Effective Hamiltonian Formalism Research Articles

Tunneling in a bistable system under rapid pumping

With this work, I demonstrate the influence of a highly oscillating external field on the coherent tunneling of a quartic double-well potential. Extensive investigation on the tunneling rate of such a quantum system is made using the effective Hamiltonian formalism of (Rahav et al 2003 Phys. Rev. Lett. 91 110404; Rahav et al 2003 Phys. Rev. A 68 013820) for rapidly driven quantum systems. The present model explains the phenomena of chaos assisted tunneling (CAT), i.e. the enhancement of the tunneling rate. The two-level system (TLS) successfully explains the coherent destruction of tunneling (CDT). The effect of dissipation on CAT and CDT are also demonstrated in the present paper.

Bifurcations in resonance widths of an open Bose-Hubbard dimer

We investigate the structure of resonance widths of a Bose-Hubbard Dimer with intersite hopping amplitude $k$, which is coupled to continuum at one of the sites with strength $\gamma$. Using an effective non-Hermitian Hamiltonian formalism, we show that by varying the on-site interaction term $\chi$ the resonances undergo consequent bifurcations. For $\Lambda=k/\gamma\geq 0.5$, the bifurcation points follow a scaling law ${\tilde \chi}_n \equiv \chi_n N/k = f_{\Lambda}(n-0.5/\Lambda)$, where $N$ is the number of bosons. For the function $f_{\Lambda}$ two different $\Lambda$ dependences are found around the minimum and the maximum bifurcation point.

Spectroscopic properties of large open quantum-chaotic cavities with and without separated time scales

The spectroscopic properties of an open large Bunimovich cavity are studied numerically in the framework of the effective Hamiltonian formalism. The cavity is opened by attaching two leads to it in four different ways. In some cases, the transmission takes place via standing waves with an intensity that closely follows the profile of the resonances. In other cases, short-lived and long-lived resonance states coexist. The short-lived states cause traveling waves in the transmission while the long-lived ones generate superposed fluctuations. The traveling waves oscillate as a function of energy. They are not localized in the interior of the large chaotic cavity. In all considered cases, the phase rigidity fluctuates with energy. It is mostly near to its maximum value and agrees well with the theoretical value for the two-channel case.

The electronic spectrum of AgCl2: Ab initio benchmark versus density-functional theory calculations on the lowest ligand-field states including spin-orbit effects

The X 2pi(g), 2sigma(g)+, and 2delta(g) states of AgCl2 have been studied through benchmark ab initio complete active space self-consistent field plus second-order complete active space multireference Moller-Plesset algorithm (CASSCF+CASPT2) and complete active space self-consistent field plus averaged coupled pair functional (CASSCF+ACPF) and density-functional theory (DFT) calculations using especially developed basis sets to study the transition energies, geometries, vibrational frequencies, Mulliken charges, and spin densities. The spin-orbit (SO) effects were included through the effective Hamiltonian formalism using the LambdaSSigma ACPF energies as diagonal elements. At the ACPF level, the ground state is 2pi(g) in contradiction with ligand-field theory, SCF, and large CASSCF; the adiabatic excitation energies for the 2sigma(g)+ and 2delta(g) states are 1640 and 18,230 cm(-1), respectively. The inclusion of the SO effects leads to a pure omega = 32(2pi(g)) ground state, a omega = 12 (66%2pi(g) and 34%2sigma(g)+) A state, a omega = 12 (34%2pi(g) and 66%2sigma(g)+) B state, a omega = 52(2delta(g))C state, and a omega = 32(99%2delta(g))D state. The X-A, X-B, X-C, and X-D transition energies are 485, 3715, 17 246, and 20 110 cm(-1), respectively. The B97-2, B3LYP, and PBE0 functionals overestimate by approximately 100% the X 2pi(g)-2sigma(g)+T(e) but provide a qualitative energetic ordering in good agreement with ACPF results. B3LYP with variable exchange leads to a 42% optimal Hartree-Fock exchange for transition energies but all equilibrium geometries get worsened. Asymptotic corrections to B3LYP do not provide improved values. The nature of the bonding in the X 2pi(g) state is very different from that of CuCl2 since the Mulliken charge on the metal is 1.1 while the spin density is only 0.35. DFT strongly delocalizes the spin density providing even smaller values of around 0.18 on Ag not only for the ground state, but also for the 2sigma(g)+ state.

Effective Hamiltonian approach to doubly degenerate electronic states

Several problems in vibronic spectroscopy are solved within the effective Hamiltonian formalism combined with Lie algebraic methods. We consider mainly vibronic interactions in orbital doublets for molecules with a principal symmetry axis of order n=4k (or n even for Dnd symmetry groups). Effective Hamiltonian models for E ⊗(b1+b2), E ⊗ (bi+aj) and E ⊗(bi+ e) Jahn-Teller dynamical systems are discussed as well as some correlations with previous studies established.

Ab initio study on the spectroscopy of CuCl2. II. Benchmark calculations on the X 2Πg-C 2Δg and X 2Πg-D 2Δg transitions

The X (2)Pi(g)-C (2)Delta(g) and X (2)Pi(g)-D (2)Delta(g) transitions on CuCl(2) have been studied using the most sophisticated nondynamic and dynamic electronic correlation treatments. We report here ab initio benchmark calculations using especially developed basis sets to study, at the complete active space self-consistent field plus second-order Møller-Plesset algorithm (CASSCF+CASPT2) and CASSCF+ACPF levels, the transition energies as well as the corresponding equilibrium geometries (ACPF-averaged coupled pair functional). The spin-orbit (SO) effects of both atoms were included in a second step through the effective Hamiltonian formalism, using the calibrated SO effective potentials developed by the Stuttgart group. Without SO at the CASSCF+ACPF level, the vertical excitation energy for the (2)Delta(g) state is 6711 cm(-1) and the symmetric stretching equilibrium Cu-Cl distance is 4.04 a.u. The inclusion of the SO effects leads to a pure (2)Delta(g) Omega=5/2C state and a Omega=3/2 (0.7% (2)Pi(g),99.3% (2)Delta(g))D state. The calculated transition energies for the C and D states are 6340 and 8020 cm(-1), in good agreement with the spin-orbit splitting recent values from gas-phase and rare-gas matrix isolation laser induced fluorescence experiments. The present benchmark results show, as was recently done for the X (2)Pi(g)-(2)Sigma(g) transition, the rather poor performance of all the density functional theory-based descriptions for the (2)Delta(g) state, which largely overestimate its T(e), systematically placing it around 19 000 cm(-1). The CASSCF+CASPT2 method also overestimates, by around 50%, the X (2)Pi(g)-(2)Delta(g) transition energy, showing that only large variational calculations can produce reliable spectroscopic results for this kind of complex systems where delicate electronic correlation effects have to be carefully dealt with.

Ab initio study on the spectroscopy of CuCl2. I. Benchmark calculations on the X2Pig - 2Sigmag+ transition.

The modern theoretical predictions on the LambdaSSigma nature of the ground state of CuCl2 have led to different answers, depending on the type (DFT-based or ab initio) and the quality of the electronic correlation treatment; for this reason the X2Pig - 2Sigmag+ transition energy has been predicted to range from -1856 to +5887 cm(-1). The physical problem at hand lies in the difficulty of accurately describing the orientation of the 3d hole on the central Cu2+(3d9)/Cu+(3d94s1) ion (in the field of both chlorine ions), which implies the need of the most sophisticated nondynamic and dynamic electronic correlation treatments. We report here ab initio benchmark calculations using especially developed basis sets to study, at the CASSCF + CASPT2 and CASSCF + ACPF levels, the transition energy as well as the corresponding equilibrium geometries. The spin-orbit (SO) effects of both atoms were included in a second step through the effective Hamiltonian formalism, using the calibrated SO effective potentials developed by the Stuttgart group. Without SO at the CASSCF + ACPF level, the ground state is X2Pig but the vertical transition energy to the 2Sigmag+ is only 99 cm(-1) at 3.95 a.u. The inclusion of the SO effects leads to a Omega = 1/2 (59% 2Pig,41% 2Sigmag+) ground state, in contradiction with the Omega experimental value of 3/2. In a last step we show that the SO effects (and therefore the final Omega ordering) are critically dependent on the LambdaSSigma electronic energies, so that it is not impossible that the Omega ordering is actually changed. For theoreticians interest in this matter is not purely academic, since many properties of organometallic complexes are linked to such delicate physical effects.

Analysis of the multireference state-universal coupled-cluster Ansatz

An algorithm for the multireference (MR) cluster analysis of configuration interaction (CI) wave functions, based on the state-universal (SU) coupled-cluster (CC) Ansatz of Jeziorski and Monkhorst, is formulated. It is illustrated on two often-studied model systems that consist of two (the so-called H4 model) and four (the H8 model), slightly stretched, hydrogen molecules and enable one to vary the degree of quasidegeneracy via a single geometric parameter. The role played by higher-than-two-body clusters is examined and their importance in the MR effective Hamiltonian formalism, particularly in the presence of intruder states, is explored in detail. It is pointed out that in contrast with the single-reference CC methods, the MR SU CC approach with singles and doubles (CCSD) cannot yield exact energies even when exact one- and two-body clusters are available. It is shown that this limitation, which is particularly crucial in the presence of intruder states, can be remedied by employing a small subset of higher-than-pair-cluster amplitudes, extracted via cluster analysis from the MR CISD wave functions that are based on the same model space as the SU CCSD method, not unlike in the reduced MR CCSD method.

Notations, Spectroscopic Parameters, Energy Levels and Their Relationship to Effective Hamiltonian Formalism of Diatomic Molecules

This paper presents a part of the culmination of the collective efforts in developing a cohesive and consolidated enunciation of the spectroscopic parameters and their relationship to effective molecular Hamiltonians for diatomic formalism, linear four-atomic formalism and quasi-linearity, Watson Hamiltonian and the complexities in symmetric and asymmetric top spectral structures. All these considered together present a beautiful and consistent picture of molecular spectroscopy. In this paper we deal with the diatomic formalism. This paper forms our tribute to Professor K. Narahari Rao and in the second part of the paper we convoy our gratitude to him with a summary of his more than half century of work and an obituary. We will deal with other forms of molecular species which will be connected to optical pumping and/or interstellar space.

Improved model-independent analysis of semileptonic and radiative rareBdecays

We update the branching ratios for the inclusive decays $B\to X_s \ell^+ \ell^-$ and the exclusive decays $B\to (K,K^*) \ell^+ \ell^-$, with $\ell=e, \m$, in the standard model by including the explicit $O(\a_s)$ and $\Lambda_{\hbox{\tiny QCD}} / m_b$ corrections. This framework is used in conjunction with the current measurements of the branching ratios for $B\to X_s \g$ and $B\to K \ell^+ \ell^-$ decays and upper limits on the branching ratios for the decays $B\to (K^*,X_s) \ell^+ \ell^-$ to work out bounds on the Wilson coefficients $C_7$, $C_8$, $C_9$ and $C_{10}$ appearing in the effective Hamiltonian formalism. The resulting bounds are found to be consistent with the predictions of the standard model and some variants of supersymmetric theories. We illustrate the constraints on supersymmetric parameters that the current data on rare B decays implies in the context of minimal flavor violating model and in more general scenarios admitting additional flavor changing mechanisms. Precise measurements of the dilepton invariant mass distributions in the decays $B \to (X_s,K^*,K) \ell^+ \ell^-$, in particular in the lower dilepton mass region, and the forward-backward asymmetry in the decays $B \to (X_s,K^*) \ell^+ \ell^-$, will greatly help in discriminating among the SM and various supersymmetric theories.

Anisotropy of the Free Exciton Emission in GaN Grown on a-Plane Sapphire

The influence of the anisotropic in-plane strain on the optical response of GaN layers grown on a-plane sapphire is investigated. A splitting of the Γ 5 states of the A and B free excitons into two components, polarized parallel and perpendicular to the strain direction, is observed. The experimental results are discussed in the framework of the effective exciton Hamiltonian formalism.

Quantization in Quantum System with Time- Dependent Boundary Condition

An approach to quantize a quantum mechanical system with time-dependent boundary condition is proposed in the framework of canonical quantization. It can be achieved by introducing the time-dependent boundary condition into the usual Lagrangian. We set up the effective Hamiltonian formalism and believe that this formalism can provide a generalized method to calculate the boundary effects.

Intermediate Hamiltonian formulation of the valence-universal coupled-cluster method for atoms

A formulation of the valence-universal coupled-cluster (VU-CC) method for atoms is presented. The method is based on Lindgren's normal-ordered exponential expansion, in which the cluster operator is restricted to its one- and two-body components represented in terms of radial amplitudes defined by the configurational excitations. The approach employs an intermediate Hamiltonian technique which replaces the effective Hamiltonian formalism used in the standard VU-CC formulation. The modified version introduces many simplifications in comparison with the standard one and, in particular, offers an efficient way of solving coupled-cluster equations. The advantages of the approach are illustrated in a calculation for the Be atom, representing one of the basic yet very challenging applications for VU-CC methods.

Decay Dynamics of the Predissociating High Rydberg States of NO

The dynamics of predissociating high molecular Rydberg states of NO below the lowest ionization threshold is computed in the presence of a weak external dc field using a quantum theory based on an effective Hamiltonian formalism. The core−electron interaction affecting the low l states (l ≤ 2) is taken into account by molecular quantum defect theory, while for the high l states (l ≥ 2), a long range multipolar expansion is used to describe the effect of the anisotropy of the molecular core. Time- and frequency-resolved ZEKE spectra are computed. In the energy range investigated, the decay kinetics of the ZEKE intensity is found to exhibit two time scales, which differ by more than an order of magnitude. The short decay constant typically falls in the submicrosecond range and is in agreement with previous experimental results and computations. In addition, our computations predict a long-time component, which decays in the tens of microseconds range. The two decay times are discussed in terms of the short and long range interseries dynamics, in terms of the strength of the external dc field, and in terms of the nature of the bottlenecks in phase space for a predissociating molecular core with a rather large rotational constant (BNO+ = 1.9842 cm-1). It is found that for this particular case where all series are coupled to the fragmentation channels through a low l bottleneck, the predissociation process does not quench the long time component in the decay kinetics. The reason is that the interplay between the interseries coupling and the external dc field leads to a shift of the decay constants to larger values together with an enhancement of the weight of the long time component in the decay kinetics. Special attention is devoted to the role of the dipolar interaction and its synergy with the external dc field.

Reduced multireference CCSD method: An effective approach to quasidegenerate states

Standard multireference (MR) coupled cluster (CC) approaches are based on the effective Hamiltonian formalism and generalized Bloch equation. Their implementation, relying on the valence universal or state universal cluster Ansatz, is very demanding and their practical exploitation is often plagued with intruder state and multiple solution problems. These problems are avoided in the so-called state selective or state specific (SS) MR approaches that concentrate on one state at a time. To preserve as much as possible the flexibility and generality offered by the general MR CC approaches, yet obtaining a reliable and manageable algorithm, we propose a novel SS strategy providing a size-extensive CC formalism, while exploiting the MR model space and the corresponding excited state manifold. This strategy involves three steps: (i) The construction of a variational configuration interaction (CI) wave function within the singly (S) and doubly (D) excited state manifold, (ii) the cluster analysis of this CI wave function providing the information about the higher than pair cluster amplitudes, and (iii) the exploitation of these amplitudes in the so-called externally corrected CCSD procedure. This approach is referred to as the reduced MR (RMR) SS CCSD method and is implemented at the ab initio level and applied to several model systems for which the exact full CI results are available. These include two four electron H4 systems (usually referred to as the H4 and S4 models), an eight electron H8 model and the singlet-triplet separation problem in CH2. It is shown that the RMR CCSD approach produces highly accurate results, is free from intruder state problems, is very general and effective and applicable to both closed and open shell systems.

Time and frequency resolved spectra of high molecular Rydberg states by dynamical computations

The absorption spectrum of bound Rydberg states which can be detected by a delayed, pulsed field ionization is computed. The spectrum, measured for various delay times, provides information on the short and the longer time dynamics of high molecular Rydberg states. A quantitative dynamical theory, based on an effective Hamiltonian formalism is applied, illustrating the role of the Rydberg electron–core coupling and of an external electrical field in the delay-time dependent spectra. The sharpening of the spectra for longer delay times is reproduced by the dynamical computations. It is found that the overall intensity, as a function of the delay time before detection, is well described by a double exponential decay where the short lifetime is primarily a manifestation of the direct autoionization to the continuum, while the long lifetime is due to interseries coupling. Both lifetimes increase with the principal quantum number of the Rydberg states. The notion of trapped “reservoir states” is illustrated by the computational results, with special reference to a kinetic model analysis. The role of the initially optically accessed state(s) and of the depth of detection, in particular with regard to the intensity, is demonstrated. The effect of varying the strength of an external dc field in the time interval prior to the detection is illustrated by the dynamical computations, with respect to both the decay kinetics and the intensity of the spectrum.

Vibrational coherence in nonadiabatic dynamics

In this paper we explore temporal vibrational coherence effects in nonadiabatic radiationless transitions between two electronic states in a large molecule or in the condensed phase, accounting explicitly for the role of the (intramolecular and/or medium) vibrational quasicontinuum of the final states. Our treatment of the time evolution of the wave packet of states and of coherence effects in the nonradiative population probabilities of the reactants and the products rests on the diagonalization of the Hamiltonian of the entire multimode system, with supplementary information being inferred from the effective Hamiltonian formalism. New features of the vibrational Franck–Condon quasicontinuum, which originate from weak, but finite, correlations between off-diagonal coupling terms, were established. The state dependence of the off-diagonal couplings Vsα between the doorway states manifold {|s〉} and the quasicontinuum {|α〉} was quantified by the correlation parameters ηss=〈VsαVαs〉/[〈Vsα2〉〈Vsα2〉]1/2, where 〈 〉 denotes the average over the relevant energy range. Calculations were conducted for a Franck–Condon four-mode system consisting of ns=100 doorway states and nα=3000 quasicontinuum states. The correlation parameters for all pairs of doorway states are considerably lower than unity (|ηss|≲0.4), obeying propensity rules with the highest values of |ηss| corresponding to a single vibrational quantum difference, while for multimode changes between |s〉 and |s〉 very low values of |ηss| are established. Quantum beats in the population probabilities of products and reactants in nonadiabatic dynamics are characterized by an upper limit for their modulation amplitudes ξ≅(Γ/ΔE)η (for ΔE/2πΓ⩾1), where Γ is the decay width of the doorway states and ΔE is their energetic spacing. These low ξ values originate from a small (∼Γ/ΔE) contribution to the off-diagonal matrix elements of the nonradiative decay matrix in conjunction with low correlation parameters. The amplitudes of the quantum beats in nonradiative temporal dynamics provide dynamic information on the larger correlation parameters ηss. Our theoretical and numerical analysis was applied for temporal coherence effects in nonadiabatic electron transfer dynamics in a Franck–Condon quasicontinuum of Mulliken charge transfer complexes [K. Wynne, G. Reid, and R. M. Hochstrasser, J. Chem. Phys. 105, 2287 (1996)]. This accounts for the “preparation” (signature of coherent excitation), for the low amplitudes of coherent temporal modulation of reactants and products (ξ≅0.05–0.06 determined by the ηss parameters) and for the dominating contributions to temporal coherence (subjected to propensity rules).

Electric Field Effects on Long Living ZEKE States

The effect of an electric field on the overall intensity of the ZEKE spectrum and on the lifetime is discussed for very long living states detected by a pulsed field ionization delayed by several microseconds or more. It is shown that the presence of a dc electrical field can shorten the very long lifetimes and that it can also reduce the overall intensity of the very long living states. The decrease in the long lifetimes of the ZEKE states is complementary to the field-induced elongation of the shorter lifetimes. The discussion is based on quantum mechanical considerations and is illustrated by detailed computational studies, for a model problem, using an effective Hamiltonian formalism for an energy range just above the lowest ionization threshold. The model allows for coupling of different Rydberg series built on different excited states of the core where the continuum corresponds to the ground state. Predissociation is not allowed for in the model Hamiltonian. The trends are in accord with the observations of Held et al. as reported in the preceding paper, and the magnitude of the measured lifetimes of the ZEKE states (dozens of microseconds) are reproduced by the computations.

Dynamics of Ultrahigh Molecular Rydbergs in Weak Electric Fields

In this paper we explore the dynamics of the mixed Stark manifold of ultrahigh Rydberg states (principal quantum number n = 50-250) of the large molecules interrogated by time-resolved ZEKE (zero-electron kinetic energy) spectroscopy. We pursue the formal analogy between the coupling, accessibility, and decay of ultrahigh Rydbergs in an external weak (F = 0.01 -1.0 Vkm) electric field and intramolecular (interstate and intrastate) relaxation in a bound level structure. The effective Hamiltonian formalism with several doorway and escape states was advanced to treat the dynamics. The theory accounts for the dilution effect, Le., the dramatic lengthening of the lifetimes of ultrahigh Rydbergs, relative to that expected on the basis of the n-3 scaling law for the decay widths. Model calculations for the field-induced (I> mixing reveal that the Rydberg time-resolved population probability is characterized by two distinct (-ns and -ps) time scales. To date, long time-resolved (10 ps-100 ns time scales) nonexponential decay of ZEKE Rydbergs was experimentally documented, in accord with our analysis. The predicted existence of the short decay times (1 - 10 ns) was not yet subjected to an experimental test. Next, we extend the model calculations to treat the mixing of several n manifolds, demonstrating that in the strong mixing limit the overall features of the temporal decay are similar to that of a single n manifold.

B. Electron Motion in Highly Excited States and Radiationless Transitions. General Dynamical Theory and Experiment. Intramolecular Dynamics and Charge Separation in Large Systems

AbstractWe adopt the theory of intramolecular dynamics to explore charge separation and recombination in two classes of ‘isolated’ solvent‐free molecular systems: (A) Supermolecules containing bridged electron donor and electron acceptor, where electron transfer occurs on a spatial scale of ≈︁ 10 Å. (B) Ultrahigh molecular Rydberg excitations with a principle quantum number n ≃ 50−300 and spatial dimensions of ≈︁ 104 Å, where relaxation processes, e.g., electron‐core recombination via internal conversion or predissociation, and charge separation by autoionization, are manifested. The molecular limit for photoinduced long‐range ET in isolated supermolecules [class (A)] is treated on the basis of the statistical limit for interstate radiationless transitions, which involve either a direct or a mode‐selective mediated coupling. The level structure, optical excitation modes and dynamics of high molecular Rydbergs [class (B)] interrogated by time‐resolved ZEKE (zero‐electron kinetic energy) spectroscopy, are treated by the effective Hamiltonian formalism. We pursue the formal analogy between the coupling, accessibility and decay of ultrahigh Rydbergs in an external weak (F = 0.01−1.0 V/cm) electric field and intramolecular (interstate and intrastate) relaxation in a bound level structure. Model calculations for the field‐induced (l) mixing reveal that the Rydberg dynamics is characterized by two distinct (≈︁ ns and ≈︁μs) time scales. Up to date, long time‐resolved (10 μs − 100 ns time scales) nonexponential decay of ZEKE Rydbergs was experimentally documented, in accord with our analysis. The predicted existence of the short decay times (1 − 10 ns) was not yet subjected to an experimental test.