Journal of Nuclear and Radiochemical Sciences,Vol. 3, No. 1, pp. 109–111, 2002

109

Atomic and Molecular Structure Calculations for Superheavy Elements
B. Fricke∗,a and V. Pershinaba
Department of Physics, University of Kassel, D-34109 Kassel, Germanyb
Gesellschaft für Schwerionenforschung, D-64220 Darmstadt, Germany
Received: November 13, 2001; In Final Form: April 12, 2002
A first prediction of the chemistry of superheavy elements has been given 30 years ago using many-electron relativistic atomic structure calculations in the quality of that time.1 The first relativistic molecular calculation for asystem with a transactinide element was published in 1977.2 Since then relativistic atomic and molecular structure
calculations have been considerably improved and applied to many atoms and molecules. In this review we try tooutline the general theory and the development of the quality of such calculations with less and less approximations
as well as better and better numerical methods and higher accuracy. But even if one has quite accurate total energies,
quantities like effective charges, overlap populations or radii of the outermost wavefunctions are still a very valuable
information for an effective prediction of the chemical behavior of the superheavy elements.

1. Introduction
The structure of the heavy elements has attracted much interest since the earlier days of quantum mechanics. It was soon realized that the chemical behavior is a result of the structure of theouter valence electrons of the atoms. The discovery and availability of the actinides and transactinides in physical and chemical research started this field in the mid forties. Theoreticalmethods became available in the early sixties which allowed topredict the electronic structure for very heavy elements. Nowadays, the analysis of many physical and chemical experiments
as well as a series of computations has lead to a considerable
understanding of these elements. Today the predictive power is
so enhanced that we are relatively sure about the magnitude ofmost of the physical quantities and of the chemical behavior ofthese elements, which are either just observed on a single atombasis (Z = 107 to 112) or will hopefully be observed in the regionof the superheavy elements in the vicinity of the nuclear charge114 in the future.

∂| Ψ
∂t
respectively (in the time-independent form)Ĥ |Ψ = −i(1a)Ĥ |Ψ = E |Ψ .

(1b)

In order to solve this equation, one has to follow various steps:— At first one has to define the Hamiltonian which one would
like to solve.
— Secondly, one has to separate the nuclear and electronic coordinates with the help of the Born-Oppenheimer approximation and— thirdly, choose an ansatz for the wavefunction Ψ.
2.1. Choice of the Hamiltonian. As a first step in all “normal” calculations only the Coulomb interaction between allcharged particles usually is taken into account.
 2NZN∇2αi∇α1 (Ze)2∼++Ĥ =∑ 2M ∑ 2m ∑ 2 |Rα − Rβ |α=1i=1β=1



Vαβ
Z



(3)

allows to separate eq 1 in two equations. (R stands for the nuclear coordinates, r for the electronic coordinates, the index e isthe electronic quantum number, and the index N is the nuclear
quantum number.)∇2∑ 2mi ψe (R, r) + [Ee (r) − Vi j − Vαi ] ψe (R, r) = 0 ,(4)i∇α∑ M ψN,e (R) + [E − Ee (r) − Vαβ] ψN,e (R) = 0 .

(5)Equation 4 is the basis for Quantum Chemistry, and eq 5 isthe equation for the nuclear motion which is the basis for
Molecular Dynamics.

We all believe in the Schrödinger- (or Dirac-) equation

i=1

Ψ(R, r) = ψe (R, r) · ψN,e(R)α2. General Theory+∑

As the next step the interaction between the magnetic moments of the spins and orbital angular momenta of the electrons(and nuclei) are taken care of (usually in the form of the Breitinteraction). Finally in the most advanced form the interaction
due to quantum electrodynamical effects will be considered.
2.2. Separation of Nuclear and Electronic Coordinates.
The Born-Oppenheimer ansatz for the total wavefunction−1 Z
Ze2e2+ ∑|rβi − Rα | 2 j=1 |rαi − rβ j | 
Vαi

2.3. Choice of a Wavefunction. The simplest approach is a
product of single particle functions ψ = Πi ϕi which is, however,
not antisymmetric (Hartree approximation). Another possibility
is the ansatz of a Slater determinant1
ψ= √n!

φ1 (n)

(2)

−∑

k=1

φ2 (n)

...
...
...

φn (1)φn (2)
..
.

.

(6)

φn (n)

Z

Coulomb Part,φ∗k (r  ) φi (r  )e2dr  φk (r) = εi φi (r)|r −r  |(7)Exchange Part.

Vi j∗Corresponding author. E-mail: fricke@physik.uni-kassel.de.
FAX: +49 561 804-4006

φ2 (1)
φ2 (2)..
.

This is an antisymmetric ansatz, and the Pauli principle is automatically included. If the ansatz 6 is plugged into eq 4, one gets
after a variational procedure the Hartree-Fock equations
n Ze2∇2 Z− i − + ∑ φ∗k (r i ) φk (r j )dr  φi (r))2m r(r−r
k=1n.

φ1 (1)
φ1 (2)..
.

Equation 7 is solved iteratively in a self-consistent field (SCF)
way. The wavefunctions in eq 7 have to be specified as atomicc 2002 The Japan Society of Nuclear and Radiochemical Sciences
Published on Web 6/30/2002

110 J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

FrickeTABLE 1: Results from atomic structure calculations for the elements 104–108.

Property

DbSg

BhHs

MCDF
6d7s2 7p

RfCCSD
6d 2 7s2

6d 3 7s2

6d 4 7s2

Excited state
6d 2 7s2
Excitation energy / eV
0.5
Ionization energy / eV
6.5
For the references see table 2 in Reference 3.

6d7s2 7p
0.3
6.01

6d 2 7s2 7p
1.9
6.89

6d 3 7s2 7p
1.64
7.10

6d 5 7s2(90%)
6d 4 7s2 7p
0.79
7.7

6d 6 7s2(91%)
6d 5 7s2 7p
0.63
7.6

Ground state

orbitals (AO) in case of an atom or molecular orbitals (MO) incase of a molecule, which themselves are usually constructed ina linear combination of atomic orbitals (LCAO)
nφk = ∑ cik χi .
i=1

The resulting matrix equations with the coefficient matrix c,the overlap matrix S and the Fock matrix F are often called
Roothaan LCAO-MO-SCF method
F c=E Sc.
These equations are the basis for most practical computational
methods in Quantum Chemistry.
2.4. Choice of a Basis Set. In all of these calculations usually the single particle wavefunctions φi are expanded in either
Gaussian-type orbitals (GTO)χuvw = N xu yv zw exp(−αr2 )where u, v, w are the non-negative integers and α is the positive
orbital exponent, orSlater-type orbitals (STO)χnlm (r, θ, ϕ) = N · rn∗ −1Z∗ rexp(− ∗) Y lm (θ, ϕ)n a0

which do not represent the inner orbitals well because they do
not have the right number of nodes, or numerical atomic orbitals
which are generated from numerical atomic structure calculations. These orbitals have the correct number of nodes and showa correct asymptotic behavior.
2.5. Correlation Energy. The problem is that HartreeFock takes into account the interelectronic repulsion only in anaverage way. To account for this deficiency one has to include
the so-called “correlation energy”. There are various methods toaccount for this:One possible method is the configuration interaction (CI).
Here (in principle) every Slater determinant ψk which can beconstructed from the occupied and unoccupied Hartree-Fock orbitals contributes to the total wavefunction∞Ψ = ∑ ck ψk .
k=1

Depending on the fact if only single (S), double (D), triple(T), etc. excited configurations are included, the method is called
CIS, CISD, CISDT, etc.
In the multi-configuration self-consistent field (MCSCF)
method a limited number of configuration state functions areused, but not only the expansion coefficients ck are determined.
In addition, the single particle wavefunctions φu which construct
the Slater determinants ψk are optimized .
Another possibility to treat correlation is the many-body
perturbation theory (MBPT). In the Møller-Plesset theory (MP)the Hamiltonian is split into the unperturbed Hartree-Fock operator H (0) and the perturbed rest H (1)Ĥ = Ĥ (0) + Ĥ (1) .

The most sophisticated description is the coupled-cluster(CC) theory. Other typical quantum chemical methods are e.g.
the zero differential overlap approximation, NDDO, CNDO,
or any kind of Hückel approximation including the extendedHückel theory. All these methods shall not be discussed here.
2.6. Density Functional Approach. Slater had the idea touse the exchange term in the Hartree-Fock equations as an effective potential. The exchange of a free electron gas is given as(α = 1).
VXα = −3α(3ρ/8π)1/3 ,where α is an adjustable parameter, and X refers to the exchange. Thus the one-electron energies and wavefunctions are
determined by solving−∇2 +Vc +VXα φi = εi φi .

In principle, this method is also exact. It is based on theHohenberg-Kohn theorem which shows that the ground state
density n corresponds uniquely to a given external potential. In
general the total energy of a system isE [n] = T [n] + E ext [n] + E h [n] + E xc [n] + Eαβ ,where T [n] is the kinetic energy, E ext [n] the electron-nucleus interaction, E h [n] the direct electron-electron interaction, E xc [n]the exchange-correlation part of the electron-electron interaction and Eαβ the nucleus-nucleus interaction energy. The problem with this method is that one does not know the exact E xc [n],but only approximations.
3. Relativistic Atomic CalculationsFirst relativistic atomic structure calculations for atoms became available at the end of the 60s. They led to a first qualitative discussion of the structure of the valence electrons in very
heavy elements. However, the quality of these results was onlyof limited overall predictive power. Later, the development ofhigh-speed computers as well as relativistic atomic codes madeit possible to approach the goal of an improved understanding ofthese very heavy elements with Z ≥ 100. A summary is given in
the review Reference 3.
An example for these calculations is presented in Table 1
which summarizes atomic structure calculations for the elements 104–108.
4. Relativistic Molecular Calculations
A new period in the study of elements started when it became
possible to perform calculations of small molecules which contain one or more atoms with a charge number above Z = 100.
To our knowledge, the first application with the newly developed discrete variational method was a calculation of (110)F6 in1978.13 In the 80s this method has been extended considerably.
During the last years, calculations of improved wave functions
became possible even for more complex molecules. Since thena relatively large number of already complex molecules of allelements have been calculated. In order to compare the qualityof the various methods and approximations discussed here one
should compare those for the same molecule. Unfortunately,

Atomic and Molecular Structure Calculations

J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

TABLE 2: Results from various calculations of diatomic Au2 .

Ref.
4555
5
6
789101011our12Method
P-HFS
SRPP-HF
SRPP-MP2
SRPP-CCSD
SRPP-CCSD (T)cp-CCSD (T)CASSCF-MRSDCIZORA-GGA
ZORA-GGA
ZORA-GGAI
ZORA-GGA2
ZORA-GGA
4-comp. RGGA
exp

Re / a.u.
4.62
4.934.64
4.74
4.73
4.70
4.76
4.75
4.76
4.76
4.75
4.88
4.67

De / eV
2.52
0.83
2.53
2.00
2.22
2.19
2.00
2.26
2.31
2.53
2.26
2.33
2.50
2.29

−1

ωe / cm
201
160
205
185
182
187
174
178
177
174
183
169
191

there is no such system besides the diatomic Au2 where we summarize the binding energy De , bond distance Re , and vibrationalfrequency ωe from several calculations. A few references aregiven in Table 2.
5. Conclusion
Many features of the transactinide elements are now relatively
well understood. The statement that relativistic effects play adominant role in the prediction of physical quantities and chemical behavior is now well established. Extended calculations willcertainly change details of the predictions for specific elementsin the future, but not the gross structure. The quality of molec-111

ular calculations has still to be improved, but for a number ofelements one already knows large details of their chemical behavior.
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