Density Matrices Of Systems Research Articles

Integrity bases for local invariants of composite quantum systems

Unitary group branchings appropriate to the calculation of local invariants of density matrices of composite quantum systems are formulated using the method of $S$-function plethysms. From this, the generating function for the number of invariants at each degree in the density matrix can be computed. For the case of two two-level systems the generating function is $F(q) = 1 + q + 4q^{2} + 6 q^{3} + 16 q^{4} + 23 q^{5} + 52 q^{6} + 77 q^{7} + 150 q^{8} + 224 q^{9} + 396 q^{10} + 583 q^{11}+ O(q^{12})$. Factorisation of such series leads in principle to the identification of an integrity basis of algebraically independent invariants. This note replaces Appendix B of our paper\cite{us} J Phys {\bf A33} (2000) 1895-1914 (\texttt{quant-ph/0001076}) which is incorrect.

Linear-scaling localized-density-matrix method for the ground and excited states of one-dimensional molecular systems

A linear-scaling localized-density-matrix (LDM) method is developed to evaluate the ground-state reduced single-electron density matrices of one-dimensional molecular systems. The new method may be combined with the existing linear-scaling LDM method for the excited states (Yokojima and Chen, Chem. Phys. Lett. 292 (1998) 379), and thus leads to a linear-scaling calculation method for the properties of both the ground and excited states. The combined method is applied to the polyacetylene oligomers and the linear-scaling of the total computational time is clearly demonstrated.

Remarks on separability of compound quantum systems and time reversal

A separability test for density matrices of compound quantum systems proposed recently by Peres is formulated in a manifestly basis independent way using antiunitary operators. We prove that the test amounts to a criterion for pure states, while it is known to fail to do so in the case of mixed states. An interesting connection with the operation of time reversal is thus exhibited.

Effective Hamiltonian for conjugated polyenes based on the ground state single-electron density matrix

A method for constructing effective Hamiltonians using the information contained in the ground state reduced single-electron density matrices for many-electron systems is proposed and applied to trans-polyacetylene oligomers C 2 n H 2 n+2 with n = 4−24. The resulting effective Hamiltonian for π electrons, expanded in the basis set of natural atomic orbitals (NAOs) with nearest neighbor transfer integrals, reproduce accurately the Hartree-Fock ground state density matrices.

Direct sampling of density matrices in field-strength bases.

A simple and efficient method for direct sampling of the density matrix of a signal mode in optical homodyne tomography is developed. The method is based on approximate, quasianalytical techniques of Fourier transform and yields the density matrix in an arbitrary field-strength basis. It is also suitable for direct sampling of density matrices of matter systems in generalized position or momentum bases. \textcopyright{} 1996 The American Physical Society.

Use of computer algebra for the study of quadrupole spin systems

Analytical computation of the effect of radiofrequency pulses and free evolution between pulses are studied in a spin system dominated by the nuclear electric quadrupole interaction using ‘symbolic computer algebra’. The program MAPLE is used to compute spin density matrices for single spin systems with spin magnitude I = 1, 3/2, 2 and 5/2. In all cases except I = 5/2, the pure nuclear electric quadrupole Hamiltonian with arbitrary asymmetry parameter η is used to obtain results for a single pulse and an evolution period following the pulse. In the spin 5/2 system η is set to zero. The calculations are done using a simple matrix representation of the density operator and the Baker-Campbell-Hausdorff formula. In addition, the validity of dropping non-secular (i.e., time-dependent) terms from the Hamiltonian for the pulse in the quadrupole interaction frame is examined for the spin 1 case using a truncated Magnus expansion.

Tensor propagator for iterative quantum time evolution of reduced density matrices. II. Numerical methodology

In a recent Letter [Chem. Phys. Lett. 221, 482 (1994)], we demonstrated that the dynamics of reduced density matrices for systems in contact with dissipative harmonic environments can be obtained in an iterative fashion by multiplication of a propagator tensor. The feasibility of iterative procedures in reduced dimension spaces arises from intrinsic features of the dissipative influence functional in Feynman’s path integral formulation of quantum dynamics. Specifically, the continuum of frequencies characteristic of broad condensed phase spectra disrupts phase coherence to a large extent, such that the dynamics of an augmented reduced density tensor becomes Markovian. In a preceding article [J. Chem. Phys. 102, 4600 (1995)] we examined in detail the formal properties of the tensor propagator. In the present paper we show that the tensor propagator can be further decomposed into a product of small rank tensors, resulting in an extremely simple and efficient numerical scheme that scales almost linearly with the dimension of the augmented reduced density tensor. Numerical application to a model electron transfer reaction is presented.

Applications of cut polyhedra — I

In this paper and in its continuation (Part II, this issue), we group, within a unified framework, many applications of the following polyhedra: cut, boolean quadric, hypermetric and metric polyhedra. We treat, in particular, the following applications: • ℓ 1- and L 1-metrics in functional analysis, • the max-cut problem, the Boole problem and multicommodity flow problems in combinatorial optimization, • lattice holes in geometry of numbers, • density matrices of many-fermions systems in quantum mechanics. We present some other applications, in probability theory, statistical data analysis and design theory. In this first part, after introducing the main definitions and operations for cut polyhedra, we describe the connections with ℓ 1-metrics and with other metric properties, and we consider, in particular, the applications to some classes of metrics arising from graphs, normed spaces and lattices.

Applications of cut polyhedra — II

This is the continuation of Part I (this issue). In this second part, we present the following applications of cut polyhedra: the max-cut problem, the Boole problem and the multicommodity flow problems in combinatorial optimization, lattice holes in geometry of numbers, density matrices of many-fermions systems in quantum mechanics, as well as some other applications, in probability theory, statistical data analysis and design theory. As we shall frequently use results, definitions and notation from Part I, the sections in this second part are numbered consecutively.

Density matrices from densities

The density matrix can be reconstructed from the density alone, if a basis set is used whose product functions are all linearly independent. This is not the case for quantum-chemical basis sets, except for small s-type sets. In case of the idempotency constraint on the density matrix - justified in the density functional approach, but in general not in standard ab initio approaches - only a few linearly indenpedent product functions are required. Then very accurate densities open the feasibility to reconstruct density matrices for systems with a single s-valence shell on each atom. For more general systems the density matrix cannot be obtained so easily from the density without exploiting additional data.

On the direct determination of constrained pure state one-electron density matrices: Part I. A new theoretical model for closed-shell systems

A new method based on the penalty-function way of satisfying equality constraints is proposed for the determination of constrained pure state one-electron density matrices for closed-shell many-electron systems. The algorithm suggested can handle many constraints simultaneously. Certain interesting features of the proposed algorithm are discussed with numerical examples.

Invariants and nonequilibrium density matrices

A new method of calculating nonequilibrium density matrices with the aid of the quantum integrals of motion is proposed. The method is shown to be very effective in the case of systems described by means of quadratic Hamiltonians. The possibility of constructing phenomenological nonstationary Hamiltonians for a wide class of dissipative systems is discussed. The exact formulas for nonequilibrium density matrices of arbitrary quadratic systems are obtained. The quantum problem of the motion of a charged particle in uniform electric and magnetic fields in the presence of a frictional force proportional to the velocity is solved exactly by means of introducing the new phenomenological Hamiltonian.

The variational calculation of reduced density matrices

We discuss the computational problem encountered in making direct variational calculations of the reduced density matrices of many-particle systems. The problem is one of minimizing a linear function within a convex domain defined by a finite set of nonlinear constraints. Two different algorithms are presented for which working programs have been written.

Masterequation,H-theorem and stationary solution for coupled quantum systems

We extend the Zwanzig projector formalism to coupled systems taking into account the mutual interactions of the reduced density matrices of both systems. In the Born- and Markoff-approximation we end up with a bilinear masterequation for occupation probabilities, in contrast to the usually studied linear equations. We derive theH-theorem for this equation and show that the stationary solution is the canonical or more generally a grand canonical density matrix.

Semiclassical treatment of bound state systems: Trajectory calculations using an analytic approximation to the quantum momentum

Using an analytic approximation to the quantum momentum previously derived to calculate single particle reduced density matrices in the semiclassical approximation, we have been able to construct equations of motion with which to perform trajectory calculations. Information calculated along these trajectories is used to calculate single particle reduced density matrices for one-dimensional systems. Charge densities for N fermions in a harmonic oscillator potential calculated using this quantum momentum approximation are in excellent agreement with the exact quantum mechanical calculations.

General Structure of the Distribution Functions for the Heisenberg Model and the Ising Model

The general structure for the distribution functions (reduced density matrices) for systems composed of a number of elements is given by taking the variation with respect to the distribution functions in the formalism of the cluster variation method. The parameters or the Lagrange multipliers occurring in the distribution functions must be determined by the reducibility condition of the distribution functions or by the stationariness condition of the free energy.

Direct Calculation of First- and Second-Order Density Matrices. The Higher RPA Method

We present a procedure for approximately determining the first- and second-order density matrices of pure-state N-fermion systems. A higher random-phase approximation is used to derive a method in which the resulting density matrices are approximated self-consistently in an iterative scheme. As test calculations, the method is applied to the ground states of the helium, lithium, and beryllium atoms. The possibility and importance of obtaining error bounds involving the density matrices are discussed.

Reduced-Density-Matrix Theory: The 2-Matrix of Four-Electron Systems

The theory of reduced density matrices for polyelectronic systems is formulated in a manner such that the reduced density matrix of any order $p$ is characterized by a coefficient matrix. This matrix of coefficients, resulting from expressing the polyelectronic wave function in the appropriate bilinear form, is sufficient to allow one to find the eigenvalues and the transformation to natural form. This formalism is a generalization of the work of L\owdin and Shull on the natural orbitals of two-electron systems. The second-order reduced density matrix, the 2-matrix, is obtained exactly from the approximate solutions $\ensuremath{\Psi}$ of the Schr\odinger equation for the Be-atom functions of Weiss, Watson, and Boys, and the LiH function of Ebbing. The important eigenfunctions and complete eigenvalue spectra of the integral operator ${\ensuremath{\Gamma}}^{(2)}$, which has the 2-matrix as kernel, are reported here. The degeneracies of the eigenvalue spectra of ${\ensuremath{\Gamma}}^{(2)}$ and the properties of the natural geminals, the eigenfunctions of ${\ensuremath{\Gamma}}^{(2)}$, are discussed in detail. The multiplicities 1, 2, 3, 4, and 6 are the only nonaccidental degeneracies that can occur in the 4-electron problem when the one-electron basis of $\ensuremath{\Psi}$ is considered in symmetry-adapted spin-orbital form. The natural geminals can always be obtained in symmetry-adapted form and can be completely described by a set of numbers ($\ensuremath{\lambda},s,{m}_{s},{m}_{l},\dots{}$), eigenvalues for the operators ${\ensuremath{\Gamma}}^{(2)},{S}^{2},{S}_{z},{L}_{z},\dots{}$, respectively. The identity of the eigenvalue spectra and the equivalence of the two operators ${\ensuremath{\Gamma}}^{(p)}$ and ${\ensuremath{\Gamma}}^{(N\ensuremath{-}p)}$ are demonstrated in the case where the one-electron basis of $\ensuremath{\Psi}$ is finite. The natural expansion of $\ensuremath{\Psi}$ is defined as the expansion in eigenfunctions of ${\ensuremath{\Gamma}}^{(p)}$ and ${\ensuremath{\Gamma}}^{(N\ensuremath{-}p)}$. In the case $2p=N$, the phase of the two sets of eigenfunctions can be chosen as equal and the signs of the natural expansion coefficients are uniquely determined by the function $\ensuremath{\Psi}$.

Simplified Derivation of the Binary Collision Expansion and Its Connection with the Scattering Operator Expansion

The binary collision expansion for the density matrix of a system of $N$ particles with pair interaction was derived by Huang, Lee, and Yang by expansion in the interaction and subsequent summation over terms represented by certain classes of diagrams. A simpler derivation has been obtained by the use of the $\frac{N(N\ensuremath{-}1)}{2}$ integral equations which relate the density matrix of the system to the density matrices of systems in which only one pair of particles interacts. Successive substitution of the integral equations into each other yields an expansion which by a trivial additional step becomes the binary collision expansion. Our derivation shows also that the binary collision expansion is the Laplace inverse of the coordinate representation of the expansion in terms of scattering operators.