Table Of ContentLecture Notes
ni Physics
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Berthold-Georg trelgnE
lacissalcimeS Theory fo Atoms
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Berlin Heidelberg NewYork London Paris Tokyo
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Berthold-Georg trelgnE
t.~tisrevinU ,nehcniLM Sektion kisyhP
Am Coulombwall ,1 D-8046 Garching, F.R.G.
ISBN 3-540-19204-2 galreV-regnirpS Berlin Heidelberg NewYork
ISBN 0-387-19204-2 galreV-regnirpS NewYork Berlin Heidelberg
Library of Congress Cataloging-in-Publication Data. Englert, B.-G. (Berthold-Georg), 1953-
Semiclassical theory of atoms / B.-G. Englert. p. cm.-(Lecture notes in physics; 300) Includes
bibliographies and indexes. ISBN 0-38?-19204-2 (U.S.) .1 Atoms-Models. 2. Thomas-Fermi
theory. .I Title. .1I Series.
O.C 173.E56 1988 539'. 14-dc 19 88-4949
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PREFACE
This book grew out of a set of notes that I supplied to the audience
of a series of lectures on "The Thomas-Fermi Method in Atomic Physics
and Its Refinements" delivered at the University of Munich in 1985.
Standard textbook material played a minor role during these lectures[
the emphasis was on the novel approach developed by Professor Julian
Schwinger and myself, beginning about eight years ago. As a con-
sequence, this book is the first complete, detailed step-by-step
presentation of our ideas and their implications.
Naturally, the work of other researchers is not ignored. In par-
ticular, I have tried to collect and organize the many pieces of
knowledge about the Thomas-Fermi model that are scattered over as
many original publications. On the other hand, my intention was not
to supply a complete list of every paper on the subject, as this
would have been of little value. Thus referencing is selective and I
cite only the most relevant papers. On a few occasions honesty de-
manded critical remarks about someone else's work[ I hope that these
comments will not be misunderstood as put-downs.
The reader is not expected to have any previous knowledge about
the subject. In addition to an open mind, the only prerequisite is a
thorough understanding of elementary quantum mechanics, and some
familiarity with the phenomenology of atoms is certainly helpful.
The text consists of a mixture of general concepts and technical
detail. Both need to be absorbed, although some of the latter can be
skipped during a first reading. I trust that readers can perform a
reasonable selection themselves.
I am grateful for the many insights gained in discussions with a
large number of people. Being afraid of forgetting somebody, I shall
not even try to list them.
It is a pleasure to thank Mrs. E. Figge, who typed the manuscript
with enviable skill.
Garching, February 1988 B.-G. Englert
ELBAT FO CONTENTS
Chapter One. Introduction ........................................ I
Atomic units .................................................. 2
Bohr atoms .................................................... 4
Traces and phase-space integrals .............................. 10
Bohr atoms with shielding ..................................... 11
The effective potential ........................................ 13
Size of atoms ................................................. 21
Problems ...................................................... 24
Chapter Two. Thomas-Fermi Model .................................. 27
General formalism ............................................. 27
The TF model .................................................. 33
Neutral TF atoms .... .......................................... 37
Maximum property of the TF potential functional ............... 39
An electrostatic analogy ...................................... 41
TF density functional ......................................... 43
Minimum property of the TF density functional ................. 45
Upper bounds on B ............................................. 46
Lower bounds on B ............................................. 50
Binding energy of neutral TF atoms ............................ 54
TF function F(x) .............................................. 56
Scaling properties of the TF model ............................ 66
Highly ionized TF atoms ....................................... 73
Weakly ionized TF atoms ........................................ 80
Arbitrarily ionized TF atoms .................................. 96
Validity of the TF model ...................................... 99
Density and potential functionals ............................. 104
Relation between the TF approximation and Hartree's method .... 119
Problems ...................................................... 124
Chapter Three. Strongly Bound Electrons .......................... 130
Qualitative argument .......................................... 130
First quantitative derivation of Scott's correction ........... 131
Scott's original argument ..................................... 138
TFS energy functional ......................................... 139
TFS density ................................................... 146
VI
Consistency ................................................... 148
Scaling properties of the TFS model ........................... 155
Second quantitative derivation of Scott's correction ........... 158
Some implications concerning energy ........................... 162
Electron density at the site of the nucleus ................... 164
Numerical procedure ............................................ 168
Numerical results for neutral mercury ......................... 170
Problems ...................................................... 173
Chapter Four. Quantum Corrections and Exchange ................... 175
Qualitative arguments .......................................... 176
Quantum corrections I (time transformation function) ........... 177
Quantum corrections II (leading energy correction) ............ 190
The yon Weizs~cker term ....................................... 194
Quantum corrections III (energy) .............................. 196
Airy averages ................................................. 199
Validity of the TF approximation ................................ 207
Quantum corrected EI(V+~) ..................................... 210
Quantum corrected density ..................................... 213
Exchange I (general) ........................................... 221
Self energy ....................................................... 225
Exchange II (leading correction) .............................. 225
History ........................................................ 230
Energy corrections for ions .................................... 232
Ionization energies .............................................. 236
Minimal binding energies (chemical potentials) ................ 239
Shielding of the nuclear magnetic moment ....................... 241
Simplified new differential equation (ES model) ............... 244
An application of the ES model. Diamagnetic susceptibilities.. 255
Improved (?) ES model. Electric polarizabilities .............. 260
Exchange III. (Exchange potential) ............................. 275
New differential equation ...................................... 278
Small distances ................................................. 282
Large distances ................................................ 283
Numerical results ............................................... 286
Problems ....................................................... 293
Chapter Five. Shell Structure .................................... 295
Experimental evidence ........................................... 297
Qualitative arguments ......................................... 299
Bohr atoms .................................................... 300
VII
TF quantization ............................................... 305
Fourier formulation ........................................... 312
Isolating the TF contribution ................................. 314
Lines of degeneracy ........................................... 315
Classical orbits .............................................. 320
Degeneracy in the TF potential ................................. 323
TF degeneracy and the systematics of the Periodic Table ....... 327
General features of N ....................................... 329
qu
General features of E ....................................... 333
qu
Linear degeneracy. Scott correction ........................... 335
Perturbative approach to E ................................. 336
osc
Z-quantized TF model ........................................... 338
j~0 terms. Leading energy oscillation ......................... 342
Fresnel integrals ............................................. 347
I oscillations ................................................ 351
v oscillations ................................................ 358
Semiclassical prediction for E ............................. 360
osc
Other manifestations of shell structure ....................... 363
Problems ...................................................... 367
Chapter Six. Miscellanea ......................................... 370
Relativistic corrections ...................................... 370
Kohn-Sham equation ............................................ 377
Wigner's phase-space functions ................................ 379
Problems ...................................................... 381
Footnotes ......................................................... 383
Chapter One ................................................... 383
Chapter Two ................................................... 384
Chapter Three ................................................. 387
Chapter Four .................................................. 389
Chapter Five .................................................. 394
Chapter Six ................................................... 395
Index ............................................................ 397
Names ......................................................... 397
Subjects ...................................................... 398
Chapter One
INTRODUCTION
Atoms that contain many electrons possess a degree of comple-
xity so high that it is impossible to give an exact answer even when
we are asking simple questions. We are therefore compelled to resort to
approximate descriptions. Two main approaches have been pursued in the-
oretical atomic physics. One is the Hartree-Fock(HF) method and its re-
finements; it can be viewed as a generalization of Schr~dinger's des-
cription of the hydrogen atom to many-electron systems; it is, by con-
struction, the more reliable the smaller the number of electrons. The
other one is the Thomas-Fermi (TF) treatment and its improvements; this
one uses the picture of an electronic atmosphere surrounding the nucleus;
it is the better the larger the number of electrons. For this reason,
the TF method is frequently called the "statistical theory of the atoms."
Throughout these lectures we shall be concerned with the TF
approach, thereby concentrating on more recent developments. The repe-
tition of material that has been presented in textbooks I already will be
limited to the minimal amount necessary to make the lectures self-con-
tained. The derivation of known results will, wherever feasible, be done
differently, and - I believe - more elegantly, than in standard texts on
the subject.
It should be realized that the methods of the TF approach are
in no way limited to atomic physics. Besides the immediate modifications
for applying the formalism to molecules or solids, there exists the pos-
sibility of employing the technics in astrophysics and in nuclear phy-
sics. The latter application naturally requires appropriate changes re-
flecting the transition from the CoUlomb interaction of the electrons
to the much more complicated nucleon-nucleon forces.
In these lectures we shall confine the discussion to atoms,
however. This has the advantage of keeping the complexity of most cal-
culations at a rather low level, so that we can fully focus on the pro-
perties of the TF method without being distracted by the technical com-
plications that arise from the considerations of molecular structure
or from our incomplete knowledge of the nuclear forces, for instance.
Restricting ourselyes to atoms is further advantageous because it en-
ables us to compare predictions of TF theory with those of other methods,
like HF calculations. The ultimate test of a theoretical description is,
of course, the comparison of its implications with experimental data.
Whenever possible, we shall therefore measure the accuracy of the TF
predictions by confronting it with experimental results.
Lack of experimental data sometimes forces us into relying
upon HF results for comparison. The same situation occurs when quanti-
ties of a more theoretical nature are discussed (as, e.g., the nonrela-
tivistic binding energy, which is not available from experiments). Such
a procedure must not be misunderstood as an attempt of reproducing HF
predictions by TF theory. The TF method is not an approximation to the
HF description, but an independent approach to theoretical atomic phy-
sics. Incidentally, it is the historically older one: TF theory origi-
nated in the years 1926 (Thomas) and 1927 (Fermi), whereas the HF model
did not exist prior to 1928 (Hartree) and 1930 (Fock). 2 The two appro-
aches should not be regarded as competing with each other, but as sup-
plementing one another. Each of the two methods is well suited for stu-
dying certain properties of atoms. For example, if one is interested in
the ionization energy of oxygen, a HF calculation will produce a reli-
able result; but if you want to know how the total binding energy varies
over the entire Periodic Table, the TF model will tell you. Tersely: the
HF method for specific information about a particular atom, the TF me-
thod for the systematics of all atoms. There is, of course, a certain
overlap of the two approaches, and they are not completely unrelated.
We shall discuss their connection to some extent in Chapter Two.
Atomic units. All future algebraic manipulations are eased significant-
ly when atomic units are used for measuring distances, energies, etc.
Let us briefly consider the many-particle Hamilton operator
N 1 +2 N Ze 2 i ~ j e2
(I-I)
=' 9=I j,k=1 rjk
of an atom with nuclear charge Ze and N electrons, each of mass m and
carrying charge -e. The third sum is primed to denote the omission of
the term with j = k. Obviously, r. stands for the distance between the
3
nucleus and the j-th electron, whereas rjk is the distance from the j-th
to the k-th electron, and pj the momentum of the j-th electron. This
Hmp is accompanied by the commutation relations
~j,Pk = i ~ ? 6jk (1-2)
and the injunctions caused by the Fermi statistics that the electrons
obey. Equations )I( and )2( contain three dimensional parameters: m, e,
~. But none of them can possibly be used as expansion variable of a
perturbation series because together they do no more than set the ato-
mic scale. To see this in detail, let us rewrite )I( and )2( with the
aid of the Bohr radius
~2
ao = ef-~--m = 0.5292 ~ (I-3)
and twice the Rydberg energy
2
Eo _ e ao _ me ~2 - 27.21eV. (I-4)
Equations )I( and )2( now appear as
/-Z 7- ,7 1>'
1 /~ ) _ + (I-5)
Hmp/E° = 3 ~(PJ a° j ~ ~ j,k (rjk/ao)
and
(rj/a )o , (pk/~{ = i 1 6jk . (I-6)
If we then introduce the dimensionless quantities ~./ao, 3 ÷ Pj/ao' and
Hmp/E o as relevant objects, all reference to m, e, and ~ disappears.
Using the same letters as for the dimensional quantities, we now have
I Z
Hmp = y pj ~ + y .... (I-7)
j j J j ,k rjk
and
÷ ~_ 9e
rj,Pk = i 1 6jk . (I-8)
Equations )7( and )8( are identical with Eqs. )I( and )2( except that
instead of the macroscopics units (cm, erg, etc.) atomic untis are used.
Formally, the transition from )I( and )2( to )7( and )8( can be done by
"setting e = { = m = I," but the meaning of this colloquial procedure
is made precise by the argument presented above.
Besides simplifying the algebra, the use of atomic units also
prevents us from trying such foolish things like "expanding the energy
in powers of ~,"a phrase that one meets surprisingly frequently in the
literature. The energy is nothing but Eo times a dimensionless func-
tion of Z and N, it depends on ~ only through Eo~I/~ .2 We shall see la-
ter, what is really meant when the foregoing phrase is used.
The many particle problem defined by Eqs. )7( and )8( cannot
be solved exactly. It is much too complicated. This is true even when
the number of electrons is only two, the situation of helium-like atoms.
There is a branch of research 3 in which rigorous theorems about the sy-
stem )7( and )8( are proved, such as (disappointingly rough) limits on
the total binding energy. One can show for example, that for N=Z÷~ the
many particle problem reduces to the original TF model, which we shall
describe in the next Chapter. In these lectures, we shall not follow
those highly mathematized lines. I prefer rather simple physical argu-
ments instead of employing the machinery of functional analysis. Also,
it is my impression that those "rigorous" methods are of little help
when it comes to improving the description by going beyond the original
TF model. Finally, let us not forget that mathematical theorems about
)7( and )8( are not absolute knowledge about real atoms, because in put-
ting down the Hamilton operator )7( we have already made physical ap-
proximations: the finite size and mass of the nucleus is disregarded;
so are all relativistic effects including magnetic interactions and
quantum electrodynamical corrections; other than electric interactions
are neglected - no reference is made to gravitational and weak forces.
Of course, both attitudes, the highly mathematical one and the more
physical one, are valuable, but there is danger in judging one by the
standards of the other.
Bohr atoms. We continue the introductory remarks by studying a very
simple model in order to illustrate a few basic concepts. This primi-
tive theoretical model neglects the inter-electronic interaction, thus
treating the electrons as independently bound by the nucleus. But even
if fermions do not interact they are aware of each other through the
Pauli principle. Therefore, such noninteractin~ electrons (NIE) will
fill the successive Bohr shells of the Coulomb potential with two elec-
trons in each occupied orbital state.
For the present purpose it would be sufficient to consider
the situation of m full Bohr shells. But with an eye on a later dis-
cussion of shell effects, in Chapter Five, let us additionally suppose
