Variational principle definition

If classical Coulombic interactions are assumed among point charges for electrostatic interactions between solute and solvent, and the term for the Cl coefficients (C) is omitted, the solvated Eock operator is reduced to Eq. (6). The significance of this definition of the Eock operator from a variational principle is that it enables us to express the analytical first derivative of the free energy with respect to the nuclear coordinate of the solute molecule R ,... 

Frieden s theory is that any physical measurement induces a transformation of Fisher information J I connecting the phenomenon being measured to intrinsic data. What we call physics - i.e. our objective description of phenomenologically observed behavior - thus derives from the Extreme Physical Information (EPI) principle, which is a variational principle. EPI asserts that, if we define K = I — J as the net physical information, K is an extremum. If one accepts this EPI principle as the foundation, the status of a Lagrangian is immediately elevated from that of a largely ad-hoc construction that yields a desired differential equation to a measure of physical information density that has a definite prior significance. 

The interelectronic interactions W are defined using constrained search [21, 22] over all A-representable 2-RDMs that reduce to R g). Since the set of 2-RDMs in the definition of W contains the AGP 2-RDM of g, that set is not empty and W is well defined. Through this construction, E still follows the variational principle and coincides with the energy of a wavefunction ip, which reproduces R g) = D[ T ] and = W[g]. The latter is due to... 

If the topological property which defines an atom is also one of physical significance, then it should be possible to obtain from quantum mechanics an equivalent mechanical definition. As demonstrated in Chapters 5 and 8, this can be accomplished through a generalization of the quantum action principle to obtain a statement of this principle which applies equally to the total system or to an atom within the system. The result is a single variational principle which defines the observables, their equations of motion, and their average values for the total system or for an atom within the system. 

From the preceding discussion, the mode of integration used in the definition of an atomic property is determined by the atomic variation principle and is the same as that used in the definition of the charge density itself. The atomic average of an observable A is given by... 

As mentioned, in order to be able to apply the variational principle in DFT, it is necessary to extend the definition of the functionals beyond the domain of v-representable densities, and the standard procedure is here to apply the Levy constrained-search procedure [17]. This has led to the functionals known as the Levy-Lieb (FL[p ) and Lieb (FL[p ) functionals, respectively, and we shall now investigate the differentiability of these functionals. This will represent the main part of our paper. 

Let us go back to Fig. 1 and consider the orbits" Ol for = 1, , HK, . These orbits" or columns appearing in Fig. 1 are made up of wavefunctions belonging to Hilbert space (at this point we just assume that they exist a formal definition is given in Section 2.6). We assume, furthermore, that these orbits" are endowed with the following characteristic no two wavefunctions belonging to the same orbit" can have the same density, i.e., there is a one to one correspondence between p r) e Af and W g (this fact is proven in Section 2.6, using local-scaling transformations). We assume, moreover, that the union of all orbits" exhausts Hilbert space. Clearly, in terms of these orbits", the variational principle can be reformulated as follows [21] ... 

The KS equations are obtained by utilizing the variation principle, which the second Hohenberg-Kohn theorem assures us applies to DFT. We use the fact that the electron density of the reference system, which is the same as that of our real system (see the definition at the beginning of the discussion of the KS energy), is given by [7]... 

Here, are the so-called natural orbitals, and their occupancies , lie in the interval [0 1]. Per definition, the improvements that are obtained when replacing Eq. (4) by Eq. (11) are the correlation effects. These may be included either through application of the variational principle or perturbatively. 

Substituting again = + with the definition (106) of H and (107) with the definition (108) of the variational principle) one obtains instead of (73) the generalized coupled HF equations.125... 

The variational principle of the energy density functional theory based on the definition (33) is a straightforward consequence of the quantum mechanical variational principle (8) and the functional mapping (13). It is clearly orbit-dependent... 

A cautionary remark has to be made. The variational principle in this section is based on the positive definiteness of the mobility matrix, i.e., for any vector F ... 

The stress theorem relies upon the variational principle applied together with a strain-scaling of the quantum system, as discussed in detail by the present authors elsewhere (Nielsen and Martin, to be published). The strain scales particle positions as x- (1 + e)x, and by definition the macroscopic stress per volume n (a and B denote cartesian coordinates) is derived from the total energy by... 

An equivalent definition of the ground-state total energy can be obtained from the variational principle, i.e. 

It should be noted here that the definition of the phenomenological coefficients is slightly different from customary ones such as the ordinary definition of thermal conductivity, which is different from by the factor T. This difference is not essential as long as the fluctuation of temperature in the s retem is not too large. However, this definition of the coefficient is essential in the derivation of the variational principle for continuous systems. 

In terms of these definitions, the S-matrix version of the Kohn variational principle is embodied in the variational functional... 

Both hypo- and hyperparathyroidism have been observed in human beings, and, if the principle of genetic gradients (p. 13) is valid, various intermediate levels of hormone activity will be found in the general population. Although the data on the subject, with the exception of that concerning anatomy, are not definitive, there would seem to be little doubt as to the existence of several-fold variation in the parathyroid activities of "normal" individuals. 

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