Reduced density-matrix discussion

Husimi, K., Proc. Phys.-Math. Soc. Japan 22, 264, "Some formal properties of the density matrix." Introduction of the concept of reduced density matrix. Statistical-mechanical treatment of the Hartree-Fock approximation at an arbitrary temperature and an alternative method of obtaining the reduced density matrices are discussed. 

However, billiard balls are a pretty bad model for electrons. First of all, as discussed above, electrons are fermions and therefore have an antisymmetric wave function. Second, they are charged particles and interact through the Coulomb repulsion they try to stay away from each other as much as possible. Both of these properties heavily influence the pair density and we will now enter an in-depth discussion of these effects. Let us begin with an exposition of the consequences of the antisymmetry of the wave function. This is most easily done if we introduce the concept of the reduced density matrix for two electrons, which we call y2. This is a simple generalization of p2(x1 x2) given above according to... 

The dynamical behaviors of p(At) v and p(At)av av, have to be determined by solving the stochastic Liouville equation for the reduced density matrix the initial conditions are determined by the pumping process. For the purpose of qualitative discussion, we assume that the 80-fs pulse can only pump two vibrational states, say v = 0 and v = 1 states. In this case we obtain... 

From the discussion so far, it is clear that the mapping to a system of noninteracting particles under the action of suitable effective potentials provides an efficient means for the calculation of the density and current density variables of the actual system of interacting electrons. The question that often arises is whether there are effective ways to obtain other properties of the interacting system from the calculation of the noninteracting model system. Examples of such properties are the one-particle reduced density matrix, response functions, etc. An excellent overview of response theory within TDDFT has been provided by Casida [15] and also more recently by van Leeuwen [17]. A recent formulation of density matrix-based TD density functional response theory has been provided by Furche [22]. 

For simplicity, we shall commonly refer to the Q-electron distribution function as the 2-density and the 2-electron reduced density matrix as the 2-ntatrix. In position-space discussions, the diagonal elements of the 2-ntatrix are commonly referred to as the 2-density. In this chapter, we will also refer to the diagonal element of orbital-space representation of the Q-vaatnx as the 2-density. 

Because polynomials of orbital occupation number operators do not depend on the off-diagonal elements of the reduced density matrix, it is important to generalize the Q, R) constraints to off-diagonal elements. This can be done in two ways. First, because the Q, R) conditions must hold in any orbital basis, unitary transformations of the orbital basis set can be used to constrain the off-diagonal elements of the density matrix. (See Eq. (56) and the surrounding discussion.) Second, one can replace the number operators by creation and annihilation operators on different orbitals according to the rule... 

In a famous after-dinner address at a 1959 conference in Boulder, Colorado, Charles Coulson [1] discussed both the promises and the challenges of using the two-electron reduced density matrix (2-RDM) rather than the many-electron wavefunction as the primary variable in quantum computations of atomic and molecular systems. Integrating the A-electron density matrix,... 

Equns. (9.15) and (9.17), which correspond to (8.14) and (8.15), restrict the number of parameters so that only eight independent parameters are required to characterise in general the reduced density matrix of the scattered electrons. In special situations the number of independent parameters can be reduced even further as discussed earlier. 

The Redfield equation describes the time evolution of the reduced density matrix of a system coupled to an equilibrium bath. The effect of the bath enters via the average coupling V = and the relaxation operator, the last sum on the right of Eq. (10.155). The physical implications of this term will be discussed below. 

He goes on to show that for the description of the energy it is sufficient to know the second-order reduced density matrix F(jc, jc2 i 2)- This was a favorite subject when he was lecturing and led to speculations about the possibility to compute the second-order density matrix directly, and discussions of the so called A/ -represent-ability problem. In spite of several attempts, this way of attacking the quantum chemical many-particle problem has so far been unsuccessful. Of special interest was the first-order reduced density matrix, 7( 1 jC ), which when expanded in a complete one-electron basis, ij/, is obtained as... 

In the final part the application of concepts from information theory is reviewed. After covering the necessary theoretical background a particular form of the Kullback-Liebler information measure is adopted and employed to define a functional for the investigation of density functions throughout Mendeleev s Table. The evaluation of the constructed functional reveals clear periodic patterns, which are even further improved when the shape function is employed instead of the density functions. These results clearly demonstrate that it is possible to retrieve chemically interesting information from the density function. Moreover the results indicate that the shape function further simplifies the density function without loosing essential information. The latter point of view is extensively treated in [64], where the authors elaborately discuss information carriers such as the wave function, the reduced density matrix, the electron density function and the shape function. 

As described in Sec. 3.1, each Hartree-Fock iteration involves the construction of the Fock matrix for a given density matrix, followed by the diagonalization of the Fock matrix to generate a set of improved spin orbitals and thus an improved density matrix. Formally, the construction of the Fock matrix requires a number of operations proportional to K4, where K is the number of atoms (because the number of two-electron integrals scales as Al4). For large systems, however, this quartic scaling with K (i.e., with system size) can be reduced to linear by special techniques, as will now be discussed. 

In the present section we first review some of the standing difficulties appearing in density matrix theory, such as the IV-representability of the the reduced 2-matrix. We also discuss the nature of the functional" v- and iV-representability problems in the Hohenberg-Kohn versions of density functional theory. We show how the neglect of functional iV-representability leads to an ill-posed variational problem. We then indicate how these difficulties can be removed from density functional theory by a reformulation based on local-scaling transformations, 21, 22]. 

Before going on to consider more complicated systems, we review here some of the basic behavior of a two-state quantum system in the presence of a fast stochastic bath. This highly simplified bath model is useful because it allows qualitatively meaningful results to be obtained from a density matrix calculation when bath correlation functions are not available in fact, the bath coupling to any given system operator is reduced to a scalar. In the case of the two-level system, analytic results for the density matrix dynamics are easily obtained, and these provide an important reference point for discussing more complicated systems, both because it is often possible to isolate important parts of more complicated systems as effective two-level systems and because many aspects of the dynamics of multilevel systems appear already at this level. An earlier discussion of the two-level system can be found in Ref. 80. The more... 

The single concept, which is defined and discussed in these articles, that without doubt has had the most penetrating impact on the whole field of quantum chemistry, is the concept of the natural spin orbitals. This concept is, in principle, very simple the set of MOs that makes the first-order density matrix diagonal. Lowdin starts out by defining a hierarchy of reduced density matrices for a general wave function ... 

In the spin-free formulations of the UGA-MRCC theories, the use of CSFs entails that both the MRCC equations are in matrix form and the associated effective Hamiltonians will involve various w-body spin-free reduced density matrices (n-RDMs). n-RDMs are product separable and hence not size-extensive. From now on, we will refer to the spin-free RDMs as simply the ROMs. When spinorbital-based RDMs are discussed, we will explicitly indicate this. So, no confusions should arise. It is non-trivial to establish the extensivity of both the cluster operators and the effective Hamiltonian in spite of the occurrence of these n-RDMs. This paper will briefly review the formulation of the UGA-MRCC theories mentioned above and will present a comprehensive account of the aspects of connectivity which leads to extensivity. Although in some of our earlier papers [47] we sketched how size extensivity emerges after the cumulant decomposition of the n-RDMs, we will present here a detailed and thorough analysis of the underlying issues. 

A comprehensive discussion of this important derivation can be found in Refs. 38 and 39. The nuclear contributions to the TM leading to vibrational fine structure in the spectra are discussed in the following section. Alternatively, the electronic part of the TM is obtained from the first-order reduced transition density matrix y as... 

To facilitate the discussion, we couch DFT in the language of p, the first-order reduced density operator of the noninteracting reference system. Consider an N electron system in a spin-compensated state and in an external potential Wext(r) (extension to spin-polarized state is trivial). The real space representation of p is the density matrix... 

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