Neel state

Because of the difficulty and complexity of the calculations of the preceding two paragraphs, suitable approximation schemes are of interest Here suitable most desirably means both simply maneagable and easily comprehendable (with physicochemical content). For alternants one such scheme involves the Neel state [42]... 

Tn magnetic transition (into Neel state) temperature... 

Two of the determinants, labeled c and Oq( are unique, and as depicted in 4 and 5 in Scheme 5.1, the electrons are arranged in these determinants in an alternant manner, spin-up—spin-down, and so on. In contrast, in the other determinants for example, fl",-, there are pairs of identical spins on adjacent carbon atoms. The spin-alternant determinants, flqc and Oqc, were earlier called the antiferromagnetic determinants, the Neel state (16), or the (QC) state (17). The corresponding wave functions in Equations 5.8 and 5.9 use the QC shorthand notation for these determinants ... 

Neel-state-based approaches do better for higher z (say for z>3, as for the... 

Thence a neat resolution arises of what at first sight might seem quite ominous (because of the great success of the Neel-state-based approaches). 

Figure 1 - Plots of resonance energies per site for several simple ground-state approximants. The single-Kekule-structure plot applies for general graphs, the Neel-state plot for general bipartite graphs, and the RVB plot for a special class [115] of benzenoid polymers.

For infinite systems there is another type of result concerning long-range magnetic order, which is implicated in making a material magnetic in the sense that there be a phase transition at the onset, as well as hysteresis. This Neel-state type of ordering concerns the spin correlation function... 

Therefore, two main different kinds of variational localized-site cluster expanded ansatze have so far been considered first the Neel-state-based ansatze (NSBA), where the Neel state is the zero order wave-function from which the trial wave-functions are generated, and second the Resonating... 

THE NEEL-STATE-BASED CLUSTER-EXPANDED ANSATZE... 

The cluster expanded wave-function ansatze in this section are based upon the Neel state as a zero-order wave-function,... 

Starting from n)> we notice that it is neither an eigenstate of a Heisenberg Hamiltonian nor an eigenstate of the total spin operator S2. For instance, the Neel state is not invariant under the action of the nearest-neighbor XY spin terms of the Heisenberg Hamiltonian, which may produce a spin-flip on two nearest-neighbor sites at a time with respect to n). These XY excitation operators from the vacuum 4>n > can be written as... 

Therefore, a family of states susceptible to mixing with the Neel state are obtained when applying to the Neel state a product of an arbitrary number of such F(ni mj) excitation operators in an unlinked way,... 

Notice that the Neel state belongs to this space, n) = F I n), where 0 stands for the empty set. 

Thence, variational localized-site cluster expansion ansdtze based upon the Neel state can be defined as... 

Three-Bond-Range RVB ansatze Following a strategy parallel to that of the Neel-state based ansatze, we try to generate a trial wave function as a superposition, now, of singlets which are obtained from a cluster-... 

Neel-State-Based Ansatz The NSBA of Subsec. 4.1, with M = 2 and M = 4, and alternate translational symmetry conditions, has been used [34] to obtain the energy as a function of the variational parameters and the interatomic distances, rq and r 2- Upon optimization, the geometry and the alternating-to-regular geometry transition have been obtain (see Ref. 34). 

Table 2 Results of Variational Localized-site Cluster expansions from either a Neel-state based ansatze or a Resonating Valence Bond ansatz. We notice that the lower level NSBA is unable of showing the dimerization of polyacetylene. rc is the critical bifurcation mean bond length, r and r2 are the optimized short and long bond distances (in A). E is the energy per carbon atom (in eV), taking the energy of the Neel state with 1.40 A equal bond lengths as zero of energy. NSB forth order perturbative and Dimer-covering second order perturbative (see Ref. 34), CEPA ab-initio estimate of Kpnig and Stollhoff [52], and the experimental results [46,47] for rx and r2 have been added for comparison.

Table 3 Optimized regular interatomic distance (in A) and gain in energy per C2H2 unit (in eV) of regular polyacetylene by report to the alternating ground state, with the Neel state based higher order and the RVB ansatze. Results from Local-density-functional approach of Mintmire and White [48], Hartree-Fock and Mpller-Plesset perturbation Theory of Suhai [50], or Ashkenazi [49] are included.

An alternative mapping between the electronic and spin Hamiltonians can be carried out by employing broken-symmetry (BS) states [14-16]. The triplet states I ft) and I U ) of the dimer are referred to as the high spin states IHS), and the Neel states I fi ) and I if ) as the broken-symmetry states IBS)... 

Expectation value of a non-Neel state Calculate the expectation value of

Heisenberg Hamiltonian with nearest neighbour interactions only for the following two cases (a) o is the Neel state of a 4 x 4 lattice. Both systems have periodic boundaries and 5 > j. 

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