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

Article and pictures stem from :
Quantum Science  Methods and Structure
A Tribute to PerOlov Löwdin
Edited by
JeanLouis Calais and Osvaldo Goscinski,
Jan Linderberg and Yngve Öhrn.
Plenum Press, New York, 1976.
Used here:
pages 111: Kimio Ohno, PerOlov Löwdin,
Conqueror of Scientific,
Educational, and Rocky Mountains.

FOREWORD
I happened to be the first Japanese scientist to meet Per
Löwdin on Japanese soil. In September 1953, an international
conference on theoretical physics was held in Japan. Professor
Kotani gave me the responsibility of meeting some foreign guests
at the Tokyo airport. Thus on one hot summer day, I met Per as
well as Professor Waller and Professor Mulliken at the airport.
At that time, I did not realize that this was to be the beginning
of a long and happy association with Per and his groups.

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 JeanLouis 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 19611962. 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 coexistence
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
PerOlov 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 solidstate research.
In 195051 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 FischerHjalmars
the StockholmUppsala 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 neverfailing 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 nonorthogonality 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 manycenter  defined over these atomic orbitals. The numerical
work constituted a feat. lt was carried out on primitive, electric
FACIT deskcalculators 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 scientiststobe. 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 abinitio
calculations, was carried out before the advent of electronic
computers and it is, indeed, the first successful abinitio
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 quantummechanical 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 MOLCAO
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 HartreeFock 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 abinitio calculations were certainly very timeconsuming
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 Slatertype 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
"nonorthogonality catastrophe", although the overlap integrals are
of essential importance (19). These orthogonal orbitals are important
in the PariserParrPople, 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 3^{rd}, 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 alternantmolecularorbitals (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 TexasSwedish 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 wellknown 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, SolidState 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
3540 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 PerOlov 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 deeprooted
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
2matrix alone evoked a long series of sophisticated researches to
obtain 2matrices directly without going through wave functions.
These include the Nrepresentability problem and general
mathematical structure of 1 and 2matrices.
Natural spin orbitals are a genuine Löwdin concept. They are
defined as the orbital bases in which the 1matrix 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 H_{2} (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 LCAOM0
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 HartreeFock (RHF) approximation.
One is the extended HartreeFock method änd the other is the
projected HartreeFock method. The fomner is a precursor of the
multiconriguration 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
E_{corr} = E_{exact}  E_{RHF}
where E_{exact} is the nonrelativistic 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
e_{1} defining a "bracketing
function"
e_{1} = f(e).
This has the property that, between the two
real numbers e and
e_{1} ‚
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 selfconsistent 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
semiempirical 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 WatsonCrick 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 innerprojections,
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 oneline 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, JeanLouis 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.
Publications of PerOlov Löwdin
19391976
The following reference list originates from :
Quantum Science  Methods and Structure
A Tribute to PerOlov Löwdin
Edited by
JeanLouis Calais and Osvaldo Goscinski,
Jan Linderberg and Yngve Öhrn.
Plenum Press, New York, 1976.
Used here:
pages 1323: Kimio Ohno(?), Publications of PerOlov Löwdin,
19391976.
1. LorentzTransforinationerna 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 c_{12} and c_{44} .
(Ark. Mat., Astr., Fys. 35A, No 30. 118, 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, 1819, 1948)
6. On the Occurrence of ManyBodyForces 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, 367370, 1950)
9. On the NonOrthogonality Problem Connected with the Use of
Atomic Wave Functions in the Theory of Molecules and Crystals
(J. Chem. Phys. 18, 365375, 1950)
10. (with S.0. Lundqyist) On the Calculation of Certain Integrals
Occurring in the Theory of Molecules, Especially ThreeCentre
and FourCentre Integrals (Ark. Fysik 3, 147154, 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, 155166, 1951)
12. Calculation of Electric Dipole Moments of Some Heterocyclics
(J. Chem. Phys. 19, 13231324, 1951)
13. A Note on the QuantumMechanical Perturbation Theory.
(J. Chem. Phys. 19, 13961401, 1951)
14. On the QuantumMechanical Calculation of the Cohesive Energy
of Molecules and Crystals. I. A General Energy Formula for
the Ground State. (J. Chem. Phys. 19, 15701578, 1951)
15. On the QuantumMechanical 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,
15791591, 1951)
16. On the Methods of Numerical Integration Used in Determining
SelfConsistent Fields. (NASONR Report from the Shelter
Island Conference in 1951, 187194, 1951)
17. On The Numerical Integration of Ordinary Differential Equations
of the First Order. (Quart. Appl. Math. 10, 97111, 1952)
18. Approximate Formulas for ManyCenter Integrals in the Theory
of Molecules and Crystals (J. Chem. Phys. 21, 374375, 1953)
19. On the MolecularOrbital Theory of Conjugated Organic Compounds
with Application to the Perturbed Benzene Ring.
(J. Chem. Phys. 21, 496515, 1953)
20. Studies of Atomic SelfConsistent Fields. I. Calculation of
Slater Functions. (Phys. Rev. 90, 120125, 1953)
21. Studies of Atomic SelfConsistent Fields. II. Interpolation
Problems. (Phys. Rev. 94‚ 16001609, 1954)
22. (with H. Sponer) Les Niveaux d'énergie électronique dans
l'éthylène (J. Phys. Rad. 15, 607611, 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, 599611, 1954)
24. A Method of Alternant Molecular Orbitals. (Symposium on
Molecular Physics at Nikko, Japan in 1953, 1316, 1954)
25. Calculations of Molecular Integrals in Uppsala. (Symposium
on Molecular Physics at Nikko, Japan in 1953, 113117, 1954)
26 Quantum Theory of ManyParticle Systems. 1. Physical Inter
pretations by Means of Density Matrices, Natural SpinOrbit
als, and Convergence Problems in the Method of Configurational
Interaction. (Phys. Rev. 97, 14741489, 1955)
27. Quantum Theory of ManyParticle Systems. II. Study of the
Ordinary HartreeFock Approximation.
(Phys. Rev. 97, 14901508, 1955)
28. Quantum Theory of ManyParticle Systems. III. Extension of
the HartreeFock Scheme to Include Degenerate Systems and
Correlation Effects. (Phys. Rev. 97, 15091520, 1955)
29. An Extension of the HartreeFock Method to Include Correla
tion Effects (Les Electrons dans les Métaux, Dixième Con
férence Solvay, Bruxelles in 1954, 7188, 1955)
30. (with I. FischerHjalmars) Report From the Symposium on
Quantum Theory of Molecules, Stockholm and Uppsala, 1955.
(Sv. Kem. Tidskrift 67, 365398, 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 SpinOrbitals for Helium.
(J. Chem. Phys. 23, 1565, 1955)
33. Quanttun Theory of Cohesive Properties of Solids.
(Adv. Phys. 5, 1172, 1956
34. (with H. Shull) Natural Orbitals in the Quantum Theory of
TwoEleCtrOn Systems (Phys. Rev. 101, 17301739, 1956).
35. (with H. Shull) Correlation Splitting in HeliumLike Ions
(J. Chem. Phys. 25‚ 10351040, 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 SelfConsistent Fields. III.
Analytic Wave Functions for the ArgonLike Ions and for the
First Row of the Transition Metals.
(Phys. Rev. 103, 17461755, 1956).
38. Present Situation of Quantum Chemistry.
(J. Phys. Chem. 61, 5568, 1957)
39. Den Kovalenta Kemiska Bindningen i Kvantmekanisk Belysning
(Elementa 40, 924, 1957).
40. Generalizations of the HartreeFock Scheme.
(Ann. Aead. Reg. Sci. Upsaliensis, 2, 127135, 1958).
41. (with H. Shull) Variation Theorem for Excited States.
(Phys. Rev. 110, 14661467, 1958).
42. Nature des Fonctions de la Mésomérie.
(Ed. du Centre Nat. Rech. Sei. LXXXII, 2337, 1958).
43. (with A.J. Freeman) Quantum Meehanical Kinetic Energy Trans
formation (Phys. Rev. 111, 12121213, 1958).
44. Spin Degeneracy Problem .
(Coll. Int. Centre Nat. Rech. Sci. 82, 23, 1958).
45. Correlation Problem in ManyEleetron Quantum Mechanies. I.
Review of Different Approaches and Discussion of Some Current
Ideas (Adv. Chem. Phys. 2, 207322, 1959).
46. Scaling Problem, Virial Theorem and Connected Relations in
Quantum Mechanics (J. Mol. Spect. 3, 4666, 1959).
47. (with H. Shull) Superposition of Configurations and Natural
Spin Orbitals. Applications to the He Problem.
(J. Chem. Phys. 30, 617626, 1959).
48. (with J.0. Hirschfelder) The LongRange Interaction of Two ls
Hydrogen Atoms Expressed in Terms of Natural Spin Orbitals
(Mol. Phys. 2, 229258, 1959).
49. (with L. Rédei) Combined Use of the Method of Superposition of
Configurations and Correlation Factor on the Ground States of
the HeliumLike Ions (Phys. Rev. 114, 752757, 1959)
50. En IterationsVariationsmetod För Att Lösa Egenvärdesproblem.
(NordSAM, Karlskrona and Lund 1959, 199209, 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, 575, 1960)
52. Expansion Theorems of the Total Wave Function and Extended
HartreeFock Schemes (Rev. Mod. Phys. 32, 328334, 1960)
53. Quantum Theory of Electronic Structure of Molecules.
(Ann. Rev. Phys. Chem. 11, 107132, 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, 461467, 1960)
55. The Principle of Causality, the Chemical Bond and Modern
Quantum Chemistry.
(Ann. Acad. Reg. Sci. Upsaliensis, 5, 6378, 1961)
56. Note an the Separability Theorem for Electron Pairs.
(J. Chem. Phys. 35, 7881, 1961)
57. Band Theory, Valence Band and TightBinding Calculations.
(J. Appl. Phys. 33, 251280, 1962
58. Exchange, Correlation and Spin Effects in Molecular and Solid
State Theory (Rev. Mod. Phys. 34, 8087, 1962)
59. (with J.L. Calais) A Simple Method of Treating Atomic Integrals
Containing Functions of r_{12}.
(J. Mol. Spect. 8, 203211, 1962)
60. (with A. Fröman) Virial Theorem and Cohesive Energy of Solids,
Particularly Ionic Crystals (J. Phys. Chem. Sol 23, 7584,
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 ClosedShell Structure.
(J. Chem. Phys. 36, 22472256, 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, 22572265, 1962
63. The Normal Constants of Motion in Quantum Mechanics Treated
by Projection Technique. (Rev. Mod. Phys. 34, 520530, 1962).
64. Studies in Perturbation Theory. IV. Solution of Eigenvalue
Problem by Projection Operator Formalism.
(J. Math. Phys. 3, 969982, 1962).
65. Studies in Perturbation Theory V. Some Aspects on the Exact
SelfConsistent Field Theory. (J. Math. Phys. 3, 11711184,
1962).
66. Studies in Perturbation Theory. 1. An Elementary Iteration
Variation Procedure for Solving the Schrödinger Equation by
Partitioning Technique (J. Mol. Spect. 10, 1233, 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, 702708, 1963).
69. Discussion on the HartreeFock Approximation.
(Rev. Mod. Phys. 35, 496498, 1963).
70. Discussion on Natural Expansions and Properties of the Chemical
Bond (Rev. Mod. Phys. 35, 629630, 1963).
71. Proton Tunnelling in DNA and its Biological Implications.
(Rev. Mod. Phys. 35, 724732, 1963).
72. Effect of Proton Tunnelling in DNA on Genetic Information and
Problems of Mutations, Aging, and Tumors.
(Bio. Symp. 1, 161181, 1964).
73. Some Aspects of Quantum Biology. (Bio. Symp. 1, 293311, 1964).
74. Molecular Orbitals in the Exact SCF Theory. (Molecular Orbitals
in Chemistry, Physics, and. Biology, Mulliken Dedicatory Volume,
Academic Press, Inc., New York, 3755, 1964).
75. Some Aspects on DNA Replication; Incorporation Errors and
Proton Transfer. (Electronic Aspects of Biochemistry, Academic
Press, Inc., New York, 167201, 1964).
76. Studies in Perturbation Theory. II. Generalization of the
BrillouinWigner Formalism. (J. Mol. Spect. 13, 326331, 1964).
77. Studies in Perturbation Theory. III. Solution of the Schrödinger
Equation Under a Variation of a Parameter.
(J. Mol. Spect. 13, 331337, 1964).
78. Studies in Perturbation Theory. VI. Contraction of Secular
Equations.(J. Mol. Spect. 14, 112118, 1964).
79. Studies in Perturbation Theory. VII. Localized Perturbation
(J. Mol. Spect. 14, 119130, 1964).
80. Studies in Perturbation Theory. VIII. Separation of the Dirac
Equation and Study of the SpinOrbital Coupling and Fermi
Contact Terms. (J. Mol. Spect. 14, 131144, 1964).
81. Angular Momentum Wavefunctions Constructed by Projection
Operators. (Rev. Mod. Phys. 36, 966976, 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, 218221, 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, 13411353, 1965).
85. Studies in Perturbation Theory. X. Lower Bounds to Energy
Eigenvalues in PerturbationTheory Ground State.
(Phys. Rev. 139, A357A372, 1965).
86. Studies in Perturbation Theory. XI. Lower Bounds to Energy
Eigenvalues, Ground State, and Excited States.
(J. Chem. Phys. 143, S175S185, 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 DNAMolecule.
(Adv. Quant. Chem. II, 1965).
88. Arvsanlagen och Deras Förändringar  Ur kvantgenetisk synpunkt.
(Sv. Naturvetenskap, 1965).
89. (with J. 0. Hirschfelder) LongRange Interaction of Two 1s
Hydrogen Atoms Expressed in Terms of Natural SpinOrbitals.
(Mol. Phys. 9‚ 4911496, 1965).
90. Some Recent Developments in the Quantum Theory of ManyElectron
Systems and the Correlation Problem.
(Adv. Chem. Phys. 8, 34, 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, 177179, 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, 255294,
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 HartreeFock Method. An Extension of the
IndependentParticle Scheme.
(Quantum Theory of Atoms, Molecules, and Solid State, 601, 1966)
96. Program. (Int. J. Quant. Chem. 1, 16, 1967)
91. Nature of Quantum Chemistry.
(Int. J. Quant. Chem. 1, 712, 1967)
98. Group Algebra, Convolution Algebra, and Applications to Quantum
Mechanics. (Rev. Mod. Phys. 39‚ 259287, 1967)
99. Quantumn Theory of TimeDependent Phenomena Treated by the
Evolution Operator Technique.
(Adv. Quant. Chem. 3, 323381, 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, 637708, 1967)
101. Eigenvalue Problem in a Linearly Dependent Basis and the Super
SecularEq.uation. (Int. J. Quant. Chem. 1S, 811827, 1967)
102. Molecular Associations in Biology  4 Brief Summary. (Molecular
Associations in Biology, Academic Press, Inc., New York,
539549, 1968)
103. (with P. Lindner) Upper and Lower Bounds in SecondOrder
Perturbation Theory and the Unsöld Approximation.
(Int. J. Quant. Chem. 2S, 161173, 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 PhysicoChemical 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, 867931, 1968)
i06. Some Comments on the Treatment of Symmetry Properties in
Perturbation Theory. (Int. J. Quant. Chem. 25, 137l50, 1968)
107. (with W.M. MacIntyre) Electronic Energy of the DNA Replication
Plane. (Int. J. Quant. Chem. 2S, 20217, 1968)
108. Some Aspects on the Correlation Problem and Possible Extensions
of the IndependentParticle 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, 767780, 1969)
110. Some Comments on the Periodic System of the Elements.
(Int. J. Quant. Chem. 3S, 331334, 1969)
111. (with 0. Goscinski) The Exchange Phenomenon, the Symmetric
Group, and the Spin Degeneracy Problem.
(Int. J. Quant. Chem. 3S, 533591, 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, 697702, 1969)
113. Some Aspects of the Hydrogen Bond in Molecular Biology. (Ann.
N. Y. Acad. Sci. 158, 8695, 1969)
114. Energia Sajátékek Alsóés Felsó Korlátjának Számítása
Perturbáiós  Zámításban Particionlálási Technikával.
(Magyar Fizikai Folyóirat XVIII, 515540, 1969)
115. On the NonOrthogonality Problem.
(Adv. Quant. Chem. 5, 185199, 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, 112135, 1970)
117. (with J. Gruninger and Y. Öhrn) Comments on the Analysis of
Atomic Correlation Energies).
(J. Chem. Phys. 52, 55515554, 1970)
118. Recent Research in the Uppsala & Florida Quantum Theory
Projects. (IBM Ludwigsburg Meeting Report, 4783, 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, 231237, 1971)
121. (with 0. Goscinski) Studies in Perturbation Theory. XIV.
Treatment of Constants of Motion, Degeneracies and Symmetry
Properties by Means of MultiDimensional Partitioning. (Int.
J. Quant. Chem. 5S, 685705, 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, 207266, 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 NonMetals 1, 6378, 1972)
125. (with P.K. Mukherjee) Some Comments on the Time Dependent
Variation Prineiple. (Chem. Phys. Lett. 14, 17, 1972)
126. (with B. Laskowski) Treatment of Constants of Motion in the
Variation Principle. Symmetry Properties of Variational Wave
Functions. (Chem. Phys. Lett. 16, 14, 1972)
127. Electronie Mobility and Transfer of Energy and Momentum as
TimeDependent Processes. (Proc. 3rd Conf. From Theoretieal
Physics to Biology, Versailles 1971, 145146 (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, 18961908, 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, 27772787, 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, 121125, 1974)
132. Some Aspects on the AmericanSwedish Exchange in Quantum
Sciences particularly the UppsalaFlorida 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)
Last updated : Apr. 8, 2002  10:30 CET