Force Theorem

Ab-initio studies of surface segregation in alloys are based on the Ising-type Hamiltonian, whose parameters are the effective cluster interactions (ECI). The ECIs for alloy surfaces can be determined by various methods, e.g., by the Connolly-Williams inversion scheme , or by the generalized perturbation method (GPM) . The GPM relies on the force theorem , according to which only the band term is mapped onto the Ising Hamiltonian in the bulk case. The case of macroscopically inhomogeneous systems, like disordered surfaces is more complex. The ECIs can be determined on two levels of sophistication ... 

A method to calculate J]j, based on the local approximation to spin density functional theory has been developed by Liechtenstein et al. [51, 52]. Using spherical charge and spin densities and a local force theorem, expression for Jjj is... 

The Derjaguin idea, a mainstay in colloid science since its 1934 publication, was rediscovered by nuclear physicists in the 1970s. In the physics literature one speaks of "proximity forces," surface forces that fit the criteria already given. The "Derjaguin transformation" or "Derjaguin approximation" of colloid science, to convert parallel-surface interaction into that between oppositely curved surfaces, becomes the physicists "proximity force theorem" used in nuclear physics and in the transformation of Casimir forces.23... 

Figure 7.3. Planar spin spiral energetics for bcc Fe, fee Co, and fee Ni directly calculated from the BGFM (filled symbols) and evaluated using the magnetic force theorem with ferromagnetic (FM-MFT, dashed lines) and disordered local moment reference states (DLM-MFT, continuous lines). Energies are measured relative to the NM energy, and the moment is fixed to the ground state moment.

Figure 7.5. Relaxation of the magnetic moment of planar spin spirals in fee Ni, fee Co, and bcc Fe. Shown are both the results of direct calculation via the BGFM (open circles) and evaluation from the m-dependent magnetic force theorem approach.

Figure 7.9. Magnetization energy of planar spin spirals at a volume of 6.78 a.u., calculated directly with the BGFM (full circles) or using the Heisenberg model with either FM (dashed lines) or DLM (full lines) reference states. The open squares and triangles represent force theorem calculations from the ferromagnetic and anti-ferromagnetic states respectively.

In Fig. (7.8) are shown direct calculations of the planar spin spirals as a function of moment and of volume, the moment is seen to increase monotonically with the volume. One can see that the spin spiral qxw is stable for low volumes and moments, while the spin spiral qrx is stable at higher volumes and moments. Now, we calculate the spin spirals for fixed moments, both with the BGFM and with with the Heisenberg model with both FM and DLM interactions, just as described above for bcc Fe, fee Ni, and fee Co. We also calculate the spin spirals directly with the force theorem without the Heisenberg expansion by doing one calculation with the potentials from either the ferromagnetic or anti-ferromagnetic solution and the new spin spiral structure. The results can be seen in Fig.(7.9). 

If we assume that the force theorem is correct for this system, the other pos-... 

The problems with performing ab initio calculations for quantum corrals is the very long computational time that is needed, especially if one would want to do big supercell calculations. Even with the help of the force theorem, and perturbative approaches to the problem, one has so far had to diagonalize very big matrices, which makes it hard to perform exhaustive searches for quantities of interest. There has also been interest in engineering quantum corrals to achieve specified electronic properties [184], and also here the problem of finding optimized quantum corral structures appears. 

The 3d transition metals have very small magnetic anisotropy energies (MAE), and their calculation is a challenge to ab initio relativistic electronic structure calculations. The MAE may be calculated by means of the so-called Force Theorem,[155-158] (FT) ... 

Using the magnetic force theorem, the single-particle energy part of the DFT grand potential gives... 

Hernandez-Trujillo and Bader studied the evolution of the electron densities of two separated atoms into an equilibrium molecular distribution, and considered a range of interactions from closed-shell with and without charge transfer, through polar-shared, to equally shared interactions. The harpoon mechanism operative in the formation of LiF was found to exert dramatic effects on the electron density and on the atomic and molecular properties. The virial, the Hellmann-Feynman and the Ehrenfest force theorems provided an imderstanding of the similarities and differences in the bonding. 

This may be derived - - from (1) by replacing rj- (1 +A)rj and Pj- (1 - A)Pj and differentiating in the limit A- 0. The usual form of the virial theorem, with the left-hand side of (7) replaced by zero, applies for an isolated system. Hov/ever, if the system is in a fixed external potential (which is not scaled), then P is the pressure exerted upon the system by the external forces and is the volume of the system. The virial theorem is analogous to the force theorem in that it gives an expression for the external pressure (i.e. the force conjugate to the volume) in terms of the internal operators of the Hamiltonian. Unlike the force theorem, however, the virial theorem involves the kinetic energies and the interactions of all particles-nuclei and electrons. 

A different approach has been shown to be capable of providing more extensive information. In the supercell force constant approach, the idea is to displace a single atom (on a single plane of atoms) and to carry out a self-consistent calculation to determine the change in electronic charge density at all points due to the displacement of the single atom or plane. By use of the force theorem, from this one calculation can be found many independent forces. If a plane of atoms is displaced, this provides planar... 

This general result is well known as the "Hellmann-Feynman" theorem when X represents the position x of a nucleus. The force F that the system exerts on the nucleus is the expectation value of minus the gradient of V(x), where V is the potential that acts on the nucleus. This theorem was originally derived by Ehrenfest (1927), and was used in Hellmann s (1937) treatise to establish the forces in a molecule. Feynman (1939) independently derived the result for molecules. We will refer to the result simply as the "force theorem". 

One application of the stress theorem is the study of elastic properties of solids, which becomes straightforward when a suitable finite macroscopic strain is applied to the solid. When the wavefunctions of the distorted solid are known, the stress tensor is evaluated with the stress theorem. In the harmonic approximation elastic constants are defined as the ratio of stress to strain, and it is furthermore possible to go to large strains to obtain all nonlinear elastic properties. In general it is necessary to be concerned with internal strains that may appear microscopically owing to the lower symmetry of the strained solid. In section 6 we show in detail how this problem is solved by combining the stress and force theorems. 

Barnett, D. M. and Lothe, J. (1974), An image force theorem for dislocations in anisotropic bicrystals. Journal of Physics. F Metal Physics A, 1618-1635. 

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