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Quaternion Symmetry in Relativistic Molecular Calculations: the Dirac–Hartree–Fock Method T

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Quaternion Symmetry in Relativistic Molecular Calculations: the Dirac–Hartree–Fock Method T Quaternion symmetry in relativistic molecular calculations: The Dirac–Hartree–Fock method T. Saue, H. Jensen To cite this version: T. Saue, H. Jensen. Quaternion symmetry in relativistic molecular calculations: The Dirac–Hartree– Fock method. Journal of Chemical Physics, American Institute of Physics, 1999, 111 (14), pp.6211- 6222. 10.1063/1.479958. hal-03327446 HAL Id: hal-03327446 https://hal.archives-ouvertes.fr/hal-03327446 Submitted on 27 Aug 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. JOURNAL OF CHEMICAL PHYSICS VOLUME 111, NUMBER 14 8 OCTOBER 1999 Quaternion symmetry in relativistic molecular calculations: The Dirac–Hartree–Fock method T. Sauea) and H. J. Aa Jensen Department of Chemistry, University of Southern Denmark—Main campus: Odense University, DK-5230 Odense M, Denmark ͑Received 5 April 1999; accepted 9 July 1999͒ A symmetry scheme based on the irreducible corepresentations of the full symmetry group of a molecular system is presented for use in relativistic calculations. Consideration of time-reversal symmetry leads to a reformulation of the Dirac–Hartree–Fock equations in terms of quaternion algebra. Further symmetry reductions due to molecular point group symmetry are then manifested by a descent to complex or real algebra. Spatial symmetry will be restricted to D2h and subgroups, and it will be demonstrated that the Frobenius–Schur test can be used to characterize these groups as a whole. The resulting symmetry scheme automatically provides maximum point group and time-reversal symmetry reduction of the computational effort, also when the Fock matrix is constructed in a scalar basis, that is, from the same type of electron repulsion integrals over symmetry-adapted scalar basis functions as in nonrelativistic theory. An illustrative numerical example is given showing symmetry reductions comparable to the nonrelativistic case. © 1999 American Institute of Physics. ͓S0021-9606͑99͒31637-8͔ I. INTRODUCTION groups are termed boson irreps. When time reversal is in- cluded in the symmetry group as well, it is no longer pos- Symmetry plays an important role both in the theoretical sible to form representations such that the product of two formulation and the computational implementation of mo- operators is represented by the product of the corresponding lecular quantum mechanics. On the one hand, it elucidates representation matrices. However, it is still possible to form theory, possibly leading to its reformulation in a more com- a system of matrices, a corepresentation, that may be broken pact form. On the other hand, it may greatly reduce the com- down to irreducible forms.1,3,4 putational effort in the implementation of theory. The full symmetry group of a molecular system within In this paper we consider the exploitation of symmetry the Born–Oppenheimer approximation consists of all unitary in relativistic molecular calculations in the finite basis ap- or antiunitary operators that commute with its electronic proximation when the Hamiltonian is totally symmetric with ͑ Hamiltonian.1 Physically, the antiunitary operators involve respect to time-reversal symmetry this excludes external ͒ time reversal, whereas rotations, reflections, and inversion of magnetic fields and nuclear spins . We shall specifically in- ͑ ͒ space and spin coordinates are unitary operators. As the vestigate the closed-shell Dirac–Hartree–Fock DHF product of two antiunitary operators is a unitary operator, we method, and we shall concentrate on the dominating compu- need to consider only one antiunitary operator, namely the tational task, namely the construction of Fock matrices. time-reversal operator itself; all others can then be expressed A straightforward implementation of symmetry in rela- as products of the time-reversal operator and a unitary op- tivistic molecular calculations would be to construct a basis erator. adapted to double group symmetry ͑see Refs. 5–8 and refer- In the nonrelativistic realm, the Hamiltonian is spin-free, ences therein͒, or, alternatively, to use projective ͑ray͒ allowing a separation of spin and spatial symmetry. In the representations.9,10 We shall, however, pursue a somewhat relativistic realm, however, spin symmetry is lost and the different approach in which we work with the irreducible single point groups are replaced by double groups.2 These corepresentations of the full symmetry group. A characteris- are constructed from the single groups by the introduction of tic feature of the symmetry scheme that we present is the an extra element E¯ , corresponding to a rotation of 2␲ about transfer of symmetry information into the algebra of the an arbitrary axis. This leads to a doubling of the number of problem at hand. Considering only time-reversal symmetry, elements, but in general not to a doubling of the number of the Dirac–Hartree–Fock problem can be efficiently treated irreducible representations ͑irreps͒. For instance, the single using quaternion algebra.11 In the nonrelativistic limit this point group D2 has four irreps, but in the corresponding corresponds to using orbitals instead of spin-orbitals. How- double group only one extra irrep appears. The extra irreps ever, as we shall see, for some point groups the problem may are spanned by fermion functions and are therefore denoted be reduced to complex or real algebra, which leads to further fermion irreps. Correspondingly, the irreps of the single considerable savings in computation time and memory re- quirements of approximately a factor two and four, respec- a͒Current address: IRSAMC—LPQ, Universite´ Paul Sabatier, 118, route de tively, compared to not invoking point group symmetry. This Narbonne, 31062 Toulouse Cedex, France. symmetry scheme allows us to work with scalar basis func- 0021-9606/99/111(14)/6211/12/$15.006211 © 1999 American Institute of Physics Downloaded 30 Oct 2002 to 130.79.34.222. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp 6212 J. Chem. Phys., Vol. 111, No. 14, 8 October 1999 T. Saue and H. J. Aa Jensen tions ͑so-called atomic orbitals͒ adapted to boson irreps quaternion algebra naturally allows exploitation of time- rather than fermion irreps, just as in nonrelativistic calcula- reversal symmetry. Finally we apply these results to the tions. This has the advantage that it allows us to work with Dirac equation given above. conventional integral packages for nonrelativistic ab initio calculations ͑although small modifications to skip the inte- gral classes not used is, of course, computationally highly A. Quaternion algebra and time-reversal symmetry beneficial͒ and, as we have seen above, there is more sym- metry to exploit. Spatial symmetry will be restricted to D2h A quaternion number is given by and subgroups thereof. This is the set of all single point 3 groups with no elements of order higher than two. We shall ϭ ϭ ϩ ϩ ˇ ϩ ˇ ෈ ` ͑ ͒ q ͚ v⌳e⌳ v0 v1ˇı v2 v3k; v⌳ R. 4 therefore denote them binary groups. The symmetry scheme ⌳ϭ0 11,12 presented in this article has been implemented in DIRAC, In his landmark paper on ‘‘magnetic electrons,’’ 14 Pauli in a a code for four-component relativistic molecular calculations footnote refers to a remark by P. Jordan, stating that the which incorporates the nonrelativistic two-electron integral algebra of the spin matrices times imaginary i is identical to generator HERMIT.13 that of the quaternion units ˇı, ˇ, and kˇ . In other words, we The article is organized as follows: In Sec. II we focus can make the isomorphous connection on time-reversal symmetry and reduce the Dirac equation to quaternion form. In Sec. III the structure of Dirac spinors is ϭ $ ␴ ϭˇ$ ␴ ϭ ˇ $ ␴ ͑ ͒ e1 ˇı i z ; e2 i y ; e3 k i x . 5 analyzed in terms of the boson irreps of the molecular point group. In Sec. IV we present the new symmetry scheme This allows the expression of quaternion numbers as based on quaternion algebra that automatically provides qϵ͓s,v͔$sϩi͑␴˜ v͒, ͑6͒ maximum time reversal and point group symmetry reduction • in computational implementations based on the finite basis where s can be considered the scalar part, v the vector part, ␴ϭ ␴ ␴ ␴ approximation, with basis functions adapted to boson irreps, and ˜ ( z , y , x). Observe that there is a certain arbitrari- as in nonrelativistic implementations. The symmetry scheme ness in the above connection in the sense that we can reas- is applied to the Dirac–Hartree–Fock problem and the com- sign the Pauli matrices to any cyclic permutation of the putational gains are illustrated by a test case. quaternion units. Accordingly, the quaternion units are equivalent, and in a complex number aϩib the imaginary i may correspond to either ˇı, ˇ,orkˇ without changing its al- gebraic properties ͓we shall later see that the specific choice II. TIME-REVERSAL SYMMETRY in Eq. ͑6͒ corresponds to quantization of spin along the ͔ ϭ Our point of departure is the time-independent Dirac z-axis . In view of this we shall term s v0 the real part and equation v1 , v2 , and v3 the i-, j-, and k-imaginary parts, respectively, of a quaternion number. ˆ ␺ϭ ␺ ˆ ϭ␤Ј 2ϩ ͑␣ ˆ ͒ϩ ˆ ͑ ͒ hD E ; hD mc c •p V, 1 The multiplication rule of the quaternion units is conse- ˆ ˆ quently in one-to-one correspondence to the multiplication where hD is the Dirac operator in the field V of fixed nuclei 15 with the zero point aligned with the nonrelativistic energy rule of the spin matrices, ␣ ␤Ј ␴ ␴ ϭ␦ ϩ ⑀ ␴ ⇒ ϭϪ␦ Ϫ⑀ ͑ ͒ scale.
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