Many-particle states

The normalization requirement of the one and many particle states is satisfied if we impose the following commutation rules on the b and d... 

Mandelstam, S., 356,371,377,381,664 Mandelstam s postulate, 376 Many particle state, 540 Margenau distribution, 49 Margenau, Henry, 49,391 Marginal densities of probability density functions, 138... 

Nevertheless, thermal averaging is widely used and we define it here explicitly. If we introduce the basis of exact time-independent many-particle states n) with energies En, the averaging over equilibrium state can be written as... 

We shall investigate the densities n(r, t) of electronic systems evolving from a. fixed initial (many-particle) state... 

There exist 21N many-particle states in a chain with N bonds, namely, r) = i where rikM = 0,1 is the number of protons in the... 

Fullerenes and their derivatives not only represent the most massive and most complex single particles in interference experiments until recently, they also mark a qualitative step towards the mesoscopic world. In many aspects they resemble rather small solids than simple quantum systems they possess collective many-particle states like plasmons and excitons, and they exhibit thermionic electron, photon and particle emission [Mitzner 1995 Hansen 1998] - which may be regarded as microscopic analogs of glow emission, blackbody radiation and thermal evaporation. Fullerenes contain about two... 

Let us now consider what is the analog of a Fermi resonance in a molecule when we consider the crystals. In going over from an isolated molecule to a crystal, the branches of optical phonons appear. In the region of overtone and sum frequencies, several bands of many-particle states arise and, if anharmonicity is sufficiently strong, bands of states with quasiparticles bound to one another (for instance, biphonons) will also appear. Thus, in crystals a large number of... 

Within the na-qmd approach, electronic (quantum) and nuclear (classical) dof are treated by combining time-dependent (td) density functional theory (dft) with molecular dynamics (md). The basic theorem of tddft [24-26] states that for a system of interacting particles the many-particle state and, thus, any observable are uniquely determined by the time-dependent single-particle density p(r, t) which can be written identically as the density... 

Repeated application of the creation operators generates a Fock space in which many-particle states are expanded. Each of the basis states of this Fock space is represented by its occupation vector n> given in the pair index notation... 

The many-particle state vector used to be constructed from simpler units— the electronic and nuclear wave functions—and the many-electron wave function consists of the one-electron wave functions, the spinorbitals general expression for the many-electron wave function in terms of the products of spinorbitals is... 

Definition Let Itfi) and ip) be normalised many-particle states, i. e. 

The re-definition of the vacuum determines also the (so-called) effective particle character of a given quantum system or model space. In the following, we shall often denote the desired many-particle states of the S5 tem (and the determinants... 

It is easy to verify that the same representation holds for many-particle states. Now consider a unitary operator, for example, a rotation R. Then,... 

Formally, the KS determinant lkinetic energy under the constraint that its density p (r) is equal to that of the exact state l P) [28], that is. 

We have adopted a transparent notation for the eigen energies of the many-particle states, and we keep the notation that... 

The last member of the first group of methods was proposed by Ziegler, Rauk and Baerends in 1977 [31] and is based on an idea borrowed from multiconfiguration Hartree-Fock. The procedure starts with the construction of many-particle states with good symmetry, by taking a finite superposition of states... 

This is not quite the end of the matter. What we have assumed in the QED approach is that the reference vacuum is that of the current guess. If we were to take an absolute reference, such as the free-particle vacuum, the normal ordering should take place with respect to this fixed vacuum, and then the QED approach would give the same results as the filled Dirac approach, in which rotations between the negative-energy states and the unoccupied electron states affect the energy. By this means a vacuum polarization term has been introduced into the procedure, but without the renormalization term. In atomic structure calculations in which QED effects are introduced, the many-particle states employed are usually the Dirac-Fock states (Mittleman 1981), that is, those that result from the empty Dirac picture. We will therefore take as our reference the QED approach with the floating vacuum —a vacuum defined with respect to the current set of spinors. 

The subscripts I and J on the strings are indices of the particular set of Kramers pairs from which the creation operators that make up the string are taken. The spinors that make up the Kramers pairs can, in this notation, be labeled A and B spinors. The many-particle state can now be written... 

We now consider the effect of time reversal on a general many-particle state (determinant), which we write in terms of A and B strings ... 

As with most Cl schemes of that period, the construction of the Hamiltonian matrix and its direct diagonalization effectively limited the size of calculations to a few thousand determinants. One possible strategy for extending the capability of this type of calculation is to introduce some sort of selection criterion for the A -particle functions, and to leave out those that do not contribute appreciably. Such methods had been developed within the framework of multireference Cl (MR-CI) calculations, and Hess, Peyerimhoff, and coworkers (Hess et al. 1982) extended this to the case of spin-orbit interactions. Their procedure was based on performing a configuration-selected non-relativistic MR-CI, followed by extrapolation to zero threshold. This technique may be applied in a one-step scheme, where selection criteria are introduced not only for the correlating many-particle states, but also for those that couple to the reference space via spin-orbit interaction. The size of the calculation that has to be performed in the double group may thereby be reduced. The errors introduced by these selection procedures appear to be small. 

In general, a truly uncorrelated many-particle state is always represented by a product of one-particle functions. Conversely, any superposition of such products represents a stale where the motion of the particles is correlated. Nevertheless, a Slater determinant - which leiniesents an antisymmetric superposition of spin-orbital products - may in some cases represent a tmly uncorrelated electronic state. Thus, in the closed-shell bonding and antibonding configurations of the hydrogen molecule, the Pauli principle is satisfied by an antisymmetrization of the spin part of the wave function... 

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