Density matrix single-particle

Fig. 3.7. Schematic comparison of the multicanonical and iterative transition-matrix methods. The light gray box indicates a single iteration of several thousand or more individual Monte Carlo steps. The acceptance criterion includes the configuration-space density of the original ensemble, p, and is presented for symmetric moves such as single-particle displacements...

In Equations 4.1 and 4.2, the numbers before the integral signs occur due to the indistinguishability of electrons and electron pairs, respectively. The single-particle density p(x) is defined as the diagonal element of the single-particle density matrix Pi(xi xi), viz.,... 

In Section 8.4.2, we considered the problem of the reduced dynamics from a standard DFT approach, i.e., in terms of single-particle wave functions from which the (single-particle) probability density is obtained. However, one could also use an alternative description which arises from the field of decoherence. Here, in order to extract useful information about the system of interest, one usually computes its associated reduced density matrix by tracing the total density matrix p, (the subscript t here indicates time-dependence), over the environment degrees of freedom. In the configuration representation and for an environment constituted by N particles, the system reduced density matrix is obtained after integrating pt = T) (( over the 3N environment degrees of freedom, rk Nk, ... 

For degenerate states a problem arises with the definition of cumulants. We consider here only spin degeneracy. Spatial degeneracy can be discussed on similar lines. For S 0 there are (2S + 1) different Afs-values for one S. The n-particle density matrix p Ms) = of a single one of these states does not... 

Equations (7) can be viewed as a formal Taylor-series expansion, around the averaged part of the one-particle density matrix, of the HF energy functional E[p] [16, 18], this defining a shell-correction series . In Eqn (13) the first-order term of this expansion is expressed in terms of the single-particle energies e,. 

The static Hartree-Fock problem assumes T-reversal invariance and T-even single-particle density matrix. In this case, Skyrme forces can be limited by only T-even densities psif) Tsif) In the case of dy-... 

Using the definition (1.10) of the single-particle density matrix and Eq. (1.11), it is easy to obtain an equation for the single-particle density matrix ... 

The essential information about transport properties in many-particle systems is given by the single-particle density matrix or by the singleparticle Wigner distribution. The equations of motion (1.18) and (1.23) for these important quantities are called kinetic equations. For the further consideration we write the latter equation in the momentum representation ... 

The averages of the operators in Eqs. (190) and (194) are the elements of the density matrix in the single-particle space... 

It is important to note, that the single-particle density matrix (195) should not be mixed up with the density matrix in the basis of many-body eigenstates. 

We can calculate the natural one-particle states from the density matrix generated by the VB wave function. However, for chemical interpretation purposes it is better to analyse the non-orthogonal singly-occupied orbitals since each one will correspond to an atomic localized electron overlapping (making a chemical bond) with another one. To illustrate the importance of a non-zero overlap among the spatial orbitals we can calculate the energy expression for this simple case ... 

Here if RQM> i) represents the quantum wavepacket value at the spatial point Rqm at time t, Rc represents the classical nuclear coordinates and Pc represents the single-particle electronic density matrix. It is important to note that Eqs. (10) and... 

Though we can compare electron densities directly, there is often a need for more condensed information. The missing link in the experimental sequence are the steps from the electron density to the one-particle density matrix f(1,1 ) to the wavefunction. Essentially the difficulty is that the wavefunction is a function of the 3n space coordinates of the electrons (and the n spin coordinates), while the electron density is only a three-dimensional function. Drastic assumptions must be introduced, such as the description of the molecular orbitals by a limited basis set, and the representation of the density by a single Slater-determinant, in which case the idempotency constraint reduces the number of unknowns... 

Besides displaying only linear local convergence, another problem with the SCI method is that the construction of the H matrix requires either the explicit construction of the single excitation states or the evaluation of four-particle density matrix type elements of the form <0 In an... 

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