Variational solution of the dirac equation within a multicentre basis set of gaussian functions

Mark, F.; Lischka, Bans; Rosicky, Franz

doi:10.1016/0009-2614(80)80214-4

Cited by 63 publications

(38 citation statements)

References 10 publications

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“…The Gaunt term also carries the full spin-spin interaction, whereas the gauge-dependent term g gauge , Equation (23), must be included for the full orbitorbit interaction. [36] The first four-component relativistic Hartree-Fock calculations based on the Dirac-Coulomb Hamiltonian and using finite basis sets [38][39][40] were flawed because the coupling of large and small components were not taken into account. [41][42][43][44] From the Dirac equation for an electron in a molecular field, Equation (16), the exact coupling is found to be given by Equation (24):…”

Section: The Electronic Hamiltonianmentioning

confidence: 99%

Relativistic Hamiltonians for Chemistry: A Primer

Saue

2011

ChemPhysChem

454

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Section: The Electronic Hamiltonianmentioning

confidence: 99%

Relativistic Hamiltonians for Chemistry: A Primer

Saue

2011

ChemPhysChem

454

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“…Fair results have been obtained for atoms [16,19] for which finite difference methods [ l l , 121 are more powerful. Molecular calculations with the expansion method [8][9][10]20-241 led to rather discouraging results. Schwarz and Wallmeier [6] coined the term "variational collapse" for the following observations.…”

Section: Variational Collapse and The Importance Of The Correct "Schrmentioning

confidence: 99%

“…A final comparison and evaluation of the merits and drawbacks is made in Sec. 8 and TableI. We shall come to the conclusion that for most applications the best approach is C2 or C3 (they differ very little), which consists of a rather simple manipulation of the matrix representation of the Dirac operator that is exact for a complete basis and that has the correct Schrodinger limit in the given basis.…”

Section: Introductionmentioning

confidence: 99%

Basis set expansion of the dirac operator without variational collapse

Kutzelnigg

1984

Int J of Quantum Chemistry

356

187

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The eigenstates of the matrix representation of the Dirac operator for c +a do not approach their nonrelativistic counterparts in the same basis. This wrong "Schrodinger limit" is shown to be the main reason for the phenomenon known as "variational collapse." After a short review of existing proposals to overcome the "variational collapse," a systematic study of the possible ways to avoid it is given. All discussed approaches are analyzed in terms of various criteria that one wants to fulfill. The most promising approach consists of a free-particle Foldy-Wouthuysen (FW) transformation on operator level and a back transformation on matrix level (approaches C2 and C3). This implies a modification of the free-electron part of the matrix representation of the Dirac operator and leads to the correct Schrodinger limit (and if one wishes even the correct Pauli limit) in the same basis (and to the exact results for a complete basis). The potential energy is unchanged, which makes the application to n -electron systems straightforward. Projection of the Dirac operator to positive energy states does not remove the variational collapse unless this is done in a very special way.

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“…Because the integral Dirac equation in momentum space is unitarily equivalent to the one in position space, it is essential to also use balanced basis in the present momentum-space approach both to achieve the desired accuracy and to avoid spurious solutions. It is to be noted that a failure to incorporate "balance" in the basis is known to erroneously produce the relativistic correction of H l that is several orders of magnitude larger than the limiting value [23,24]. In a related work on the integral DF equation in momentum space, Rosicky proposed that each of the four components of the Dirac 4-spinors be expanded in terms of the same set of scalar basis functions (181.…”

Section: Vo(p)+(p) = Jd'qv(p -Q)+(s)mentioning

confidence: 99%