density-functional theory
Density functional theory is an extremely successful approach for the description of ground state properties of metals, semiconductors, and insulators. The success of density functional theory (DFT) not only encompasses standard bulk materials but also complex materials such as proteins and carbon nanotubes.
The main idea of DFT is to describe an interacting system of fermions via its density and not via its many-body wave function. For N electrons in a solid, which obey the Pauli principle and repulse each other via the Coulomb potential, this means that the basic variable of the system depends only on three -- the spatial coordinates x, y, and z -- rather than 3*N degrees of freedom.
An overview over the basic principles of DFT and some neat applications of DFT to real life problems is given in the section ``Basics of Density Functional Theory.'' This section contains a talk that I gave at the Nuclear Physics Seminar at OSU and at the physics colloquium of the University of Braunschweig, Germany. Some application examples are currently studied at OSU (defects in Si and GaN), others (proteins and carbon nanotubes) are taken from the literature.
While DFT in principle gives a good description of ground state properties, practical applications of DFT are based on approximations for the so-called exchange-correlation potential. The exchange-correlation potential describes the effects of the Pauli principle and the Coulomb potential beyond a pure electrostatic interaction of the electrons. Possessing the exact exchange-correlation potential means that we solved the many-body problem exactly, which is clearly not feasible in solids.
A common approximation is the so-called local density approximation (LDA) which locally substitutes the exchange-correlation energy density of an inhomogeneous system by that of an electron gas evaluated at the local density. While many ground state properties (lattice constants, bulk moduli, etc.) are well described in the LDA, the dielectric constant is overestimated by 10-40% in LDA compared to experiment. This overestimation stems from the neglect of a polarization-dependent exchange correlation field in LDA compared to DFT. The section Density Polarization Functional Theory analyzes the properties of this field for real materials.
Rather than approximating exchange and correlation as a functional of the system density, we can also determine the exchange potential exactly. The section Exact-Exchange Density Functional Theory and Quasiparticle Calculations discusses such density functional schemes and their relevance for quasiparticle calculations, that is, computational many-body theory.
An apparent failure of density functional theory (DFT) with the local density approximation (LDA) or generalized gradient approximations (GGA) is the overestimation of the electronic static dielectric constant compared to experiment. Since the dielectric constant is determined by the second derivative of the ground-state energy with respect to an applied electric field, it is a ground state property and can in principle be calculated exactly within DFT. However, for medium- and small-band-gap materials, the dielectric constant is overestimated by at least 10% in LDA-based calculations. A slight reduction occurs in the case of silicon using GGA rather than LDA. The discrepancy between experiment and LDA- or GGA-based calculations of the dielectric constant indicates a principal failure inherent to both approximations. This failure can be traced back to the neglect of the exchange-correlation energy's polarization dependence in LDA and GGA. The talk and the paper quantify the relevance of the neglect of polarization dependence for medium- and wide-gap semiconductors.
The description of excited states in solids is a challenging and important problem in condensed matter physics since excited states are needed to determine transport and optical properties of materials. Untill the mid-eighties, a commonly used way to determine, for example, band structures of materials was based on density functional theory in the Kohn-Sham formulation using the local density approximation. This approach maps the many-body problem onto a system of non-interacting, fictitious Kohn-Sham particles. While the eigenvalues that result from the solution of the single-particle Kohn-Sham equations have no physical meaning within the frame work of the theory they compare rather well with experiment. The most notable exception is the band gap of insulators which is generally 0.5 - 2.0 eV too small compared to experiment.
Very recently, new density functional methods have been presented in the literature whose energy gaps generally agree much better with experiment than the LDA. One such method, which has been studied in much detail for semiconductors, is the so-called exact-exchange density functional theory which allows the determination of the exact local Kohn-Sham potential. In particular, self-interaction errors due to incomplete cancellation of the self-Hartree and the self-exchange potentials are eliminated in exact exchange density functioanl theory. Using exact-exchange calculations in combination with LDA or generalized gradient approximation (GGA) correlation functionals leads to energy gaps and structural properties (lattice constant, cohesive energy, bulk modulus, etc.) of standard semiconductors that agree well with experiment.
An alternative way to describe the excited states of materials is provided by compuational many-body theory using the dynamically screened or GW approximation (GWA). In the GWA, the electronic self-energy, which describes exchange and correlation beyond the Hartree approximation, is expressed as the product of a single-particle propagator G and a screened interaction W. In principle, the GWA requires the self-consistent solution of a set of four coupled integral equations as is discussed in more detail in the paper. However, GWA calculations that are non-self-consistent, use LDA energies and wave functions as input, and neglect the imaginary part of the self-energy describe the experimental electronic structure of sp bonded semiconductors and insulators generally to within 0.1 to 0.5 eV.
Here we test the accuracy of wave functions and energies obtained using exact-exchange plus LDA correlation density functional calculations as input for quasiparticle calculations. We choose eight standard sp bonded semiconductors for our test. These provide a hard test, since LDA plus GWA calculations work very well in these systems. Overall, we find electronic energies that agree with LDA plus GWA calculations and experiment.
I suppose I am not the first Nobelist who, on the occasion of receiving this Prize, wonders how on earth, by what strange alchemy of family background, teachers, friends, talents and especially accidents of history and of personal life he or she arrived at this point. I have browsed in previous volumes of "Les Prix Nobel" and I know that there are others whose eventual destinies were foreshadowed early in their lives – mathematical precocity, champion bird watching, insatiable reading, mechanical genius. Not in my case, at least not before my late teens. On the contrary: An early photo of my older sister and myself, taken at a children's costume party in Vienna – I look about 7 years old – shows me dressed up in a dark suit and a black top hat, toy glasses pushed down my nose, and carrying a large sign under my arm with the inscription "Professor Know-Nothing".
Here then is my attempt to convey to the reader how, at age 75, I see my life which brought me to the present point: a long-retired professor of theoretical physics at the University of California, still loving and doing physics, including chemical physics, mostly together with young people less than half my age; moderately involved in the life of my community of Santa Barbara and in broader political and social issues; with unremarkable hobbies such as listening to classical music, reading (including French literature), walking with my wife Mara or alone, a little cooking (unjustifiably proud of my ratatouille); and a weekly half hour of relaxed roller blading along the shore, a throwback to the ice-skating of my Viennese childhood. My three daughters and three grandchildren all live in California and so we get to see each other reasonably often.
I was naturalized as an American citizen in 1957 and this has been my primary self-identity ever since. But, like many other scientists, I also have a strong sense of global citizenship, including especially Canada, Denmark, England, France and Israel, where I have worked and lived with a family for considerable periods, and where I have some of my closest friends.
My feelings towards Austria, my native land, are – and will remain – very painful. They are dominated by my vivid recollections of 1 1/2 years as a Jewish boy under the Austrian Nazi regime, and by the subsequent murder of my parents, Salomon and Gittel Kohn, of other relatives and several teachers, during the holocaust. At the same time I have in recent years been glad to work with Austrians, one or two generations younger than I: Physicists, some teachers at my former High School and young people (Gedenkdiener) who face the dark years of Austria's past honestly and constructively.
On another level, I want to mention that I have a strong Jewish identity and – over the years – have been involved in several Jewish projects, such as the establishment of a strong program of Judaic Studies at the University of California in San Diego.
My father, who had lost a brother, fighting on the Austrian side in World War I, was a committed pacifist. However, while the Nazi barbarians and their collaborators threatened the entire world, I could not accept his philosophy and, after several earlier attempts, was finally accepted into the Canadian Infantry Corps during the last year of World War II. Many decades later I became active in attempts to bring an end to the US-Soviet nuclear arms race and became a leader of unsuccessful faculty initiatives to terminate the role of the University of California as manager of the nuclear weapons laboratories at Los Alamos and Livermore. I offered early support to Jeffrey Leiffer, the founder of the student Pugwash movement which concerns itself with global issues having a strong scientific component and in which scientists can play a useful role. Twenty years after its founding this organization continues strong and vibrant. My commitment to a humane and peaceful world continues to this day. I have just joined the Board of the Population Institute because I am convinced that early stabilization of the world's population is important for the attainment of this objective.
After these introductory general reflections from my present vantage point I would now like to give an idea of my childhood and adolescence. I was born in 1923 into a middle class Jewish family in Vienna, a few years after the end of World War I, which was disastrous from the Austrian point of view. Both my parents were born in parts of the former Austro-Hungarian Empire, my father in Hodonin, Moravia, my mother in Brody, then in Galicia, Poland, now in the Ukraine. Later they both moved to the capital of Vienna along with their parents. I have no recollection of my father's parents, who died relatively young. My maternal grandparents Rappaport were orthodox Jews who lived a simple life of retirement and, in the case of my grandfather, of prayer and the study of religious texts in a small nearby synagogue, a Schul as it was called. My father carried on a business, Postkartenverlag Brueder Kohn Wien I, whose main product was high quality art postcards, mostly based on paintings by contemporary artists which were commissioned by his firm. The business had flourished in the first two decades of the century but then, in part due to the death of his brother Adolf in World War I, to the dismantlement of the Austrian monarchy and to a worldwide economic depression, it gradually fell on hard times in the 1920s and 1930s. My father struggled from crisis to crisis to keep the business going and to support the family. Left over from the prosperous times was a wonderful summer property in Heringsdorf at the Baltic Sea, not far from Berlin, where my mother, sister and I spent our summer vacations until Hitler came to power in Germany in 1933. My father came for occasional visits (the firm had a branch in Berlin). My mother was a highly educated woman with a good knowledge of German, Latin, Polish and French and some acquaintance with Greek, Hebrew and English. I believe that she had completed an academically oriented High School in Galicia. Through her parents we maintained contact with traditional Judaism. At the same time my parents, especially my father, also were a part of the secular artistic and intellectual life of Vienna.
After I had completed a public elementary school, my mother enrolled me in the Akademische Gymnasium, a fine public high school in Vienna's inner city. There, for almost five years, I received an excellent education, strongly oriented toward Latin and Greek, until March 1938, when Hitler Germany annexed Austria. (This so-called Anschluss was, after a few weeks, supported by the great majority of the Austrian population). Until that time my favorite subject had been Latin, whose architecture and succinctness I loved. By contrast, I had no interest in, nor apparent talent for, mathematics which was routinely taught and gave me the only C in high school. During this time it was my tacit understanding that I would eventually be asked to take over the family business, a prospect which I faced with resignation and without the least enthusiasm.
The Anschluss changed everything: The family business was confiscated but my father was required to continue its management without any compensation; my sister managed to emigrate rather promptly to England; and I was expelled from my school.
In the following fall I was able to enter a Jewish school, the Chajes Gymnasium, where I had two extraordinary teachers: In physics, Dr. Emil Nohel, and in mathematics Dr. Victor Sabbath. While outside the school walls arbitrary acts of persecution and brutality took place, on the inside these two inspired teachers conveyed to us their own deep understanding and love of their subjects. I take this occasion to record my profound gratitude for their inspiration to which I owe my initial interest in science. (Alas, they both became victims of Nazi barbarism).
I note with deep gratitude that twice, during the Second World War, after having been separated from my parents who were unable to leave Austria, I was taken into the homes of two wonderful families who had never seen me before: Charles and Eva Hauff in Sussex, England, who also welcomed my older sister, Minna. Charles, like my father, was in art publishing and they had a business relationship. A few years later, Dr. Bruno Mendel and his wife Hertha of Toronto, Canada, took me and my friend Joseph Eisinger into their family. (They also supported three other young Nazi refugees). Both of these families strongly encouraged me in my studies, the Hauffs at the East Grinstead County School in Sussex and the Mendels at the University of Toronto. I cannot imagine how I might have become a scientist without their help.
My first wife, Lois Kohn, gave me invaluable support during the early phases of my scientific career; my present wife of over 20 years, Mara, has supported me in the latter phases of my scientific life. She also created a wonderful home for us, and gave me an entire new family, including her father Vishniac, a biologist as well as a noted photographer of pre-war Jewish communities in Eastern Europe, and her mother Luta. (They both died rather recently, well into their nineties).
After these rather personal reminiscences I now turn to a brief description of my life as a scientist.
When I arrived in England in August 1939, three weeks before the outbreak of World War II, I had my mind set on becoming a farmer (I had seen too many unemployed intellectuals during the 1930s), and I started out on a training arm in Kent. However, I became seriously ill and physically weak with meningitis, and so in January 1940 my "acting parents", the Hauffs, arranged for me to attend the above-mentioned county school, where – after a period of uncertainty – I concentrated on mathematics, physics and chemistry.
However, in May 1940, shortly after I had turned 17, and while the German army swept through Western Europe and Britain girded for a possible German air-assault, Churchill ordered most male "enemy aliens" (i.e., holders of enemy passports, like myself) to be interned ("Collar the lot" was his crisp order). I spent about two months in various British camps, including the Isle of Man, where my school sent me the books I needed to study. There I also audited, with little comprehension, some lectures on mathematics and physics, offered by mature interned scientists.
In July 1940, I was shipped on, as part of a British convoy moving through U-boat-infested waters, to Quebec City in Canada; and from there, by train, to a camp in Trois Rivieres, which housed both German civilian internees and refugees like myself. Again various internee-taught courses were offered. The one which interested me most was a course on set-theory given by the mathematician Dr. Fritz Rothberger and attended by two students. Dr. Rothberger, from Vienna, a most kind and unassuming man, had been an advanced private scholar in Cambridge, England, when the internment order was issued. His love for the intrinsic depth and beauty of mathematics was gradually absorbed by his students.
Later I was moved around among various other camps in Quebec and New Brunswick. Another fellow internee, Dr. A. Heckscher, an art historian, organized a fine camp school for young people like myself, whose education had been interrupted and who prepared to take official Canadian High School exams. In this way I passed the McGill University junior Matriculation exam and exams in mathematics, physics and chemistry on the senior matriculation level. At this point, at age 18, I was pretty firmly looking forward to a career in physics, with a strong secondary interest in mathematics.
I mention with gratitude that camp educational programs received support from the Canadian Red Cross and Jewish Canadian philanthropic sources. I also mention that in most camps we had the opportunity to work as lumberjacks and earn 20 cents per day. With this princely sum, carefully saved up, I was able to buy Hardy's Pure Mathematics and Slater's Chemical Physics, books which are still on my shelves. In January 1942, having been cleared by Scotland Yard of being a potential spy, I was released from internment and welcomed by the family of Professor Bruno Mendel in Toronto. At this point I planned to take up engineering rather than physics, in order to be able to support my parents after the war. The Mendels introduced me to Professor Leopold Infeld who had come to Toronto after several years with Einstein. Infeld, after talking with me (in a kind of drawing room oral exam), concluded that my real love was physics and advised me to major in an excellent, very stiff program, then called mathematics and physics, at the University of Toronto. He argued that this program would enable me to earn a decent living at least as well as an engineering program.
However, because of my now German nationality, I was not allowed into the chemistry building, where war work was in progress, and hence I could not enroll in any chemistry courses. (In fact, the last time I attended a chemistry class was in my English school at the age of 17.) Since chemistry was required, this seemed to sink any hope of enrolling. Here I express my deep appreciation to Dean and head of mathematics, Samuel Beatty, who helped me, and several others, nevertheless to enter mathematics and physics as special students, whose status was regularized one or two years later.
I was fortunate to find an extraordinary mathematics and applied mathematics program in Toronto. Luminous members whom I recall with special vividness were the algebraist Richard Brauer, the non-Euclidean geometer, H.S.M. Coxeter, the aforementioned Leopold Infeld, and the classical applied mathematicians John Lighton Synge and Alexander Weinstein. This group had been largely assembled by Dean Beatty. In those years the University of Toronto team of mathematics students, competing with teams from the leading North-American Institutions, consistently won the annual Putman competition. (For the record I remark that I never participated). Physics too had many distinguished faculty members, largely recruited by John C. McLennan, one of the earliest low temperature physicists, who had died before I arrived. They included the Raman specialist H.L. Welsh, M.F. Crawford in optics and the low-temperature physicists H.G. Smith and A.D. Misener. Among my fellow students was Arthur Schawlow, who later was to share the Nobel Prize for the development of the laser.
During one or two summers, as well as part-time during the school year, I worked for a small Canadian company which developed electrical instruments for military planes. A little later I spent two summers, working for a geophysicist, looking for (and finding!) gold deposits in northern Ontario and Quebec.
After my junior year I joined the Canadian Army. An excellent upper division course in mechanics by A. Weinstein had introduced me to the dynamics of tops and gyroscopes. While in the army I used my spare time to develop new strict bounds on the precession of heavy, symmetrical tops. This paper, "Contour Integration in the Theory of the Spherical Pendulum and the Heavy Symmetrical Top" was published in the Transactions of American Mathematical Society. At the end of one year's army service, having completed only 2 1/2 out of the 4-year undergraduate program, I received a war-time bachelor's degree "on – active – service" in applied mathematics.
In the year 1945-6, after my discharge from the army, I took an excellent crash master's program, including some of the senior courses which I had missed, graduate courses, a master's thesis consisting of my paper on tops and a paper on scaling of atomic wave-functions.
My teachers wisely insisted that I do not stay on in Toronto for a Ph.D, but financial support for further study was very hard to come by. Eventually I was thrilled to receive a fine Lehman fellowship at Harvard. Leopold Infeld recommended that I should try to be accepted by Julian Schwinger, whom he knew and who, still in his 20s, was already one of the most exciting theoretical physicists in the world.
Arriving from the relatively isolated University of Toronto and finding myself at the illustrious Harvard, where many faculty and graduate students had just come back from doing brilliant war-related work at Los Alamos, the MIT Radiation Laboratory, etc., I felt very insecure and set as my goal survival for at least one year. The Department Chair, J.H. Van Vleck, was very kind and referred to me as the Toronto-Kohn to distinguish me from another person who, I gathered, had caused some trouble. Once Van Vleck told me of an idea in the band-theory of solids, later known as the quantum defect method, and asked me if I would like to work on it. I asked for time to consider. When I returned a few days later, without in the least grasping his idea, I thanked him for the opportunity but explained that, while I did not yet know in what subfield of physics I wanted to do my thesis, I was sure it would not be in solid state physics. This problem then became the thesis of Thomas Kuhn, (later a renowned philosopher of science), and was further developed by myself and others. In spite of my original disconnect with Van Vleck, solid state physics soon became the center of my professional life and Van Vleck and I became lifelong friends.
After my encounter with Van Vleck I presented myself to Julian Schwinger requesting to be accepted as one of his thesis students. His evident brilliance as a researcher and as a lecturer in advanced graduate courses (such as waveguides and nuclear physics) attracted large numbers of students, including many who had returned to their studies after spending "time out" on various war-related projects.
I told Schwinger briefly of my very modest efforts using variational principles. He himself had developed brilliant new Green's function variational principles during the war for wave-guides, optics and nuclear physics (Soon afterwards Green's functions played an important role in his Nobel-Prize-winning work on quantum electrodynamics). He accepted me within minutes as one of his approximately 10 thesis students. He suggested that I should try to develop a Green's function variational method for three-body scattering problems, like low-energy neutron-deuteron scattering, while warning me ominously, that he himself had tried and failed. Some six months later, when I had obtained some partial, very unsatisfactory results, I looked for alternative approaches and soon found a rather elementary formulation, later known as Kohn's variational principle for scattering, and useful for nuclear, atomic and molecular problems. Since I had circumvented Schwinger's beloved Green's functions, I felt that he was very disappointed. Nevertheless he accepted this work as my thesis in 1948. (Much later L. Fadeev offered his celebrated solution of the three-body scattering problem).
My Harvard friends, close and not so close, included P.W. Anderson, N. Bloembergen, H. Broida (a little later), K. Case, F. De Hoffman, J. Eisenstein, R. Glauber, T. Kuhn, R. Landauer, B. Mottelson, G. Pake, F. Rohrlich, and C. Slichter. Schwinger's brilliant lectures on nuclear physics also attracted many students and Postdocs from MIT, including J. Blatt, M. Goldberger, and J.M. Luttinger. Quite a number of this remarkable group would become lifelong friends, and one – J.M. "Quin" Luttinger – also my closest collaborators for 13 years, 1954-66. Almost all went on to outstanding careers of one sort or another.
I was totally surprised and thrilled when in the spring of 1948 Schwinger offered to keep me at Harvard for up to three years. I had the choice of being a regular post-doctoral fellow or dividing my time equally between research and teaching. Wisely – as it turned out – I chose the latter. For the next two years I shared an office with Sidney Borowitz, later Chancellor of New York University, who had a similar appointment. We were to assist Schwinger in his work on quantum electrodynamics and the emerging field theory of strong interactions between nucleons and mesons. In view of Schwinger's deep physical insights and celebrated mathematical power, I soon felt almost completely useless. Borowitz and I did make some very minor contributions, while the greats, especially Schwinger and Feynman, seemed to be on their way to unplumbed, perhaps ultimate depths.
For the summer of 1949, I got a job in the Polaroid laboratory in Cambridge, Mass., just before the Polaroid camera made its public appearance. My task was to bring some understanding to the mechanism by which charged particles falling on a photographic plate lead to a photographic image. (This technique had just been introduced to study cosmic rays). I therefore needed to learn something about solid state physics and occasionally, when I encountered things I didn't understand, I consulted Van Vleck.
It seems that these meetings gave him the erroneous impression that I knew something about the subject. For one day he explained to me that he was about to take a leave of absence and, "since you are familiar with solid state physics", he asked me if I could teach a course on this subject, which he had planned to offer. This time, frustrated with my work on quantum field theory, I agreed. I had a family, jobs were scarce, and I thought that broadening my competence into a new, more practical, area might give me more opportunities.