Lagrangian spin

Time-differentiating RR = i yields RR + RR =0, therefore the Eulerian spin o) and Lagrangian spin co, which is the pull-back of the Eulerian spin into the undeformed body, can be defined by... 

A similar effect is obtained by using the spin-constrained UHF method (SUHF). In this method, the spin contamination error in a UHF wave function is constrained by the use of a Lagrangian multiplier. This removes the spin contamination completely as the multiplier goes to infinity. In practice, small positive values remove most of the spin contamination. 

Ehf from equation (1-20) is obviously a functional of the spin orbitals, EHF = E[ XJ]. Thus, the variational freedom in this expression is in the choice of the orbitals. In addition, the constraint that the % remain orthonormal must be satisfied throughout the minimization, which introduces the Lagrangian multipliers e in the resulting equations. These equations (1-24) represent the Hartree-Fock equations, which determine the best spin orbitals, i. e., those (xj for which EHF attains its lowest value (for a detailed derivation see Szabo and Ostlund, 1982)... 

Note, that we studied (Nasriddinov, 1998) the t —> ir riuT decay of the r lepton in the framework of this method with taking into account the isotopic spin violation of chiral symmetry in the Oakes scheme.In this case the 7r° — rj - mixing Lagrangian has the form... 

The Wess-Zumino term in Eq. (11) guarantees the correct quantization of the soliton as a spin 1/2 object. Here we neglect the breaking of Lorentz symmetries, irrelevant to our discussion. The Euler-Lagrangian equations of motion for the classical, time independent, chiral field Uo(r) are highly non-linear partial differential equations. To simplify these equations Skyrme adopted the hedgehog ansatz which, suitably generalized for the three flavor case, reads [40] ... 

FM at some density 1. One of the essential points we learned here is that we need no spin-dependent interaction at the original Lagrangian to see SSP. We can see a similar phenomenon in dealing with nuclear matter within the relativistic mean-field theory, where the Fock interaction can be extracted by way of the Fierz transformation from the original Lagrangian [11],... 

The local spin values obtained in a decomposition scheme may either be local expectation values as evident from the last section. Since we are interested merely in Slater determinants describing local high-spin centers, it is most suitable to employ local (Sza) values as properties to be constrained in such a multiplier scheme. If we recall that the (Sza) expectation values should be equal to the ideal Msa and denote ideal values as M al values, the Lagrangian can be chosen to be (135,136)... 

There are N spin orbitals, so there are N(N+ 1)/2 independent constraints (note a b) = (6 a) ), so we need that many undetermined multipliers in our lagrangian function, for which we present... 

The antiferromagnetic antiferromagnetic spin fluctuations which result in d-pairing in cuprate superconductor are described by the Lagrangian in lattice representation [5-7] ... 

Because the many-particle Lagrangian density L° reduces to a sum of singleparticle operators, one may define an effective single-particle Lagrangian density iP°(r, t) by the usual recipe of summing over all spins and integrating over the coordinates of all the electrons but one, a step which expresses the result in terms of the charge density p(r, t) as... 

Introducing these expressions into the general Lagrangian and neglecting all spin contributions gives... 

Recall that the Lagrangian triads was introduced in (2.55). Let us also define the spin of iV, by... 

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