PER-OLOV LÖWDIN CONQUEROR OF SCIENTIFIC, EDUCATIONAL AND ROCKY MOUNTAINS - 1976 Authored by : Kimio Ohno Department of Chemistry, Faculty of Science, Hokkaido University, Sapporo 060, Japan

Four years later, in the summer of 1957, I was working with Roy McWeeny at Keele. Professor Daudel organized on international colloquium on quantum chemistry in Paris in October of that year. I took advaxitage of attending this colloquium by extending the trip to Uppsala in late September. The quantum chemistry group was busy in preparation for its move from Kemikum to a new building at Rundelsgränd 2A, but Per took very good care of us. I had been able to meet Professor Kotani at the Stockholm station, and we spent five nights in the group's apartment at Luthagesplanaden. The weather was miserable and it even sleeted on one day. Nevertheless I enjoyed my first visit to Uppsala immensely. The group members I remember having met on this occasion include Anders Fröman, Klaus and Marianne Appel and Jean-Louis Calais. Although tickets were bought quite independently in England and Sweden, Per and I were in the same compartment in the continental train from Copenhagen to Paris. We had plenty of time for discussion and when we arrived at Paris, I must confess that I was quite exhausted from the brain labour to which I had been subjected in the train.

The next summer, on my way back to Japan, I visited the Uppsala Quantum Chemistry Group again, at Vålådalen this time. Natural surroundings were beautiful and although I stayed only a week I was greatly impressed by the vigorous scientific activities and warm international friendship of the first, now farnous, Summer School.

With this background, it was no wonder that I gave a most enthusiastic "yes" when Professor Kotani passed on to me Per's invitation to Uppsala in May, 1961. The invitation was the oppor tunity for me to serve as the Assistant Director of the academic year 1961-1962. lt was indeed a great honour for me to follow the excellent series of Harrison Shull, Ruben Pauncz and Roy McWeeny.

I stayed in Uppsala for a little more than two years and moved to Gainesville to spend a year with the Quantum Theory Project. Through the long association with Per, what I learned from him was certainly not limited to science. Mutual understanding of people of different nations is absolutely necessary for peaceful co-existence of different nations and different ideologies. The efforts of Per to this direction is clear - he never gets tired of organizing Summer Schools and Winter Institutes - thus he is not only spreding quantum chemistry among scientists but is also making a definite contribution towards true peace in the world.

To me Per Löwdin is an inspired, great teacher and a close, personal friend. However, in what follows, I try to be impersonal and to present a review of what he has achieved from a scientists point of view.



I. A BRIEF BIOGRAPHY


Per-Olov Löwdin was born on October 28, 1916, at Uppsala where he also attended school and in 1935 started his studies in natural sciences at the university. He obtained the Swedish degree of Filosofie licentiat in 1942 and the same year he became a lecturer in Mechanics and Mathematical Physics at Uppsala University. During a six months' visit to Pauli's group in l946 he studied some problems in quantum electrodynamics, a field appropriate to his keen interest in mathematics. The influence of Pauli's crystal clear thinking is perceivable in Lowdin's later work, writing and teaching. In 1948 he presented his doctoral thesis devoted to quite a different subject, namely a theoretical investigation of ionic crystals. He was appointed Docent at the Uppsala University and the following year he spent five months with N. F. Mott in England continuing his solid-state research.

In 1950-51 he worked in USA together with Hertha Sponer at Duke University, Robert S. Mulliken at Chicago and John C. Slater at MIT and he attended the 1951 Shelter Island Conference. This visit to USA seems to have greatly influenced Lowdin's research interest. With a vigour and capability seldom found in a single person he was instrumental in establishing quantum chemical research in Sweden.

In 1955 he organized together with Inga Fischer-Hjalmars the Stockholm-Uppsala symposium and also launched the Quantum Chemistry Group with himself as a captain. During 1955 to 1960 he held a Laboratorship at the Swedish Natural Sciences Research Council and on May 13, 1960, he was appointed the first professor of quantum chemistry at the Uppsala University. This formal recognition of quantum chomistry in the Swedish University system represents a milestone in Lowdin's effort in strengthening this field of research. Not content with this achievement he found an outlet for his never-failing enthusiasm and energy in activities on the international level. He has been and still is the organizer and driving force in numerous Summer and Winter Institutes, he serves on the editorial board of several international journals, and by extensive travelling he has made the whole world his lecture room.

In 1960 he started the Quantum Theory Project at the University of Florida where he is Graduate Research Professor of Physics and Chemistry. He became a member of the Royal Swedish Academy of Sciences in 1969 and now serves on the Academy's Nobel Committee for physics.

For an active member of the international science community a brief note like this one has to be incomplete. But after all, the important aspects of a scientist's work are his contributions with scientific ideas, methode and concepts leading to an increased knowledge of nature. This is the thema of the following sections.



II. CRYSTAL PHYSICS


The first major scientific papers by Lowdin contain an investigation of the ionic crystals (2, 3, 4). This topic was suggested by his teacher Waller, who apparently had in his mind an analysis of the failure of the Cauchy relations and a calculation of the lattice dynamics. The main emphasis of Löwdin's work, however, turned out to be on cohesion and high pressure properties. The calculation was big and difficult. A number of mathematical and numerical techniques, particularly in connection with the overlap and non-orthogonality problem, had to be developed. In addition, he stood in need of the atomic orbitals and a great number of molecular integrals - overlap, kinetic, Coulomb, exchange, hybrid, and many-center - defined over these atomic orbitals. The numerical work constituted a feat. lt was carried out on primitive, electric FACIT desk-calculators by a group of students. Some of them later became professors. When Mulliken visited Uppsala and was introduced to the students working in the basement at Trädgårdsgatan, his opening remark was "So this is the famous student computer". Learning to appreciate proper nummerical techniques is a good training for scientists-to-be. Löwdin handled this part of his work with elegance and efficiency and he cultivated a lasting interest in practicai calculations.

The calculation on ionic crystals, a landmark in ab-initio calculations, was carried out before the advent of electronic computers and it is, indeed, the first successful ab-initio calculation on crystals. The following quotations may suffice to show how the calculations were evaluated. Kittel wrote in his "Introduction to Solid State Physics" (1^st ed., 1953), that "The most basic quantum-mechanical discussion of ionic crystals has been made by Löwdin". Similarly in "Perspectives in Material Research" (1963) the committee chaired by Brooks stated: "The most satisfactory cohesive energy calculation is probably that carried out for the alkali halides by Löwdin". When writing on the development of solid state physics (Intern. J. Quantum Chem. 1, 31 (1961)), Slater mentioned the stimulus resulting from the invention in 1948 of the transistor and also Löwdin's thesis work which "gave physicists a new hope that same of the problems that seemed too hard before the war might be capable of being handled". These three comments are certainiy very unusual appraisals of a doctoral thesis.

After the work on ionic crystals,the general energy formuia for the graund state of molecules and crystals in the MO-LCAO scheme was further developed (14). He then embarked on the adventure af applying this technique to metallic sodium (15). This required handling of much larger overlap than the one in ionic crystals. The calculation was made at the restricted Hartree-Fock level and it gave a good result for the cohesive energy. lt was clear that there was some cancellation af errors since the correlation energies of the crystal and the free atoms would not cancel each other for the alkali metals. This interested him in the correlation problem in general which became one of his main research interests.These kinds of ab-initio calculations were certainly very time-consuming and tedious. He always insists on a clear connection with the Schrödinger equation, and he dislikes to neglect complicated terms by assuming,without check, that they are not important. These are among his outstanding scientific characteristics.

In 1956 Löwdin published an often quoted review paper on the quantum theory of cohesive prpperties of solids (33). Many of the ingenious techniques developed earlier then became available to a wider public. The readers were also confronted with projection operators and density matrices, two of the main fields of Löwdin's research interests at that time. During later years he published papers on cohesive properties (59, 89, 119) and on correlation and exchange phenomena in solids (56, 57, 116, 117), but these papers do not appear to have had such a great impact as the early work on ionic crystals.



III. EARLY WORK ON MOLECULES


Löwdin treated crystals as large molecules, therefore his method was directly applicable to molecules. He developed a method of fitting the numerical Hartree Fock atomic orbitals with analytic functions (20, 21, 37). This was a continuation of Slater's pioneering study and was followed by works of Watson, Roothaan, Clementi and others. Molecular integrals were evaluated by expanding the atomic orbitals at various centers in spherical harmonics around one and the same center (2,3,4,10,25,33). The same basic idea was applied to integrals defined over Slater-type orbitals by Barnett and Coulson.

Among his work related to molecules, the Löwdin orthogonalization (2, 3, 4, 9, 33) has probably the greatest influence on the present day molecular calculations. By using the orthogonal atomic orbitals obtained by his transformation, he showed that there is no "non-orthogonality catastrophe", although the overlap integrals are of essential importance (19). These orthogonal orbitals are important in the Pariser-Parr-Pople, CNDO, INDO, and many other NDO schemes. One of the basic assumptions in these schemes is the neglect of differential overlap. This seemingly crude procedure can be justified, to some extent at least, by assuming that the basic orbitals are Löwdin orthogonalized A0's. Löwdin pointed this out in a letter to Parr dated December 3rd, 1952 (cf also 30). Correlation effects are important also in molecules. At the Nikko Symposium on Molecular Physics in 1953, he proposed a simple, intuitive but ingenious method to deal with the effect (24 see also 28, 29, 33, 44).

This is the alternant-molecular-orbitals (AMO) method. In on alternant system, the electrons having different spins may accumulate on different subsystems and their separation may be regulated by one or more parameters to be determined by the variational principle. This is an example of using different orbitals for electrons with different spins. The AMO method has been successfully applied to conjugated systems by Löwdin himself (60, 61), Itoh, Yoshizumi, Pauncz, de Heer and others, and to the alkali metals by Calais and collaborators. At the same time, the AM0 method opened the door to a number of important theoretical questions. A single Slater determinant with different orbitals for different spins usually contains a set of different spin multiplets. The question No 1 is what are the weights of these multiplets. The question No 2 is what we should do if we want to recover a pure spin multiplici ty. These questions were duely solved mainly by Löwdin (28, )42), and his collaborators.




IV. UPPSALA QUANTUM CHEMISTRY GROUP AND QUANTUM THEQRY PROJECT IN GAINESVILLE


In 1955, the Uppsala Quantum Chemistry Group was founded by Löwdin. In the beginning the Group had its office in Kemikum overlooking the English Park with the playing squirrels. In autumn of 1951 the Group moved to the present modern building at Rundelsgränd. The international character of the Group was distinct from the beginning. Among the first members, we find Roberto Fieschi and Hiroyuki Yoshizumi in addition to the Swedes; Anders Fröman, Klaus and Marianne Appel. Fröman recalls the following episode: One day at a formal lunch at Flustret a university official commented that he found it most remarkable that on Italian could work in Sweden as a Texas-Swedish Cultural Foundation Fellow.

By that time Löwdin had done extensive travelling abroad. He attended a large number of international conferences held in Europe, U.S.A. and Japan. He was bringing back ideas and news as well as distinguished scientists to Uppsala. When the first Summer School was held at Vålådalen in 1958, Pauling, Mulliken, Matsen, Preuss, Jansen, Shull, Hall, and McWeeny were among the visitors and guest lecturers.

The Summer Schools, which have been held almost every year since 1958, are now well-known by their heavy schedule - both intellectual and physical - and the warm international atmosphere. Mouatain climbing has been on important ingredient of the school's physical activities in recent years.

The Quantum Theory Project was established in the winter of 1960 at the University of Florida. At the very beginning a large part of the staff consisted of visiting scientists from the Uppsala Quantum Chemistry Group. The Project has developed rapidly and it has now four professors, four associate professors and one visiting associate professor.

Corresponding to the Summer Schools, the Winter Institutes have been held each winter since 1961. international Syinposia in Atomic, Molecular, Solid-State Theories have also been organized following the Institute at the Sanibel Island. The Symposium in 1963 was dedicated to E.A. Hyllerass for his pioneering work on the atomic and molecular theory. Since then R.S. Mulliken (1965), J.C. Slater (1967), H. Eyring (1969), J.H. Van Vleck (1971), E.U. Condon (l973), and L.H. Thomas (l975) were honored in the odd year symposia.

Counting both Summer Schools, Winter Institutes and Sanibel Symposia, there must be between 4000 and 5000 scientists from 35-40 different countries all over the world, who have participated in these activities. There have been students of students of students and so on, participating so that academically fifth or sixth generation students are now being sent to these institutes. lt is clear that these schools and symposia, which were organized by Per-Olov Löwdin, have had great impact on quantum chemistry and related fields. By bringing together people from all over the world, these meetings also encouraged the mutual understanding and generosity for the building of a better world which is a deep-rooted concern of Löwdin.



Va. QUANTUM THEORY OF MANY PARTICLE SYSTEMS


Löwdin published a series of three papers on the subject in the Physical Review in 1955 (26, 27, 28). In these papers a number of important concepts and methods concerning the many body problem was proposed and discussed. First of all the use of reduced density matrices in analysing general wave functions was advocated. Although there had been several authors who had explicitly or implicitly used these concepts, he and McWeeny were the first quantum chemists who emphasized their use and potentialities in the quantum theory of many body problems at absolute zero temperature. Their observation that the energy can be determined from the reduced 2-matrix alone evoked a long series of sophisticated researches to obtain 2-matrices directly without going through wave functions. These include the N-representability problem and general mathematical structure of 1- and 2-matrices.

Natural spin orbitals are a genuine Löwdin concept. They are defined as the orbital bases in which the 1-matrix has a diagonal form. It is argued that they lead to a configuration interaction expansion of most rapid convergence. While the knowledge of the wave function is required, the general shape and occupation number characteristics of the natural orbitals derived from different approximate wave functions enable us to make a meaningful comparison which otherwise would elude us. The earliest example of this are found in Löwdin and Shull's papers on He and H₂ (32, 34, 35, 46). The natural spin orbitals thus perform a function very similar to that the charge and bond order matrix serves in a LCAO-M0 calculation. An iterative scheme using the natural spin orbitals are also widely used in numerical computations to speed up the convergence of CI expansions. There is again a great number of papers concerning properties of natural spin orbitals and natural orbitals. Their symmetry properties and various modifications are discussed and utilized in a wide range of the theory of many electron systems.

In the series of Phys. Rev. papers, Löwdin proposed two extensions of the restricted Hartree-Fock (RHF) approximation. One is the extended Hartree-Fock method änd the other is the projected Hartree-Fock method. The fomner is a precursor of the multi-conriguration SCF method and he derived the equation, which determines the optimal orbitals, in terms of the density matrix. The latter started from on optimal single determinant, not RIfF, which in general does not have syrrimetry. The symmetry properties were reinstated by applying appropriate projection operators. The projected HF method goes beyond the RHF limit but it preserves the physical simplicity and visuality of RHF. Löwdin insisted that we should not look at a projeoted determinant as a sum of deterininants, but as an entity, just as we think of the determinant as an entity and not as a sum of products. The AMO method which was mentioned in section II is an example of the projected HF method.

Projection operators are one of the most used tods in his work. The use is made in two ways. One way is its use in actual calculations such as in the construction of eigenfunctions of spin and space angular manenta (42, 50, 78) and for the construction of symmetry functions for finite groups (62, 92). The other way is more abstract. That is on introduction of the concept "projection on a manifold", which was used extensively in the development of the perturbation theory and the theory of bounds (cf section VI).

It was already mentioned that one important factor which was missing in his Na metal calculation was correlation between electrons and that the correlation problem became a center of his research interests and is probably so even now. He wrote a treatise on the subject which was published in 1959 (44). In this article he proposed the definition of the correlation energy E, that is



        
                       Ecorr = Eexact - ERHF
 


where E_exact is the non-relativistic exact energy of the system and E_RHF obtained by the RHF approximation. The correlation energy has now become one of the most important concepts in quantum chemistry. It has a tremendous practical value too. Being combined with Nesbet and Sinanoglu's idea of the pair correlation energy, it can give a fair estimate of E_exact for most systems from the RHF calculation.

The Virial theorem is a unique relation in that it holds rigorously for the exact wave function and also for any scaled approximate wave function. He has emphasized its importance and has tried to make every use of it (45, 47, 59, 111, 116, 117).



Vb. PERTURBATION THEORY


There is another lang series of papers by Löwdin entitled "Studies in Perturbation Theory" (65, 73, 74, 63, 64, 75, 76, 77, 80, 81, 82, 87, 99 and 115). The starting point of the whole approach is older and is found in a paper published in 1951 (13), in which the solution of the eigenvalue problem by means of "partitioning" was proposed. The technique has been reformulated in terms of two associated variables e and e₁ defining a "bracketing function" e₁ = f(e). This has the property that, between the two real numbers e and e₁ ‚ there is at least one eigenvalue E. The bracketing theorem alters successful methods for solving eigenvalue problems and also constitutes the basis for bounding the eigenvalues. The very significant concept of a reduced resolvent was introduced. A meticulous study of the properties of the reduced resolvents was made and various approximations for the resolvents were put forward. The strongest tool was the inner projection. It has the form

A' = A^1/2 O A^1/2

where A is a positive definite operator and 0 is a projection operator. It can be shown that the expectation values of A' are smaller or equal to those of A. By appropriate choice of 0, e.g. the projection on a linear manifold, a number of interesting applications to concrete problems were made. For example, a number of numerical calculations of lower bounds were carried out in the Florida Quantum Theory Project and the Uppsala Quantum Chemistry Group.

The perturbation theory has been extended in many directions. Simple and unified treatments of degenerate eigenvalues and linearly dependent reference functions were developed. Bounding of physical properties in terms of others and the reduction of the Dirac equation to the Pauli form were accomplished. Contact with Padé approximant theory and continued fractions was clarified. The exact self-consistent field theory was proposed and the symmetry- adapted perturbation theory was established. The idea of inner- projections has had on impact in the field of propagator theory when discussing decoupling schemes.

These developments are rather formal and may seem too abstract. However, there are singular merits in Löwdin's persistenee in having on exact, closed form, as a basis for physical and also semi-empirical discussions. The elegant and powerful formalism makes it feasible to compare in detail most of the various, often bewildering, perturbation methods in quantum mechanics. Being written in a typically Löwdin style - stringent and transparent reasoning, logical sequence, natural language - this series on perturbation theory is the best guide for anyone who wishes to understand the essentials of modern theory in this field of study.



VI. PROTON TUNNELING IN DNA


Löwdin became interested in biophysics rather suddenly around 1961. Biophysics was an entirely new field for him. With his characteristic energy and concentration, he read through books and papers and listened to lectures and seminars. Two years later in 1963, he proposed the theory of proton tunneling in DNA and then discussed its possible consequences (66, 68, 70, 72, 79, 83, 86, 88, 94, 98, 101, 107). Based upon the Watson-Crick model of DNA, he pointed out that there is a certain intrinsic probability of a proton movement in DNA. Namely the proton in one or more of the hydrogen bonds between a base pair changes its position in time from the most favourable position to the next most favourable position. This spontaneous shift of the position, which is characteristic and inherent to a quantum mechanical particle, transforms a base to its tautomeric form. The tautomeric form can make a pair only with a base different from the normal partner. This would cause on error in the genetic code and accumulated errors of this kind could be responsible for mutation, aging and spontaneous tumors. At the present time, as for as the writer of this article is aware, this interesting hypothesis has been neither proved nor disproved. It remains to be seen. There is no doubt, however, that the proposal is a major philosophical contribution in that it brings together the concept of quantum mechanics and the question of mutation. It is also clear that this bold attempt provided a great stimulus to these scientists interested in quantum biology.



VII. THE SCIENTIST AND THE TEACHER


One of the roots of Löwdin's scientific strength is undoubtedly his mastery of the mathematical "craft". This, in addition to his background and experience, makes it possible to "see through" a paper or a seminar very quickly and to put his finger on a "sore point" if there is any. This also gives him the ability to translate different scientific languages to a unified one. His insistence of finding good notations is closely connected with this. Other roots of his scientific strength, which are perhaps more basic, are his enthusiasm and devotion to research and his enormous working capabity. In the winter of 1963, when he was working on the lower bound er the energy and got the idea of inner-projections, he continued to give two seminars per week for a few months on the same subject. Each seminar contained one or more new steps towards his goal and it was fascinating to watch on evolution of the new theoretical approach to this difficult problem.

In all his scientific work, there is a very obvious desire to present the material in a pedagogical fashion. Per Löwdin is undoubtedly an outstanding scientist but as a teacher he is also uniquely successful. He has an unusual ability to enthusiastically lead and york with younger colleagues and never get tired of teaching his methods and mode of thinking. As as teacher, he is clear and deceivingly simple in his approach, which allows him to present even very sophisticated ideas to beginning students. He likes and always strives for "the one-line proof" and a short and elegant way to a result is made as important as the result itself. Thus comes his insistence on clear notations, simple algebra, and a complete bibliography.

Another activity of his, which should be mentioned, is his extensive editorial commitments. He is the chief editor of the International Journal of Quantum Chemistry and the Advances in Quantum Chemistry. He has also been serving as a member of the advisory Editorial Board of a large number of international journals. He has edited several books including "Molecular Orbitals in Chemistry, Physics, and Biology - Mulliken Dedicatory Volume - " and "Quantum Theory of Atoms, Molecules, and the Solids - Slater Dedicatory Volume - ". Per Löwdin's name is closely associated with quantum chemistry. Quantum chemistry, however, is perhaps a too narrow concept to use in describing his scientiric work, which has had a much wider impact. He is and will be remembered by his colleagues, friends and students as on excellent, original scientist, an inspiring and tireless teacher, and a warm and sympathetic person combined in one.



ACKNOWLEDGEMENT


The article would never have been written without the help from Anders Fröman, Jean-Louis Calais, Jan Linderberg, Yngve Öhrn and Osvaldo Goscinski to whom the author expresses his sincere thanks and great indebtedness. Nikolaj Stepanov has read a part of the manuscript and has offered a number 0±2 useful oorn- ments which are very muoh appreciated. Re stayed with our little group in Sapporo recently and this is an example of many products produced by the catalytic activity of Per Löwdin in the inter- national cooperation.

The following reference list originates from :

Quantum Science - Methods and Structure
A Tribute to Per-Olov Löwdin

Edited by
Jean-Louis Calais and Osvaldo Goscinski,
Jan Linderberg and Yngve Öhrn.
Plenum Press, New York, 1976.

Used here:
pages 13-23: Kimio Ohno(?), Publications of Per-Olov Löwdin, 1939-1976.

 
        1.   Lorentz-Transforinationerna och den Kinematiska Relativitets-
             principen (Elementa 22, 161 - 169, 1939)
         
        2.   A Quantum Mechanical Calculation of the Cohesive Energy, the
             Interionic Distance, and the Elastic Constants of Some Ionic
             Crystals. I. (Ark. Mat., Astr., Fys. 35A,  No 9, 1 - 10, 1947)
         
        3.   A Quantum Mechanical Calculation of the Cohesive Energy, the
             Interionic Distance, and the Elastic Constants of Some Ionic
             Crystals. II. The Elastic Constants c12 and c44 .
             (Ark. Mat., Astr., Fys. 35A, No 30. 1-18, 1948) 12 44
         
        4.   A Theoretical Investigation into Some Properties of Ionic
             Crystals (Thesis, Almqyist & Wiksells, Uppsala, 1948)
         
        5.   A Quantum Mechanical Calculation of the Cohesive Energy, the
             Interionic Distance, and the Elastic Constants of Some Ionic
             Crystals (Reports from the Conference of the Swedish National
             Committee for Physics in 1947. Ark. Mat., Astr., Fys. 34A,
             No 29, 18-19, 1948)
         
        6.   On the Occurrence of Many-Body-Forces in Molecules and in
             Crystals. (Reports from the conference of the Swedish National
             Committee for Physics in 1948. Ark.  Fysik 1, 543, 1949)
         
        7.   The Band Theory of Metals and the Importance of the Overlap
             Integrals. (Reports from the Conference of the Swedish
             National Committee for Physics in 1949.  Ark. Fysik 2, 220,
             1950)
      
        8.   A Note on the Method of Steepest Descents with a Remark on
             T. Ljunggren's Paper "Contributions to the Theory of
             Diffraction of Electromagnetic Waves by Spherical Particles."
             (Ark.  Fysik 2,  367-370, 1950)
         
        9.   On the Non-Orthogonality Problem Connected with the Use of
             Atomic Wave Functions in the Theory of Molecules and Crystals
             (J. Chem.  Phys. 18, 365-375, 1950)
         
        10.  (with S.0. Lundqyist) On the Calculation of Certain Integrals
             Occurring in the Theory of Molecules, Especially Three-Centre
             and Four-Centre Integrals (Ark.  Fysik  3, 147-154, 1951)
         
        11.  (with A Sjölander) A Note on the Numerical Calculation of
             Asymptotic Phases with a Numerical Study of Hulthén's Varia-
             tional Principle (Ark. Fysik 3, 155-166, 1951)
         
        12.  Calculation of Electric Dipole Moments of Some Heterocyclics
             (J. Chem.  Phys. 19, 1323-1324, 1951)
         
        13.  A Note on the Quantum-Mechanical Perturbation Theory. 
             (J. Chem. Phys. 19, 1396-1401, 1951)
         
        14.  On the Quantum-Mechanical Calculation of the Cohesive Energy
             of Molecules and Crystals. I. A General Energy Formula for
             the Ground State. (J. Chem. Phys. 19, 1570-1578, 1951)
         
       15.   On the Quantum-Mechanical Calculation of the Cohesive Energy
             of Molecules and Crystals. II. Treatment of the Alkali Metals
             with Numerical Applications to Sodiu.m. (J. Chem. Phys. 19,
             1579-1591, 1951)
         
       16.   On the Methods of Numerical Integration Used in Determining
             Self-Consistent Fields. (NAS-ONR Report from the Shelter
             Island Conference in 1951, 187-194, 1951)
         
       17.   On The Numerical Integration of Ordinary Differential Equations
             of the First Order. (Quart. Appl. Math. 10, 97-111, 1952)
         
       18.   Approximate Formulas for Many-Center Integrals in the Theory
             of Molecules and Crystals (J. Chem. Phys. 21, 374-375, 1953)
         
       19.    On the Molecular-Orbital Theory of Conjugated Organic Compounds
              with Application to the Perturbed Benzene Ring.
              (J. Chem.  Phys. 21, 496-515, 1953)
         
       20.    Studies of Atomic Self-Consistent Fields. I. Calculation of
              Slater Functions. (Phys. Rev. 90, 120-125, 1953)

       21.    Studies of Atomic Self-Consistent Fields. II. Interpolation
              Problems. (Phys. Rev. 94‚ 1600-1609, 1954)
         
       22.   (with H. Sponer) Les Niveaux d'énergie électronique dans
              l'éthylène (J. Phys. Rad. 15, 607-611, 1954)
         
       23.   Recent Simplifications in the Molecular Orbital Theory of
             Calculating Energy Levels. (Proceedings of the International
             Conference of the Theoretical Physics at Kyoto and Tokyo,
             Japan in 1953, 599-611, 1954)
         
       24.   A Method of Alternant Molecular Orbitals. (Symposium on
             Molecular Physics at Nikko, Japan in 1953, 13-16, 1954)
         
       25.   Calculations of Molecular Integrals in Uppsala. (Symposium
             on Molecular Physics at Nikko, Japan in 1953, 113-117, 1954)
         
       26    Quantum Theory of Many-Particle Systems. 1. Physical Inter-
             pretations by Means of Density Matrices, Natural Spin-Orbit-
             als, and Convergence Problems in the Method of Configurational
             Interaction. (Phys. Rev. 97, 1474-1489, 1955)
         
       27.   Quantum Theory of Many-Particle Systems. II. Study of the
             Ordinary Hartree-Fock Approximation. 
             (Phys. Rev. 97, 1490-1508, 1955)
         
       28.   Quantum Theory of Many-Particle Systems. III. Extension of
             the Hartree-Fock Scheme to Include Degenerate Systems and
             Correlation Effects. (Phys. Rev. 97, 1509-1520, 1955)
         
       29.   An Extension of the Hartree-Fock Method to Include Correla-
             tion Effects (Les Electrons dans les Métaux, Dixième Con-
             férence Solvay, Bruxelles in 1954, 71-88, 1955)
         
       30.   (with I. Fischer-Hjalmars) Report From the Symposium on
             Quantum Theory of Molecules, Stockholm and Uppsala, 1955.
             (Sv. Kem. Tidskrift  67, 365-398, 1955; esp. 369,370,373,375,
             379,380,383)
         
       31.   (with H. Shull) Role of the Continuu.m in Superposition of
             Configurations. (J. Chem. Phys. 23, 1362, 1955)
         
       32.   (with H. Shull) Natural Spin-Orbitals for Helium.
             (J. Chem. Phys. 23, 1565, 1955)
         
       33.   Quanttun Theory of Cohesive Properties of Solids. 
             (Adv. Phys. 5, 1-172, 1956
         
       34.   (with H. Shull) Natural Orbitals in the Quantum Theory of
             Two-EleCtrOn Systems (Phys. Rev. 101, 1730-1739, 1956).
         
       35.   (with H. Shull) Correlation Splitting in Helium-Like Ions
             (J. Chem. Phys. 25‚ 1035-1040, 1956).
         
       36.   Electronic Correlation in the Theory of Molecular Energy Levels.
             (A Report from the Molecular Quantum Mechanics Conference in
             Austin, Texas in 1955, 30; Texas J. Sci. 8, 163, 1956).
         
       37.   (with K. Appel) Studies of Atomie Self-Consistent Fields. III.
             Analytic Wave Functions for the Argon-Like Ions and for the
             First Row of the Transition Metals. 
             (Phys. Rev. 103, 1746-1755, 1956).
         
       38.   Present Situation of Quantum Chemistry.
             (J. Phys. Chem. 61, 55-68, 1957)
         
       39.   Den Kovalenta Kemiska Bindningen i Kvantmekanisk Belysning
             (Elementa 40, 9-24, 1957).
         
       40.   Generalizations of the Hartree-Fock Scheme.
             (Ann. Aead. Reg. Sci. Upsaliensis, 2, 127-135, 1958).
         
       41.   (with H. Shull) Variation Theorem for Excited States.
             (Phys. Rev. 110,  1466-1467, 1958).
         
       42.   Nature des Fonctions de la Mésomérie.
             (Ed. du Centre Nat. Rech. Sei. LXXXII, 23-37, 1958).
         
       43.   (with A.J. Freeman) Quantum Meehanical Kinetic Energy Trans-
             formation (Phys. Rev. 111, 1212-1213, 1958).
         
       44.   Spin Degeneracy Problem .
             (Coll. Int. Centre Nat. Rech. Sci. 82, 23, 1958).
         
       45.   Correlation Problem in Many-Eleetron Quantum Mechanies. I.
             Review of Different Approaches and Discussion of Some Current
             Ideas (Adv. Chem. Phys. 2, 207-322, 1959).
         
       46.   Scaling Problem, Virial Theorem and Connected Relations in
             Quantum Mechanics (J. Mol. Spect. 3,  46-66, 1959).
         
       47.   (with H. Shull) Superposition of Configurations and Natural
             Spin Orbitals. Applications to the He Problem.
             (J. Chem. Phys. 30, 617-626, 1959).
         
       48.   (with J.0. Hirschfelder) The Long-Range Interaction of Two ls
             Hydrogen Atoms Expressed in Terms of Natural Spin Orbitals
             (Mol. Phys. 2, 229-258, 1959).

       49.   (with L. Rédei) Combined Use of the Method of Superposition of
             Configurations and Correlation Factor on the Ground States of
             the Helium-Like Ions (Phys. Rev. 114, 752-757, 1959)
         
       50.   En Iterations-Variationsmetod För Att Lösa Egenvärdesproblem.
             (Nord-SAM, Karlskrona and Lund 1959, 199-209, 1959)
         
       51.   Some Aspects on the Recent Development of the Theory of the
             Electronic Structure of Atoms. (Proceedings of the Robert A.
             Welch Foundation Conferences an Chemical Research. II. Atomic
             Structure, 5-75, 1960)
         
       52.   Expansion Theorems of the Total Wave Function and Extended
             Hartree-Fock Schemes (Rev. Mod. Phys. 32, 328-334, 1960)
         
       53.   Quantum Theory of Electronic Structure of Molecules.
            (Ann. Rev. Phys. Chem. 11, 107-132, 1960)
         
       54.   (with R. Pauncz and J. de Heer) On the Calculation of the
             Inverse of the Overlap Matrix in Cyclic Systems.
             (J. Math. Phys 1,  461-467, 1960)
         
       55.   The Principle of Causality, the Chemical Bond and Modern
             Quantum Chemistry.
             (Ann. Acad. Reg. Sci. Upsaliensis, 5, 63-78, 1961)
         
       56.   Note an the Separability Theorem for Electron Pairs.
             (J. Chem. Phys. 35, 78-81, 1961)
         
       57.   Band Theory, Valence Band and Tight-Binding Calculations.
             (J. Appl. Phys. 33, 251-280, 1962
         
       58.   Exchange, Correlation and Spin Effects in Molecular and Solid-
             State Theory (Rev. Mod. Phys. 34, 80-87, 1962)
         
       59.   (with J.-L. Calais) A Simple Method of Treating Atomic Integrals
             Containing Functions of r12.
             (J. Mol. Spect. 8, 203-211, 1962)
         
       60.   (with A. Fröman) Virial Theorem and Cohesive Energy of Solids,
             Particularly Ionic Crystals (J. Phys. Chem. Sol 23, 75-84,
             1962)
         
       61.   (with R. Pauncz and J. de Heer) Studies an the Alternant
             Molecular Orbital Method. I. General Energy Expression for
             an Alternant System with Closed-Shell Structure.
             (J. Chem. Phys. 36, 2247-2256, 1962)
         
       62.   (with R. Pauncz and J. de Heer) Studies an the Alternant
             Molecular Orbital Method. II. Application to Cyclic Systems.
             J. Chem. Phys. 36, 2257-2265, 1962
        
       63.   The Normal Constants of Motion in Quantum Mechanics Treated
             by Projection Technique. (Rev. Mod. Phys. 34, 520-530, 1962).
         
       64.   Studies in Perturbation Theory. IV. Solution of Eigenvalue
             Problem by Projection Operator Formalism. 
             (J. Math. Phys. 3, 969-982, 1962).
         
       65.   Studies in Perturbation Theory V. Some Aspects on the Exact
             Self-Consistent Field Theory. (J. Math. Phys. 3, 1171-1184,
             1962).
         
       66.   Studies in Perturbation Theory. 1. An Elementary Iteration-
             Variation Procedure for Solving the Schrödinger Equation by
             Partitioning Technique (J. Mol. Spect. 10, 12-33, 1963).
         
       67.   Quantum Genetics (Int. Sci. Tech. 17, 64, 1963).
         
       68.   Wave and Reaction Operators in the Quantum Theory of Many-
             Particle Systems. (Rev. Mod. Phys. 35, 702-708, 1963).
         
       69.   Discussion on the Hartree-Fock Approximation. 
             (Rev. Mod. Phys. 35, 496-498, 1963).
         
       70.   Discussion on Natural Expansions and Properties of the Chemical
             Bond (Rev. Mod. Phys. 35, 629-630, 1963).
         
       71.   Proton Tunnelling in DNA and its Biological Implications. 
             (Rev. Mod. Phys. 35, 724-732, 1963).
         
       72.   Effect of Proton Tunnelling in DNA on Genetic Information and
             Problems of Mutations, Aging, and Tumors. 
             (Bio. Symp. 1, 161-181, 1964).
         
       73.   Some Aspects of Quantum Biology. (Bio. Symp. 1, 293-311, 1964).
         
       74.   Molecular Orbitals in the Exact SCF Theory. (Molecular Orbitals
             in Chemistry, Physics, and. Biology, Mulliken Dedicatory Volume,
             Academic Press, Inc., New York, 37-55, 1964).
         
       75.   Some Aspects on DNA Replication; Incorporation Errors and
             Proton Transfer. (Electronic Aspects of Biochemistry, Academic
             Press, Inc., New York, 167-201, 1964).
         
       76.   Studies in Perturbation Theory. II. Generalization of the
             Brillouin-Wigner Formalism. (J. Mol. Spect. 13, 326-331, 1964).
         
       77.   Studies in Perturbation Theory. III. Solution of the Schrödinger
             Equation Under a Variation of a Parameter. 
             (J. Mol. Spect. 13, 331-337, 1964).
         
       78.   Studies in Perturbation Theory. VI. Contraction of Secular
             Equations.(J. Mol. Spect. 14, 112-118, 1964).
         
       79.   Studies in Perturbation Theory. VII. Localized Perturbation
             (J. Mol. Spect. 14, 119-130, 1964).
         
       80.   Studies in Perturbation Theory. VIII. Separation of the Dirac
             Equation and Study of the Spin-Orbital Coupling and Fermi
             Contact Terms. (J. Mol. Spect. 14, 131-144, 1964).
         
       81.   Angular Momentum Wavefunctions Constructed by Projection
             Operators. (Rev. Mod. Phys. 36, 966-976, 1964).
         
       82.   Datamaskinupprustning på universitetsområdet.
             (ULF, organ för Universitetslärarförbundet, 1, 1965).
         
       83.   Isotope Effect in Tunneling and its influence on Mutation
             Rates. (Mutation Research 2, 218-221, 1965).
         
       84.   Studies in Perturbation Theory. IX. Connection Between Various
             Approaches in the Recent Development. Evaluation of Upper
             Bounds to Energy Eigenvalues in Schrödinger's Perturbation
             Theory. (J. Math. Phys. 6, 1341-1353, 1965).
         
       85.   Studies in Perturbation Theory. X. Lower Bounds to Energy
             Eigenvalues in Perturbation-Theory Ground State.
             (Phys. Rev. 139, A357-A372, 1965).
         
       86.   Studies in Perturbation Theory. XI. Lower Bounds to Energy
             Eigenvalues, Ground State, and Excited States. 
             (J. Chem. Phys. 143,  S175-S185, 1965).
         
       87.   Quantum Genetics and the Aperiodic Solid. Some Aspects on the
             Biological Problems of Heredity, Mutations, Ageing, and Tumors
             in View of the Quantum Theory of the DNA-Molecule.
             (Adv. Quant. Chem. II, 1965).
         
       88.   Arvsanlagen och Deras Förändringar - Ur kvantgenetisk synpunkt.
             (Sv. Naturvetenskap, 1965).
         
       89.   (with J. 0. Hirschfelder) Long-Range Interaction of Two 1s-
             Hydrogen Atoms Expressed in Terms of Natural Spin-Orbitals.
             (Mol. Phys. 9‚ 491-1496, 1965).
         
       90.   Some Recent Developments in the Quantum Theory of Many-Electron
             Systems and the Correlation Problem.
             (Adv. Chem. Phys. 8, 3-4, 1965).
        
       91.   Some Aspects of the Biological Problems of Heredity, Mutations,
             Ageing and Tumours in View of the Quantum Theory of the DNA
             Molecule. (Adv. Chem. Phys. 8, 177-179, 1965)
         
       92.   The Calculation of Upper and Lower Bounds of Energy Eigenvalues
             in Perturbation Theory by Means of Partitioning Techniques.
             (Perturbation Theory and its Application in Quantum Mechanics,
             Ed., C.H. Wilcox, Proceedings of Madison Symposium, 255-294,
             John Wiley and Sons, Inc., 1966)
         
       93.   Quantum Genetics and the Aperiodic Solid. Some Aspects on the
             Biological Problems of Heredity, Mutations, Ageing and Tumours
             in View of the Quantum Theory of the DNA Molecule.
             (Adv. Quant. Chem. 2,  213, 1965)
         
       94.   Comments on Professor John C. Slater's Paper, "Cohesion in
             Monovalent Metals".
             (Quantum Theory of Atoms, Molecules, Solid State, 15, 1966)
         
       95.   The Projected Hartree-Fock Method. An Extension of the
             Independent-Particle Scheme. 
             (Quantum Theory of Atoms, Molecules, and Solid State, 601, 1966)
         
       96.   Program. (Int. J. Quant. Chem. 1, 1-6, 1967)
         
       91.   Nature of Quantum Chemistry.
             (Int. J. Quant. Chem. 1, 7-12, 1967)
         
       98.   Group Algebra, Convolution Algebra, and Applications to Quantum
             Mechanics. (Rev. Mod. Phys. 39‚  259-287, 1967)
         
       99.   Quantumn Theory of Time-Dependent Phenomena Treated by the
             Evolution Operator Technique.
            (Adv. Quant. Chem. 3, 323-381, 1967)
         
       100.  Some Properties of the Hydrogen Bonds in Biochemistry with
             Particular Reference to the Stability of the Genetic Code.
             (Pontificiae Academiae Scientiarum Scripta Varia 31,
             "Semaine d'Etude sur les Forces Moléculaires, 637-708, 1967)
         
       101.  Eigenvalue Problem in a Linearly Dependent Basis and the Super
             Secular-Eq.uation. (Int. J. Quant. Chem. 1S,  811-827, 1967)
         
       102.  Molecular Associations in Biology - 4 Brief Summary. (Molecular
             Associations in Biology, Academic Press, Inc., New York,
             539-549, 1968)
         
       103.  (with P. Lindner) Upper and Lower Bounds in Second-Order
             Perturbation Theory and the Unsöld Approximation.
             (Int. J. Quant. Chem. 2S, 161-173, 1968)

       104.  Some Aspects on the Possible Importance of the Reading
             Mechanism of DNA in Carcinogenesis. (Proceedings of the Israel
             Academy of Sciences Symposium on Physico-Chemical Mechanism
             of Carcinogenesis, Jerusalem in 1968)
         
       105.  Studies in Perturbation Theory. XIII. Treatment of Constants
             of Motion in Resolvent Method, Partitioning Technique, and
             Perturbation Theory. (Int. J. Quant. Chem. 2, 867-931, 1968)
         
       i06.  Some Comments on the Treatment of Symmetry Properties in
             Perturbation Theory. (Int. J. Quant. Chem. 25, 137-l50, 1968)
         
       107.  (with W.M. MacIntyre) Electronic Energy of the DNA Replication
             Plane. (Int. J. Quant. Chem. 2S, 20-217, 1968)
         
       108.  Some Aspects on the Correlation Problem and Possible Extensions
             of the Independent-Particle Model. (Proceedings of Frascati
             Summer School on the Correlation Problem, 1961; Eds. R.
             Lefebvre and C. Moser, Interscience, 1968;
             Adv. Chem. Phys. 14,  283, 1969)
         
       109.  (with M. Berrondo) The Projection Operator for a Space Spanned
             by a Linearly Dependent Set. 
             (Int. J. Quant. Chem. 3, 767-780, 1969)
         
       110.  Some Comments on the Periodic System of the Elements.
             (Int. J. Quant. Chem. 3S, 331-334, 1969)
         
       111.  (with 0. Goscinski) The Exchange Phenomenon, the Symmetric
              Group, and the Spin Degeneracy Problem. 
             (Int. J. Quant. Chem. 3S, 533-591, 1969)
         
       112.  (with T.K. Lim) Calculation of Lower Bounds to Energy Eigen-
              values by Reduced Density Matrices and the Representability
              Problem. (Int. J. Quant. Chem. 3S, 697-702, 1969)
         
       113.  Some Aspects of the Hydrogen Bond in Molecular Biology. (Ann.
             N. Y. Acad. Sci. 158, 86-95, 1969)
         
       114.  Energia Sajátékek Alsóés Felsó Korlátjának Számítása
              Perturbáiós - Zámításban Particionlálási Technikával.
             (Magyar Fizikai Folyóirat XVIII, 515-540, 1969)
         
       115.  On the Non-Orthogonality Problem. 
             (Adv. Quant. Chem. 5, 185-199, 1970).
         
       116.  Same Aspects on the Research Process in the Natural Sciences.
             (Scientists at Work. Festschrift in Honour of Herman Wold,
             Almqvist and Wiksells, Uppsala, 112-135, 1970)

       117.  (with J. Gruninger and Y. Öhrn) Comments on the Analysis of
             Atomic Correlation Energies).
             (J. Chem. Phys. 52,  5551-5554, 1970)
         
       118.  Recent Research in the Uppsala & Florida Quantum Theory
             Projects. (IBM Ludwigsburg Meeting Report, 47-83, 1970)
         
       119.  Theoretical Treatment of Impurities in Solid State Physics
             and Quantum Biology. (Proceedings of the Seminar on
             Impurity Centers in Crystals at Tallinn, USSR in 1970)
         
       120.  Some Properties of Inner Projections.
             (Int. J. Quant. Chem. 4S, 231-237, 1971)
         
       121.  (with 0. Goscinski) Studies in Perturbation Theory. XIV.
             Treatment of Constants of Motion, Degeneracies and Symmetry
             Properties by Means of Multi-Dimensional Partitioning. (Int.
             J. Quant. Chem.  5S,  685-705, 1971)
         
       122.  Treatment of Exchange and Correlation Effects in Crystals.
             An Introduction. (Proc. Wildbad Symposium, 1971)
         
       123.  Quantum Chemistry and Molecular Physics of Solids.
             (Coll. Int. C. Nat. Res. Sci. 197, 207-266, 1971)
         
       124.  (with J.-L. Calais and M.R. Hayns) A Theoretical Study of the
             Behaviour of Solids under High Pressure and the Borelius' Law,
             with Applications to the Alkali Halides. 
             (J. of Non-Metals 1, 63-78, 1972)
         
       125.  (with P.K. Mukherjee) Some Comments on the Time Dependent
             Variation Prineiple. (Chem. Phys. Lett. 14, 1-7, 1972)
         
       126.  (with B. Laskowski) Treatment of Constants of Motion in the
             Variation Principle. Symmetry Properties of Variational Wave
             Functions. (Chem. Phys. Lett. 16, 1-4, 1972)
         
       127.  Electronie Mobility and Transfer of Energy and Momentum as
             Time-Dependent Processes. (Proc. 3rd Conf. From Theoretieal
             Physics to Biology, Versailles 1971, 145-146 (Karger, Basel
             1973))
         
       128.  (with T. Ahlenius and J.-L. Calais) Some Comments on the
             Construetion of an Orthonormal Set of LCAO Basis Functions for
             Crystals. (J. Phys. C: 6, 1896-1908, 1973)
         
       129.  (with B. Laskowski, J.-L. Calais and P.V. Leuven) Electron
             Gas Test for the Alternant Molecular Orbital Method. 
             (J. Phys. C: 6, 2777-2787, 1973).

       130.  Josef - Maria Jauch, In Memoriam.
             (Eur. Phys. News. 5, No. 12, 7, 1974
         
       131.  Människan och hennes psyke i den moderna kvantteorins världs-
             bild (Forskning och Praktik 6, 121-125, 1974)
         
       132.  Some Aspects on the American-Swedish Exchange in Quantum
             Sciences particularly the Uppsala-Florida Exchange Project.
             (Uppsala TN 470, to be published, l976)
         
       133.  Internationella aktiviteter å Kvantkemiska Institutionen
             och Forskargruppen vid Uppsala Universitet. 
             (Universen, Uppsala Universitet informerar 1/76, 1976)
         
       134.  Set Theory and Linear Algebra. 
             (Uppsala TN 472, unpublished lecture notes, 1976)