Quaternions Dirac

Hence, the relativistic analog of the spin-restriction in nonrelativistic closed-shell Hartree-Fock theory is Kramers-restricted Dirac-Hartree-Fock theory. We should emphasize that our derivation of the Roothaan equation above is the pedestrian way chosen in order to produce this matrix-SCF equation step by step. The most sophisticated formulations are the Kramers-restricted quaternion Dirac-Hartree-Fock implementations [286,318,319]. A basis of Kramers pairs, i.e., one adapted to time-reversal s)mimetry, transforms into another basis under quatemionic unitary transformation [589]. This can be exploited not only for the optimization of Dirac-Hartree-Fock spinors, but also for MCSCF spinors. In a Kramers one-electron basis, an operator O invariant under time reversal possesses a specific block structure. 

The second term on the right-hand side of the equation gives for point nuclei directly the one-electron spin-orhit operator (2) of the Breit-Pauli Hamiltonian and can he eliminated to give a spin-free equation that becomes equivalent to the Schrddinger equation in the non-relativistic limit. In a quaternion formulation of the Dirac equation the elimination becomes particularly simple. The algebra of the quaternion units is that of the Pauli spin matrices... 

In the quaternion modified Dirac equation the spin-free equation is thereby obtained simply by deleting the quaternion imaginary parts. For further details, the reader is referred to Ref. [13]. 

We then turn to the question of how to eliminate the spin-orbit interaction in four-component relativistic calculations. This allows the assessment of spin-orbit effects on molecular properties within the framework of a single theory. In a previous publication [13], we have shown how the spin-orbit interaction can be eliminated in four-component relativistic calculations of spectroscopic properties by deleting the quaternion imaginary parts of matrix representations of the quaternion modified Dirac equation. We show in this chapter how the application of the same procedure to second-order electric properties takes out spin-forbidden transitions in the spectrum of the mercury atom. Second-order magnetic properties require more care since the straightforward application of the above procedure will extinguish all spin interactions. After careful analysis on how to proceed we... 

Table A2.3. Multiplication table for the Dirac characters of the quaternion group Q.

Table A2.4 Characteristic determinants obtained in the diagonalization of the Dirac characters for the quaternion group Q.

Table A2.6. Character table for the quaternion group Qfound by the diagonalization of the MRs of the Dirac characters. ...

The aim of this volume is twofold. First, it is an attempt to simplify and clarify the relativistic theory of the hydrogen-like atoms. For this purpose we have used the mathematical formalism, introduced in the Dirac theory of the electron by David Hestenes, based on the use of the real Clifford algebra Cl(M) associated with the Minkwoski space-time M, that is, the euclidean R4 space of signature (1,3). This algebra may be considered as the extension to this space of the theory of the Hamilton quaternions (which occupies an important place in the resolution of the Dirac equation for the central potential problem). 

The subalgebra Cl+(E3) of Cl(E3) is the field of the Hamilton quaternions. We see that this field plays an important role in the theory of the Dirac electron in a central potential. 

Note. In what follows, the identification of the Hamilton quaternion q and the Hestenes spinor tjj with the Pauli spinor and the Dirac spinor P is based on the articles [28,44]. Perhaps, it is not the shortest one, but it allows a step-by-step conversion of P into vice versa when P is expressed by means of its four complex components. 

The DIRAC package [55,56], devised by Saue and collaborators, rather than exploiting the group theoretical properties of Dirac spherical 4-spinors as in BERTHA, treats each component in terms of a conventional quantum chemical basis of real-valued Cartesian functions. The approach used in DIRAC, building on earlier work by Rosch [80] for semi-empirical models, uses a quaternion matrix representation of one electron operators in a basis of Kramers pairs. The transformation properties of these matrices, analysed in [55], are used to build point group transformation properties into the Fock matrix. 

This requirement does not fix the Dirac matrices uniquely, and thus the whole Dirac theory and all systematic approximations to it could equally well be formulated in terms of general four-dimensional quaternions, which are independent of a special representation and rely only on the algebraic properties of the Clifford algebra [8-10]. Such an implementation of the Dirac theory is known to speed up diagonalisation procedures significantly, and has successfully been employed in modem four-component relativistic program packages like Dirac... 

The development of MOLFDIR came to an end in 2001 and some of the developers of this program joined forces with a new Scandinavian program, Dirac, that emerged in the mid 1990s [518]. Dirac contains an elegant implementation of Dirac-Hartree-Fock theory as a direct SCF method [317] in terms of quaternion algebra [318,319]. For the treatment of electron correla-... 

T. Saue, H. J. A. Jensen. Quaternion symmetry in relativistic molecular calculations The Dirac-Hartree-Fock method. /. Chem. Phys., 111(14) (1999) 6211-6222. 

L. Visscher, T. Saue. Approximate relativistic electronic structure methods based on the quaternion modified Dirac equation. /. Chem. Pkys., 113(10) (2000)3996-4002. 

The overlap and potential energy matrices are matrices of dimension for each component, both in the modified and unmodified Dirac method, and both are Hermitian. The unmodified Dirac matrices are quaternion matrices, and there are n 2n — 1) unique quantities (real numbers) in each matrix. The modified spin-free Dirac matrices are real, with n n + l)/2 unique quantities. The reduction in the number of unique quantities is a factor of approximately 4. 

The kinetic energy matrix is a full matrix of dimension rp- in both cases. As for the other two, the unmodified Dirac kinetic energy matrix is a quaternion matrix, with ArP unique quantities, whereas the spin-fl-ee modified Dirac kinetic energy matrix is a real matrix with rp unique quantities, and the resultant reduction is a factor of 4. However, if an uncontracted basis is used, the spin-free modified Dirac kinetic energy matrix is symmetric, and is the same as the pseudo-large-component overlap matrix. 

It is ironic that spin, which is the only non-classical attribute of quantum mechanics, is absent from the pioneering formulations of Heisenberg and Schrodinger. Even in Dirac s equation, the appearance of spin is ascribed by fiat to Lorentz invariance, without further elucidation. In reality, both Lorentz invariance and spin, representing relativity and quantum mechanics, respectively, are properties of the quaternion field that underpins both theories. 

We used the DIRAC program suite. Time-reversal symmetry [30] and Abelian point groups, including C2v, are fully exploited in the DIRAC program with the help of quaternion algebra. [31] We briefly summarize quaternion algebra for the case of C2v... 

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