Computer Physics Communications 133 (2001) 396–406 www.elsevier.nl/locate/cpc
An atomic program for energy levels of equivalent electrons: lanthanides and actinides Sverker Edvardsson a,∗ , Daniel Åberg a,b a Department of Physics, Mid Sweden University, S-851 70 Sundsvall, Sweden b Condensed Matter Theory Group, Uppsala University, Box 530, S-751 21 Uppsala, Sweden
Received 26 February 1999; received in revised form 30 May 1999
Abstract A program written in C is presented to carry out brute force calculations in order to derive energy levels for an equivalent electronic configuration. Relativistic effects are partly neglected except for the spin-orbit interaction. Since the main relativistic effects are indirect, i.e. causing a contraction of the core which in turn causes the outer shells to expand, they are included to a high degree through the use of appropriate Slater integrals. The program is especially useful for primarily unfilled f -shells of the rare-earth or actinide ions. Modifications of the program to include spin−spin, spin−other orbit, Breit interaction etc. is straight forward. The program is also general in the sense that there is no need to find out or generate any Racah coefficients of fractional parentage. The complete energy matrix is diagonalized with all operators interacting simultaneously thus allowing mixing of all quantum numbers. This result in all energy eigenvalues and eigenvectors that in turn for example are partly responsible for the polarized dipole, quadrupole, . . . transitions within the unfilled shell. Free ion configuration interaction is accounted for through the use of standard CI operators. The Stark splitting can be studied via the standard crystal field Hamiltonian. Magnetic field influence on the energy levels may also be studied. 2001 Elsevier Science B.V. All rights reserved. PACS: 71.20.Eh; 78.20.Bh
PROGRAM SUMMARY
Computers: SUN SPARC Ultra 10 Installations: IBM SP, PC’s running Solaris for Intel or Linux
Title of program: Lanthanide Catalogue identifier: ADMZ
Operating systems or monitors under which the program has been tested: SUN Solaris 2.6 & 2.7, IBM AIX 4, Linux
Program Summary URL: http://cpc.cs.qub.ac.uk/summaries/ADMZ
Programming language used: gcc version 2.8.1
Program obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ireland
Memory required to execute with typical data: 8 Mwords
* Corresponding author.
E-mail address:
[email protected] (S. Edvardsson). 0010-4655/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 0 - 4 6 5 5 ( 0 0 ) 0 0 1 7 1 - 5
S. Edvardsson, D. Åberg / Computer Physics Communications 133 (2001) 396–406 No. of bits in a word: 32 No. of processors used: 1 Has the code been vectorized or parallelized?: no No. of bytes in distributed program, including test data, etc.: 80 521 Distribution format: tar gzip file Keywords: Rare-earth ions, crystal field energy levels, configuration interaction, electron correlation, effective operators, magnetic field, three-particle operators, determinantal product states Nature of physical problem Diagonalizations of various one-, two- and three-particle operators that result in energy eigenvalues and their eigenfunctions. The various operator parameters have to be calculated ab initio or taken from experiment; these are used as input to the program. The approach is non-relativistic (it is empirically known that the main relativistic and correlation effects can be absorbed in the input parameters) making the program mainly useful for systems such as studies of energy structure of open valence shells of rare-earth or actinide atoms. Method of solution First a basis set of determinantal product states are generated for a pure electronic configuration according to the exclusion principle of Pauli. Using this basis, the matrix elements of the upper triangle of the Hermitean matrix are calculated. The matrix elements of all operators are added. The complete matrix is subsequently diagonalized to give the various eigenvalues and eigenvectors. Configuration interaction effects are accounted for by using various standard twoand three-particle operators [1–3]. Restriction on the complexity of the problem The Hamiltonians are all classical so this program is inappropriate (without appropriate modifications) for relativistic systems with energy levels that have angular dependencies that cannot be absorbed into the used classical operators. Other practical restrictions are mainly three: computing time, matrix size and disk storage. Other than that there are no remaining limitations built into the
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software. An example of these three practical limitations are the electron configuration f 7 . In this case the size of the energy matrix is 3432 × 3432 which place a great demand on primary memory (approx. 200 Mb in total) and takes several hours to diagonalize on a fast workstation. However the main problem is not the diagonalization, in fact the calculation of the three-particle matrix elements is even more demanding. After several optimizations (application of several symmetries, code and compiler optimizations), this part of the calculation still takes approximately a day on a SUN Ultra 10. Fortunately, the calculations of the three-particle matrix elements only need to be made once and are then saved to the local disk. Other files are storage of the Hermitean matrix and subsequently its eigenvectors; these can become quite large. This problem is solved within the program since these files are written in a compressed format (zlib) causing a size reduction of approximately a factor of ten. Typical running time The running times range from a few minutes (f 2 ) up to ∼ a day (f 7 ), i.e. depending on the problem and whether reading of previously stored matrix elements is used or not. Unusual features of the program The linear-algebra operations (zhpeev: complex packed format diagonalization routine) are performed by the standard numerical libraries CLAPACK and CBLAS [4]. Several files that otherwise could be huge are saved in gz-format using the library zlib [5]. Rules for fast calculations of three-particle matrix elements are derived (described below) and implemented in the program. The program does not need or use any tables of Racah coefficients of fractional parentage.
References [1] [2] [3] [4] [5]
K. Rajnak, B.G. Wybourne, Phys. Rev. 132 (1963) 280. K. Rajnak, J. Opt. Soc. Amer. 55 (1965) 126. B.R. Judd, Phys. Rev. 141 (1966) 4. http://www.netlib.org/clapack/. http://sunsite.doc.ic.ac.uk/packages/zip/zlib/zlib.html. .
LONG WRITE-UP 1. Theoretical background and optimization The complexity of the f -elements of the periodic system is generally known to be tremendous. Even with the fast computers of today, accurate ab initio calculations have still only been able to attack the f 2 electronic configuration, e.g., the term structure was calculated using the multiconfiguration Dirac–Fock program GRASP [1]. Improved results were later obtained using the relativistic coupled-cluster method [2]. For three or more f -
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S. Edvardsson, D. Åberg / Computer Physics Communications 133 (2001) 396–406
electrons we are still obliged to phenomenological model Hamiltonians who nevertheless are well motivated in higher order perturbation theory. These parameters can also be calculated ab initio, see e.g. [3]. The use of determinantal product states as basis is clearly an disadvantage for the group theoretical aspects of the energy levels, but is indeed appealing with regard to the simplicity in calculating the matrix elements for any number of electrons. This approach also means that there is no need to find out or generate any fractional parentage coefficients of Racah. These considerations have also been used and appreciated earlier in practical calculations by Garcia and Faucher [4]. The mathematical expressions in this section will be presented as short and concise as possible. The operators that are considered in Lanthanide are either of the type one-, two- or three-particle operators. The usual rules to calculate the matrix elements of the first two types may be found in any quantum mechanics textbook, e.g., Judd [5]. The non spherical part of the Hamiltonian used in the program may be written: X X b 2 + β G(G b 2 ) + γ G(R b 7) b= F k (nf, nf )fˆk + ξ(ri ) lˆi · sˆi + α L H k=2,4,6
+
X
i=2,3,4,6,7,8
i
XX → Xˆ btp (θi , φi ) + µB − li + 2 sˆi . T tˆi + (1 − σt )Atp rit C B · i
i
tp
(1)
i
The F k and ξ(r) are the electrostatic and spin-orbit parameters. fk are angular parts of the electrostatic interactions. α, β, γ are the Trees parameters associated with the two-body correction terms for free ion b 7 ) are Casimir’s operators for groups G2 and R7 , b 2 ) and G(R configuration interaction effects [6,7]. G(G respectively. Other free-ion CI effects are included by means of three-body operators. Judds investigations of these resulted in the operators tˆi with corresponding parameters T i [8]. Atp represents the crystal environment which is related to the crystal field parameters (CFP) through Btp = (1 − σt )Atp r t and σt are the shielding factors by Sternheimer [9]. However, the shielding parameters are unnecessary if a cluster calculation is performed since the adjustments of the rare-earth wavefunctions would then already been taken into account. In the electrostatic approach Atp may be calculated according to: Z ρ(Er 0 ) p+1 Ct −p (α 0 , β 0 ) d rE0 (2) Atp = (−1) 0 r t +1 with ρ(Er 0 ) being the external charge densities and Ct −p (α 0 , β 0 ) is a spherical tensor of rank t and projection −p. A broadening of the basis set (i.e. CI) would imply that also correlation crystal field is accounted for. In future calculations we will try to get access to the density ρ(Er 0 ) using density functional theory. The last term is the standard magnetic field Hamiltonian. Most of the operators are standard and the derivations of their matrix elements will not be repeated here. However, the matrix elements of the operators associated with configuration interaction deserve some attention since those are nontrivial to calculate.
2. Two-particle operators of Trees, Wybourne and Rajnak b2 operator, it needs to be expressed differently because To enable calculation of the matrix elements for the α L 2 b may be written: of the basis used here. Using unit tensor operators, α L b2 = αl(l + 1)(2l + 1) U 1 · U 1 , αL P where U k = i uki and the summation is over all electrons within the f -shell. It is interesting to note that X X uki · uki + 2 uki · ukj . Uk · Uk = i
i
