Exact Density Matrix Research Articles

Direct and indirect DIET and DIMET from semiconductor and metal surfaces: what can we learn from 'toy models'?

Desorption induced by electronic transitions (DIET) and its variant DIMET (M = 'Multiple'), are among the simplest possible "reactions" of ad-species involving ultra-short lived electronically excited states at surfaces. The non-adiabatic bond-cleavage can be enforced, for example, with laser irradiation or with electrons or holes emitted from the tip of a scanning tunnelling microscope (STM). The transient creation of excited intermediates can proceed directly (localised to the adsorbate-substrate complex), or indirectly (i.e., through the substrate). To understand the basic processes, simple one-mode two-state "toy models" such as the Menzel-Gomer-Redhead (MGR) or the Antoniewicz scenarios have proven very useful in the past. We adopt and extend MGR- and Antoniewicz-type models together with numerically exact open-system density matrix theory to address a few actual problems/experiments in DI(M)ET: (1) Direct, laser-induced desorption of H(D) from Si(100) surfaces which has been realised in the continuous-wave DIET regime only recently [T. Vondrak and X.-Y. Zhu, Phys. Rev. Lett., 1999, 82, 1967], is studied and compared to so-far hypothetical femtosecond laser desorption. The possibility of controlling the reaction by shaping the laser pulses is addressed. (2) For the same system, temperature effects are studied for electron- or hole-stimulated desorption with an STM [T. C. Shen, C. Wang, G. C. Abeln, T. R. Tucker, J. W. Lyding, Ph. Avouris and R. E. Walkup, Science, 1995, 268, 1590; C. Thirstrup, M. Sakurai, T. Nakayama and K. Stokbro, Surf. Sci., 1999, 424, L329]. A modified version of Gadzuk's "sudden transition and averaging" approach is adopted which accounts for temperature dependent excited state lifetimes. (3) For photodesorption of NO from Pt(111), based on quantum dynamical simulations possible experimental tests involving static electric fields are suggested to address the relevance of the recently challenged [F. M. Zimmermann, Surf. Sci., 1997. 390, 174], "negative ion resonance" model of the Antoniewicz type.

Possibilities for a density matrix theory

Two possible routes are considered to arrive at a one-particle reduced density matrix formulation of electronic structure theory. In the first scheme, an extended Fock matrix H is defined that has twice the dimension of the one-particle basis set. The corresponding Green’s function, defined as the upper left block of (ω1−H)−1, yields the exact one-particle density matrix and energy. The poles of the Green’s function are precisely the ionization potentials and electron affinities of the extended Koopmans theorem. In the second scheme, a generalized Fock equation [F(ρ),ρ]=X is derived that is satisfied by the exact non-idempotent one-particle density matrix. The antisymmetric matrix X on the right-hand side is obtained from the irreducible part of the two-particle reduced density matrix, while F is the usual Fock matrix defined using the correlated one-matrix. The generalized Fock equation is a necessary condition but does not determine ρ uniquely. Alternatively, the one-matrix can be obtained from the irreducible part of the two-matrix directly, using a sum rule. The analysis leads to some additional desiderata and separability properties that may be imposed on traditional wave function based approaches. Possibilities for practical computational schemes are addressed briefly.

Crossover between ordinary and normal transitions in two dimensional critical Ising films.

We investigate two dimensional critical Ising films of width L with surface fields H(1)=H(L) in the crossover between ordinary (H(1)=0) and normal (H(1)=infinity) transitions. Using exact transfer-matrix diagonalization and density matrix renormalization-group (DMRG) methods, we calculate magnetization profiles m(z), the excess magnetization Gamma, and the analog of the solvation force f(solv) as functions of H1 for several L. Scaling functions of the above quantities deviate substantially from their asymptotic forms at fixed points for a broad region of the scaling variable LH21 approximately L/l(1), where l(1) is the length induced by the surface field H1. The scaling function for /f(solv)/ has a deep minimum near LH(2)(1)=1, which is about one order of magnitude smaller than its value at both fixed points (the "Casimir" amplitude). For weak H1 (l(1)>L) the magnetization profile has a maximum at the center of the film, and f(solv) decays much faster than L-2. For stronger H1 (1<l(1)<L), the magnetization has two maxima at a distance approximately l(1) from the walls, and the solvation force decays much slower than L-2. For L>>l(1) the solvation force decays according to the universal power law f(solv) approximately L(-2). The results of the approximate DMRG method show remarkable agreement with the exact ones.

Soliton bound states in the Raman spectrum of pure and doped spin-Peierls chains

We investigate the occurrence of singlet bound states in the Raman spectrum of dimerized spin 1/2 chains by exact diagonalization and density matrix renormalization-group techniques. We predict that several bulk $\ensuremath{\delta}$ peaks could be observed in pure systems. Furthermore, we show that additional low-energy lines arise from nonmagnetic impurity doping. These features are interpreted in terms of soliton-antisoliton and soliton-impurity bound states, respectively. Energies and spectral weights associated with these bound states are sensitive to lattice relaxation effects. Our results are discussed in the context of the inorganic spin-Peierls compound ${\mathrm{CuGeO}}_{3}$ and quantitatively compared to recent Raman experiments.

Exact solution of the Jaynes-Cummings model with cavity damping

Operating in Laplace language and making use of a representation based on photon-number states, we find the exact solution for the density operator that belongs to the Jaynes-Cummings model with cavity damping. The detuning parameter is set equal to zero and the optical resonator does not contain any thermal photons. It is shown that the master equation for the density operator can be replaced by two algebraic recursion relations for vectors of dimension 2 and 4. These vectors are built up from suitably chosen matrix elements of the density operator. By performing an iterative procedure, the exact solution for each matrix element is found in the form of an infinite series. We demonstrate that all series are convergent and discuss how they can be truncated when carrying out numerical work. With the help of techniques from function theory, it is proved that our solutions respect the following conditions on the density operator: conservation of trace, Hermiticity, convergence to the initial state for small times, and convergence to the ground state for large times. We compute some matrix elements of the density operator for the case of weak damping and find that their analytic structure becomes much simpler. Finally, it is shown that the exact atomic density matrix converges to the state of maximum von Neumann entropy if the time, the square of the initial electromagnetic energy density, and the inverse of the cavity-damping parameter tend to infinity equally fast. The initial condition for the atom can be chosen freely, whereas the field may start from either a coherent or a photon-number state.

Exact and semiclassical density matrix of a particle moving in a barrier potential with bound states

We present a barrier potential with bound states that is exactly solvable and determine the eigenfunctions and eigenvalues of the Hamiltonian. The equilibrium density matrix of a particle moving at temperature T in this nonlinear barrier potential field is determined. The exact density matrix is compared with the result of the path integral approach in the semiclassical approximation. For opaque barriers the simple semiclassical approximation is found to be sufficient at high temperatures while at low temperatures the fluctuation paths may have a caustic depending on temperature and endpoints. Near the caustics the divergence of the simple semiclassical approximation of the density matrix is removed by a nonlinear fluctuation potential. For opaque barriers the improved semiclassical approximation is again in agreement with the exact result. In particular, bound states and the form of resonance states are described accurately by the semiclassical approach.

Local moments coupled to a strongly correlated electron chain.

A 1D model hamiltonian that is motivated by the recent discovery of the heavy-fermion behavior in the cuprates of the $Nd_2CuO_4$ type is studied. It consists of $t-J$ interacting conduction electrons coupled to a lattice of localized spins through a Kondo exchange term $J_K$. Exact diagonalization and density matrix renormalization group methods are used. The latter method is generalized to arbitrary densities. At half-filling, a spin gap opens for all $J_K>0$. Away from half-filling$,$ it is shown that, at strong $J_K$% , the ground state is an unsaturated ferromagnet . At weak $J_K$ the system is in a paramagnetic phase with enhanced RKKY correlations. The importance of self-screening of the local moments in the depletion regime is discussed. We argue that these findings transcend the specifics of the model.

Exchange energy functional from a Pauli correlationfactor model

The spherical average of the Hartree-Fock second-order density matrix is approximated through an ansatz similar to that proposed by McWeeny for the exact second-order density matrix. This leads to the definition of a Pauli correlation factor which produces the exchange hole. In this work, the Pauli correlation factor is modeled in order to obtain an exchange energy density functional. The factor satisfies the normalization condition of the second-order density matrix. Numerical results are presented for the atoms He-Ar.

On the relationship between single-particle overlap functions, natural orbitals and the one-body density matrix for many-fermion systems

We consider the groundstate of a system of A interacting fermions and construct relationships connecting the asymptotic behaviour of the one-body density matrix with the overlap functions of the eigenstates in the ( A−1)-particle system. It is shown that the knowledge of the exact one-body density matrix is sufficient to determine the overlap functions, spectroscopic factors and separation energies of the bound ( A−1)-particle eigenstates. We also derive sumrules linking the overlap functions with the natural orbitals of the system.

Coherent population trapping: asymptotical behaviour of the atomic density matrix

An asymptotically exact density matrix is derived for atoms, which are cooled by a resonant optical field, with the method of the one-dimensional velocity-selective coherent populaation trapping scheme. The width of the atomic distribution function decreases as t - 1 2 with increasing of duration t of the interaction with the field, so that at large values of t the distribution function tends to the δ-function. The quantity of cooled atoms does not depend on the initial distribution and is determined only by the intensity of the field.

Exact density matrix for the degenerate-Raman-coupled model in the presence of collisions.

Exact expressions are obtained for the density matrix in the operator form and for the quasiprobabilities for the system of an atom undergoing Raman transitions between two degenerate levels on interaction with a quantized field in a lossy cavity driven by an external field. The model is generalized to include the effects of atomic dephasing collisions. The spectral properties of the radiation are discussed, and the equation of motion of the density matrix is derived in the presence of collisions and an external drive. The equation is found to be of second order with respect to time in the absence of the driving field and also analytically solvable.

Full Potential Electronic Structure Calculations and the Concept of Stress Fields and Energy Densities for Total Energy Calculations

ABSTRACTWith the availability of reliable full atomic cell orbitals the possibility arises to calculate pressure or stress, restoring or relaxation driving Hellmann -Feynman forces, and total energies (especially of defects) alternatively and directly via stress tensor fields and energy densities, two local quantities. Although quantum mechanical stress field and energy density can not be defined uniquely, there is a recent interest in these quantities, because integrals with physical meaning are gauge invariant.The mentioned fields can be defined (i) for the full many-body description with the exact one-particle density matrix and pair distribution function as well as (ii) for the Kohn-Sham one-particle description with LDA or beyond (gradient expansion approximation). If the local stress field for a special system once is constructed, then the global stress tensor and /or forces on nuclei can be calculated via the stress theorem and the force theorem by means of unit cell surface integrals. The energy density can be derived from the terms of the stress field by taking the trace and can be used to calculate defect energies without bothering about the thermodynamic limit.

Multipole theory of soft pulses in nmr of quadrupolar solids

By making use of a multipole operator decomposition of the spin density matrix the spin dynamics of quadrupolar nuclei under the influence of soft RF pulses is studied. In particular, exact density matrix solutions for any arbitrary sequence of two soft pulses on an axially symmetric quadrupolar spin I = 3/2 are obtained analytically in terms of all usual experimental pulse parameters. The theory is shown to give good quantitative agreement with the observed cosine and derivative echoes of the FID for a system of I = 3/2 in a single crystal of NaNO 3. The theory then provides some extension of the multipole theory of multiple-hard-pulse sequences. For systems of arbitrary spin magnitudes, the limiting case in which the RF pulse dominates is analyzed and the spin density matrix is obtained perturbatively in the multipole basis. A simple description follows in which all multipole polarizations are rotated plus terms in which the tensor rank is mixed from κ to k ± 1 to first order in the quadrupole coupling strength. This result can be readily implemented to extend other treatments of higher spins.

Density matrix and entropy of scattered light in the resonance fluorescence

The Bethe ansatz approach to the problem of multiphoton scattering for the many-particle Schrödinger equation allows one to calculate the exact density matrix and entropy of scattered light.

Distribution function for the orientational order parameters in solid H2 and D2.

A model for the distribution function P(\ensuremath{\sigma},\ensuremath{\eta}) for the axial and eccentric orientational order parameters \ensuremath{\sigma} and \ensuremath{\eta} is proposed for a system of quadrupoles which represents the situation in solid ${\mathrm{H}}_{2}$ and ${\mathrm{D}}_{2}$. It is based on a plausible approximation to the exact one-particle orientational density matrix for J=1 molecules in the randomly distributed (J${=1)}_{X}$(J${=0)}_{(1\mathrm{\ensuremath{-}}X)}$ solid, where X is the molar concentration of the J=1 component. The properties of P(\ensuremath{\sigma},\ensuremath{\eta}) are described, and can be expressed as a function of a single parameter D. Three-dimensional representations of P(\ensuremath{\sigma},\ensuremath{\eta}) for several values of D are shown, where D\ensuremath{\ll}1 and D\ensuremath{\gg}1 represent the extreme situations for zero and complete ordering. By means of the P(\ensuremath{\sigma},\ensuremath{\eta}) model, the NMR line shape for an ortho-para-mixture polycrystalline sample is calculated and compared with experiments in ${\mathrm{H}}_{2}$. The correlation of the parameter D with a given ordering state in the temperature-concentration plane for ${\mathrm{H}}_{2}$, the effect of zero-point motion, and the effect of crystalline anisotropy on the line shape are discussed.

Correlation functions of the nonstationary quantum singular oscillator

The exact density matrix elements and the correlation functions of the one-dimensional non-stationary singular oscillator (the potential U(x,t)= omega 2(t)x2/2+g/x2) excited from the thermodynamic equilibrium states are obtained. Generating functions for rho nm and also for transition probabilities are constructed.

Computer simulations of multiple-quantum NMR experiments. II. Selective excitation

The effects of selective multiple-quantum pulse sequences have previously been analyzed by coherent averaging theory. However, convergence of the Magnus expansion used for those calculations is questionable in the experimentally important region of long cycle times. Exact density matrix evolutions are calculated here to show when the coherent averaging calculations will be reliable. In addition, simple selective sequences which cannot be treated by coherent averaging theory are also analyzed.

Multiple phase-coherent laser pulses in optical spectroscopy. II. Applications to multilevel systems

The effects of intense laser pulse trains in coupled multilevel systems (such as pure or mixed molecular crystals) are analyzed by calculating exact density matrix evolutions. It is shown that two-level approximations are inadequate. The contributions of exchange couplings, inhomogeneous broadening, permanent multipole interactions and transition multipole interactions to absorption and photon echo line shapes are calculated. The absorption line shape of 1,4-dibromonaphthalene (DBN) is shown to be predominantly an isotopic substitution effect, as our Monte Carlo results give quantitative agreement with experiment for this model. Average t-matrix approximations to the Green’s function, which have been used to propose a different mechanism for the DBN line shape are shown to be qualitatively inadequate. Dipole–dipole interactions are shown to be an important photon echo decay mechanism in mixed crystals, with the relative importance of permanent and transition multipole interactions dependent on the resonance frequency distribution. Multiple pulse trains, including multiple pulse echoes and optical multiple-quantum sequences, are shown to be capable of distinguishing different types of interactions in the molecular Hamiltonian and reducing optical density effects. Specific pulse sequences are proposed and their effects are calculated.

New derivation and a k‐particle generalization of SCF‐type theories

AbstractA hierarchy of necessary conditions that an exact density matrix of a pure state or an ensemble has to satisfy is derived, namely the hermiticity of certain operators F(k). For k = 1 this reduces to the well‐known Hartree‐Fock condition. It is then shown that the kth set of conditions is equivalent to stationarity of the energy with respect to unitary k‐particle transformations. k‐Particle generalizations of Hartree‐Fock theory are then discussed both in the spirit of k‐particle pseudoeigenvalue equations and in the framework of a Newton–Raphson‐type constructive scheme.

Nuclear relaxation and anisotropic molecular reorientation of o-dichlorobenzene

Proton recovery curves are reported for nonselective 180°-τ-90°-FT spin-lattice relaxation of a dilute solution of o-dichlorobenzene in carbon disulfide. An exact density matrix analysis, combined with measured 13C relaxation rates, is used to obtain the principal reorientational correlation times. Differences between these values and those found previously for 1,2,3-trichlorobenzene in the same solvent can be rationalized in terms of molecular shape. Exact density matrix calculations of proton relaxation are compared with an approximate treatment in which dipolar cross correlation and nonlinear mixing effects of strong pulses are ignored. It is shown that simple formulae arising from the approximate treatment successfully account for the average relaxation rate of a given multiplet, but fail to account for differential relaxation of lines within a multiplet.