DFT and orbital energies are much too high (5-6 eV) compared to the exact values. 

In this chapter, a short introduction to DFT and to its implementation in the so-called ab initio molecular dynamics (AIMD) method will be given first. Then, focusing mainly on our own work, applications of DFT to such fields as the definition of structure-activity relationships (SAR) of bioactive compounds, the interpretation of the mechanism of enzyme-catalyzed reactions, and the study of the physicochemical properties of transition metal complexes will be reviewed. Where possible, a case study will be examined, and other applications will be described in less detail. 

In the following text we present a very short synopsis both of the DFT approach and the ab initio molecular dynamics (AIMD) method that can by no means be considered as an introduction to the use of the computational tools based on them. The interested reader will find exhaustive treatment of these arguments elsewhere in this book (Chapter 1). 

the difficulty is that p is a many-body electronic wavefunc-tion, which depends on the coordinates of all the n electrons. 

In DFT, a different approach is followed. Rather than focusing on y/ one focuses on the single particle density p(r), which is a quantity related to y/ by the equation  

The density p is a much simpler quantity than y/, because it depends on one spatial coordinate only. 

We know a number of things about the electron density P(r). First of all, if we integrate it over space, we get the number of electrons, m. This follows simply from the definition 

However, physicists and material scientists needed to study metals, semiconductors, etc. and it was shown in the above paper that there was, in fact, a way around this bottleneck. Computer packages for solids and semiconductors, such as CASTEP used by P. Hu (Phys. Rev. Lett. 91 (2003) 266102), implement many of the concepts in the Car and Parrinello paper. 

in any ab initio quantum mechanical calculation, a starting geometry is supplied either by giving the x-, y- and -coordinates of each atom or by supplying the same data in the form of internal coordinates, that is, bond lengths, bond angles and dihedral (twist) angles. 

In addition, it is necessary to state the charge on the molecule, its spin multiplicity (singlet, triplet, etc.) and the basis set. 

This last requirement essentially tells the computer which orbitals (s, p, d and /) are to be included in the calculation and to what degree of accuracy they are going to be represented, that is, there are minimal basis sets for not so accurate calculations and very high level basis sets for more refined calculations. 

the above procedure can also be very time-consuming when carrying out calculations on extended arrays of atoms as in crystals and other solids since convergence is often very slow and many iterations are required before the optimisation criterion is satisfied. So, these conventional methods of optimisation also became difficult on large systems. 

Such p-based theories date back, in fact, to the early days of quantum mechanics, and the pioneering work of Thomas [137] and Fermi [138] provides a method of statistically describing the distribution of the electrons in an atom. Without going into detail, the Thomas-Fermi (TF) total energy of an atom with a nucleus charged Z may be directly given as 

The curly brackets around the electron density p(r) indicate that the total energy is afunctional of the electron density, and this anticipates an important 

Another use of the electron density was introduced by Slater [140] quite early on, as a method of approximating the nonlocal exchange potential which is so tedious to calculate within the framework of Hartree-Fock theory (see Section 2.11.3). With reference to an electron-gas model of uniform density. Slater replaced the nonlocal exchange potential by the so-called Xot local potential 

A little consideration is needed at this point. Recall that within Thomas-Fermi theory, the kinetic energy (and, therefore, the total energy) can be di- 

Consequently, DFT is restricted to ground-state properties. For example, band gaps of semiconductors are notoriously underestimated [142] because they are related to the properties of excited states. Nonetheless, DFT-inspired techniques which also deal with excited states have been developed. These either go by the name of time-dependent density-functional theory (TD-DFT), often for molecular properties [147], or are performed in the context of many-body perturbation theory for solids such as Hedin s GW approximation [148]. 

4 Solvent Effects on the Nuclear Spin-Spin Coupling Constants. - Aut- 

Method MP5 MP6 CCSD(T) CX SDT-1 QCISD(T) CCSDT CISDTQ CCSDTQ 

The electronic wave function of an n-electron molecule depends on 3ra spatial and n spin coordinates. Since the Hamiltonian operator (15.10) contains only one- and two-electron spatial terms, the molecular energy can be written in terms of integrals involving only six spatial coordinates (Problem 15.82). In a sense, the wave function of a many-electron molecule contains more information than is needed and is lacking in direct physical significance. This has prompted the search for functions that involve fewer variables than the wave function and that can be used to calculate the energy and other properties. 

What is a functional Recall that a function f x) is a rule that associates a number with each value of the variable x for which the function / is defined. For example, the function f(x) = jc + 1 associates the number 10 with the value 3 of x and associates a number with each other value of x. A functional F[f] is a rule that associates a number with each function /. For example, the functional F[f] = / / (x)/(x) dx associates a number, found by integration of /p over all space, with each quadratically 

The proof of the Hohenberg-Kohn theorem is as follows. The ground-state electronic wave function ipQ of an n-electron molecule is an eigenfunction of the purely electronic Hamiltonian of Eq. (13.5), which, in atomic units, is 

The quantity u(rj), the potential energy of interaction between electron / and the nuclei, depends on the coordinates Xj, y Zi of electron i and on the nuclear coordinates. Since the electronic Schrodinger equation is solved for fixed locations of the nuclei, the nuclear coordinates are not variables for the electronic Schrodinger equation. Thus, u(r,) in the electronic Schrodinger equation is a function of only Xi,y,-, Zi, which we indicate by using the vector notation of Section 5.2. In DFT, n(r,) is called the extamal potential acting on electron i, since it is produced by charges external to the system of electrons. 

Thus any other interaction effects, such as exchange, will tend to be reduced (that is, screened) by the correlation hole. 

One can now clearly see why the Hartree-Fock approach fails for solids firstly the exchange interaction should be screened by the correlahon hole rather than acting in full, and secondly the binding between the correlation hole and electron is ignored. Nevertheless she Hartree-Fock approach gives quite creditable results for small molecules. This is because there are far fewer electrons involved than in a solid, and so correlation effects are minimal compared to exchange effects. 

The breakthrough was made by Hohenberg and Kohn [12]. They proved two significant theorems. The first theorem is the total ground-state energy of a many-electron system is a unique functional. of the electron density n(r). Value of n is measured in electrons per unit of volume and depends on coordinates. 

The density functional theory that has been formulated by Hohenberg and Kohn is an exact theory of many-body systems. The theory asserts that the ground-state electron density rio(r) determines all properties of a system in the ground state and in excited states. 

the term inside the summation is the probability that an electron with an eigenfunction ipi(r) is located at position r. The summation goes over all the individual electron eigenfunctions. The factor of 2 appears because electrons can have two different spins. The point is that the electron density n(r), which is a function of only three coordinates, contains a great amount of information that is actually physically observable. 

NICS(0)and NICS(l) (in parentheses) are the total (isotropic) absolute shielding computed at the centre of the ring or cage and at lA away from the ring centre, respectively. 

Ever since the advent of quantum mechanics attempts have been made to have a classical interpretation of the same. Reducing the quantum mechanics to a 3D vector space became very important and p(f) being a 3D function has become the immediate choice.  

A decomposition of E[p] in terms of the Hohenberg-Kohn universal functional F[p] would convert eqn (14) to 

Kohn and Sham eonsidered a system of non-interacting particles moving under an elfeetive potential, x)eff r)and the density is obtained by solving the following Kohn-Sham equations, 

In eqn (17) the last term in the right hand side is the exchange-correlation potential which is the functional derivative of the exchange-correlation energy functional, E c p] with respect to p f). Various approximate forms for E c[p] like PBE, B3PW91, BLYP, B3LYP etc. are available. Once p r) is known all physico-chemical properties may be calculated as the density functionals. 

The ground-state electronic energy Eq is thus a functional of the function po(r), which we write as Eq = [po], where the v subscript emphasizes the dependence of Eq on the external potential v(r), which differs for different molecules. 

Structures were fully optimized, within symmetry constraints, using a Broyden-Fletcher-Goldfarb-Shanno (BFGS) [7] algorithm. Optimization was considered to be complete when the residual forces were below 1.0 x 10 Hartrees Bohr V The 

In order to obtain the potential energy surface, one needs of course to solve an electronic structure problem. Several methods have been used, ranging from methods treating the electrons of the metal as a free electron gas to traditional ab initio calculations. One of the simplest models treats the solid as a jellium, i.e., a uniform elecron gas on a positive background. The hamiltonian for the electrons is then just the kinetic energy — ( /2m, )V. For z 0 the one-electron wavefunctions are approximated by 

In the effective medium theory, the electronic interaction between an atom and the solid is replaced by that of the atom and a homogeneous electron gas. The approach is based upon density functional theory, but since the solid is not really behaving as a homogeneous electron gas, gradient corrections to the theory have been devised [210]. Simple model hamiltonians have also been used to study chemisorption of atoms and molecules at metals. Here we mention the Anderson-Grimley-Newns model hamiltonian of the type 

The general theory for electron density, density functional theory (DFT) is due to Hohenberg and Kohn [211]. It was shown that the ground state energy of the many-particle system was related to the density distribution of the electrons. The proof of the theory was later extended and simplified by Levy [214]. Kohn and Sham [215] derived a system of self-consistent equations for the effective one-particle potential 

TABLE 4.6 Comparison of Results for Bond Lengths re, Vibrational Frequency a ei and Atomization Energies Ed for Some Molecules 

HF denotes Hartee Fock theory, LSD the local spin density with no correlation but exchange according to a free electron gas, and GC has exchange according to ref. [230] and correlation according to ref. [229]. Thus LSD refers to the simplest and GC to a more state-of-the-DFT theory. Data adapted from ref. [236]. 

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The statistical mechanical approach, density functional theory, allows description of the solid-liquid interface based on knowledge of the liquid properties [60, 61], This approach has been applied to the solid-liquid interface for hard spheres where experimental data on colloidal suspensions and theory [62] both indicate 0.6 this... 

S. Chains in the S phase are also oriented normal to the surface, yet the unit cell is rectangular possibly because of restricted rotation. This structure is characterized as the smectic E or herringbone phase. Schofield and Rice [204] applied a lattice density functional theory to describe the second-order rotator (LS)-heiTingbone (S) phase transition. 

Molecular dynamics and density functional theory studies (see Section IX-2) of the Lennard-Jones 6-12 system determine the interfacial tension for the solid-liquid and solid-vapor interfaces [47-49]. The dimensionless interfacial tension ya /kT, where a is the Lennard-Jones molecular size, increases from about 0.83 for the solid-liquid interface to 2.38 for the solid-vapor at the triple point [49], reflecting the large energy associated with a solid-vapor interface. 

Density functional theory from statistical mechanics is a means to describe the thermodynamics of the solid phase with information about the fluid [17-19]. In density functional theory, one makes an ansatz about the structure of the solid, usually describing the particle positions by Gaussian distributions around their lattice sites. The free... 

The entropically driven disorder-order transition in hard-sphere fluids was originally discovered in computer simulations [58, 59]. The development of colloidal suspensions behaving as hard spheres (i.e., having negligible Hamaker constants, see Section VI-3) provided the means to experimentally verify the transition. Experimental data on the nucleation of hard-sphere colloidal crystals [60] allows one to extract the hard-sphere solid-liquid interfacial tension, 7 = 0.55 0.02k T/o, where a is the hard-sphere diameter [61]. This value agrees well with that found from density functional theory, 7 = 0.6 0.02k r/a 2 [21] (Section IX-2A). 

Ira N. Levine, Quantum Chemistry, 4th ed., Prentice-Hall, Englewood Cliffs, NJ, 1991. (Source for density functional theory.)... 

Parr R G and Yang W 1994 Density-Functional Theory of Atoms and Molecules (New York Oxford)... 

A comprehensive treatment of density functional theory, an idea that is currently very popular in quantum chemistry. 

N -Fle [, ], Fle-F and Ne-F [Ml- Density-functional theory [ ] is currently unsuitable for the calculation of van der Waals interactions [90], but the situation could change. 

Parr B 2000 webpage http //net.chem.unc.edu/facultv/rap/cfrap01. html Professor Parr was among the first to push the density functional theory of Hohenberg and Kohn to bring it into the mainstream of electronic structure theory. For a good overview, see the book ... 

The Flohenberg-Kohn theorem and the basis of much of density functional theory are treated ... 

Becke A D 1995 Exchange-correlation approximations in density-functional theory Modern Eiectronic Structure Theory vol 2, ed D R Yarkony (Singapore World Scientific) pp 1022-46... 

Dunlap B I 1987 Symmetry and degeneracy in Xa and density functional theory Advances in Chemicai Physics vol LXIX, ed K P Lawley (New York Wiley-Interscience) pp 287-318... 

Parr R G 1983 Density functional theory Ann. Rev. Phys. Chem. 34 631 -56 Salahub D R, Lampson S FI and Messmer R P 1982 Is there correlation in Xa analysis of Flartree-Fock and LCAO Xa calculations for O3 Chem. Phys. Lett. 85 430-3... 

Janak J F 1978 Proof that dEldn- = r . in density-functional theory Phys. Rev. B 18 7165-8... 

Perdew J P, Parr R G, Levy M and Balduz J L Jr 1982 Density-functional theory for fractional particle number derivative discontinuities of the energy Phys. Rev. Lett. 49 1691-4... 

Fattebert J-L and Bernholc J 2000 Towards grid-based 0(N) density-functional theory methods optimized nonorthogonal orbitals and multigrid acceleration Phys. Rev. B 62 1713-22... 

Wang Y A and Carter E A 2000 Orbital-free kinetic-energy density functional theory Theoretical Methods in Condensed Phase Chemistry (Progress in Theoretical Chemistry and Physics Series) ed S D Schwartz (Boston Kluwer) pp 117-84... 

Wimmer E, Fu C L and Freeman A J 1985 Catalytic promotion and poisoning all-electron local-density-functional theory of CO on Ni(001) surfaces coadsorbed with K or S Phys. Rev. Lett. 55 2618-21... 

Hammer B, Hansen L B and Norskov J K 1999 Improved adsorption energetics within density functional theory using revised Perdew-Burke-Enerhof functionals Phys. Rev. B 59 7413-21... 

Dreizier R M and Gross E K U 1990 Density Functional Theory an Approach to the Quantum Many-body Problem (Berlin Springer)... 

Car R and Parrinello M 1985 Unified approach for molecular dynamics and density functional theory Phys. Rev. Lett. 55 2471... 

Yethira] A and Woodward C E 1995 Monte Carlo density functional theory of nonuniform polymer melts J Chem. Phys. 102 5499... 

Kierlik E and Rosinberg M L 1993 Perturbation density functional theory for polyatomic fluids III application to hard chain molecules in slitlike pores J Chem. Phys. 100 1716... 

Fraai]e J G E M 1993 Dynamic density functional theory for micro-phase separation kinetics of block copolymer melts J. Chem. Phys. 99 9202... 

Figure Cl. 1.6. Minimum energy stmctures for neutral Si clusters ( = 12-20) calculated using density functional theory witli tire local density approximation. Cohesive energies per atom are indicated. Note tire two nearly degenerate stmctures of Si g. Ho K M, Shvartsburg A A, Pan B, Lu Z Y, Wang C Z, Wacher J G, Fye J L and Jarrold M F 1998 Nature 392 582, figure 2.

Massobrio C, Pasquarello A and Corso A D 1998 Structural and electronic properties of small Cu clusters using generalized-gradient approximations within density functional theory J. Chem. Phys. 109 6626... 

To use direct dynamics for the study of non-adiabatic systems it is necessary to be able to efficiently and accurately calculate electronic wave functions for excited states. In recent years, density functional theory (DFT) has been gaining ground over traditional Hartree-Fock based SCF calculations for the treatment of the ground state of large molecules. Recent advances mean that so-called time-dependent DFT methods are now also being applied to excited states. Even so, at present, the best general methods for the treatment of the photochemistry of polyatomic organic molecules are MCSCF methods, of which the CASSCF method is particularly powerful. 

Handy, N.C. Density functional theory. In Quantum mechanical simulation methods for studying biological systems, D. Bicout and M. Field, eds. Springer, Berlin (1996) 1-35. 

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