Density matrix properties

In the HF LCAO method, (4.57) for the periodic systems replaces (4.33) written for the molecular systems. In principle, the above equation should be solved at each SCF procedure step for all the (infinite) fe-points of the Brillouin zone. Usually, a finite set kj j = 1, 2,..., L) of fe-points is taken (this means the replacing the infinite crystal by the cyclic cluster of L primitive cells). The convergence of the results relative to the increase of the fe-points set is examined in real calculations, for the convergent results the interpolation techniques are used for eigenvalues and eigenvectors as these are both continuous functions of k [84]. The convergence of the SCF calculation results is connected with the density matrix properties considered in Sect. 4.3... 

The study of the approximate density matrix properties allowed the implementation of the cyclic cluster model in the Hartree- Fock LCAO calculations of crystalUne systems [100] based on the idempotency relations of the density matrix. The results... 

Flead and Silva used occupation numbers obtained from a periodic FIF density matrix for the substrate to define localized orbitals in the chemisorption region, which then defines a cluster subspace on which to carry out FIF calculations [181]. Contributions from the surroundings also only come from the bare slab, as in the Green s matrix approach. Increases in computational power and improvements in minimization teclmiques have made it easier to obtain the electronic properties of adsorbates by supercell slab teclmiques, leading to the Green s fiinction methods becommg less popular [182]. 

N() -e that the summations are over the N/2 occupied orbitals. Other properties can be cali ulated from the density matrix for example, the electronic energy is ... 

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. 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

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... 

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]. 

To study the structure of the exchange-correlation energy functional, it is useful to relate this quantity to the pair-correlation function. The pair-correlation function of a system of interacting particles is defined in terms of the diagonal two-particle density matrix (for an extensive discussion of the properties of two-particle density matrices see [30]) as... 

M. Rosina and M. V. Mihailovic, The determination of the particle—hole excited states by using the variational approach to the ground state two-body density matrix, in International Conference on Properties of Nuclear States, Montreal 1969, Les Presses de I Universite de Montreal, 1969. 

Both the energy as well as the one- and two-electron properties of an atom or molecule can be computed from a knowledge of the 2-RDM. To perform a variational optimization of the ground-state energy, we must constrain the 2-RDM to derive from integrating an A -electron density matrix. These necessary yet sufficient constraints are known as A -representability conditions. 

K. Husimi, Some formal properties of the density matrix. Proc. Phys. Math. Soc. Japan 22, 264 (1940). 

A direction for improving DPT lies in the development of a functional theory based on the one-particle reduced density matrix (1-RDM) D rather than on the one-electron density p. Like 2-RDM, the 1-RDM is a much simpler object than the A-particle wavefunction, but the ensemble A-representability conditions that have to be imposed on variations of are well known [1]. The existence [10] and properties [11] of the total energy functional of the 1-RDM are well established. Its development may be greatly aided by imposition of multiple constraints that are more strict and abundant than their DPT counterparts [12, 13]. 

Levy identified the unknown part of the exact universal D functional as the correlation energy Ed D] and investigated a number of properties of c[ D], including scaling, bounds, convexity, and asymptotic behavior [11]. He suggested approximate explicit forms for Ec[ D] for computational purposes as well. Redondo presented a density-matrix formulation of several ab initio methods [26]. His generalization of the HK theorem followed closely Levy s... 

The structure of the reduced density matrix follows from the symmetry properties of the Hamiltonian. However, for this case the concurrence C iJ) depends on ij and the location of the impurity and not only on the difference i—j as for the pure case. Using the operator expansion for the density matrix and the symmetries of the Hamiltonian leads to the general form... 

Ramasesha, S., Pati, S.K., Krishnamurthy, H.R., Shuai, Z., Bredas, J.L. Low-lying electronic excitations and nonlinear optic properties of polymers via symmetrized density matrix renormalization group method. Synth. Met. 1997, 85(1-3), 1019. 

If we are interested only in properties that can be expressed in terms of q-electron operators, then it is sufficient to work with the th-order reduced-density matrix rather than the A -electron wavefunction [122-126]. In this section, we consider links between the r- and p-space representations of reduced-density matrices. In particular, we show that if we need the th-order density matrix in p space, then it can be obtained from its counterpart in r space without reference to the /-electron wavefunction in p space. 

Two-step calculation of molecular properties. To evaluate one-electron core properties (hyperfine structure, P,T-odd effects etc.) employing the above restoraton schemes it is sufficient to obtain the one-particle density matrix, Dpq, after the molecular RECP calculation in the basis of pseudospinors p. At the same time, the matrix elements Wpq of a property operator W(x) should be calculated in the basis of equivalent four-component spinors p. The mean value for this operator can be then evaluated as ... 

---