Lowest Average Energy Research Articles

Exact ground states and domain walls in one dimensional chiral magnets

We determine exactly the phase structure of a chiral magnet in one spatial dimension with the Dzyaloshinskii-Moriya (DM) interaction and a potential that is a function of the third component of the magnetization vector, n3, with a Zeeman (linear with the coefficient B) term and an anisotropy (quadratic with the coefficient A) term, constrained so that 2A ≤ |B|. For large values of potential parameters A and B, the system is in one of the ferromagnetic phases, whereas it is in the spiral phase for small values. In the spiral phase we find a continuum of spiral solutions, which are one-dimensionally modulated solutions with various periods. The ground state is determined as the spiral solution with the lowest average energy density. As the phase boundary approaches, the period of the lowest energy spiral solution diverges, and the spiral solutions become domain wall solutions with zero energy at the boundary. The energy of the domain wall solutions is positive in the homogeneous phase region, but is negative in the spiral phase region, signaling the instability of the homogeneous (ferromagnetic) state. The order of the phase transition between spiral and homogeneous phases and between polarized (n3 = ±1) and canted (n3 ≠ ±1) ferromagnetic phases is found to be second order.

ChemInform Abstract: The Large closo-Borane Dianions, BnHn2- (n = 13-17) Are Aromatic, Why Are They Unknown?

The relative stabilities of the unknown larger closo-borane dianions B(n)()H(n)()(2)(-) (n = 13-17), were evaluated at the B3LYP/6-31G level of density functional theory by comparing the average energies, E/n, and also by the energies using the model equation: B(n)()(-)(1)H(n)()(-)(1)(2)(-) + B(6)H(10) --> B(n)()H(n)()(2)(-) + B(5)H(9) (n = 6-17). Starting with the small closo-borane, B(5)H(5)(2)(-), the sequential addition of BH groups is represented by formal transfer from B(6)H(10) to build up larger and larger clusters. Most of the energies for these sequential steps are exothermic, but not for the B(12)H(12)(2)(-) to B(13)H(13)(2)(-) and the B(14)H(14)(2)(-) to B(15)H(15)(2)(-) stages. The cumulative total energies (DeltaH(add)) of these BH group additions, based on B(5)H(5)(2)(-) as the reference zero, tend to increase with increasing cluster size. DeltaH(add) indicates that the larger unknown closo-boranes B(13)H(13)(2)(-) to B(17)H(17)(2)(-) are more stable than B(9)H(9)(2)(-), B(10)H(10)(2)(-), and B(11)H(11)(2)(-); this agrees with E/n and with Lipscomb's earlier conclusion based on the PRDDO average energies. B(13)H(13)(2)(-), B(14)H(14)(2)(-), and B(15)H(15)(2)(-) are less stable than B(12)H(12)(2)(-), which has the lowest average energy on a per vertex basis among the closo-borane dianions. However, the total DeltaH(add) treatment indicates the larger B(16)H(16)(2)(-) and B(17)H(17)(2)(-) to be favorable relative to B(12)H(12)(2)(-), because of the larger number of vertexes. The formation of B(13)H(13)(2)(-) from B(12)H(12)(2)(-) is especially unfavorable. The further formation of B(14)H(14)(2)(-) and B(15)H(15)(2)(-) via BH transfer also is endothermic. These are not the only thermodynamic difficulties in building up large closo-borane dianions beyond B(12)H(12)(2)(-). The highly exothermic disproportionation of larger and smaller closo-borane dianions, e.g., B(12+)(n)()H(12+)(n)()(2)(-) + B(12)(-)(n)()H(12)(-)(n)()(2)(-) --> 2B(12)H(12)(2)(-) (n = 1-5), also indicate possible synthetic problems in preparing larger closo-boranes with more than 12 vertexes under condition where smaller boranes are present. All the larger closo-B(n)()H(n)()(2)(-) (n = 13-17) cluster exhibit three-dimensional aromaticity, judging from the computed Nucleus Independent Chemical Shifts (NICS), which range from -30.9 to -36.5 ppm. The trends in NICS values are similar to the variations in the bond length alternations, Deltar. Thus, the qualitative relationships between geometric and magnetic criteria of aromaticity found earlier for the smaller clusters extends to the larger closo-borane dianions, B(n)()H(n)()(2)(-) (n = 13-17).

The Large closo-Borane Dianions, B(n)()H(n)()(2-) (n = 13-17) Are Aromatic, Why Are They Unknown?

The relative stabilities of the unknown larger closo-borane dianions B(n)()H(n)()(2)(-) (n = 13-17), were evaluated at the B3LYP/6-31G level of density functional theory by comparing the average energies, E/n, and also by the energies using the model equation: B(n)()(-)(1)H(n)()(-)(1)(2)(-) + B(6)H(10) --> B(n)()H(n)()(2)(-) + B(5)H(9) (n = 6-17). Starting with the small closo-borane, B(5)H(5)(2)(-), the sequential addition of BH groups is represented by formal transfer from B(6)H(10) to build up larger and larger clusters. Most of the energies for these sequential steps are exothermic, but not for the B(12)H(12)(2)(-) to B(13)H(13)(2)(-) and the B(14)H(14)(2)(-) to B(15)H(15)(2)(-) stages. The cumulative total energies (DeltaH(add)) of these BH group additions, based on B(5)H(5)(2)(-) as the reference zero, tend to increase with increasing cluster size. DeltaH(add) indicates that the larger unknown closo-boranes B(13)H(13)(2)(-) to B(17)H(17)(2)(-) are more stable than B(9)H(9)(2)(-), B(10)H(10)(2)(-), and B(11)H(11)(2)(-); this agrees with E/n and with Lipscomb's earlier conclusion based on the PRDDO average energies. B(13)H(13)(2)(-), B(14)H(14)(2)(-), and B(15)H(15)(2)(-) are less stable than B(12)H(12)(2)(-), which has the lowest average energy on a per vertex basis among the closo-borane dianions. However, the total DeltaH(add) treatment indicates the larger B(16)H(16)(2)(-) and B(17)H(17)(2)(-) to be favorable relative to B(12)H(12)(2)(-), because of the larger number of vertexes. The formation of B(13)H(13)(2)(-) from B(12)H(12)(2)(-) is especially unfavorable. The further formation of B(14)H(14)(2)(-) and B(15)H(15)(2)(-) via BH transfer also is endothermic. These are not the only thermodynamic difficulties in building up large closo-borane dianions beyond B(12)H(12)(2)(-). The highly exothermic disproportionation of larger and smaller closo-borane dianions, e.g., B(12+)(n)()H(12+)(n)()(2)(-) + B(12)(-)(n)()H(12)(-)(n)()(2)(-) --> 2B(12)H(12)(2)(-) (n = 1-5), also indicate possible synthetic problems in preparing larger closo-boranes with more than 12 vertexes under condition where smaller boranes are present. All the larger closo-B(n)()H(n)()(2)(-) (n = 13-17) cluster exhibit "three-dimensional aromaticity", judging from the computed Nucleus Independent Chemical Shifts (NICS), which range from -30.9 to -36.5 ppm. The trends in NICS values are similar to the variations in the bond length alternations, Deltar. Thus, the qualitative relationships between geometric and magnetic criteria of aromaticity found earlier for the smaller clusters extends to the larger closo-borane dianions, B(n)()H(n)()(2)(-) (n = 13-17).

Hund's rules

We review the present state of our undertanding of Hund's first and second rules, their domains of validity, and of generalizations in cases where the rules in their original form are invalid. These exceptions occur mainly in atomic configurations with more than one open shell withl ≥ 1, but also in cases with large orbital angular momentaL. We present a derivation of thealternating rule, which is, in some sense, a generalization of Hund's first rule, and present new rules, which generalize Hund's second rule. The importance of the concept ofunnatural parity states for an understanding of these rules is stressed. It is demonstrated that the lowest singlet-triplet average energy for the sameL corresponds to a pair of unnatural parity states (theunnatural parity rule). For sufficiently smalll 1 andl 2, the lowest average energy is that of the pair of states with the maximum possibleL (maximum-L rule), though exceptions are found already for moderately large values ofl 1 andl 2. Our analysis indicates that the validity of Hund's second rule is to some extent an accidental consequence of the minimisation of an elementary function ofl 1,l 2, andL — which does not depend monotonically onL — over states of unnatural parity, and that a more general and more fundamental rule should be formulated in terms of this function. We also discuss the generalisation of Hund's rules to molecules, as well as violations of them, with particular emphasis on the inversion of Hund's first rule by spin-polarization in molecules.

The singularities of the t-matrix after a canonical transformation

In the case of a pair-interaction, a geom of the canonical transformation method is given, and a criterion is deduced to distinguish among their solutions that one which gives the lowest average energy in the quasi-particle vacuum. After this transformation, a t-matrix can be defined, which is non-singular if all quasi-particle energies are positive.