Density matrices definition

P is the total spinless density matrix (P = P + P ) and P is the spin density matrix (P = p" + P ). For a closed-shell system Mayer s definition of the bond order reduces to ... 

The original definition of natural orbitals was in terms of the density matrix from a full Cl wave function, i.e. the best possible for a given basis set. In that case the natural orbitals have the significance that they provide the fastest convergence. In order to obtain the lowest energy for a Cl expansion using only a limited set of orbitals, the natural orbitals with the largest occupation numbers should be used. 

The F matrix elements in eqs. (15) and (16) are formally the same as for closed-shell systems, the only difference being the definition of the density matrix in eq. (17), where the singly occupied orbital (m) has also to be taken into account. The total electronic energy (not including core-core repulsions) is given by... 

This follows from the definition (35) of fragment density matrices P that implies exact additivity of these fragment density matrices, i.e., they add up to the density matrix P of the complete molecule,... 

We shall start with the definition of density matrix [82-84]. For this purpose, we consider a two-state system. According to the expansion theorem we have... 

There are situations in which a definite wave function cannot be ascribed to a photon and hence cannot quantum-mechanically be described completely. One example is a photon that has previously been scattered by an electron. A wave function exists only for the combined electron-photon system whose expansion in terms of the free photon wave functions contains the electron wave functions. The simplest case is where the photon has a definite momentum, i.e. there exists a wave function, but the polarization state cannot be specified definitely, since the coefficients depend on parameters characterizing the other system. Such a photon state is referred to as a state of partial polarization. It can be described in terms of a density matrix... 

Newton, the limit h —> 0 is singular. The symmetries underlying quantum and classical dynamics - unitarity and symplecticity, respectively - are fundamentally incompatible with the opposing theory s notion of a physical state quantum-mechanically, a positive semi-definite density matrix classically, a positive phase-space distribution function. 

A quantum system of N particles may also be interpreted as a system of (r — N) holes, where r is the rank of the one-particle basis set. The complementary nature of these two perspectives is known as the particle-hole duality [13, 44, 45]. Even though we treated only the iV-representability for the particles in the formal solution, any p-hole RDM must also be derivable from an (r — A)-hole density matrix. While the development of the formal solution in the literature only considers the particle reduced Hamiltonian, both the particle and the hole representations for the reduced Hamiltonian are critical in the practical solution of N-representability problem for the 1-RDM [6, 7]. The hole definitions for the sets and are analogous to the definitions for particles except that the number (r — N) of holes is substituted for the number of particles. In defining the hole RDMs, we assume that the rank r of the one-particle basis set is finite, which is reasonable for practical calculations, but the case of infinite r may be considered through the limiting process as r —> oo. 

As shown in the second line, like the expression for the energy as a function of the 2-RDM, the energy E may also be expressed as a linear functional of the two-hole reduced density matrix (2-HRDM) and the two-hole reduced Hamiltonian K. Direct minimization of the energy to determine the 2-HRDM would require (r — A)-representability conditions. The definition for the p-hole reduced density matrices in second quantization is given by... 

These definitions are easily generalized from a pure state, described by to ensemble states, described by a system density matrix V, for which an expectation value is... 

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... 

Note that this ensemble includes contributions from different choices of ( ii I I Note also that the N-electron density matrix has been defined so that it is normalized to one.) This convex set of N-electron density matrices can be reduced to a convex set of /f-electron reduced density matrices using the definition... 

Using the definition < A > = Tr(pA), we can express all the matrix elements in the density matrix in terms of different spin-spin correlation functions [62] ... 

To complete the definition of the renormalization step for the left block, we also need to construct the new matrix representations of the second-quantized operators. In the product basis Z <8> p, matrix representations can be formed by the product of operator matrices associated with left, p j and the partition orbital p separately. Then, given such a product representation of O say, the renormalized representation O in the reduced M-dimensional basis / of LEFIi. p is obtained by projecting with the density matrix eigenvectors L defined above,... 

Many problems appear to be ripe for a more quantitative discussion. What is the error involved in the introduction of unstable states as asymptotic states in the frame of the 5-matrix theory 16 What is the role of dissipation in mass symmetry breaking What is the consequence of the new definition of physical states for conservation theorems and invariance properties We hope to report soon about these problems. We would like, however, to conclude this report with some general remarks about the relation between field description and particles. The full dynamical description, as given by the density matrix, involves both p0 and the correlations pv. However, the particle description is expressed in terms of p (see Eq. (50)). Now p has only as many elements as p0. Therefore the... 

This expression is actually sometimes used to define the first and second order density matrix. It is anyway useful to know that the first order density matrix elements are equal to the coefficients for the corresponding one-electron integrals in the energy expression and similarly for the second order density matrix elements. The definition of the third order density matrix is,... 

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 ... 

For a pulse-type NMR experiment, the assumption has a straightforward interpretation, since the pulse applied at the moment zero breaks down the dynamic history of the spin system involved. The reasoning presented here, which leads to the equation of motion in the form of equation (72), bears some resemblance to Kaplan and Fraenkel s approach to the quantum-mechanical description of continuous-wave NMR. (39) The crucial point in our treatment is the introduction of the probabilities izUa which are expressed in terms of pseudo-first-order rate constants. This makes possible a definition of the mean density matrix pf of a molecule at the moment of its creation, even for complicated multi-reaction systems. The definition of the pf matrix makes unnecessary the distinction between intra- and inter-molecular spin exchange which has so far been employed in the literature. 

In accordance with this definition the Heisenberg operators ca(t), cjj(f), etc. are equal to the time-independent Schrodinger operators at some initial time to. ca(to) = ca, etc. Density matrix of the system is assumed to be equilibrium at this time p(to) = peq. Usually we can take to = 0 for simplicity, but if we want to use to 0 the transformation to Heisenberg operators should be written as... 

Using the definition of the Wigner transform [5] of the density matrix,... 

The definition of the density matrix p(X) for the system with Hamiltonian Hn is... 

The density matrix method is useful in treating relaxation processes, linear and non-linear laser spectroscopies and non-equilibrium statistical mechanics. In this chapter, the definition of density matrix and the equation of motion (EOM) it follows are introduced. The projection operator technique, which makes the density matrix method a very powerful tool in non-equilibrium statistical mechanics, is presented. 

Before presenting the applications, the theoretical background for the density matrix method should be constructed in molecular terms. For this purpose, a detailed definition of the density matrix of the system should be discussed. Let us consider a molecule at position Ry interacting with an applied laser field of bandwidth 8co and corresponding wave-vector width 8k — 8co/c. Now, it is assumed that the laser beam spot is similar to 8k and a system has only two states described by n> and m ) with respective energies hcontt and ha, within 8co. In this case, the dynamics of the system can be described as... 

Figure 10 Comparison of individual (left) and average (right) density matrices in the case of a non-mutual exchange. (A) Definition of spin set or spin system (see Figure 9). (B) Basis functions (lines) and intramolecular single quantum coherences (arrows) defined by the spin set or spin system. (C) Elements corresponding to the intramolecular single quantum coherences in the density matrix.

This definition shows that C(r t) is the density matrix on the basis of the eigenfunctions of the Hamiltonian. Substituting Equation (51) into Equation (44) and rearranging the terms gives ... 

The expressions eqs. (1.197), (1.199), (1.200), (1.201) are completely general. From them it is clear that the reduced density matrices are much more economical tools for representing the electronic structure than the wave functions. The two-electron density (more demanding quantity of the two) depends only on two pairs of electronic variables (either continuous or discrete) instead of N electronic variables required by the wave function representation. The one-electron density is even simpler since it depends only on one pair of such coordinates. That means that in the density matrix representation only about (2M)4 numbers are necessary to describe the system (in fact - less due to antisymmetry), whereas the description in terms of the wave function requires, as we know n 2m-n) numbers (FCI expansion amplitudes). However, the density matrices are rarely used directly in quantum chemistry procedures. The reason is the serious problem which appears when one is trying to construct the adequate representation for the left hand sides of the above definitions without addressing any wave functions in the right hand sides. This is known as the (V-representability problem, unsolved until now [51] for the two-electron density matrices. The second is that the symmetry conditions for the electronic states are much easier formulated and controlled in terms of the wave functions (Density matrices are the entities of the second power with respect to the wave functions so their symmetries are described by the second tensor powers of those of the wave functions). 

You will find that many of the sources do not use exactly the same matrix representations for some of the product operators and rotation matrices. The exact form of the density matrix depends on the numbering of the spin states and on certain conventions that are not consistent in the literature. In the above examples, the definitions are consistent with the product operator methods and with themselves. 

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