Invariance of the Density

The starting point of this third derivation of the OPM integral equation is the identity of the KS density n-g with the density n of the interacting system [42,43], 

Note that the relation (2.36) relies on the complete framework of the Hohen-berg-Kohn and KS formalism. In particular, it implies the application of the minimum principle for the total energy, which underlies all of ground-state DFT. This common DFT background provides the link between (2.36) and the arguments of Sects. 2.2.1 and 2.2.2. 

The KS density n-g can now be written in terms of the Green s function Gg of the KS system, while the interacting n can be expressed in terms of the 1-particle Green s function G of the interacting system. 

Here indicates an infinitesimal positive time-shift of t, i.e. = lime o(t + e ). The KS and the full many-body Green s function are defined by the ground-state expectation values of the time-ordered product of the corresponding field operators, and 

Here Go represents the Green s function of electrons which just experience the external potential riextj T is the full self-energy of the interacting system. 

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If the x-z plane is chosen to be the scattering plane the invariance of the density matrix upon reflection yields... 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

Invariance of the trace of a matrix under unitary transformation corresponds to the invariance of phase density under canonical transformation in classical theory. 

M. D. Benayoun and A. Y. Lu, Invariance of the cumulant expansion under l-particle unitary transformations in reduced density matrix theory. Chem. Phys. Lett. 387, 485 (2004). 

Of all the macroscopic quantities in our model, the hydrodynamic density p, flow velocity vector u = (ua), and thermodynamic energy E, have the unique property of being produced by additive invariants of the microscopic motion. The latter, also called sum functions4 and summation invariants,5 occur at an early stage in most treatments. The precise formulation follows. 

Above we have looked at the translational spectra of binary systems. These are obtainable experimentally at sufficiently low densities, especially when absorption of the infrared inactive gases is studied. An induced spectrum may be considered to be of a binary nature if the integrated intensity varies as density squared. In that case, the shape of the densities-normalized absorption coefficient, cc/q Q2, is invariant, regardless of the densities employed. ... 

We now derive the expression for the fluorescence signal in terms of the doorway and window wavepackets instead of the four-point correlation function. We start with Eq. (3.1) and write the four-point correlation function F(4) explicitly as the trace with respect to the equilibrium density matrix. We then use the cyclic invariance of the trace and obtain... 

One may obtain now (see [7]) the general expression of the density matrix of multi-dimensional system that is invariant to relation of the unitary transformation S ... 

Because field quantization falls outside the scope of the present text, the discussion here has been limited to properties of classical fields that follow from Lorentz and general nonabelian gauge invariance of the Lagrangian densities. Treating the interacting fermion field as a classical field allows derivation of symmetry properties and of conservation laws, but is necessarily restricted to a theory of an isolated single particle. When this is extended by field quantization, so that the field amplitude rjr becomes a sum of fermion annihilation operators, the theory becomes applicable to the real world of many fermions and of physical antiparticles, while many qualitative implications of classical gauge field theory remain valid. 

As a consequence of the translational invariance of the matrix e, K ) is an eigenstate, so that the induced dipoles build up a plane wave K>. The macroscopic polarization-density vector is given by... 

This real matrix may be diagonalized to give the principal axis of curvature and the trace of the hessian matrix i.e., the laplacian of the density, is an invariant. 

Quite generally, it must be stated that some additional effort is required to develop the RDFT towards the same level of sophistication that has been achieved in the nonrelativistic regime. In particular, all exchange-correlation functionals, which are available so far, are functionals of the density alone. An appropriate extension of the nonrelativistic spin density functional formalism on the basis of either the time reversal invariance or the assembly of current density contributions (which are e.g. accessible within the gradient expansion) is one of the tasks still to be undertaken. 

Reflection invariance provides further constraints, the total wave function of the two colliding particles retaining reflection symmetry through the collision plane. Since we have already assumed that electron spin plays no role in the collision, this means that the reflection symmetry of the excited P state is the same as that for the original S state, i.e. symmetric. Thus the coefficient of the antisymmetric py) orbital in the expansion of the excited state must vanish. From (8.6) we see that this requires /ii = —/i-i. In general reflection symmetry requires that the elements of the density matrix satisfy the condition (Blum and Kleinpoppen, 1979)... 

It should be noted that, because of the invariance properties of the density function with respect to a unitary transformation among its orbitals, the a-n description of double or triple bonds in terms of nonequivalent orbitals is not the only possible one a description in terms of two or three equivalent bent banana bonds is possible as well. It is... 

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