Translational invariance, orbital

The simplest form of spin-orbitals for a system with translational invariance are plane waves (PW) 9k r,a) = exp[ fe r]. This form was used in the first QMC study of metallic hydrogen [33]. It is particularly appealing for its simplicity and still qualitatively correct since electron-electron and electron-proton correlations are considered through the pseudopotential . The plane waves orbitals are expected to reasonably describe the nodal structure for metallic atomic hydrogen, but no information about the presence of protons appears in the nodes with PW orbitals. 

One reason for choosing the tails of muffin-tin orbitals as solutions of the translationally invariant Helmholtz wave equation (5.7) is the extremely simple expansion theorem... 

Let us also imagine that each of the translationally invariant H atoms in Scheme 2.1 possesses an (invisible) subscript, ranging from 1 to oo, which characterizes its position in the chain in terms of the translational vector T. Each translation is a multiple of the interatomic distance a, namely T = n a, and each H atom comes with a single Is atomic orbital, designated [Pg.68]

Whereas current computational techniques based on gauge-including atomic orbitals meet the requirement of translational invariance of calculated magnetic properties, they do not necessarily guarantee current/charge conservation. On the other hand, CTOCD schemes account for the fundamental identity between these constraints, which are expressed by the same quantum mechanical sum rules. 

The first term of (3.289) represents a translational Stark effect. A molecule with a permanent dipole moment experiences a moving magnetic field as an electric field and hence shows an interaction the term could equally well be interpreted as a Zeeman effect. The second term represents the nuclear rotation and vibration Zeeman interactions we shall deal with this more fully below. The fourth term gives the interaction of the field with the orbital motion of the electrons and its small polarisation correction. The other terms are probably not important but are retained to preserve the gauge invariance of the Hamiltonian. For an ionic species (q 0) we have the additional translational term... 

One may also wish to impose an additional requirement on the connection, namely that it is translationally and rotationally invariant. This may seem to be a trivial requirement. However, a connection is conveniently defined in terms of atomic Cartesian displacements rather than in terms of a set of nonredundant internal coordinates. This implies that each molecular geometry may be described in an infinite number of translationally and rotationally equivalent ways. The corresponding connections may be different and therefore not translationally and rotationally invariant. In other words, the orbital basis is not necessarily uniquely determined by the internal coordinates when the connections are defined in terms of Cartesian coordinates. Conversely, a rotationally invariant connection picks up the same basis set regardless of how the rotation is carried out and so the basis is uniquely defined by the internal coordinates. [For a discussion of translationally and rotationally invariant connections, see Carlacci and Mclver (1986).]... 

The symmetry of equation (3.1), which causes the non-hyperbolic eigenvalue zero to be triply degenerate, gives rise to a three-dimensional invariant manifold of (pt u ). This is the center manifold which coincides with the group orbit of u, namely SE(2)u = pgU, g 6 SE 2). Thus the center manifold of u, is simply the set of all translations and rotations of the initial rotating wave u. See [72] for this result and some generalizations. 

When it comes to crystals, it is clear that the system under study is trans-lationally invariant in all three spatial directions, and Bloch s theorem utilizes the translational s)mimetry to generate the crystal s wave function, composed of crystal orbitals which are also called electronic bands. We therefore imagine an idealized solid-state material whose electronic potential V possesses the periodicity of the lattice, expressed by a lattice vector T, that is... 

The crucial point in Bloch s theory of one-dimensional (ID) systems is that, at point (r) related by direct space lattice translation (ja j = integer, a = length of the ID unit cell) electron densities (P) are invariant and one-electron wave functions or "orbitals" (()>) only differ by an exponential of unity modulus and of imaginary argument (ikja) ... 

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