Density matrix diagonal elements

At each stage of the refinement of a new set of parameters, the hat matrix diagonal elements were calculated in order to detect the influential observations following the criterium of Velleman and Welsh [8,9]. The inspection of the residues of such reflections revealed those which are aberrant but progressively, these aberrations disappeared when the pseudo-atoms model was used (introduction of multipoler coefficients). This fact confirms that the determination of the phases in acentric structures is improved by sophisticated models like the multipole density model. 

The CNDO/INDO, MINDO/3, Z3NDO/1, and ZINDO/S methods might be expected to imply an even simpler equation for the electron density than the above. For example, a rigorous complete neglect of CNDO approximation, suggests that equations (87) and (88) should be replaced by expressions with a sum only over diagonal elements of the density matrix. This would represent a molecular charge density that is the exact sum of atomic densities. Alter-... 

In a ZDO computation where the overlap matrix is assumed to be the unit matrix, the diagonal elements of the density matrix... 

The simplest scheme that accounts for the destruction of phase coherence is the so-called stochastic interruption model [Nikitin and Korst 1965 Simonius 1978 Silbey and Harris 1989]. Suppose the process of free tunneling is interrupted by a sequence of collisions separated by time periods vo = to do After each collision the system forgets its initial phase, i.e., the off-diagonal matrix elements of the density matrix p go to zero, resulting in the density matrix p ... 

The density matrix p describes the pure state, as seen from the equality p = p, while p does not. The transition from (2.35a) to (2.35b) describes a strong collision , which fully localizes the particle, but in general the off-diagonal elements may not completely vanish. This however does not affect the qualitative picture. 

More accurately, one rewrites the problem in terms of the coordinates Q+ and Q. The probability to be at a certain point is given by the diagonal element of the density matrix... 

The electron density is the diagonal element of the number density matrix N(r,r ), i.e the first order redueed density matrix after integration over the spin coordinates, ... 

We notice that neither the momentum distribution nor the reciprocal form factor seems to carry any information about the translational part of the space group. The non diagonal elements of the number density matrix in momentum space, on the other hand, transform under the elements of the space group in a way which brings in the translational parts explicitly. 

Density functions can be obtained up to any order from the manipulation of the Slater determinant functions alone as defined in section 5.1 or from any of the linear combinations defined in section 5.2. Density functions of any order can be constructed by means of Lowdin or McWeeny descriptions [17], being the diagonal elements of the so called m-th order density matrix, as was named by Lowdin the whole set of possible density functions. For a system of n electrons the n-th order density function is constructed from the square modulus of any n-electron wavefunction attached to the n-electron system somehow. 

It should be obvious that the diagonal elements of this matrix (i. e., for x, = x and x2 = x2) bring us back to our pair density p2(x1 x2) defined above. If we now look at the special situation that x, = x2, that is the probability that two electrons with the same spin are found within the same volume element, we find that... 

In general the spectral one-particle density matrix p(r, rE) describes the mutual coherence of the wave field of high-energy electrons at the points r and r. For the simplest case of time-independent interaction potential the diagonal elements of... 

In general, the equations for the density operator should be solved to describe the kinetics of the process. However, if the nondiagonal matrix elements of the density operator (with respect to electron states) do not play an essential role (or if they may be expressed through the diagonal matrix elements), the problem is reduced to the solution of the master equations for the diagonal matrix elements. Equations of two types may be considered. One of them is the equation for the reduced density matrix which is obtained after the calculation of the trace over the states of the nuclear subsystem. We will consider the other type of equation, which describes the change with time of the densities of the probability to find the system in a given electron state as a function of the coordinates of heavy particles Pt(R, q, Q, s,...) and Pf(R, q, ( , s,... ).74,77 80... 

In coordinate space, the diagonal elements of the canonical density matrix in the Fourier path integral representation are given by [20]... 

If we are interested in the off-diagonal elements of the density matrix, that is p(x, x j3), the paths are no longer cyclic but begin at x and terminate at x1. Then the Fourier representation of the path is... 

The diagonal element of the density matrix, W(n) — a nan is the probability that a system chosen at random from the ensemble occurs in the state characterized by n, and implies the normalization... 

Construction of the density operator can also not be achieved without assumption of an additional axiom All quantum states of a system compatible with the knowledge revealed by macroscopic measurement have equal a priori probabilities and random a priori phases. This axiom implies that for a system as defined above all diagonal elements of the density matrix q belonging to the ith cell must be equal. Hence... 

In the case of an equilibrium system the Hamiltonian is the same as that of an ensemble of conservative systems in statistical equilibrium. If the energy of the system is measured to lie between Ek and EK + AE, then the representative ensemble is also restricted to the energy shell [AE K. From the hypotheses of equal a priori probabilities and random a priori phases it then follows that the diagonal elements of the density matrix lying inside [AE]k are all equal and that all others vanish. The density matrix of the quantum statistical microcanonical ensemble is thereby determined as... 

Beginning way back in the 20s, Thomas and Fermi had put forward a theory using just the diagonal element of the first-order density matrix, the electron density itself. This so-called statistical theory totally failed for chemistry because it could not account for the existence of molecules. Nevertheless, in 1968, after years of doing wonders with various free-electron-like descriptions of molecular electron distributions, the physicist John Platt wrote [2] We must find an equation for, or a way of computing directly, total electron density. [This was very soon after Hohenberg and Kohn, but Platt certainly was not aware of HK by that time he had left physics.]... 

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.,... 

If the external magnetic field B(r), and m(r) have only a nonvanishing Z-component, B(r) = (0,0, B(r)) and m(r) = (0,0, m(r)), the universal functional F[p, m] may then be considered as a functional of the spin densities ps(r) and p(r), F[ps(r), p(r)], because the spin density is proportional to the z-component of the magnetization m(r) = p-bPsW P-b is the electron Bohr magneton. It is of worth mentioning that it is possible to define two spin densities that are the diagonal elements of the density matrix introduced by von Barth and Hedin [3]. These correspond to the spin-up (alpha) electrons density pT(r), and the spin-down (beta) electrons density p (r). In terms of these quantities, the electron and spin densities can be written as... 

---