Hamiltonian pure spin

It is evident that a 2-RDM that corresponds to a Hamiltonian eigenstate also corresponds to a pure-spin state. However, when one is working with an approximated RDM, it is important that this RDM should correspond to a spin eigenstate. 

The spin Hamiltonian is an artificial but useful concept. It is possible that more than one spin Hamiltonian will fit the data. Further, we should note that in solving Eq. (48), we start with pure + and — spin functions and talk about the upper and lower states as being pure spin states. This is not the true case for the ion, as has already been noted in Eq. (38). As regards the Zeeman interaction, however, the final state behaves as a pure spin state, except that we must assign g values different from that of the free electron. 

As in all perturbational approaches, the Hamiltonian is divided into an unperturbed part and a perturbation V. The operator is a spin-free, one-component Hamiltonian and the spin-orbit coupling operator takes the role of the perturbation. There is no natural perturbation parameter X in this particular case. Instead, J4 so is assumed to represent a first-order perturbation The perturbational treatment of fine structure is an inherent two-step approach. It starts with the computation of correlated wave functions and energies for pure spin states—mostly at the Cl level. In a second step, spin-orbit perturbed energies and wavefunctions are determined. 

Now we are concerned with how the VB model (18) can be solved for a given conjugated system. In fact, the model Hamiltonian (18) actually acts on the space of pure spin functions, either of the Weyl-Rumer (WR) form [34] or the simple product of one-electron spin functions. The matrix element between any two WR functions can be obtained by using Pauling s graphical rules [4], while the matrix element between two simple spin products is easily available using the following expression... 

The procedure leading from the exact /V-electron Hamiltonian (2) to the Heisenberg Hamiltonian matrix (47) is very instructive, but it is rather lengthy. Much simpler is the use of effective Hamiltonians which in the space of N-electron eigenfunctions of S2 and S2 are represented by the same matrix. Furthermore, using the effective Hamiltonians may bring another insight into the nature of the interactions described by the model. The simplest effective Hamiltonian in the pure spin is... 

The Hamiltonian (8) can also be rewritten in a pure spin form if the spin space of double dimensionality is taken into account... 

It is a fundamental fact of quantum mechanics, that a spin-independent Hamiltonian will have pure spin eigenstates. For approximate wave functions that do not fulfill this criterion, e.g. those obtained with various unrestricted methods, the expectation value of the square of the total spin angular momentum operator, (5 ), has been used as a measure of the degree of spin contamination. is obviously a two-electron operator and the evaluation of its expectation value thus requires knowledge of the two-electron density matrix. 

Inclusion of Pauli s exclusion principle leads to the standard methods of ab initio computational chemistry. Within these methods, molecular systems containing the same nuclei and the same number of electrons, but having a different total electronic spin, can roughly speaking be said to be different systems. Thus, matrix elements of the Hamiltonian between Slater determinants corresponding to different spin states will all be zero, and they will not interact or mix at all. The wavefunctions obtained will be pure spin states. ... 

For identical hydrons, the symmetry postulate of identical particles has to be fulfilled. For protons and tritons this means that the overall wave function must be antisymmetric under particle exchange and for deuterons it must be symmetric under particle exchange. Due to this correlation of spin and spatial state, the energy difference A between the lowest two spatial eigenstates can be treated as a pure spin Hamiltonian, similar to the Dirac exchange interaction of electronic spins. 

As a result we find that in both the hydrogen and the deuterium case the ground state tunneling is describable by a pure spin tunnel Hamiltonian, which describes the tunnel splitting between the spatial pair of states of different symmetry. The implications are discussed in detail in Ref. [83]. 

The Hamiltonian is invariant under lattice translations, if V r) is invariant, even with inclusion of the SO term. The eigenfunctions will be of the Bloch form, but they will in general not correspond to pure spin states a or the spin functions which diagonalize (the z-axis is taken as quantization axis). Often one labels the Bloch function by arrows f i,... 

Fig, 5. In the pure-spin basis one singlet level is coupled by t>ST to a number of triplet levels 7/. When the molecular Hamiltonian is diagonalized, there results mixed eigenstates. S has primarily singlet character T and T are mostly triplets, while r/, and 7," remain pure triplets. The observed low-pressure coUisional processes are governed by the mixed states. 

In this section I will outline the different methods that have been used and are currently used for the computation of parity violating effects in molecular systems. First one-component methods will be presented, then four-component schemes and finally two-component approaches. The term one-component shall imply herein that the orbitals employed for the zeroth-order description of the electronic wavefunction are either pure spin-up spin-orbitals or pure spin-down spin-orbitals and that the zeroth-order Hamiltonian does not cause couplings between the two different sets ( spin-free Hamiltonian). The two-component approaches use Pauli bispinors as basic objects for the description of the electronic wavefunction, while the four-component schemes employ Dirac four-spinors which contain an upper (or large) component and a lower (or small) component with each component being a Pauli bispinor. 

For a 4ff, rare earth ion such as gadolinium(III), its magnetic behavior is contributed only by a pure spin state S = 7/2. The exchange coupling interaction with other paramagnetic centers can be described by the well-known Heisenberg-Dirac-Van Vleck (HDW) spin Hamiltonian... 

ROHF means a single-determinant wavefunction with maximal spin projection that is automatically eigenfunctions of S with the maximal spin projecton value S = ris/2. So, for the ROHF method projection on a pure spin state is not required. The space sjonmetry of the Hamiltonian in the ROHF method remains the same as in the RHF method, i.e. coincides with the space symmetry of nuclei configuration. The double-occupancy constraint allows the ROHF approach to obtain solutions that are eigenfunctions of the total spin operator. The molecular orbitals diagram for the ROHF half-closed shell is given in Fig. 4.1, (left). 

Because of the simple form (18) of the hamiltonian, a last approximation can be adopted in the molecules with an even number of electrons, the ground state of which is a singlet state, the n = 2v electrons are arranged in pairs, with antiparallel spins, in the v molecular orbitals of lowest energy. This is a simple way of making the function represent a state of pure spin. ... 

In the strongly-correlated limit, when the electron delocalization (i.e. the tpq terms) becomes smaller than the electron repulsion U, an appropriate description of the lowest states is provided by the neutral VB determinants only, i.e. those in which each carbon p bears one unpaired electron in its n atomic orbital (AO, hereafter labelled 5 ). The n electron systems behaves as a pure spin system, obeying a Heisenberg Hamiltonian [22,23]. The inter-atomic delocalization, i.e. the interaction between the neutral VB distributions and the ionic ones, results in an antiferromagnetic spin coupling on each bond. One of the below-discussed rules, known as the Ovchinnikov s mle [21], has been derived from this magnetic approach. Numerous works [24] have shown the relevance of magnetic descriptions... 

We introduce the dimensionless bending coordinates qr = t/XrPr anti qc = tAcPc ith Xt = (kT -r) = PrOir, Xc = sJ kcPc) = Pc nc. where cor and fOc are the harmonic frequencies for pure trans- and cis-bending vibrations, respectively. After integrating over 0, we obtain the effective Hamiltonian H = Ho + H, which is employed in the perturbative handling of the R-T effect and the spin-orbit coupling. Its zeroth-order pait is of the foim... 

This is the most general form of a spin orbital, but if the Hamiltonian does not contain the spin explicitly, it may be more convenient to try to introduce simplified spin orbitals which contain only one nonvanishing component and hence are of either pure a or character. Corresponding to the idea of the doubly occupied orbitals, the spin orbitals are often constructed in pairs simply by multiplying the same orbital tp(r) with a and ft, respectively. 

For the evaluation of magnetically split Mossbauer spectra within the spin-Hamiltonian formalism, the purely -dependent Hamiltonian must be extended by an appropriate... 

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