Spin Scaling Relations

Spin scaling relations can be used to convert density functionals into spin-density functionals. 

For example, the non-interacting kinetic energy is the sum of the separate kinetic energies of the spin-up and spin-down electrons  

The corresponding density functional, appropriate to a spin-unpolarized sys-  

For example, we can start with the local density approximations (1.110) and (1.49), then apply (1.126) and (1.127) to generate the corresponding local spin density approximations. 

Because two electrons of anti-parallel spin repel one another coulombi-cally, making an important contribution to the correlation energy, there is no simple spin scaling relation for Ec- 

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The exchange functionals appropriate to spin-compensated and to spin-polarized systems are related to each other by the spin-scaling relation [86]... 

Density Functionals for Non-relativistic Coulomb Systems 1.4.3 Spin Scaling Relations... 

For exchange, this is easily verified by applying the spin-scaling relation of (1.127) to (1.185) and (1.183). 

Now we turn to the construction of a GGA for the exchange energy. Because of the spin-scaling relation (1.127), we only need to construct which must be of the form of (1.205). To recover the good LSD description of the linear response of the uniform gas (Sect. 1.5.4), we choose the gradient coefficient for exchange to cancel that for correlation, i.e., we take advantage of (1.194) to write... 

Clearly, the correlation energy of (1.206) can be written in the form of (1.217). To get the exchange energy into this form, apply the spin-scaling relation (1.127) to (1.205), then drop small Vs contributions to find... 

Thus it is of some interest to know if the Stoll ansatz is correct, especially since several other ansUtze have been developed and used as input to a spin-resolved pair correlation function. For example, Perdew and Wang (PW) [51,52] proposed a scaling relation... 

In spite of problems as discussed above to locate the static in the zero field, numerous attempts have been made to estimate critical exponents for spin glasses. Table 1 contains a list of values for critical exponents obtained from measurements on various spin glasses either directly or via scaling relations. Monod and Bouchiat (1982) show in fig. 73 that their field-cooled magnetization data of the spin glass AgMn 10.6% (with 7 f = 37.4K) indeed exhibit a quadratic variation of MIH in H above T thus t i(7 ) is defined according to eq. 85. 

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