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Considerations on describing non-singlet spin states in variational second order density matrix methods
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10.1063/1.3672087
/content/aip/journal/jcp/136/1/10.1063/1.3672087
http://aip.metastore.ingenta.com/content/aip/journal/jcp/136/1/10.1063/1.3672087

Figures

Image of FIG. 1.
FIG. 1.

The v2DM(PQG) energy, shown for the atoms boron, fluorine, nitrogen, and oxygen, is a convex function of the expectation value under both pure spin state and ensemble spin state conditions. The maximal spin projection has the highest energy, even when only the expectation value is imposed. The energy without additional spin constraints is equivalent to the energy of a spin unconstrained problem, due to the spin independence of the Hamiltonian.

Image of FIG. 2.
FIG. 2.

The differences between the v2DM potential energy graph of the oxygen dimer under pure spin state conditions (Sec. ???) for different spin projections M of the same spin state S are remarkable. In the dissociation limit, the difference in the energy for different spin projections seems primarily attributable to the different expectation value of the dissociated atoms because their energies are very similar to atomic energies obtained under the same expectation value only (Table I).

Image of FIG. 3.
FIG. 3.

Differences between the v2DM potential energy graph of the carbon dimer under pure spin state conditions (Sec. ???) for different spin projections M of the same spin state S are remarkable. In the dissociation limit, the difference in the energy for different spin projections seems primarily attributable to the different expectation value of the dissociated atoms. Only the S = 2, M = 1 state gives atomic energies in the dissociation limit that are slightly higher than those obtained under .

Image of FIG. 4.
FIG. 4.

Although both the v2DM(PQG) triplet and quintuplet potential energy graph (black, red) for the oxygen dimer should converge to the same dissociation limit, only the same spin projections converge to a very similar dissociation limit, which practically coincides with the dissociation limit under a constraint on expectation value only, (green lines).

Image of FIG. 5.
FIG. 5.

Although both the v2DM(PQG) triplet and quintuplet potential energy graph (black, red) for the carbon dimer should converge to the same dissociation limit, only the same spin projections converge to a very similar dissociation limit, which practically coincides with the dissociation limit under a constraint on expectation value only, (green lines).

Image of FIG. 6.
FIG. 6.

The pure spin state conditions for the zero spin projection v2DM(PQG) potential energy graph (solid lines) of the carbon dimer singlet yield lower energies than those for the triplet around equilibrium bond length. Yet FCI(FC) calculations (dotted lines) prove that their relative order should be exactly opposite.

Image of FIG. 7.
FIG. 7.

The pure spin state conditions for the zero spin projection v2DM(PQG) potential energy graph (solid lines) treat all different spin states of the oxygen dimer equivalently in the dissociation limit. Yet they do not give a fully satisfying picture of its properties; the quintuplet state becomes much too strongly binding compared to FCI(FC) calculations (dotted lines).

Image of FIG. 8.
FIG. 8.

The v2DM(PQG) potential energy graph (solid lines) for the maximal spin projections of the singlet, triplet, and quintuplet of the carbon dimer under pure spin state conditions (Sec. ???) are not equivalent in the dissociation limit. Nonetheless, they do give the correct order of singlet and triplet potential energy graph, similar to that in FCI(FC) calculations (dotted lines).

Image of FIG. 9.
FIG. 9.

The v2DM(PQG) potential energy graph (solid lines) for the maximal spin projections of the singlet, triplet, and quintuplet of the oxygen dimer under pure spin state constraints (Sec. ???) are not consistent: they do not converge to equivalent dissociated states. Moreover, they give a singlet-triplet gap that is much too small compared to FCI(FC) data (dotted lines).

Image of FIG. 10.
FIG. 10.

When subspace constraints are imposed on the singlet, triplet, and quintuplet v2DM(PQG) potential energy graph under pure spin state conditions (Sec. ???), both the violation of size-consistency and the absence of degeneracy among dissociated states with different spin projections (solid lines) are corrected in the resulting potential energy graph (dotted lines). The shapes of the different spin surfaces remain poor, though, as can be seen by comparing to the FCI energies in Figure 9.

Tables

Generic image for table
Table I.

The properties of the dissociated atoms in the oxygen dimer in the dissociation limit, denoted by the superscript “atom,” are remarkably similar for molecular states that lead to dissociated atoms with the same spin projection, both when pure spin state conditions are imposed and when only spin projection is specified (column “not fixed”). This illustrates the inability of the pure state spin constraints to treat spin in a size-consistent manner.

Generic image for table
Table II.

The pure state spin constraints are much stronger when imposed on the triplet atoms separately, as shown in this table, than when they are imposed on the dissociated singlet, triplet or quintuplet oxygen dimer (shown in Table I), even though they should be equivalent to be size-consistent.

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/content/aip/journal/jcp/136/1/10.1063/1.3672087
2012-01-05
2014-04-24
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Considerations on describing non-singlet spin states in variational second order density matrix methods
http://aip.metastore.ingenta.com/content/aip/journal/jcp/136/1/10.1063/1.3672087
10.1063/1.3672087
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