Euler-Lagrange equation density

As indicated in Fig. 7, the next step after either an explicit or an implicit energy density functional orbit optimization procedure. For this purpose, one introduces the auxiliary functional Q[p(r) made up of the energy functional [p(r) 9 ]. plus the auxiliary conditions which must be imposed on the variational magnitudes. Notice that there are many ways of carrying out this variation, but that - in general - one obtains Euler-Lagrange equations by setting W[p(r) = 0. 

The total energy E of the system is also a functional of the density distribution, E = [] (r)]. Therefore, if the form of this functional is known, the ground-state electron density distribution n t) can be determined by its Euler-Lagrange equation. However, except for the electron gas of almost constant density, the form of the functional [ (r)] cannot be determined a priori. 

The Lagrangian (824), which is the same as the Lagrangian (839), gives the inhomogeneous equation (826) using the same Euler-Lagrange equation (843). Therefore the photon mass can be identified with the vacuum charge-current density as follows (in SI units) ... 

The Levy construction [222] can be used to prove Hohenberg-Kohn theorems for the ground state of any such theory. It should be noted that any explicit model of the Hohenberg-Kohn functional F[p] implies a corresponding orbital functional theory. The relevant density function p(r) is that constructed from an OFT ground state. This has the orbital decomposition , as postulated by Kohn and Sham [205]. Unlike the density p,, for an exact A-electron wave function T, which cannot be determined for most systems of interest, the OFT ground-state density function is constructed from explicit solutions of the orbital Euler-Lagrange equations, and the theory is self-contained. 

This defines the fermion contribution to an isovector gauge current density. Although the Euler-Lagrange equation is gauge covariant by construction, this fermion gauge current is not invariant, because the matrix r does not commute with the 5(7(2) unitary transformation matrices. It will be shown below that the... 

After substituting these functional derivatives, the set of the nonlinear Euler-Lagrange equations is solved for the density profile of the segments. For a linear chain, the density profile of a segment a is (Jain et al., 2007)... 

It is assumed that Ec is so defined that the functional E is minimized for ground states. Ground-state orbital functions and the density function are determined by Euler-Lagrange equations expressed in terms of functional derivatives of E. For any density or orbital functional, with fixed n, infinitesimal orbital variations determine the functional variation... 

Hardness has been calculated in various other ways. For example, a five-point finite difference formula has been used [55] to approximate (d2E/BN2). The equality of chemical potential with the total electrostatic potential at the covalent radius [56-58] has been made use of in calculating rj. The electron density required for this work [56] has been obtained from a self-consistent numerical solution of a quadratic Euler-Lagrange equation [59,60]. Orsky and Whitehead [61] have proposed another defi-... 

Let us consider now the energy density functional [/j(r,a) W] of Eq. (50). In this functional, p r, 5) stands for any one of the final densities generated from the initial density pg(r,s) coming from the orbit-generating wavefunction W. Clearly, the extremum of this functional is attained at the optimal density /> t(r,s). This density satisfies the Euler-Lagrange equation arising from the variation of the functional [/>(r,a) M] - subject to the normalization condition f d xp(r,s) = N -with respect to the one-particle density p(r, s). Explicitly, this amounts to varying the auxiliary functional... 

The use of functionals and their derivatives is not limited to density-functional theory, or even to quantum mechanics. In classical mechanics, e.g., one expresses the Lagrangian C in terms of of generalized coordinates q(x,t) and their temporal derivatives q(x,t), and obtains the equations of motion from extremizing the action functional 4[g] = J C q, q t)dt. The resulting equations of motion are the well-known Euler-Lagrange equations 0 = = fy — > which are a special case of Eq. (14). 

The electronic density of the ground state is determined fi om the Euler-Lagrange equation that results from the variation of Eq. (1) with respect to p(r), subject to the condition that the integral of the latter over the whole space must be equal to the total number of electrons N,... 

The chemical potential p, = 6J f6p enters the respective Euler-Lagrange equation obtained by minimizing the grand ensemble thermodynamic potential — p J pd a , which defines the equilibrium particle density distribution... 

First-order terms are added as well when Eq. (32) is varied with respect to p the respective Euler-Lagrange equation can be used to compute first-order correction to the density profile, which we shall not need. 

Electronic structure calculation for the ground state of an N-clectron chemical system may be performed by solving the Euler-Lagrange equation of density functional theory [1,2]... 

One of the striking eonsequcnces of the Euler-Lagrange equations is a remarkable distribution law for the NO occupations k of a stationary density matrix ... 

Note that the most common constraints in Euler-Lagrange equation take the form C[n] = 0, where C[n] is some density functional. For instance, the constraint C[n] = f n r)dr - N = 0 is used in the derivation of the Kohn-Sham equations. Additional constraints expressed as C[n] =0 are also used in some computational schemes such as the procedure to generate diabatic electronic states for the evaluation of the rate of the electron-transfer reaction [7],... 

The Euler-Lagrange equation arising from the minimization of fg with respect to the density profile, n(z), is... 

Next the connection between the local and global sensitivity indices on the DFT will be exposed in a manner which should allow an explicit implementation of the electronic densities. It is start from a generalized form of the Euler-Lagrange equation (4.165) ... 

The torque acting on the director is given by n x h, where h is called the molecular field, which can be derived from the Euler-Lagrange equation. The energy density corresponding to the flexoelectric polarization is given by — Pfl E and the molecular field can be expressed in the form ... 

The equilibrium density profile p z) is the one that minimises the surface tension functional thus by the standard methods of calculus of variations we can write down an Euler-Lagrange equation that the density profile must satisfy ... 

For the equations of motion to be covariant, must be a Lorentz scalar density. In this case the equations of motion, the Euler-Lagrange equations,... 

Flere p r) denotes the electron local density. T[/3(r)] is the free energy functional, and as stated above it can be split into the noninteraction part and the excess part accounting for the electron—electron interaction. Note that here the bulk chemical potential p is not an independent thermodynamic parameter and it is determined by the electron total number. Namely, the bulk chemical potential involved in the Euler—Lagrange equation for p r) should satisfy the electron number conversation equation... 

Again, for simplicity, we have written the energy expression as applied to atoms. Finding the electron density that makes Eq. [51] a minimum and conserves the number of electrons, Eq. [37] leads to the Euler-Lagrange equation... 

It is possible to determine the relations among the kinetic coefficients from the Euler-Lagrange- equations belonging to the variation of i-th coordinate jL and of the heat and a-th chemical component current densities... 

With variation of j-th coordinate j j of electric current density we get the following Euler-Lagrange-equations... 

The density and order profiles shown in Figure 1.7 are obtained by solving the corresponding Euler-Lagrange equations under appropriate boundary conditions. The surface tension is extra energy cost for the formation of an interface (density and order profile shown in Figure 1.7) and is defined as... 

The model (2)-(3) is symmetric with respect to the solid and fluid densities. At a given quenched solid density distribution, minimization of the functional with respect to the fluid density distribution leads to the Euler-Lagrange equation in the form ... 

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