Two-particle interactions

Andersen,E.,andUHLHORN,U., r n K>m 13,165/ Approach to the quantum mechanical many-body problem with strong two-particle interaction/ ... [Pg.357]

Of particular interest is a system of two particles interacting by way of a time-independent potential V(r — r2 that only depends on the relative coordinate r= (ri - r2). The classical Hamiltonian of the system is given by... [Pg.334]

As we show later, the energy of the state of any system of N indistinguishable fermions or bosons can be expressed in terms of the Hamiltonian and D (12,1 2 ) if its Hamiltonian involves at most two-particle interactions. Thus it should be possible to find the ground-state energy by variation of the 2-matrix, which depends on four particles. Contrast this with current methods involving direct use of the wavefunction that involves N particles. A principal obstruction for this procedure is the A-representability conditions, which ensure that the proposed RDM could be obtained from a system of N identical fermions or bosons. [Pg.4]

We can justify the above conclusion as follows. If H involves at most two-particle interactions, it is expressible as... [Pg.7]

Theorem 1 The 2-RDM for the antisymmetric, nondegenerate ground state of an unspecified N-particle Hamiltonian H with two-particle interactions has a unique preimage in the set of N-ensemble representable density matrices D. [Pg.171]

The reconstruction functionals may be understood as substantially renormalized many-body perturbation expansions. When exact lower RDMs are employed in the functionals, contributions from all orders of perturbation theory are contained in the reconstructed RDMs. As mentioned previously, the reconstruction exactly accounts for configurations in which at least one particle is statistically isolated from the others. Since we know the unconnected p-RDM exactly, all of the error arises from our imprecise knowledge of the connected p-RDM. The connected nature of the connected p-RDM will allow us to estimate the size of its error. For a Hamiltonian with no more than two-particle interactions, the connected p-RDM will have its first nonvanishing term in the (p — 1) order of many-body perturbation theory (MBPT) with a Hartree-Fock reference. This assertion may be understood by noticing that the minimum number of pairwise potentials V required to connectp particles completely is (p — 1). It follows from this that as the number of particles p in the reconstmcted RDM increases, the accuracy of the functional approximation improves. The reconstmction formula in Table I for the 2-RDM is equivalent to the Hartree-Fock approximation since it assumes that the two particles are statistically independent. Correlation corrections first appear in the 3-RDM functional, which with A = 0 is correct through first order of MBPT, and the 4-RDM functional with A = 0 is correct through second order of MBPT. [Pg.178]

Rosina s theorem states that for an unspecified Hamiltonian with no more than two-particle interactions the ground-state 2-RDM alone has sufficient information to build the higher ROMs and the exact wavefunction [20, 51]. Cumulants allow us to divide the reconstruction functional into two parts (i) an unconnected part that may be written as antisymmetrized products of the lower RDMs, and (ii) a connected part that cannot be expressed as products of the lower RDMs. As shown in the previous section, cumulant theory alone generates all of the unconnected terms in RDM reconstruction, but cumulants do not directly indicate how to compute the connected portions of the 3- and 4-RDMs from the 2-RDM. In this section we discuss a systematic approximation of the connected (or cumulant) 3-RDM [24, 26]. [Pg.179]

In what follows a two-particle interacting system having a hxed and weU-dehned number of particles N will be considered. It will also be considered that the one-electron space is spanned by a hnite basis set of 2K orthonormal spin orbitals. Under these conditions the 1-RDM and 2-RDM elements are dehned in second... [Pg.207]

If 5(2) is restricted to be an anti-Hermitian operator with no more than two-particle interactions, the variational degrees of freedom of 5(2) can be... [Pg.334]

This entropic force is important where adsorption of polymers occnrs on colloidal particles. This is due to interaction between polymer chains on the interacting particles As the particles approach each other to the point where the polymer chains of the two particles interact, there is an decrease in entropy due to confinement of the chains, in an analogous manner to the solution species discussed earher, with the same result— repulsion. This is the basis of polymeric stabilization of colloids it is generally undesirable in CD, since adhesion and aggregation are preferred in this case. However, in view of the fact that the presence of such polymers (and other stabilizing adsorbates) may prevent the aggregation needed to build up a CD layer, it is important to be aware of the effect. [Pg.36]

Hamiltonian for this problem, in terms of one- and two-particle interactions is... [Pg.339]

In these relationships the summation is carried out using (6.25), (6.26) and (6.18). Let us now introduce averaged submatrix elements of two-particle interactions separately for singlet states... [Pg.136]

The Umn s of Eq. (3) contain two-particle interactions, including Coulomb, exchange, and dipole-dipole contributions, which are parameterized according to semi-empirical functional forms [61]. The parameters are adapted to PPV and are then transposed by scaling to other polymer species. [Pg.192]

The form of the potential for the system under study was discussed in many publications [28,202,207,208]. Effective pair potentials are widely used in theoretical estimates and numerical calculations. When a many-particle interatomic potential is taken into account, the quantitative description of experimental data improves. For example, the consideration of three-body interactions along with two-particle interactions made it possible to quantitatively describe the stratification curve for interstitial hydrogen in palladium [209]. Let us describe the pair interaction of all the components (hydrogen and metal atoms in the a. and (j phases) by the Lennard Jones potential cpy(ry) = 4 zi [(ff )12- / )6], where Sy and ai are the parameters of the corresponding potentials. All the distances ry, are considered within c.s. of radius r (1 < r < R), where R is the largest radius of the radii of interaction Ry between atoms / and /). [Pg.422]

The second term in our new total energy expression is a short-range repulsive two-particle interaction and contains a correction for double counting the electrons in the band energy. It is equal to E = - E. Symbolically, the new total energy expression can, therefore, be written as ... [Pg.238]

Here, attention will be drawn to the important work of Rosina [79] and to the subsequent discussion of his study by Mazziotti [80], As summarized in Ref. [80], Rosina showed that the ground-state 2 DM for a quantum system completely determines the exact N-electron ground-state wave function without any specific knowledge of the exact Hamiltonian except that it has no more than two-particle interactions. Mazziotti [80] points out that a consequence of this theorem is that any ground-state electronic 2 DM precisely determines within the ensemble N-representable space a unique series of higher p-DMs where 2 < p < N. He asserts that these results provide important justification for the functional description of the higher DMs in terms of the 2 DM. [Pg.220]

Three types of two-particle interactions can occur. Each is represented by a horizontal dashed interaction line. For the nucleus-nucleus interaction this line is terminated by open dots, o, at both end, that is... [Pg.46]

The two-particle interactions can be subdivided into those between nuclei, those between nuclei and electrons and those between electrons. [Pg.47]

Figure 9. Second-order diagrams involving a one- and two-particle interactions in the perturbation theory of nuclei and electrons.

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