Superheavy elements a prediction of their chemical and physical properties
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ABSTRACT: Relativistic basis sets of double-zeta, triple-zeta, and quadruple-zeta quality have been optimized for the 6d elements Rf–Cn. The basis sets include SCF exponents for the occupied spinors and for the 7p shell; exponents of correlating functions for the valence shell, the 6s and 6p shells, and the 5f shell; and exponents of functions for dipole polarization of the 6d shell. A finite nuclear size was used in all optimizations. Prescriptions are given for constructing contracted basis sets. The use of the basis sets is demonstrated for some atomic and molecular systems. The basis sets are available as an Internet archive and from the Dirac program Web site, http://dirac.chem.sdu.dk.Theoretical Chemistry Accounts 01/2011; 129(3):603-613. · 2.14 Impact Factor
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ABSTRACT: Die Stabiliät der Oxidationsstufe +4 nimmt in der Gruppe 14 von Silicium zum Element 114 ab, wie relativistische und nichtrelativistische Rechnungen an den Hydriden, Fluoriden und Chloriden der Elemente der Gruppe 14 ergaben (siehe Gang der Energien der Zerfallsreaktion (1) im Bild). Damit erscheint es unwahrscheinlich, daß das superschwere Element 114, das wegen der magischen Protonen- und Neutronzahl stabile Isotope aufweisen könnte, durch Atom-at-a-time-Chemie in der Oxidationsstufe +4 untersucht werden kann.Angewandte Chemie 09/1998; 110(18):2669-2672.
- Angewandte Chemie. 01/2009; 121(19):3456-3467.
A Prediction of Their Chemical and Physical Properties
Gesamthochschule Kassel, D-3500 Kassel, Heinrich-Plett-Str. 40, Germany and
Gesellschaft für Schwerionenforschung, D-6100 Darmstadt, Postfach 541, Germany
Table of Contents
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .90
11. Predictions of Nuclear Stability .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92
111. Basis für the Predictions of Chemical and Physical Properties
1. The Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
a) Continuation of the Periodic Table
b) Ab-initio Atomic Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Trends of the Chemical and Physical Properties . . . . . . . . . . . . . . . . . . . . . . . . . ..
a) Trends Emerging from the Calculations
b) Empirical and Semi-empirical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
IV. Discussion of the Elements
1. The 6d transition Elements Z = 104 to 112 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
2. The 7p and 8s Elements Z = 113 to 120 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
3. The 5g and 6/ Elements Z = 121 to 154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
4. The Elements Z = 155 to 172 and Z = 184. .. .. . .. . . . . . .. . . . .. . . . . . . . . . . . .
V. Critical Analysis of the Predictions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .137
VI. Application of the Chemical Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..139
Very recently Oganesian, Flerou and coworkers (1) in Dubna announced the
discovery of element 106. Although they observed less than 100 fission tracks of
the decaying nuclei of this element, formed after the heavy-ion bombardment of
Crionson Pb, they were able to measure a half-life of about 20 msec for one isotope
and 7 msec for another. At about the same time Ghiorso and coworkers (2) in Ber-
keley found a new alpha activity for which they established the genetic link
with the previously identified daughter and grand-daughter nuclides
This evidence indicates that a new element has been added to the periodic
table, thus presenting a new challenge to scientists.
Until 1940 the heaviest known element was uranium with the atomic number
92, and at that time (about 1944) the actinide concept of Seaborg (3) was just a
hypothesis. It is thus apparent that great progress has been made since then.
Fourteen new elements have been added to the periodic system and much chem-
ical and physical information has been gathered concerning this region of ele-
ments. Hence we can expect that element 106 is probably not the last element
but only a step toward an even longer periodic table. The approach used in the
experiments up to now to produce even heavier transuranium elements has been
to proceed element by element into the region of atomic numbers just beyond the
heaviest known by bombarding high-Z atoms with small-Z atoms. There have
been very difficult and laborious attempts to proceed even further (4, 5). The
upper limit of this method is determined by experimental feasibility; it cannot
now be predicted with certainty but will be about element 108 or 109. The other
way to proceed is to bombard two very heavy elements with each other, thus
producing superheavy elements directly. This method will probably overlap
with the first method at its lower end.
This second method, which must be the result of bombardments with relatively
high Z heavy ions, is still in preparation at several places in the world, i. e. Dubna
in the USSR, Berkeley in the USA, Orsay in France, and Darmstadt in Germany.
If this method is successful, it should lead to the nearly simultaneous discovery
of a number of new elements.
There is general agreement that theoretical predictions of nuclear stability,
which we discuss briefly in the next paragraph, define a range of superheavy
elements with sufficiently long half-lives to allow their study, provided they can
be synthesized. What cannot be predicted is whether there exist nuclear reactions
for such synthesis in detectable amounts on earth.
The known elements heavier than uranium are usually called by the very
unspecific name of transuranium elements. In the upper range this term is ex-
pected to overlap with the equally ill-defined expression superheavy elements.
To clarify the situation from a nuclear physics point of view, one may define
the end of the transuranium elements and the beginning of the superheavy ele-
ments as the element where the nuclear stability of the longest-lived isotope in-
creases again with increasing Z. The observed strong decrease of the half-Jives of
the transuranium elements known up to now can be seen in Fig. 1. The question
is where and if this trend to even smaller half-lives is likely to end.
From a chemical point of view the elements, including the unknown super-
heavy elements, are well defined by their location in the periodic table. The ele-
ments up to 103 are the actinides or the 51 transition elements. Chemical reviews
of these are given by Seaborg (6, 7), Cunningham (8), Asprey and Pennemann (9),
and Keller (10). The 6d transition series starts with element 104. Of course, the
first chemical question to be answered is whether this simple series concept of the
periodic table still holds for the superheavy elements. A very comprehensive review
of elements 101 to 105, discussing the nuclear stability and chemical behavior of
the predicted elemcnts, was given by Seaborg (5) in 1968. Several other articles
dealing mainly with the chemical behavior of superheavy elements, the search for
superheavy elements in nature, and the electronic structure of these elements
have since been published. The references are given in the discussion below.
In this summary of the very quickly developing field of the superheavy ele-
ments, the main emphasis lies on the prediction of their chemical properties. Apart
from the general interest of the question, this knowledge is expected to be very
important because chemical separation will be one of the methods used to detect
11. Predictions of Nuclear Stability
Like the well-known effect of the elosed shells in the atomie eleetron eloud at
Z ==2, 10, 18,36,54,86, whieh is the physieal basis for the strueture of the periodie
table, the effeet of elosed nueleon shells together with a large separation to the next
unoccupied shell also makes for considerable nuclear stability. The nucleus
consists of two kinds of particles, protons and neutrons, so that we have two
series of so-ealled magie numbers. These are for protons 2, 8, 20, 28, 50, 82, and
for neutrons 2, 8, 20, 28, 50, 82, and 126. Nuelei where both protons and neutrons
are magie, (160, 40Ca or 208Pb, for example) are called double-magie nuclei and
are partieularly stable. As we go to even heavier nuclei, the effect whieh most
heavily influenees stability to ex-decay or fission (the most important decay modes)
is the inereasingly large repulsion of the nueleonie eharges against the attraetive
nuelear forces, which severely shortens the half-lives of the nuclei (11), as can be
seen from Fig. 1. This suggests the question: Is the stabilizing effect of the next
1 0 - 1 2 ~ " " " " " - " " ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' - - & o . . . I - . L - J . - ~ . . . . . \ - o I - - I - . . I . . . . J . . : . . I
Fig. 1. The longest-lived isotopes of transuranium elements as a function of Z for spontaneous
fission and a decay (11)
double-magie configuration large enough to counteract this repulsion and to
lengthen the half-lives yet again?
Because it was assumed that the next protonic magie number was 126 (by
analogy with the neutrons), early studies of possible superheavyelements did not
receive much attention (12-15), since the predicted region was too far away to
be reached with the nuclear reactions available at that time. Moreover, the ex-
istence of such nuclei in nature was not then considered possible. The situation
changed in 1966 when Meldner and Röper (16, 17) predicted that the next proton
shell closure would occur at atomic number 114, and when Myers and Swiatecki
(18) estimated that the stability fission of a superheavy nucleus with elosed
proton and neutron shells might be comparable to or even higher than that of
many heavy nuclei.
These results stimulated extensive theoretieal studies on the nuclear properties
of superheavy elements (19). The calculations published so far have been based
on a variety of approaches. Most ealeulations were performed by using a phenom-
enologieal deseription within the deformed shell model (20-23). In this model
the nucleons are considered to move in an average potential and the shape of the
potential and other parameters are chosen by fitting single-partiele levels in well-
investigated spherical or deformed nuclei. Regardless of the approach followed,
the authors agree in predicting a double-magie nueleus298114, although several
other magie proton and neutron numbers near these values have been diseussed.
There are also several self-consistent calculations (17, 24-27) but suitable
parameters have to be used, beeause the nueleon-nueleon force is not known from
general considerations. Most authors also accept the magie numbers Z === 114 and
N === 184.
In addition to the proton magie number 114, a seeond superheavy magic proton
number was investigated at Z === 164 (23, 28). Although the realization of such a
nueleus seems to be far from any praetieal possibility at the moment, one should
bear this region in mind because many most interesting questions could be an-
swered if it were possible to produce these elements. One way to actually proeeed
28 50 82126184196318
Fig. 2. Schematic drawing of the stability of the nuclei as a function of the number of protons
and neutrons. The expected islands of stability can be seen near Z = 114 and Z = 164 (29)
into this region is the observation of the X-rays from the quasi-moleeular systems
whieh are transiently formed during heavy ion eollission (109).
These predietions are depieted very schernatically in Fig. 2 in an allegorieal
fashion (29). The long peninsula eorresponds to the region of known nuelei. The
grid lines represent the magie numbers of protons (Z) and neutrons (N). The third
dimension represents the stability. The magie numbers are shown as ridges and
the double-magie nuelei, like 208Pb, are represented as mountains. The two regions
near Z ==114 and Z ==164 show up in Fig. 2 as "islands of stability" within the
large "sea of instability".
The detailed ealeulations quoted above predict potential barriers against
fission, i. e. the total energy of the nucleus is caleulated as a funetion of the
deformation, beeause a deformation parameter describes at the one extreme the
spherieal nucleus and at the other the two separated nuclei after fission. All of
these ealeulations indieate a maximum (or two) at a small deformation, whereas
we get a dip of a few MeV at zero deformation and a trough of a few hundred MeV
for very large deformations. The result of such a ealeulation is shown in Fig. 3
T 8 MeV
ENERGY RELEASE ~
Fig. 3. Total energy as a funetion of the deformation of the expeeted double-magie nueleus
298114. The small minimum at the deformation zero is expeeted to be the reason for the very
long lifetime of this nucleus (32)
for the expected double-magie nucleus298114. This small minimum at zero de-
formation plays an important role ; it keeps the nucleus in spherical shape and
prevents rapid deeay in the fission path. Spontaneous fission ean oeeur only by the
extremely slow proeess of tunneling through the several MeV high barrier.
Thus, the height and width of this barrier playamost important role in the
predietion of the half-lives against fission (31). For the double-magie nueleus298114
a height of between 9 and 14 MeV is predieted, depending on the method used.
This yields spontaneous fission half-lives of between 107and 1015years.
These first results were very promising and stimulated a very extensive but
up to now unsueeessful seareh for superheavy elements in nature. A most corn-
prehensive review of this subjeet was given by G. Herrmann (32). But, besides
spontaneous fission, a nueleus ean deeay by other deeay modes like o: deeay,
ß deeay, or eleetron eapture. The most eomprehensive study of half-lives in the
first superheavy island was performed by Fiset and Nix (33). Figure 4 is taken
from their work.
Fig. 4. Summary of predictions of the half-lives of the nuclei at the :first island of stability.
(a) spontaneous-:fission half-Iives, (b) cc-decay half-lives, (c) electron-capture and ß-decay
half-lives, and (d) total half-lives. The nurnbers give the exponent of 10 of the half-lives in
The results show that, as one moves away from the double elosed-shell nueleus
298114, the ealeulated spontaneous fission half-lives in Fig. 4a deerease from 1015
y for nuclei on the inner contour to 10-5y (about 5 min) for nuclei on the outer
contour. With respect to spontaneous fission, the island of superheavy nuclei is
a mountain ridge running north and south, with the descent being most gentle in
the northwest direction. The calculated «-decay half-lives in Fig. 4b, however,
decrease, rather smoothly with increasing proton number from 105y for nuclei
along the bottom contour to 10-15y (about 30 nsec) for nuclei along the top
contour. The discontinuities arise from shell effects. The ß-stability valley crosses
the island from the southwest to the northeast direction.
The calculated ß-decay and electron-capture half-lives in Fig. 4c decrease
from 1 y for nuclei along the inner contour to 10-7y (about 3 sec) for nuclei at
the outer contour. The total half-lives in Fig. 4d are obtained by taking into ac-
count all three decay modes. The longest total half-life of 109years is found for the
nucleus294110. A three-dimensional plot of these results (35) is given in Fig, 5,
where the island character of this region of relative stability is beautifully demon-
Fig. 5. Same as Fig. 4 d in a three-dimensional plot (35)
In considering such results (34), one should be aware of the great uncertainties
associated with the extrapolation of nuclear properties into the unknown region.
The calculations are associated with large errors. The total uncertainty for all
three decay modes discussed here is as large as 1010for the half-lives.
The half-lives for the second island of stability are even smaller. In the most
optimistic estimates, they are not more than a few hours, and the uncertainty
of 1010brings them down into the region of nsec. The second assumption of course,
which has still to be proved, is that these nuclei can in fact be produced.
In conclusion, one may say that there is general agreement that theoretical
predictions of nuclear stability define a range of superheavy elements in the vicin-
ity of element 114with sufficiently long half-lives to allow their study, provided
they can be synthesized.
111. Basis for the Predictions of Chemical and Physical Properties
1. The Electronic Structure
When M endeleev constructed his periodic system in 1869 he had actually found the
most general and overall systematics known in science. He developed this table
from his comparison of the chemieal and physical properties of the elements,
without knowing the underlying reason for it. Since the early stages of quantum
mechanics in the 1920's, it has become clear that the similarity of the properties
of the elements depends strongly on the outer electronic structure. The filled-shell
concept is in accord with the periodicity of the chemical properties that formed
the basis for the concept of the periodic table.
Thus it is obvious that the first step toward predicting chemical and physical
properties is to predict the electronic structure of the superheavy elements. An
excellent review article on this subject will be published by}. B. Mann (35) in
the near future.
a) Continuation of the PeriodicTable. As early as 1926M adelung (36) found the
empirical rules for the electron-shell filling of the ground-state configurations of
the neutral atoms. His rules are simple:
1. electron shells fill in order of increasing value of the quantum number sum
(n+l), where nis the principalquantumnumber and lthe orbitalquantum number;
2. for fixed (n+l), shells fill in order of increasing n.
In Fig. 6 we show one of the many published schemes based on these rules,
which demonstrates the filling of the electrons. This systematics provides an
almost correct explanation of all known neutral atomic configurations in the
known region of elements. This simple law was therefore used by Gol'danskii (37)
and by Seaborg (5) to predict the electron structure of the superheavy elements.
Seaborg designated the 32 elements of the Sg and 61 shells as the "superactinide
series" and placed them as elements 122 to 153 by analogy with the actinide series
13 3p 18
81 6p 86
112 113 7p
37 55 38
87 75 88
119 85 120
21 3d 30
39 4d 48
71 5d 80
157 "f 70
89 5f 102 103 6d
597d8P 1681 169 95 170 I
Fig. 6. The filling of the electron shells according to the simple rule of Madelung (36)
90 to 103 following actinium. Since there were already in the known region of
elements a few deviations from M adeiung's simple rules, especially in the lanth-
anides and actinides, Chaikkorskii (38) and later Taube (39) tried to predict these
anticipated deviations. In Table 1 we show the predictions of Gol'danskii, Seaborg,
Table 1. Predictions of the ground-state configurations of Gol'danskii (37), Chaikkorskii (38),
Taube (39) and Seaborg (5) for elements 121 to 127 and 159 to 168, using the principle of the
extrapolation within the periodic table. The main quantum numbers (5g, 61, 7d, 85) are not
shown. This table is taken from Mann (35)
g2 j2 d52
S52spn (n = 1 - 6) for all columns
Chaikkorskii and Taube for elements 121 to 127 and 159 to 168. Apart from small
discrepancies in these somewhat uncertain regions, there was general agreement
that the unfinished 8th row of the periodic table would be finished by the 6d
elements ending at element 112 and the 7p elements at 118. From a conservative
point of view, every extrapolation into the region starting with element 121 is
expected to be very speculative. Nevertheless, the reliability of the location of the
elements in the periodic table seems to be relatively unambiguous.
b) Ab-initio Atomic Calculations. The prediction ofthe electronicconfigurations
of the superheavy elements became much more reliable when ab-initio atomic
calculations became available and accurate enough to be used in the field of
the superheavy elements.
In the following paragraph we give a very brief description of the principles
used in the calculations. For the details, especially the exact formulas used, we
refer to the literature. All the calculations that are useful in this connection are
based on the calculation of the total energy ET of the electronic system, given by
where 1jJ is the total wave function and H the Hamiltonian of the system. The
physical solution is found when ET is at the total minimum after the variation of
Depending on the ansatz for the total wave functions 1jJ and the Hamiltonian
H of the system, this minimalization of the total energy leads to a set of different,
usually coupled differential equations. The solution oi these equations gives the
total wave function and hence the total energy. These methods, usually called
Hartree and Hartree-Fock methods, are described in detail in various texts and
papers (40). For those planning to do suchcalculations, R.D. Hartree's "Calculation
of Atomic Structure" (41), and "Atornic Structure Calculations" (42) by F. Herr-
mann and S. Skillman are recommended. A review article by J. P. Grant (43)
gives an excellent description of relativistic methods. A good summary is also
given by]. B. Mann (35).
Let us discuss very briefly the various methods that have been used. The first
group of calculations is done by using the non-relativistic Hamiltonian (ignoring
Here the first term with v the Nabla operator is the kinetic energy, the second
is the potential energy due to the nuclear charge, and the last term is the total
electrostatic interaction energy over all pairs of electrons.
Hartree's method (H) considers the total wave function to be a product of
one-electron wave functions 1p === II f[Ji; this leads, after the variation of the total
energy, to a set of second-order homogeneous differential equations that have
to be solved for the radial wave functions of the electrons of each shell. The last
term due to the interaction of the electrons is given by the potential generated
by all the other electrons. In this respect the set of the differential equations is,
of course, already coupled. This was the basic method used by Larson et al. (44)
for the first atomic claculations in the region of superheavy elements Z === 122 to
Hartree-Fock method (HF). Here the total wave function is assumed to be an
antisymmetric sum of Hartree functions and can be represented by a Slater
1p === (N 1)- t If[J1(1) f[Jl(2) •.••• f[JN(N) r
which automatically obeys the Pauli principle.
The effect of the determinantal wave function is to greatly complicate the
resulting differential equations by adding exchange potential terms, giving rise to
an inhomogeneous equation, for which the correct solutions becomes much more
difficult and time-consuming. For the exact equations, see for example]. B. Mann
(35). A program using this method was developed by C. Froese-Fischer (45).
Hartree-Fock-Slater method (HFS). In this method the inhomogeneous parts
of the equations used in the Hartree-Fock method are approximated by a local
potential, as proposed in 1951 by Slater (46). This approximation yields much
simpler homogeneous differential equations in which the potential terms are
identical for every orbital of the atom, which makes the actual computation
less time-consuming by a factor of about 5 to 10, although the results are nearly
as good as with the Hartree-Fock method,
For heavy elements, all of the above non-relativistic methods become in-
creasingly in error with increasing nuclear charge. Dirac (47) developed a relativist-
ic Hamiltonian that is exact for a one-electron atom. It includes relativistic mass-
velocity effects, an effect named after Darwin, and the very important interaction
that arises between the magnetic moments of spin and orbital motion of the elec-
tron (called spin-orbit interaction). A completely correct form of the relativistic
Hamiltonian for a many-electron atom has not yet been found. However, excellent
results can be obtained by simply adding an electrostatic interaction potential of
the form used in the non-relativistic method, This relativistic Hamiltonian has
(i c rx(k) . \l (k) - ß(k) c2-
H = ~~ )
where o:and ßare 4 x4 matrices and \l is the Nabla operator. Using the variation-
al method in the same manner as before and taking a Slater determinant as the
wave function, one obtains two sets of first-order inhomogeneous differential
equations to be solved for all electrons of the atom. This most complicated ver-
sion of atomic calculations is called the relativistic Hartree-Fockmethod or Dirac-
Fock method (DF) (48). Various papers calculating the ground-state configurations
of superheavy atoms by this method have been published since 1969 (49-51). A
complete discussion is given by J. B. Mann (35).
These very complicated inhomogeneous coupled differential equations can
again be simplified by using Slater's approximation. This method is therefore called
the relativisticHartree-Fock-SlaterorDirac-Fock-Slater (DFS) (52-53) calculations,
and they have also been done by several authors for the superheavy elements
The results for the ground-state configurations of all superheavy elements up
to 172 and for element 184 are given in Table 2 (35,50,56-60). In only very few
cases are the results different for the two best methods, DF and DFS, but the
differences are so small that no final decision can be made.
The first difference that becomes obvious in comparison to the empirical
continuation of the electron filling discussed above (29, 37-39) occurs at elements
110 and 111. The calculated ground-state is s2d8and s2d9, respectively, which is
not at all common in the homologs of the two elements.
Also, beginning with element 121, every element has a different ground-state
configuration than that predicted by simple extrapolations. The main reason for
this behavior is that, unexpectedly, an 8p electron state becomes occupied at
element 121, and at least one of these electrons remains bound through all the
following elements. In the 160 region the difference between the simple predictions
and the results of the calculations is already so large that the position of the
elements in the periodic table is changed drastically. (For an overview and com-
Table 2. Atomic ground-state configurations for the neutral elements 103 to 172 and 184 according to Mann (35) and Pricke and Waber (85,60), using
self-consistent Dirac-Fock calculations
Rn core + 5/14+
Z = 120 core +
Z = 120 core +
+ 8pr/2 +
Z = 120 core +
+ 8pr/2 5g1S6/14+
8p 7d 6/
8p 7d 6/25g2
7d109529pr/2 8 P ~ / 2
7d109529pr/2 8 P ~ / 2
Z = 172 core +
~ 4 6
8p2 6/37d 5g11
8p2 6/37d 5g12
parison, see the periodic table in Fig. 21, incorporating the results of the prediction
of the elements up to 172 taken from Fricke et al. (56)). This disagreement with
the results expected from a simple continuation of the periodic table is of course,
a result of the interpretation of the periodic system in terms of chemical behavior,
but the primary reason is the surprising order of filling of the outer electron shells
in this region.
If we try to proceed to even heavier elements, the calculations come to a halt
at Z = 174 because at this element the ls level reaches the negative continuum of
the electrons at a binding energy of 2mec2 ==1 MeV and the calculation breaks
down. To proceed further, Fricke (61) introdueed a phenomenologieal description
of the quantum-eleetrodynamical effects into the SCF caleulations (60), which
shifts the binding energies of the inner electrons back to lower values. Using this
method, he was then able to study the electron configurations of the elements
beyond Z == 174.
2. Trends of the Chemieal and Physical Properties
The detailed and sophistieated caleulations of the electronic-ground states of the
atoms are very worthwhile as an important, though only the first step toward
predieting the ehemical and physical properties of superheavy elements, beeause
chemistry consists not only of the properties of the atoms but also of the molecules
and their behavior. Ab-initio ealeulations of molecules were introduced for small
moleeules and small Z, and the state of the art is still far away from the point
that allows actual calculations of the chemieal properties of superheavy molecules.
A first step in this direction has been taken by Averill et al, (62), who ealeulated
the wave function of (110)F 6 using a muffin-tin method.
a) Trends Emerging from the Calculations. Althoughwe are not able to ealeulate
the properties of superheavy moleeulesat the present time, the atomic calculations
give us more than just the electronie structure of the neutral elements.
One has to bear in mind that two elements from the same ehemieal group,
which often have the same outer eleetronie strueture will be chemically and physi-
eally slightly different. This can be to some extent explained as the effeet of their
somewhat different sizes, changed ionization potential, and the different energies
and radial distributions of the wave functions between analogous shells. These
quantities are also determined directly by the atomic calculations. The size of the
atom or ion correlates strongly with the principal maximum of the outermost
electronicshell, as found by Slater (63),thus giving a first estimate of this important
magnitude. Sometimes the expectation value of <r> of the outermost shell is
used as the radius, but the agreement with experiment is not so good.
There is eonsiderable agreement that the ionization potentials have to be
ealculated in the adiabatie approximation, in which it is assumed that during the
removal of an electron sufficient time elapses for the other electrons to rearrange
themselves, so that the ionization potentialis given by thedifference in total energy
of the two calculations with m and m-l electrons. The other method, taking the
calculated energy eigenvalues (64), can only be used as an approximation to this
In the first part of the periodic table it is relatively easy to make the connection
between these quantities and a chemical interpretation because of the few shells
involved and their large separation. Moreover, the influence of the inner electron
shells is rather small so that the outer electron configurations are very similar in
the same chemical group at different periods. When we proceed to higher elements
at the end of the periodic table, the number of shells increases, the binding energy
of the last electrons decreases, and there is competition between shells; hence the
influence of the inner electrons becomes more significant. This rather complex
behavior is further complicated by the fact that relativistic effects now begin
to be important and the coupling between the angular momenta of the elec-
trons changes from LS to intermediate or j-j coupling. All these effects and their
relative influences are taken into account in the ab-initio calculations. Of course,
to prove their reliability in the superheavy region of elcmcnts, they have to
reproduce the complex structure and its relationship to chemical behavior in the
known part of the periodic table, which in all cases is done for example for the
groundstate configurations of the atoms. The main change due to relativistic effects
is the splitting of all shells with l i=0 into two subshells with j ===l+1/2 and
j ==l -1/2. This means that, for example the pstate splits into the Pl/2 subshell
DIRAC- SLATER- EIGENVALUES
VALENCE ELECTRONS FORsp'p CONFIG.
25 5075 100125150
PROTON OR ATOMIC NUMBER
8 d ~
Fig. 7. Comparison of the eigenvalues of the ns, npl/2 and np3/2 electrons in the group-IVA
elements using DFS calculations. This figure illustrates the very strong dependence of the spin-
orbit splitting between the two p states as a function of the atomic number. For element 164,
the 9s and 8d 3/2 levels are also drawn (85)