Spin-opposite-scaled

Grimme " proposed an empirical variant of MP2 that generally provides improved energies. This is the spin-component-scaled MP2 (SCS-MP2) that scales the terms involving the electron pairs having the same spin (SS) differently than those with opposite spins (OS). The SCS-MP2 correlation correction is given as... 

Just as one can use spin-component scaling to improve the performance of MP2, one can multiply the contribution of the same-spin terms to the CCSD energy by an empirical parameter and multiply the contribution of the opposite-spin terms by another parameter. This gives the SCS-CCSD method [T. Takatani et al., J. Chem. Phys., 128,124111 (2008)]. The same-spin and opposite-spin parameter values 1.13 and 1.27, respectively, were found by fitting a set of known reaction energies. SCS-CCSD performs quite well for intermo-lecular interactions. 

However, the intra-atomic Coulomb interaction Uf.f affects the dynamics of f spin and f charge in different ways while the spin fluctuation propagator x(q, co) is enhanced by a factor (1 - U fX°(q, co)) which may exhibit a phase transition as Uy is increased, the charge fluctuation propagator C(q, co) is depressed by a factor (1 -H UffC°(q, co)) In the case of light actinide materials no evidence of charge fluctuation has been found. Most of the theoretical effort for the concentrated case (by opposition to the dilute one-impurity limit) has been done within the Fermi hquid theory Main practical results are a T term in electrical resistivity, scaled to order T/T f where T f is the characteristic spin fluctuation temperature (which is of the order - Tp/S where S is the Stoner enhancement factor (S = 1/1 — IN((iF)) and Tp A/ks is the Fermi temperature of the narrow band). 

Jung et al. [81] reported a variant of this approach that used the multiplicative factor of zero for the SS component and 1.3 for the OS component. This calculation, SOS-MP2 (scaled opposite-spin MP2), can be performed with only an 0(n4) operation cost when combined with Almlof s Laplace transform technique [82], The SOS approximation can be applied to CIS(D) [69], A similar simplification was often adopted in the GW method under the name COHSEX approximation [32] also partly from an operation cost consideration. 

Let me first describe the situation without umklapp (g3 = 0). For positive j, the fixed points are at g = 0 and 2 = 82 i/2. For negative x, the g s scale to the strong-coupling values f = -2, = 82 i/2 — 1. This can be understood in the following way. gi is known from boson-ization to be associated with the spin excitations. If gx is positive, the electrons will make use of the Pauli principle to avoid one another efficiently. Parallel-spin (triplet) correlations will be favored on which has no effect. For gx < 0, the opposite situation prevails. The electrons want to take advantage of the interaction and will thus prefer singlet correlations in which there is nothing to prevent an intimate contact. A gap opens up in the spin excitations. 

Using an LG03 doughnut the theoretical prediction is that the frequencies should scale as 2 3 4 when going fi-om opposite helicity Poynting vector and spin to linearly polarised helical wave fi ont through to the same helicities. This was verified experimentally (see Figure 5). 

Positron Annihilation Lifetime Spectroscopy (PALS) provides a measure of free volume holes or voids, free volume, and free volume distribution, at an atomic scale. The technique exploits the fact that the positively charged positron (e" ), the antiparticle to the electron, preferentially samples regions of low positive charge density. When injected in a polymer matrix, thermalized positrons can combine with an electron to form a bound state, known as positronium (Ps). This species can only exist in a void and it rapidly annihilates on contact with the electron cloud of a molecule. For polymer studies using PALS, it is ortho-positronium (oPs, a triplet state) which is of interest. The oPs spin exchanges with electrons of opposite spin on the walls of the cavity and it is annihilated. Thus, the oPs lifetime, 13, gives a measure of the mean free volume cavity radius, whereas the relative intensity of... 

The pure Hartree-Fock approximation works fairly well for predicting the geometries of metal complexes. Vibrational frequencies are overestimated while formation energies are underestimated. These errors are rather systematic so that, for example, it is standard practice to scale calculated frequencies some empirical factor (Pople et al. 1993) to predict experimental values. The problem is that the single-Slater-determinant approximation to the wavefunction is too restrictive it doesn t allow the electrons with opposite spin to avoid each other as much as they would like. In accordance with the variational principle, the Hartree-Fock total energy is always too... 

The scaled-opposite-spin MP2 (SOS-MP2) method omits the E l contribution entirely and takes E = 1.3 os [Y. Jung et al., /. Chem. Phys., 121, 9793 (2004) J. Comput. Chem., 28, 1953 (2007)]. The SOS-MP2 method gives results slightly less accurate than the SCS-MP2 method, but is faster and can be applied to larger molecules than SCS-MP2. Of course, the use of empirical parameters makes the SCS-MP2 and SOS-MP2 methods no longer strictly ab initio methods. 

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