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 solid-state calculations involving these atoms.

In the molecular domain, Dr. Beck has developed better inter-molecular potentials to describe the transport of natural gas. He used many-body 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 anion-atom 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 “f-block” 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 1s22s22p2 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 1s22s22p63s23p2, more succinctly by 3s23p2, or in its simplest form just 3p2.

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 f-blocks, 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 3d54s rather than 3d44s2 as expected by the general trend. Similarly, the configuration for Cu is 3d104s rather than 3d94s2. In the f-block 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 4f75d6s2 rather than 4f86s2. In Pd and Th, this swapping is doubled as indicated by the thicker border: they have configurations of 4d10 (no 5s) and 6d27s2 (no 5f). Finally, the unique case of Lr (dark blue) prefers to add a 7p electron rather than the expected 6d, resulting in a 5f147s27p 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 f-shell 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 two-dimensional 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 4f46s2 and anion 6p attachment (4f46s26p). 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 fn 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 low-lying neutral levels, this restriction also optimizes the anion calculations. By retaining only the few important fn configurations the size of the calculations is dramatically reduced, with the most relative savings occurring for the difficult center-row 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 two-dimensional matrix calculations are more complicated near the center of the row, the size of the matrix is roughly ten times bigger than the end-row 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 “open-s” 4f75d26s levels, resulting in anion states with both 4f75d6s26p and 4f75d26s2 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+1XJ, 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):

[color portrait][black and white portrait][color landscape][black and white landscape]






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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.