Orbital Euler-Lagrange equations

With fixed n, 0. for variations of occupied orbitals that are unconstrained in the orbital Hilbert space, the variational condition implies orbital Euler-Lagrange equations 

The orbital functional derivative here defines an effective Hamiltonian 

The theory is usually expressed in terms of canonical equations 

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Orbital Euler-Lagrange equations are determined by functional derivatives... 

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. 

Euler-Lagrange Equation for Intra-Orbit Optimization of p(r) 206... 

Euler-Lagrange Equations for the Intra-Orbit Optimization of... 

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 computational effort of solving orbital Euler-Lagrange (OEL) equations is significantly reduced if the generally nonlocal exchange-correlation potential vxc can be replaced or approximated by a local potential vxc(r). A variationally defined optimal local potential is determined using the optimized effective potential (OEP) method [380, 398]. This method can be applied to any theory in which the model... 

A modified effective Hamiltonian Goep is defined by replacing vxc by a model local potential vxc(r). The energy functional is made stationary with respect to variations of occupied orbitals that are determined by modified OEL equations in which Q is replaced by Goep- 84>i is determined by variations 8vxc(r) in these modified OEL equations. To maintain orthonormality, <5, can be constrained to be orthogonal to all occupied orbitals of the OEP trial state , so that (r) = J]a(l — na)i). First-order perturbation theory for the OEP Euler-Lagrange equations implies that... 

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... 

Euler-Lagrange equation for intra-orbit optimization of p(r, s)... 

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... 

Euler-Lagrange equations for the intra-orbit optimization of N orthonormal orbitals Kohn-Sham-like equations... 

Orbit Variational Principle and Euler-Lagrange Equation... 

As the electronic energy is a function of the nuclear positions as well as function of the orbitals (/ , its derivative is taken with respect to the nuclear positions but also with respect to the wavefunction. The Euler-Lagrange equations then read... 

The equations of motion corresponding to eqn [9] are obtained from the associated Euler-Lagrange equations, that for the orbitals read... 

Here, is the expectation value of the hamiltonian operator with a single-determinant wave function. The second term comes from the orthonormalization condition of the molecular orbitals, while the third one arises from each population restriction. The Euler-Lagrange equations from this problem lead to a matrix problem similar to Eq. (15) ... 

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