Dr. Beck and his group concentrate on understanding the most complicated
atoms, the
transition metal,
lanthanide, and
actinide series.
These pose a
large computational challenge to existing theory and require
the simultaneous inclusion of relativistic and
correlation
(many body) effects. These atoms are technologically important
in solid state and fusion (plasma) devices. Current properties
of interest include electron affinities, hyperfine structure,
and lifetimes (see below); these are among the most difficult properties
to describe accurately. The methodology used is called relativistic
configuration interaction (RCI).
The work of the Beck research group has led to the discovery
of an entirely new
anion
(negatively charged ion), Be^{−},
and has been the first to characterize the lanthanide and actinide
anions
Ce^{−}, Pr^{−},
Nd^{−}, Pm^{−},
Sm^{−}, Eu^{−},
Gd^{−}, Tb^{−},
Dy^{−}, Ho^{−}, Er^{−},
Th^{−}, Pa^{−},
U^{−}, Np^{−},
Pu^{−}, Cm^{−}, Bk^{−},
Cf^{−}, and Es^{−} with
ab initio
calculations.
They have also been able to understand and remove
the systematic discrepancy between theory and experiment for
hyperfine structure in transition metal atoms. Much of what the Beck
group has learned can be transported to molecular and solidstate
calculations involving these atoms.
In the molecular domain, Dr. Beck has developed better
intermolecular potentials to describe the transport of natural gas.
He used manybody theory to obtain potential energy surfaces
of substances such as methane, ethane, and propane. He then did
Monte
Carlo simulation studies to predict various thermophysical
properties of the gas.
Currently the group has two PCs with 2.5 GHz AMD processors
dedicated to this work,
which is supported by the National Science Foundation and
until recently the Department of Energy. Typically, there are two
graduate students and one postdoctoral research associate working
with Prof. Beck.
Electron
affinities indicate how strongly an anion
is bound relative to its neutral atom ground
state. The lower the anion energy, the greater affinity that atom has
for an extra electron.
EA’s are calculated by
direct comparison of the total energies from two separate RCI
calculations (one of the anion, one of the neutral ground
state or an excited state threshold where appropriate) that have
been equally treated with respect to correlation.
Many anions have more than one bound state. In these cases physicists
refer to the “binding energy” of each state, and the largest
binding energy (for the lowest lying state) is the atom’s EA.
Transition
probabilities are a measure of the strength of the transition
between two atomic states. Other terminologies used more or less
interchangeably for this property are
“oscillator strengths” or “f values.”
TP’s are calculated as an absorption event; an atom or ion in an
initial state absorbs a photon of a particular energy, leaving it in a
excited final state.
Knowing all possible TP’s with the same final state allows one to
calculate the lifetime of that state, which is the inverse of the sum
of those TP’s (similar to the combination of parallel resistors
in electronics).
The energy difference between levels indicates the
color (wavelength) of the absorbed photon (or emitted in the case of
a decay), and the TP affects the width and intensity of a line in an
emission spectrum as in the neutral Fe case below:
Hyperfine
structure (normally abbreviated “hfs,” but labeled
as “HS” in the table below)
A and B constants are measures of the amount of
splitting of a
level due to the magnetic dipole and electric quadrupole
moments
of the nucleus. The energy differences between levels of an atom or
ion that are calculated in the RCI methodology, the fine
structure of the system, are typically orders of magnitude greater than
the hfs. The hfs A’s and B’s are actually computed
as a small perturbation at the end of an RCI calculation with the
atomic wavefunctions determined in the fine structure calculation.
The hfs A’s and B’s are difficult to calculate because
they are often affected by electron
correlation of the atom or ion’s core. Additional configurations
(single electron core replacements) that might not normally be needed
to accurately place the energy levels in the spectrum are often
required, which enlarges the size of the calculation.
In addition to the above, several other atomic properties may be
calculated. While each relativistic calculation has a unique total
J (total angular momentum), it is often useful to characterize
an atomic state by its total L (orbital angular momentum of all
electrons,
analogous to the planets’ revolutions about the sun) and total
S (spin angular momentum of all electrons, analogous to the
planets rotating about their axes).
A property that is related to J, L, and S is the
Landé
g value. From the computational perspective it
can be easily obtained once one has the atomic wavefunctions from the
fine structure calculation, and from the experimental
point of view, often this is the quantity that is directly measured
while L and S are derived from analysis of the g
values of multiple states of the same J.
More recently the Beck research group has focused on
photodetachment
cross
sections. These calculations are an indication of the probability
that an incident photon of a given wavelength will eject an electron
from a particular anion state leaving it in a particular final state
of the neutral atom. If one knows the largest cross sections of the
important anionatom level combinations, one can apply this knowledge
to assist the analysis of experimental data. The experimenters shine
a laser at the anion, which strips it of an electron. Peaks in the
photodetached electron kinetic energy spectrum then correspond to the
energy differences between anion and neutral atom states.
Such calculations are typically made in conjunction with an EA study.
Finally, the group has also performed
autodetachment
calculations.
These involve metastable anion states that lie above one or more neutral
atom levels. An anion in such a state will spontaneously detach an
electron and emit a photon, rather than absorb a photon as in the
bound state (stable) case. Under certain conditions studied by
Dr. Beck’s research group the lifetime of these states could be
long enough that experimenters could see photodetachments from
these metastable states
and mistake them for transitions from bound anion states.
The following
periodic
table shows the elements that Dr. Beck and
his research group have studied over the years.
The color coding indicates which of the three main properties has
been the focus of
the calculations for each atom or ion that has been studied (with
secondary colors representing multiple properties). In many
cases multiple ionization stages have been explored; see the
list of publications or
time line for more information.
H 1 

He 2 
Li 3 
Be 4 

EA 
TP 
HS 
EA TP 
TP HS 
HS EA 

B 5 
C 6 
N 7 
O 8 
F 9 
Ne 10 
Na 11 
Mg 12 

Al 13 
Si 14 
P 15 
S 16 
Cl 17 
Ar 18 
K 19 
Ca 20 
Sc 21 
Ti 22 
V 23 
Cr 24 
Mn 25 
Fe 26 
Co 27 
Ni 28 
Cu 29 
Zn 30 
Ga 31 
Ge 32 
As 33 
Se 34 
Br 35 
Kr 36 
Rb 37 
Sr 38 
Y 39 
Zr 40 
Nb 41 
Mo 42 
Tc 43 
Ru 44 
Rh 45 
Pd 46 
Ag 47 
Cd 48 
In 49 
Sn 50 
Sb 51 
Te 52 
I 53 
Xe 54 
Cs 55 
Ba 56 

Hf 72 
Ta 73 
W 74 
Re 75 
Os 76 
Ir 77 
Pt 78 
Au 79 
Hg 80 
Tl 81 
Pb 82 
Bi 83 
Po 84 
At 85 
Rn 86 
Fr 87 
Ra 88 

Rf 104 
Db 105 
Sg 106 
Bh 107 
Hs 108 
Mt 109 
Ds 110 
Rg 111 
Uub 112 
Uut 113 
Uuq 114 
Uup 115 
Uuh 116 
Uus 117 
Uuo 118 



La 57 
Ce 58 
Pr 59 
Nd 60 
Pm 61 
Sm 62 
Eu 63 
Gd 64 
Tb 65 
Dy 66 
Ho 67 
Er 68 
Tm 69 
Yb 70 
Lu 71 



Ac 89 
Th 90 
Pa 91 
U 92 
Np 93 
Pu 94 
Am 95 
Cm 96 
Bk 97 
Cf 98 
Es 99 
Fm 100 
Md 101 
No 102 
Lr 103 

[This portion of the site was originally linked from the March 12,
2009 issue of
Tech Today and has been more recently mentioned in the
Michigan Tech Research Magazine.]
In order to understand the computational complexities of the lanthanide
and actinide
elements (the bottom two rows of the periodic chart above), it is first
necessary to consider a brief review of the notations
atomic physicists use to characterize the levels of atoms and ions.
Atomic physicists characterize the electrons in an atom or ion by
principle
quantum number n and
orbital
quantum number
l (a measure
of the electron’s angular momentum about the nucleus).
The layered
shells
of an atom are numbered by n=1, 2, 3, etc., with
each layer containing subshells for l=0 up to
l=n−1, and each of
these subshells can hold up to 2(2l+1) electrons.
Historically, the notation for the values of l have been given letter
designations: s (0), p (1), d (2), f (3), etc.
(students are often taught a mnemonic based on the shapes of the first
three of these; spherical, pear, and
dumbell).
The above quantum mechanical rules then mean that the first shell of
electrons is made up of a single 1s subshell, the second shell is made
up of 2s and 2p,
the third is made up of 3s, 3p, and 3d, etc.
Typically, a large periodic chart of the elements will include the
ground state (lowest level)
configuration
of the atom. This is
essentially a list of the number of electrons in each subshell.
The somewhat unconventional
wide
chart below shows the general trend of
added electrons. Each subsequent element has one more proton in its
nucleus and one more electron than the previous element.
The periodicity of the chart breaks the elements into
blocks
associated with each type of subshell.
The two columns on the left (alkali metals and alkaline earth metals,
color coded yellow here)
represent sequential filling of s subshells (maximum
of two electrons), and He has been moved from its usual position with
the other noble gases to reflect this point. On the right side of the table
(nonmetals, halogens, noble gases, etc.; color coded green here)
p subshells (maximum of six electrons)
are filled as one moves across the table. Transition metal series
(blue elements below) represent filling up to ten d electrons,
and the lanthanides and actinides (red elements below) fill
the fourteen possible electrons of the f subshells. In this
chart the “fblock” containing the lanthanides and
actinides which are marked by the grey dots has
been reinserted into the main table, while in the chart above it is
extracted to the bottom in the more traditional representation.
In a configuration, the number of electrons of a subshell
is denoted by a superscript.
For example, the C (carbon, Z=6) ground state has six
electrons in the
1s^{2}2s^{2}2p^{2}
configuration (think of “loading” two 1s electrons with
H and He, two 2s electrons with Li and Be, one 2p electron
with B, and the second 2p with C).
Often, closed subshells or completed rows are omitted
from a configuration and assumed. For example, Si
(silicon, Z=14 just below C) has a ground state configuration
that could be described fully by
1s^{2}2s^{2}2p^{6}3s^{2}3p^{2}, more succinctly by 3s^{2}3p^{2},
or in its simplest form just 3p^{2}.
Note also, that the table does not fill beyond the first two rows in what is
perhaps the obvious order, quantum number n
followed by quantum number l, but in a manner such that d and
f electrons “lag behind.” For example, the second from the
bottom row, which contains the lanthanide series, fills the 6s,
4f, 5d, and 6p subshells.
To further complicate things, in certain cases in the d
and fblocks, the near
degeneracy
(same energy) of Nd subshells
with (N+1)s and
(N−1)f subshells results in a “trade off.”
These spots are indicated in the table above by the dark outlines. For example,
Cr and Cu in the top blue row prefer to have a half full
or completely full
3d subshell. The ground state configuration of Cr is
3d^{5}4s rather than
3d^{4}4s^{2} as expected by the
general trend. Similarly, the configuration for Cu is
3d^{10}4s rather than
3d^{9}4s^{2}.
In the fblock the trade off is between d and f
electrons. For example, Gd in the middle of the lanthanides prefers a half
full 4f subshell with a configuration of
4f^{7}5d6s^{2}
rather than 4f^{8}6s^{2}.
In Pd and Th, this swapping is doubled as indicated by the thicker border:
they have
configurations of 4d^{10} (no 5s) and
6d^{2}7s^{2} (no 5f).
Finally, the unique case of Lr (dark blue) prefers to add a 7p
electron rather than the expected 6d, resulting in a
5f^{14}7s^{2}7p ground
state.
Because of the trends shown in the above chart,
early theoretical attempts on the lanthanides focused on
trying to add a 4f electron, but large, incorrect results were
obtained.
In the early 1990’s experimental work done at the University of Toronto
detected a number of the lanthanide anions and estimated their
EA’s to be quite small, but those techniques could not further
characterize them.
A theoretician at Toronto proposed that these lanthanides were created by
the attachment of a 6p (or in rare cases a 5d) electron.
Subsequent calculations by several different research groups, including
Dr. Beck’s (see papers by D. Datta, K. D. Dinov, and D. R. Beck in
the list of publications),
on the simpler atoms at the ends of the lanthanide and actinide rows
indicated these EA’s were consistent with existing
observation and have been confirmed in detail in a series of experiments
from 1998 to the present by a group of physicists at the University of
Nevada, Reno.
One case in particular, a detailed analysis of Ce^{−}
calculations
by O’Malley and Beck in 2006, has served as a model for interaction
between experiment and theory to remove some of the previously mentioned
discrepancies in EA’s as a reinterpretation of earlier
data from the Reno group was later confirmed by experimenters at Denison
College in Ohio.
Since the early 1990’s the Beck research group has pushed toward
the center of the lanthanide row one element at a time, expanding the
RCI methodology to accommodate ever increasingly difficult problems.
In addition to quantum numbers n and l, there are others
that provide even further detail of the state of electrons in a
configuration (yes, the above discussion is actually the simplified version).
Without getting into further detail of the quantum physics, the placement
of each electron within the subshell further differentiates electronic
configurations from one another. To use an every day example, consider
the possible ways to place eggs in an egg carton, in particular one of
the smaller half dozen sized cartons to provide an analogy to a p
subshell which also has a maximum of six “bins” to place
electrons.
There is only one way to distribute six eggs in a half dozen carton,
all bins full (and the same is true for distributing zero eggs, all bins
empty, of course). These
cases are depicted in the red cartons below.
If one places a single egg or five eggs there are six different bins to
either hold the egg or to be the empty spot in the latter case. These
are the blue carton cases below.
The closer the carton is to half full the more ways there are to pack it,
and with a little work one can count up twenty different ways to place
three eggs as in the yellow cartons
(assuming the eggs, like electrons, are
indistinguishable from one another).
Now consider a carton with ten bins, analogous to a d subshell.
There are ten ways to place one or nine eggs, but 252 ways to pack the
carton with five eggs (which is already becoming too tedious to bother
with graphics).
The corresponding values for the fourteen bin fshell analogue are
fourteen (almost empty or almost full) and 3432 (seven eggs).
Not only are the number of configurations increasing, but the ratio
of complexity of the ends of the row to the center (getting back to the
red lanthanide and actinide rows in the periodic chart above)
is also more pronounced.
Furthermore, because the quantum mechanical calculation is essentially
a twodimensional matrix problem this ratio is squared, meaning
lanthanide calculations near the center of the row are, in principle,
tens of thousands of times more complex than the ends!
By 2007 the Beck research group was working on methods of simplifying
calculations of Nd^{−},
the most complex anion RCI calculation at the time,
with a neutral ground state of
4f^{4}6s^{2} and anion
6p attachment
(4f^{4}6s^{2}6p).
These calculations took about six months of human effort to set up many
separate calculations, the largest of which took about a day of CPU time
on a 2.4 GHz PC.
The breakthrough in the methodology occurred with the development of an
algorithm to preselect and restrict the configurations within the
f^{n} group of electrons that
best described the lowest few levels of the neutral atom.
Since the anion states are created by attaching an electron to one of
these lowlying neutral levels, this restriction also optimizes the
anion calculations.
By retaining only the few important
f^{n} configurations
the size of the calculations is dramatically reduced, with the most
relative savings occurring for the difficult centerrow elements.
With this new methodology and and improved data preparation codes,
the Beck group has automated much of the process to the point that the human
time involved for each anion has been reduced to a few weeks.
While the individual pieces of the twodimensional matrix calculations
are more complicated near the center of the row, the size of the matrix
is roughly ten times bigger than the endrow elements rather than
tens of thousands as would otherwise be the case.
This savings is even more critical in cases as described above with
the “extra” d electron.
For example, Gd^{−} was found to have both
6p attachments to it ground state and 6s attachments
to higher “opens”
4f^{7}5d^{2}6s
levels, resulting in anion states with both
4f^{7}5d6s^{2}6p and
4f^{7}5d^{2}6s^{2}
configurations.
Without these new approximations there would be insufficient memory or
computing power to tackle these problems on the group’s
current PCs, but even if the memory and speed were increased several
times over, the largest individual calculations would take months
to a year of CPU time (compared to less than a day) with very little
effect on the binding energies.
The most recent results of the O’Malley and Beck
lanthanide and actinide anion studies are presented
graphically below. These data are from PRA 78, 012510 (2008);
PRA 79, 012511 (2009); and
PRA 80, 032514 (2009); see the
list of publications.
Many of these recent anion
EA’s, Nd^{−} through Er^{−}
and Np^{−} through Es^{−},
are the first available ab initio calculations, and most of the
118 lanthanide and 41 actinide
predicted bound states are also unknown experimentally.
The neutral energy levels presented below
are from:
Atomic Energy Levels − The
Rare Earth Elements, edited by W. C. Martin, R. Zalubas, and L.
Hagan, Natl. Bur. Stand. Ref. Data Ser. Natl. Bur. Stand. (U.S.) Circ.
No. 60 (U.S. GPO, Washington, DC, 1978).
Energy Levels and Atomic Spectra of Actinides,
edited by J. Blaise and J.F. Wyart,
International Tables of Selected Constants 20, Paris (1992).
They can also be obtained from the
NIST Atomic Spectra
Database and
NIST Handbook of
Basic Atomic Spectroscopic Data. Neutral excited states of Lr are
from the following calculations:
A. Borschevsky, E. Eliav, M. J. Vilkas, Y. Ishikawa, and U. Kaldor,
Eur. Phys. J. D 45, 115 (2007).
S. Fritzsche, C. Z. Dong, F. Koike, and A. Uvarov,
Eur. Phys. J. D 45, 107 (2007).
Y. Zou and C. Froese Fischer,
Phys. Rev. Lett. 88, 183001 (2002).
E. Eliav, U. Kaldor, and Y. Ishikawa,
Phys. Rev. A 52, 291 (1995).
Click any plot below for a larger version with detailed labeling of
neutral and anion configurations incuding
term symbols,
which indicate the levels’ total J, L, and
S in the form
^{2S+1}X_{J}, where X is the
capital letter for L corresponding
to the single electron l lower case letters:
S (0), P (1), D (2), F (3),
G (4), H (5), etc.
Neutral spectra are shown for energies up to
1 eV. For
comparison of this unit, light of the
visible
spectrum (red to violet)
has photon energies of 1.7 to 3.2 eV. The
plots that are opened are best viewed full size, the details may be
illegible if your browser shrinks them to fit its window.
These plots are also
available in the following pdf files (the landscape
versions are condensed, and the black and white versions are more
printer friendly):
email Dr. Beck
back to the Beck
group’s main page
back to the MTU Physics page
The Beck research group gratefully acknowledges current funding by the
National Science Foundation (1981 to
present) and prior funding from the
Department of Energy (1992 to 2007).
Any opinions, findings, and conclusions or recommendations expressed on
this website are those of the Beck research group and do not necessarily
reflect the views of NSF or DOE.