QMMS Lecture #4 (Venables/Heggie)

Notes for QMMS Lecture 4 (Venables/Heggie)

Lecture notes by John A. Venables and Edward Hernández. Lecture scheduled for 25 Oct 05. Latest version 31st August 2005.

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4. Exchange-correlation: Density functional theory and LDA methods

4.1 Hartree-Fock and exchange-correlation

Broadly speaking there are two families of first-principles methods for the calculation of the electronic structure of atoms, molecules and solids, namely methods based on Density Functional theory (DFT), and methods based on Hartree-Fock (HF) theory. Density Functional methods have been extensively used in studying solids, because this methodology is much simpler and computationally cheaper. The Computational Chemistry community has traditionally favoured Hartree-Fock based methods, although in recent years there is a growing interest in Density Functional Theory methods there also.

HF-type methods are not commonly used in the context of solids, partly due to the high computational costs involved; but even if only for reference, we will give here the briefest of introductions. If you wish to pursue this further, you can explore the methods via problem 4.4.1.

The starting point in HF theory is the assumption that electrons in a molecule or solid can be treated as independent particles. This is not the same thing as neglecting their interaction, but rather it means that the interaction between electrons is going to be accounted for in some kind of approximate way: it will be assumed that each electron moves in an effective field resulting from the presence of the other electrons in the system. Of course, each electron is also subject to the Coulomb field due to the presence of the nuclei, but this does not complicate matters very much, because by virtue of the Born-Oppenheimer approximation we are assuming that nuclei are fixed in space anyway.

Rather than going into the details of a methodology that we are not going to use in practice, it is more instructive to consider why HF theory is only an approximation. Electrons are really not moving independently from one another. Their motion is correlated, and this correlated motion results in a lowering of the energy with respect to the ficticious uncorrelated situation. The energy difference between correlated and uncorrelated situations is (not surprisingly!) called the correlation energy.

Another fact to remember is that electrons carry a spin of 1/2, which makes them Fermions. Fermions obey Pauli's Exclusion Principle, which implies that the wave function describing the electronic system must be anti-symmetric (change sign) upon exchange of any two electrons. This anti-symmetry of the wave-function results in another contribution to the total energy of the system, called the exchange energy. While HF theory can fully account for the exchange energy of an electronic system, it completely fails to account for the correlation energy, because it assumes that electrons are independent from each other.

Correlation energy is normally a small part of the total energy of a system of nuclei and electrons (ca. 1%), but this does not mean that it is negligible! Usually chemical reactions involve fairly small energy changes when compared to this 1%; so if one wishes to predict chemistry accurately by means of electronic structure calculations it is necessary to do better than this! The situation can be improved by using extensions of the Hartree-Fock method, such as Configuration Interaction (CI) or Møller-Plesset (MP) many-body perturbation theory, but these increase the computational demands significantly. When heavy atoms are involved, one needs the relativistic version of HF theory, known as Dirac-Fock theory. For calculating overlap integrals within DF theory, one needs to have access to a Relativistic Integrals Program, known as RIP; we rest our case!

In the rest of this lecture we discuss an alternative approach, which also treats electrons as a collection of mutually independent particles, but is nevertheless capable of accounting for both exchange and correlation energies, albeit in an approximate way.

4.2 The uniform electron gas and the jellium model

4.2.1 Introduction

Free electron models of metals have a long history, going back to the Drude model of conductivity which dates from 1900 (Ashcroft and Mermin chapter 1, Sutton chapter 7). The partly true, partly false predictions of this classical model were important precursors to quantum mechanical models based on the Fermi-Dirac energy distribution (Ashcroft and Mermin chapter 2, Sutton chapter 8). One puzzle left over from this previous era is why free electron methods work so well, when they appear to neglect completely all the important interactions between the electrons with each other, and with the ions. All that has been included in lectures 2 and 3 is the notion of a potential 'box' or well, which is deep enough to retain these electrons. Obviously some serious averaging has been going on, or we have been making implicit assumptions which need to be unmasked. These problems will be discussed in this lecture and the next.

Modern calculations start from a description of the electron density, r-(r) (r(z) in 1D) in the presence of a uniform density r+(r or z) of metal ions. This is the jellium model, where the positive charge is smeared out uniformly; it is a function of only one parameter, rs, which is the radius of the sphere which contains one (free) electron. At a later stage we can add the effects of the ion cores Dr+(r) by pseudopotentials or other approximations. This complication introduces one or more competing length scales into the description, and so the results become more specific to the particular material. We will start that discussion in the next lecture.

4.2.2 Jellium, exchange-correlation and surfaces

In the jellium model, the division into a uniform r+, with a step function to zero at the surface, allows us to consider the electron density r-(r) as the response to this discontinuity. Clearly, a long way inside jellium, r- = r+, and there is overall charge neutrality. But at the surface there is a charge imbalance, and the electrostatic potential V varies as a function of z.

To calculate this response, and hence the work function of a metal, was a major challenge which was solved by Lang and Kohn in the early 1970's. I have described this work in some detail in my Surfaces and Thin Films course, section A1, but am not expecting you to go into this level of detail unless you are specifically interested in clusters and/or surfaces. A few years ago Ben Saubi and I did a project on this topic which has produced some diagrams which have been incorporated into my own book.. You can see how this goes - try to do projects which can be used for more than one purpose!

The main aspect of the Density Functional Theory (DFT) methods is the replacement of the (insoluble) many electron N-body problem by N one-electron problems with an effective potential, Veff(r or z), which is a functional of the electron density. In general, as described in section 4.3 this potential contains the original electron-nuclei and electron-electron terms, and also has a term to describe exchange (x) and correlation (c) between electrons. In the special case of jellium, the first two terms cancel out in the bulk, because both positive and negative charges densities are uniform. The last two terms are usually lumped together (xc), and have been worked out precisely for a uniform electron gas, corresponding to the interior of jellium, so that explicit, numerical values can be given to these energies as a function of electron density. These numbers seem as banal as they are mysterious, and are typically quoted in atomic units, e.g. by Pettifor, sections 2.5 and 5.6 as

Uxc = -0.916/rs - (0.115-0.0313 lnrs),

where the first and second terms represent exchange and correlation respectively; Exc is a notation widely used, and a certain investment of time is needed to see where these terms come from! Follow this up via problem 4.4.2, and see how far you get.

Indeed, for theorists especially, one also needs time to learn 'atomic units', which corresponds to putting h, e, and m all equal to 1. This makes the equations look simple, but the cost is that no-one can check the units, i.e. beware gigo (garbage in, garbage out) ALL the time. Look at this via problem 4.4.3.

Some results of Lang and Kohn’s work on jellium are indicated on figures given in the project, and these can be shared in the lecture and/or looked up in your own time. The key point is that the electron density, the electrostatic potential and effective potential all have oscillations normal to the surface in the self-consistent solution obtained; there are substantial cancellations between the various terms.

The work function of these model alkali metals varies weakly from Li (rs about 3.3) to Cs (rs about 5.6), whereas the individual components of the work function vary quite a lot, as shown in Table 4.1 below. This model was the first to get the order of magnitude, and the trends with rs correct: a big achievement.

From this section, we need to remember that the position of the ions do not enter this model at all: everything is due to the electron gas, and the importance of the exchange-correlation term Uxc, and the variation of the electrostatic contribution, are evident in the following Table 4.1. taken from Lang and Kohn's original work in 1971.  

Table 4.1 The work function of jellium and its components 
Columns 2 and 3 represent the kinetic and exchange-correlation energy respectively;
Column 4 = column 2 + column 3; Column 5 = the electrostatic potential due to the surface dipole,
and Column 6 = column 4 + column 5 is the resulting work function (after Lang & Kohn 1971).
rs
kF2/2
Uxc (or mxc)
mbar
DV
f
2
12.52
-9.61
2.91
6.80
3.89
3
5.57
-6.75
-1.18
2.32
3.50
4
3.13
-5.28
-2.15
0.91
3.06
5
2.00
-4.38
-2.38
0.35
2.73
6
1.39
-3.76
-2.37
0.04
2.41

The ripples in the electron density are called Friedel oscillations; these occur when a more or less localized change in the positive charge density (the discontinuity at the jellium model surface being an extreme case) is coupled with a sharp Fermi surface. In other words, they are a feature of defects in metals in general, not just surfaces, and are an expression of Lindhard screening, which is screening in the high electron density limit. Screening in metals is so effective that there are ripples in the response, corresponding to overscreening. These ripples are most dramatically seen in scanning tunneling microscope (STM) pictures of metal surfaces. Many of you will have seen the stunning pictures of Quantum Corrals from Don Eigler's group (IBM Almaden Lab, California), and we can discuss qualitatively the causes of such images in the lecture. These are some of the many STM sites which can be visited from my page on Web-based resources.

In the thirty-five years since Lang and Kohn’s initial work, there have been major developments within the jellium model. As computers have improved, this method has also been applied to clusters, especially of alkali metals, of increasing size. There are many interesting points which can be followed up via the review literature on jellium clusters (Brack 1993), or via the web-based project.

4.3 Density functional theory: LDA and beyond

The trick now is to apply these same numerical recipes developed for jellium to non-uniform densities, whence the term local density approximation (LDA). There are many further methods which try to correct LDA for non-local effects and density gradients, such as the generalized gradient approximation (GGA), but it is not clear that they always produce a better result. In any case, we are now getting into the realm of arguments between specialists, which we will ignore at this stage.

Although we introduced density functional theory (DFT) in section 4.2 in the context of Lang and Kohn’s work on metal surfaces, the concept itself is much broader. It consisted of setting up a general single particle method in two stages to solve the Schrödinger equation for the ground state of a many electron system by:

The main non-relativistic scheme in use is due to Kohn and Sham (1965), but many others have been proposed and tried out. The pervasiveness of these methods was recognized in 1998 by the award of the Nobel prize for Chemistry to Walter Kohn. The other half-share of the prize awarded to John Pople recognised his pioneering work in computational studies of cluster chemistry, which is behind many of the packages which are now widely available.

Writing down too many equations specifically here will take too much space, and may encourage you to believe that the method is simpler than it actually is. Sutton has an introduction in chapter 11, and Pettifor discusses these methods only in outline on page 46-48, at the end of an all-encompassing chapter 2! Three key review articles have been already been cited, and we can discuss which others might suit your specific needs. So many words have already be spilt on the topic, the methods are so widespread, and yet no-one can give a measure of just how good an approximation DFT represents, or say categorically whether further developments such as GGA necessarily improve matters, that there is no sense in which I should try to confuse you further. However, we will go through it in outline in the lectures, and can have a discussion, based on a handout which makes points which are close to Sutton's pages 205-209. For those of you who will specialise on this topic, the more recent books by Martin, Kaxiras and others given in the reference list should be consulted.

One further conceptual aid is to discuss how we can visualize the exchange-correlation term in real space. In an electron gas of the alkali metals, we have 1 electron per Wigner-Seitz sphere, which has radius rs, i.e. it is of ~atomic dimensions. Thus the ‘electron’ or quasi-particle, when it moves, carries around a sphere of about this size which is deficient in electrons, due to their mutual interaction, i.e. the electron position and motions are correlated. Moreover, this ‘sphere of influence’ has Friedel oscillations associated with it, and depends on the electron spin, like spins repelling each other via the Pauli exclusion principle, and unlike spins ignoring each other. This exchange effect does not have to be added in separately, it is already there: on average in LDA, and explicitly in the Local Spin Density (LSD) models which are used to discuss magnetic materials.

For the record, essentially all the above material is in my book, at roughly the same level of detail. Updates are possible, particularly in chapter 6.

4.4 Problems relating to this topic

4.4.1: Hartree-Fock derivation of exchange and correlation terms

If you want to learn more about Hartree-Fock theory and the methods that derive from it, a good reference which keeps the level of treatment fairly basic is the book by Atkins and Friedman, Molecular Quantum Mechanics. Explain how the various terms arise and can be compared with experiment in the case of either a) the Helium atom, or b) a simple molecule of interest to you.

4.4.2: Original derivation of jellium exchange and correlation

Use the review article references provided for this lecture to find the original paper(s) where the numbers for the exchange and correlation terms for jellium were first calculated. Note that these journals are in the CPES library, but must NOT be borrowed; for this problem it is NOT worth using either your own or your supervisor's budget to copy such references, with the possible exception of the final paper when or if you find it.

Note: there is a description of the mathematics of exchange, and a discusion of correlation, in C. Kittel, Quantum Theory of Solids, chapter 5, pages 86-97, and chapter 6. This 1963 John Wiley book (reprinted with solutions to problems in 1987) gives an idea of how the problem was approached before the famous Kohn papers. At a more survey level, Ashcroft and Mermin, chapter 17 is also useful for perspective on this topic, particularly for the emphasis it gives to screening in general, and to the distinction between the low density (Thomas-Fermi) and high density (Lindhard) limits.

4.4.3: Atomic units in theoretical papers

In 'atomic units', we put h, e and m all equal to 1. This means that there are corresponding units for length, time, energy, momentum, etc., and that constants such as the fine structure constant a, and the velocity of light, have a particular value. By considering the actual units of these three fundamental units (or otherwise), and the formula for a, work out the value of these units, and find out what they are called. Derive the form of the Schrödinger equation in atomic units.


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