Principle of the energy minimization

Therefore the weak form and the variational principle (i.e., the principle of the energy minimization) of the parabolic PDE are obtained as follows ... 

The change in state from liquid to solid as a material crystallizes, is driven by thermodynamics and the principle of free energy minimization, in turn this results from a trade between the total enthalpy and entropy of a system. 

Then energy and density come from the variational principle of the energy, while Fukui function and hardness are obtained from the minimization of the hardness functional. 

To illustrate the equivalence of the energy minimization principle advanced above and the linear elastic equilibrium equations of section 2.4.2, we resort to evaluating the functional derivative of eqn (2.77) and setting it equal to zero, or in mathematical terms 8Y /8ui = 0. Equivalently, working in indicial notation, we find... 

Most older theories rely more or less explicitly on the principle of free-energy minimization. This principle is well-known but, in order to apply it to polymer theory, we shall reformulate it precisely. 

The equilibrium state of a partially wetting droplet put in contact with a flat solid surface can be derived by using the principle of free-energy minimization. The optimum state is a hemispherical cap (see Figure 7.33), with a contact angle given by Young s equation ... 

For each configuration of the nuclei, minimization of tlie total energy with respect to the electron density yields the instantaneous value of a potential energy fiinction V(/ ), and the corresponding forces on the nuclei. In principle,... 

The scope of the early papers was to use the SK approach to accurately interpolate the results of first principles calculations of the energy bands and densities of states. An important characteristic of these calculations is that the first, second, and third nearest neighbor interactions are treated as independent parameters, which is advantageous for minimizing the rms deviation from the first principles bands. 

A key to the application of DFT in handling the interacting electron gas was given by Kohn and Sham [51] who used the variational principle implied by the minimal properties of the energy functional to derive effective singleparticle Schrodinger equations. The functional F[ ] can be split into four parts ... 

We now return to Equation A2.3 and substitute into it Equation A2.4. We then obtain A2.7, where N is the total number of basis orbitals being used. The variation principle now has to be applied to A2.7 to find the values of the c s which will give the best iji s possible with the chosen basis. The energy is minimized simultaneously with respect to all the c s by carrying out a partial differentiation with respect to each c and making the derivatives of the energy satisfy A2.8. The result, after some manipulations, is a set of N equations of the form A2.9, where the index i takes a different value for each equation. 

There is some evidence in support of the view that an electron-domain s effective volume, if not its shape, is approximately transferable from one system to another. Compress a Sidgwick-type unshared electron on one side and it appears to expand elsewhere, particularly on the opposite (trans) side of the kernel, much as one might expect from the form of the kinetic energy operator and the energy minimization principle, which, taken together, require smooth changes in electron density, within a domain. 

Chemistry remains a mystery without understanding three basic attributes of matter cohesion, structure and affinity, each of them shaped by an extremum principle. The principle of minimum energy regulates chemical cohesion, which results from the interaction between atoms, also known as chemical bonding. Minimization of angular momentum dictates the three-dimensional arrangement of atoms in chemical substances, which defines their... 

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