Absolute energy minimum principle

W. Yang Absolute-energy-minimum principles for linear-scaling electronic-structure calculations, Phys. Rev. B 56, 9294-9297 (1997). 

Absolute energy minimum principle. Kohn developed an absolute energy minimum principle in terms of density matrix also for the case of noninteracting electrons. The method is based on a penalty functional for imposing the idem-potency condition. It does not require a series of calculations with increasing values of the coefficient for the penalty functional. One fixed value of the coefficient suffices. Kohn proved that for a nonself-consistent Hamiltonian h, the ground-state... 

Absolute energy minimum principle. Yang developed two absolute energy minimum principles for first-principle linear-scaling electronic structure calculations. One is with a normalization constraint and the other without any constraint. The density matrix is represented by a set of nonorthogonal localized orbitals [i/] and an auxiliary matrix X which at the minimum becomes the (l)-inverse of the overlap matrix S of the localized orbitals. The number of localized orbitals is allowed to exceed the number of occupied orbitals. 

It is evident from the form of Equation [17] that the two-photon tensor and hence the extent of two-photon absorption, is significantly enhanced if the molecule possesses an electronic excited state of an energy close to E + fuo, for then, in the sum over r, the term that state contributes has an absolute minimum value for the denominator. This situation, typical of multiphoton processes, is known as resonance enhancement, and its physical basis can be understood in terms of the time-energy uncertainty principle ... 

A classical mechanical system at equilibrium is at rest in a (local) minimum on the PES and will have a potential energy given by the minimum value, E ". A (real) quantum system will not, in general, be able to attain such an absolute potential energy minimum. This is due to the Heisenberg uncertainty principle, which prescribes that the uncertainty in the position of a particle. Ax, is related to the uncertainty in the momentum, Ap, of the particle ... 

As might be expected, this is easiest to visualize in two dimensions, but the principles apply directly to three dimensions as well. For sodium chloride, at low temperatures, a plot qualitatively similar to Figure 2.5a is obtained. In this two-dimensional section [taken through the (100) plane], the center square corresponds to the Wulff construction described above and it is the absolute minimum energy. The 100 facets are lower in energy than the 110 or 111 facets, the latter not shown in this 2D section. Accordingly, in three dimensions, a cube represents the lowest-energy crystal form for sodium chloride at low temperatures. 

Stable A term describing a system in a state of equilibrium corresponding to a local minimum of the appropriate thermodynamic potential for the specified constraints on the system. Stability cannot be defined in an absolute sense, but if several states are in principle accessible to the system under given conditions, that with the lowest potential is called the stable state, while the other states are described as metastable. Unstable states are not at a local minimum. Transitions between metastable and stable states occur at rates that depend on the magnitude of the appropriate activation energy barriers that separate them. 

Bohr has succeeded in overcoming these difficulties by rejecting the classical principles in favour of the quantum principles discussed in 1 and 2. He postulates the existence of discrete stationary states, fixed by quantum conditions, the exchange of energy between these states and the radiation field being governed by his frequency condition (1), 2. The existence of a stationary state of minimum energy, which the atom cannot spontaneously abandon, provides for the absolute stability of atoms which is required by experience. Further,... 

---