Concepts in Computational MaterialsScience

Lecture CourseTechnical University of Berlin

Summer 2010

Volker BlumApril 24, 2010

2

Title image: Corrugated “reconstructed” (100)-oriented surface of a Pt crystal. The ar-rangement of atoms in the surface plane would be a simple square lattice if the “periodic”array of atoms deep inside the crystal were continued as is right up to the edge. Instead,the atoms in the topmost surface plane form a very nearly ideal hexagonal lattice, whichis wavy, or “buckled” (due to the mismatch with the layers underneath). This surfacerestructuring was a surprise when first discovered (in 1965). The exact structure shownhere was computed from quantum-mechanical first principles, using density-functionaltheory in the so-called local density approximation. The structure in question contains656 atoms, and a total of 51168 electrons, in a periodic geometry, and is close to thelimit of what modern computer codes and modern computational hardware can routinelyhandle. Only ten years ago, a direct calculation of this size was unthinkable.

3

Contents

1 Introduction: Computational Materials Science 4

1.1 The Many-Body Schrodinger Equation . . . . . . . . . . . . . . . . . . 4

1.2 The “Difficulty” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 The Many-Body Schrodinger Equation 12

2.1 Discussion of the Many-Body Wave Function . . . . . . . . . . . . . . . 12

2.2 Separating electrons and nuclei: Born-Oppenheimer Approximation . . . 14

Bibliography 17

Index 17

4

Chapter 1

Introduction: ComputationalMaterials Science

1.1 The Many-Body Schrodinger Equation

“The underlying physical laws necessary for the mathematical theory of alarge part of physics and the whole of chemistry are thus completely known,and the difficulty is only that the exact application of these laws leads toequations much too complicated to be soluble. It therefore becomes de-sirable that approximate practical methods of applying quantum mechanicsshould be developed, which can lead to an explanation of the main featuresof complex atomic systems without too much computation.” P.A.M.Dirac, 1929

Computational Materials Science is a fast-growing field in the literature: The studyof materials properties (solids, surfaces, biomolecules and -materials, ...) by theoreticalmeans, but with the goal of making accurate predictions of the properties of specificmaterials, if necessary with the help of a computer. In the context of this course,the definition will even be a bit more demanding: In principle, the theory of quantummechanics gives us a complete recipes to make such predictions based on “first principles”of theoretical physics only. The catch is that such predictions must actually be feasiblewithin the means defined by real computers, and yet with a well-controlled accuracyapproaching that of the full underlying theory.

The present course will introduce some key theoretical concepts that make “computa-tional materials science” feasible today. In many cases, these concepts originate directlyfrom theoretical solid state physics, and are now used and developed in an active andgrowing field at the boundary between solid state physics, chemistry, and biophysics.

Paul Dirac wrote the now-famous quote cited at the beginning of this chapter in his intro-duction to an article on the “Quantum Mechanics of the Many-Electron System.”[Dirac(1929)]His declaration comes at a remarkably early point in time. The “quantum” phenomenapuzzling physicists had been known for some decades, and had led to the developmentof the phenomenological quantum theories of Einstein, Bohr, Sommerfeld and manyothers. But the main steps that provided a mathematically solvable background “from

1.1. The Many-Body Schrodinger Equation 5

first principles” had been taken in only a few years before 1929:

• Schrodinger’s Equation, which provided a clear mathematical recipe to derive somecentral observations of the earlier, “old” quantum mechanics from a set of “firstprinciples” had only been out since 1926 (Ref. [Schrodinger(1926a)], with anEnglish summary in Ref. [Schrodinger(1926b)]).

• Schrodinger’s Equation had formally failed to account for relativity, and thus alsolacked a formally rigorous inclusion of electromagnetism (as well as, it turns out, agood reason for the existence of the spin of electrons, among other things). Dirac’srelativistic generalization [Dirac(1928), Dirac(1930)], the “Dirac Equation,” set-tled this problem in 1928.

• The first important applications of the new theory to multi-electron systems(Hartree, 1928?), polyatomic systems [Heitler & London(1927)] and crystallinesystems [Bloch(1928)] were only beginning to appear, albeit at a rapid pace.

Of course, in 1929 there was (and there still is) much more exciting physics to come forwhich the “underlying physical laws” were not known: Quantum field theory, “weak”and “strong” interactions, quantum theory of gravity etc. In the context of matter andmaterials composed of electrons and nuclei, though, Dirac’s statement was essentiallycorrect. One can indeed argue that the “underlying physical laws” for this part of physicswere “completely known” in 1929, describing, for example,

• atoms and ions

• molecules

• liquids

• solids (crystalline, amorphous, etc.)

• etc.

with remarkable accuracy for the phenomena that occur at the temperature and pressurefound on Earth up to the interior of planets. With the exception of a few phenomena(e.g., nuclear fission, fusion, the conditions at the boundary between the atmosphereand space, the conditions inside a star), matter and materials are covered.

As the basic equation that enables all this, we take here the many-body SchrodingerEquation, here written in its time-independent form.

HΨn ({RI}, {rk}) = EnΨn ({RI}, {rk}) . (1.1)

Ψn is the many-body wave function (nth eigenstate) for a system composed of M nucleiwith coordinates RI (I=1,...,M) and N electrons with coordinates rk (k=1,...,N). En

is the total energy (an actual number that can be calculated!) of the system in the purequantum state n. The many-body Hamiltonian H consists of a few basic ingredientsthat are all well known:

1. The kinetic energy of the electrons:

6 Chapter 1. Introduction: Computational Materials Science

T e =N∑

k=1

p2k

2me, pk →

~i∇k (1.2)

2. The kinetic energy of the nuclei:

T nuc =M∑

I=1

P 2I

2MI

, P I →~i∇I (1.3)

3. The Coulomb interactions between electrons:

V e-e =1

2

1

4πε0

N,N∑k 6=k′

e2

|rk − rk′|(1.4)

4. The Coulomb interactions between nuclei:

V nuc-nuc =1

2

1

4πε0

M,M∑I 6=J

ZIZJe2

|RI −RJ |(1.5)

5. The Coulomb interactions between nuclei and electrons:

V e-nuc =1

4πε0

M∑I=1

N∑k=1

ZIe2

|RI − rk|(1.6)

The full Hamiltonian thus reads

H = T e + T nuc + V e-e + V nuc-nuc + V e-nuc , (1.7)

and we have some residual explaining left to do. pk and P I are the momentum operatorsof electrons and nuclei, respectively, and me and MI are their respective masses. ZI arethe atomic number of each nucleus.

We also note that Eqs. (1.2-1.6) above are written in SI units, so that the usual factors~, me, or 1/(4πε0) appear explicitly. For most of this course, we will instead use atomicunits that are not just simpler to write, but also simplify a computational implementation,by setting:

~ → 1 (1.8)

e → 1 (1.9)1

4πε0

→ 1 (1.10)

me → 1 (1.11)

If one goes through this definitions, the result is that all lengths used in the equationsare now given in units of the Hydrogen atom,

1bohr = a0 = 0.52917720859(36) nm. (1.12)

1.2. The “Difficulty” 7

Similarly, all computed energies are given in “Hartree” units,

1Ha = 27.21138386(68) eV. (1.13)

The numbers given above can be derived directly in terms of fundamental constantsfrom Schrodinger’s Equation. However, even our values for fundamental constants mustbe measured and updated routinely, and this task falls to standards organizations suchas the American National Institute of Standards (NIST). The above values for 1 bohrand 1 Ha can be found athttp://physics.nist.gov/cuu/Constants/index.html ,and are current as of 2010. The numbers in brackets refer to the current experimentaluncertainty of the last two digits. One might think that one could safely leave awayanything but the first few digits, but for practical calculations on a computer, this is atleast an unnecessary, if not a dangerous assumption to make. In the best case, nothinghappens (work load does not change). However, if one is unlucky, leaving away toomany digits in the wrong place may cause a surprising amount of scatter in calculatedresults, since often, the last few digits of a total energy contain the physically relevantenergy differences between (say) two different structures of a molecule or solid.

We still have to note what was already stated above, and that is that the full answer isactually not given by Schrodinger’s Equation Eqs. (1.1-1.7) above. For electrons, the fineprint is actually given by Dirac’s Equation, which sports a four-component wave function.The Hamilton operator becomes a 4×4 matrix, is first order in space and time, and thewhole arrangement is Lorenz invariant. Dirac’s Equation thus accounts in a natural wayfor electromagnetic fields, for the changing electron mass at high kinetic energies, and,somewhat surprisingly at first, also yields the reason for the spin of the electron. Forthe rest of this course, we will still mostly talk about Schrodinger’s Equation, but it isimportant to stress that Dirac’s Equation should remain somewhere in the back of ourheads. The reason that Schrodinger’s Equation is acceptable is that most of the effectsof Dirac’s Equation that concern us in practice can be “added on” to Schrodinger-likeEquations in some way. To throw out a few important keywords, “scalar-relativistic”approximations or pseudopotentials take care of the electron mass for valence electrons(responsible for the chemical bond), and the Pauli Equations could take care of spin.Nonetheless, without such modifications, relativity is not a small effect. For example,the color of gold is “gold” only when relativity is accounted for in some way. In the deepCoulomb potential near the heavy nucleus, the electrons are so “fast” that they require arelativistic treatment. If one forgets about this effect, the energies of the electron stateswhich are responsible for the optical properties of gold come out too high.

1.2 The “Difficulty”

So far, life seems easy, but there is still the slightly troubling fact that Dirac’s 1929quote did happen contain the word “difficulty”:

8 Chapter 1. Introduction: Computational Materials Science

Material Al Pb MgB2

Superconducting Tc (expt.) 1.175 K 7.196 K 39 K

Superconducting Tc (calc.) 0.90 K 5.25 K 34.1 K

Table 1.1: Experimental critical temperatures of some “conventional” superconductorsvs. calculated values from a recent first-principles approach [M. Luders et al.(2005)].The theoretical values are taken from Refs. [M. A. L. Marques et al.(2005)] (Al),[C. Bersier et al.(2009)] (Pb), and [A. Floris et al.(2005)] (MgB2). Experimental valuestaken from http://www.superconductors.org/.

“... the difficulty is only that the exact application of these laws leads toequations much too complicated to be soluble. It therefore becomes de-sirable that approximate practical methods of applying quantum mechanicsshould be developed, which can lead to an explanation of the main featuresof complex atomic systems without too much computation.” P.A.M.Dirac, 1929

In this exact spirit, explaining “the main features of complex atomic systems” startingfrom only the laws of quantum mechanics is a success story that has been going onfor more than eighty years, with many highlights in chemistry and solid state physics.Spectacular examples include the entire theory of semiconductors, superconductivity,the nature of the chemical bond, the formation of structure in molecules and solids, orchemical reactions.

Strictly speaking, the examples listed above are successes of brilliant “qualitative” appli-cations of the laws of quantum mechanics, explaining the features of a physical effect,but often without making a clear prediction (from theoretical “first principles” alone,without invoking an experiment) for which particular material(s) we should expect agiven effect. Of course, making accurate predictions for specific systems has been agoal for at least the same amount of time. Atoms and ions were tackled first (in the1920s and 1930s), mainly through the work of Hartree, Slater, and others. In the early1950’s, the field of “quantum chemistry” nucleated, allowing to treat increasingly largermolecular systems (containing a finite number of nuclei and electrons), step by step,simply by finding good approximations to the many-body wave function Ψn (or, at least,its electronic part, to be introduced in the next chapter).

In contrast, accurate predictive treatments of solids were much slower in coming. Onemay even argue that only just before 1980 did the first truly “predictive” calculations forsimple solids become available. The reason is that, even with some kind of crystallineperiodicity, when one might be able to restrict the problem of O(1023) electrons and nu-clei to O(1023) identical unit cells, the electrons in neighboring unit cells are still coupledthrough the wave function. The theory that finally allowed to do accurate calculationsfor solids is called density-functional theory, and was placed on a rigorous footing byPierre Hohenberg and Walter Kohn only in 1964. Their proof of the Hohenberg-Kohntheorem, and many related insights to make this theory practical garnered Kohn theNobel prize in 1998.

???

1.3. Further reading 9

Today, there are many striking examples for the ongoing success (and, in my view, thebright future) of first-principles based Computational Materials Science. One particularlystriking example of how far the underlying methods can be pushed today is given in Table1.1. This table lists the critical temperatures of some (conventional) superconductorsas computed by “superconductor density functional theory” (SCDFT) in comparisonwith the experimental values at zero external magnetic field. The important pointhere is not that an understanding of conventional superconductivity can be achievedin principle—of course, this understanding has been achieved long ago. However, evengiven the mechanism in principle, it is far from trivial to predict a specific material’ssuperconducting critical temperatures (if that even exists) and other properties of thesuperconducting state with any accuracy at all. As we will note again in a moment, thereason is that superconductivity is a phenomenon that would not work at all without thecorrelated motion of “electrons” and “nuclei” in the many-body Schrodinger Equation.Seemingly in contrast, most of our first-principles theories are based of effective single-particle models, where those single particles are formally non-interacting. It would seemto be counterintuitive that this type of theory should capture a phenomenon such assuperconductivity at all. Nonetheless, this is the case. Although we will not treatsupercoductivity from first principles in this course, we will see why a scheme basedon effective, non-interacting single particles scheme can be expected to account evenfor superconductivity in principle. As a side note, the practical method that led to theresults of Table 1.1 was developed over the course of several years, and mostly here inBerlin.

In the remainder of this course,we will introduce some key physical concepts that enablecomputational materials science today—their derivation, but importantly also the limitsof these approximations.

• The first part of the course focuses on what we can do by simplifying Schrodinger’sEquation itself, and mainly for the electrons. Here, we will be discussing approxi-mations that are at least in principle exact—and how to solve them in practice.

• The second part of the course will focus on what we can do with the results of thefirst part. The specific examples include the motion of atoms, molecules and solids(“molecular dynamics”) and the use of “first principles” to make accurate statisti-cal mechanics predictions for real materials. To overcome the computational limitsthat we face with pure first-principles methods, this part also exemplifies how to“coarse-grain” the results of quantum mechanical calculations into model Hamil-tonians with a much simpler form, but which should then reproduce the propertiesof a specific system with the same accuracy as if one performed the full quantummechanical calculation.

1.3 Further reading

The essentials of the course will be summarized in this script, but of course, many goodtextbooks exist that cover similar subjects, and which are excellently suited for furtherreading on a specific subject. To my knowledge, there is no single textbook that covers

10 Chapter 1. Introduction: Computational Materials Science

the entirety of this course, but then, I have not read every single textbook out there inits entirety. Certainly, there are a lot of new and good textbooks currently appearing. Ifnothing else, this shows what a thriving and fast-moving field “Computional MaterialsScience” is at this point.

The list given below names only a few books that I find particularly relevant, but thereis—obviously—no claim that this list is in any way complete, or even contains “the best”textbooks that one can find. If you know or hear of any book that is equally or bettersuited, please inform me, and I will happily put it on the list.

Some basic textbooks (concepts):

Familiarity with the basics of Quantum Theory is assumed for this course. The followinglist is by no means exhaustive. I am simply naming some particularly influential books.There are many others.

• Franz Schwabl, Quantum Mechanics. Springer, Berlin (2007). [4th ed.]

• Neil W. Ashcroft, N. David Mermin, Solid State Physics. Harcourt, Fort Worth(1976).

• Bernard Diu, Claudine Guthmann, Danielle Lederer, Bernard Roulet, Grundlagender Statistischen Physik. de Gruyter, Berlin (1994).

Electronic Structure Theory:

• Jorge Kohanoff, Electronic Structure Calculations for Solids and Molecules. Cam-bridge University Press, Cambridge (2006).Personally, I like the style and choice of material of this book best of all thespecialized books that I have seen.

• Efthimios Kaxiras, Atomic and Electronic Structure of Solids. Cambridge Univer-sity Press, Cambridge (2003).

• Richard M. Martin, Electronic Structure: Basic Theory and Practical Methods.Cambridge University Press, Cambridge (2004).

• David S. Sholl, Janice Steckel, Density Functional Theory. John Wiley and Sons,Hoboken (2009).

• R.M. Dreiszler, E.K.U. Gross, Density Functional Theory: An Approach to theQuantum Many-Body Problem. Springer, Berlin (1990).Despite its relative age, an excellent book on the formal foundations of densityfunctional theory.

• Attila Szabo, Neil S. Ostlund, Modern Quantum Chemistry. McGraw-Hill, NewYork (1989).This book is certainly advanced, and not what is usually covered in physics. In“physics” thinking, actual wave function are often discussed in principle, and then

1.3. Further reading 11

forgotten—already because there are other, more convenient conceptual formu-lations around for the solid state. In molecular “quantum chemistry”, however,the front door to accurate numbers is still the actual many-body wave functionitself. In order to get an exact number where this is possible, wave function basedapproaches are arguably still, unrivalled. This book is perhaps the definitive deriva-tion of the (almost) purely wave function based approaches of quantum chemistry,and is thus included here.

Molecular Simulations and Computational Algorithms—Molecular Dynamics,Monte Carlo, etc.

• Daan Frenkel, Berend Smit, Understanding Molecular Simulation. Academic Press,San Diego (2002).

• Mark Newman, Gerard Barkema, Monte Carlo Methods in Statistical Physics.Clarendon Press, Oxford (2002).

• J. M. Thijssen, Computational Physics. Cambridge University Press, Cambridge(2003). [with corrections]

12

Chapter 2

The Many-Body SchrodingerEquation

2.1 Discussion of the Many-Body Wave Function

For the duration of this course, we will mostly be busy with dissecting the formally exactmany-body Schrodinger Equation, Eq. (1.1), to get from a formally exact recipe to aseries of controlled approximations that we can handle in practice. To begin with, wewill analyze the contents of Eq. (1.1) a little further, in order to better understand theprecise nature of the proble before us.

We begin by writing down (again) the many-body Schrodinger Equation and the wavefunction that results from it:

HΨn = EnΨn (2.1)

Ψn ≡ Ψn({RI}, {rk})

The time-independent ground state is usually labelled as n=0, and minimizes E0. Allhigher n correspond to excited states.1

The basic trouble is that there is a priori nothing that tells us the precise form of Ψn

as a function of all its coordinates. And in the case of a realistically large piece ofmatter, there may be O(1023) of these coordinates. Of course, we did throw in O(1023)individual electrons and nuclei, identified by those coordinates. Yet, all these particlesinteract, and annoyingly, they do so by way of the Coulomb interaction ∼ 1/r. Asanyone who has ever had their hair stand on end after brushing it can testify, this isa rather long-ranged interaction. So after putting those O(1023) individual electronsand nuclei together, there is nothing to indicate they might still behave like individualparticles afterwards. For all we know, the combined system before us might even behavelike one particle with 1023 coordinates—at any rate, these coordinates are coupled.

To illustrate that this problem is not entirely hypothetical, let us revisit one more time

1We here leave aside the question how meaningful “the ground state” is for Schrodinger’sequation when it includes the nuclei, and whether we should not really be looking at the time-dependent version instead. We may come back to this issue at a later point in the course, though.

2.1. Discussion of the Many-Body Wave Function 13

the example of superconductivity. Again, we will here do so only qualitatively (the actualtheory of superconductivity will not be part of the present course).

One of the many properties of a superconductor is that there exists a critical temperatureTc below which its electrical resistance approaches zero. The conductivity becomes infi-nite, and once induced, a current will flow for an essentially infinite amount of time, forexample in a ring.2 At least for “conventional” superconductors, the qualitative reasonbehind this effect can be understood as follows.

• Among the conduction electrons of the system, two electrons each experiencean effective attraction, and at low enough temperature, are able to form stable(bound) pairs, so-called “Cooper pairs”.

• While we shall soon learn that individual “electrons” (spin 1/2) behave as Fermionsand cannot occupy the exact same single-particle like state within the many-bodysystem (Pauli principle), the combined spin S of each bound pair is an integer (inpractice, S=0 for conventional superconductors). In principle, many Cooper pairscould occupy the same “pair state”.

• It turns out that the excitation of such a Cooper pair, or breaking it up into twoseparate “electrons”, costs a finite amount of energy. As a consequence, the scat-tering events that are responsible for the resistivity of a “normal” material can notexert the same effect on a system of Cooper pairs. Once the full system is in agiven many-body state (say, current in a ring), it stays there.

• However, two electrons usually repel each other pretty strongly due to their neg-ative charges. An effective attraction must be mediated by something else. That“something else” turns out to be the nuclei of the superconductor, and their col-lective vibrations (phonons).

It is dangerous to take the crude outline above too literally, but the key point is that,strictly, all of this has to happen within the same many-particle wave function Ψn, re-quiring us to correlate a macroscopic number of electron and nuclear coordinates withone another. First, the picture of individual electrons, even some kind of effective singleelectrons, no longer describes the nature of the actually observed charge carriers evenapproximately. There must be two of them, bound in a pair. Then, many of these pairsmust occupy what looks like the same pair wave function. [How this could come topass in the macroscopic many-body wave function is described by the “Bardeen-Cooper-Schrieffer” (BCS) theory of the 1950’s, specifically the BCS wave function.] Finally, thewhole arrangement can not hold even approximately without the simultaneous, coordi-nated “motion” of the nuclei. Indeed, a significant part of the many-body wave functioncomes together to behave like one large, macroscopic entity.3

In summary, we are faced with a complex many-body problem. For all that we know,even thinking in terms of individual particles might not make sense at all. Do we reallyhave to address all O(1023) at once?

2At any rate, existing experiments do not seem to give an upper limit for this time.3One should also state here that the actual description of superconductivity, especially once the

nuclei come into play, is not usually done by analyzing the many-body wave function itself. Onesuccessful description post the stroke of insight(s) by BCS use Green’s function based formalisms.

14 Chapter 2. The Many-Body Schrodinger Equation

Fortunately, we will find that most of the successful strategies pursued in solid statephysics do approach the problem by separating the many-body wave function into indi-vidual, effective particles. In fact, the success of single-particle like approximations wasa source of puzzlement to the early solid-state physicists.4

For all the possible complexity of Ψn, it is important to remember what we want outof the many-body Schrodinger Equation. Only very rarely (if at all) are we interestedin the wave function itself, which is not experimentally observable after all. Rather,we are usually after the actual quantum-mechanical observables of a material, and theirconsequences. Quantities of interest could be the total energy or some excitation energy,the elastic properties of a material, the magnetization of a ferromagnet, the dipolemoment of a molecule, etc. If we could somehow devise a rigorous approximation thatobtains these quantities for us without having to solve for the full many-body wavefunction first, we would certainly be interested. Later on, we shall be doing exactly that.

2.2 Separating electrons and nuclei:Born-Oppenheimer Approximation

Apart from the philosophical issues with the many-body wave function discussed in theprevious section, we face a very practical problem: We would actually like to write awave function down. It turns out that this task is a little tedious for Ψn in the generalform of Eq. (2.1) already for a rather simple reason: There are electrons and nuclei inthe game. We would have to form a multi-component wave function. On top of that (aswe will see a little further below), the symmetry properties of the multicomponent wavefunction depend on whether its members are Fermions or Bosons—and nuclei could beboth.

On the other hand, in most “normal” states of matter that we are used to (crystals,simple molecules), intuitively the nuclei seem to mostly “sit around” in fairly well definedlocations, while the electrons form a much more delocalized “gas” of particles that takescare of bonding etc. While a multicomponent wave function could still be written down,it would be most convenient if we could get rid of the nuclei for the moment, and focusonly on the electrons. Can we?

The central approximation that allows us to separate electrons and nuclei is known asthe Born-Oppenheimer (BO) approximation or “adiabatic approximation”. While we willhere treat it as intuitive, we have already seen that such a separability can not alwayshold. The BO approximation, therefore, is one systematic approximation to Eq. (1.1)

4It is further worth pointing out that, in the field of quantum chemistry, this separation strategyinto individual effective particles is pursued much less, if at all—although the scientists involveddeal with the same underlying Equation, Eq. 1.1. In my view, the reason is as much philosophicalas it is practical. The traditional methods of quantum chemists focus directly on the many-bodywave function, which can be treated for a reasonably small, finite-sized molecule. On top of that,many phenomena in molecules do look a lot less like effective single particles than in standardsolids (metals or semiconductors). For example, the correlated motion of electrons and nuclei canbe much more important for the electrical current happening in a molecule than it normally is forelectrical current in (say) a bulk semiconductor.

2.2. Separating electrons and nuclei: Born-Oppenheimer Approximation 15

that we will almost always make, but that we should still keep in the back of our heads,just the case.

Qualitatively, the Born-Oppenheimer approximation is based on the observation thatMI � me for all possible nuclei I in the periodic table—the smallest MI being thatof the proton, MH/me ≈1836. As far as the kinetic energy is concerns, this meansthat electrons will respond to an external perturbation much faster that the nucleipossibly will. For example, for a given placement of the nuclei, the electronic part ofthe many-body wave function should “almost always” assume its instantaneous groundstate, before the nuclei even have time to move appreciably far and alter the shape ofthe electronic ground state appreciably.

Can we formulate this in a more rigorous way? We can, although we will here only per-form some qualitative reasoning, and then move on quickly to the problem of treatingthe electrons, which will occupy us mostly. We will, however, return to the BO approx-imation later. For those who are curious now, very good treatments can of course befound in the literature, for example in the book by Kohanoff.

To begin with, no one can prevent us from defining a subspace of electron many-bodywave functions for fixed nuclear coordinates, i.e., treating the nuclei as an externalpotential. For this purpose, we return to the Schrodinger Equation Eqs. (1.1-1.7), andsingle out only the parts that act on the electron coordinates rk in some way. We obtain

He[{RI}]Φν({RI}, {rk}) = EνΦν({RI}, {rk}) (2.2)

He = T e + V e-nuc + V nuc-nuc (2.3)

It is important to stress that we are absolutely allowed to write down this definition ofan electronic sub-problem, even if it should later turn out to be useless for some otherreason (but this will not be the case). We have not yet made any approximation byintroducing our new electron-only many-body wave function Φν({RI}, {rk}) and itsassociated eigenenergy Eν . {Φν({RI}, {rk})} are called the adiabatic electronic eigen-states. The Hamiltonian He does not act on the nuclear coordinates in any non-trivialway, so these are mere parameters, but Eν will still depend on them, i.e., it becomes afunction Eν({RI}): the “Born-Oppenheimer (energy) surface” for the electrons, definedas though the nuclei did not move.

The “adiabatic approximation” states that we hope that the electrons always remain inthe same adiabatic eigenstate Φν({RI}, {rk}) which corresponds to the instantaneousnuclear coordinates, regardless of what the nuclear wave function is.5 In particular, oncewe had the electronic subsystem in its ground state ν=0, without external exitations theelectrons would remain in their adiabatic ground state {Φ0({RI}, {rk}) forever.

In the following, let us see why the adiabatic approximation does not hold exactly. Infact, we could go further along the same lines and find out how to go beyond theadiabatic approximation in quantitative way, but we shall not do this yet. Consider thefollowing ansatz for a full many-body wave function, where the electrons are forced to

5Transitions from one electronic state to another could still happen through the interactionwith an external electromagnetic field (e.g., emission or absorption of a photon, not consideredhere), but not “just like that” (radiationless).

16 Chapter 2. The Many-Body Schrodinger Equation

remain in the same adiabatic state, regardless of the nuclear wave function:

ΨBO = Λ({RI}) · Φν({RI}, {rk}) (2.4)

Could an eigenstate of the full many-body Hamiltonian have this form? We find:

HΨBO = [Hnuc + He][Λ({RI})Φν({RI}, {rk})] , (2.5)

whereHnuc = T nuc + V nuc-nuc . (2.6)

Clearly,

He[Λ({RI})Φν({RI}, {rk})] = Λ({RI}) · HeΦν({RI}, {rk}) (2.7)

= Eν({RI}) · Λ({RI}) .

Similarly,

V nuc[Λ({RI})Φν({RI}, {rk})] = Φν({RI}, {rk}) · V nucΛ({RI}) . (2.8)

However, the nuclear kinetic energy does something more complicated:

T nuc[Λ({RI})Φν({RI}, {rk})] = Φν · [∑

I

∇2I

2MI

Λ] (2.9)

+∑

I

1

2MI

[∇IΦν ][∇IΛ] + Λ[∑

I

∇2I

2MI

Φν ] .

If it were not for the second line of the preceding equation, we could have eliminated Φν

from Eq. (2.5) in order to find:[Hnuc + Eν({RI})

]Λ({RI}) = EΛ({RI}) . (2.10)

This equation would have looked like a system of interacting nuclei moving around in apotential Eν({RI}), defined by the Born-Oppenheimer surface. However, as it stands,Eq. (Eq:BO-nuclei) is not equivalent to the full many-body Schrodinger equation. Forthis to happen, we would have to neglect the following two terms:∑

I

1

2MI

[∇IΦν ][∇IΛ] + Λ[∑

I

∇2I

2MI

Φν ] . (2.11)

As it stands, Eq. (2.4) with only a single, adiabatic electron state can not be turned intoan eigenstate of the full many-body Schrodinger Equation. Instead, the kinetic energyoperator of the nuclei ensures that different adiabatic electron states will be mixed ina full eigenstate Ψn. This does not mean that the adiabatic electron eigenstates areuseless. For example, we could still expand the full many-body wave function in termsof the adiabatic electron states, in the following way:

Ψn({RI}, {rk}) =∑

ν

Λn,ν({RI})Φν({RI}, {rk}) (2.12)

With this ansatz, which is still exact, we could then tackle the problem of how differentadiabatic electron states are “mixed” by the nuclei, and under which circumstances thiswould be the case. For the moment, we will not pursue the matter further, except fornoting that, due to our use of atomic units, the ion mass MI in T nuc is implicitly givenin units of the electron mass. Thus, as MI/me → ∞, we would indeed find the BOapproximation to hold exactly. For real systems, this is not necessarily the case.

17

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Following pages: Index

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