A simple and efficient evolution operator for time-dependent Hamiltonians: the Taylor expansion

No compact expression of the evolution operator is known when the Hamiltonian operator is time dependent, like when Hamiltonian operators describe, in a semiclassical limit, the interaction of a molecule with an electric field. It is well known that Magnus [N. Magnus, Commun. Pure Appl. Math. 7, 649 (1954)] has derived a formal expression where the evolution operator is expressed as an exponential of an operator defined as a series. In spite of its formal simplicity, it turns out to be difficult to use at high orders. For numerical purposes, approximate methods such as "Runge-Kutta" or "split operator" are often used usually, however, to a small order (<5), so that only small time steps, about one-tenth or one-hundredth of the field cycle, are acceptable. Moreover, concerning the latter method, split operator, it is only very efficient when a diagonal representation of the kinetic energy operator is known. The Taylor expansion of the evolution operator or the wave function about the initial time provides an alternative approach, which is very simple to implement and, unlike split operator, without restrictions on the Hamiltonian. In addition, relatively large time steps (up to the field cycle) can be used. A two-level model and a propagation of a Gaussian wave packet in a harmonic potential illustrate the efficiency of the Taylor expansion. Finally, the calculation of the time-averaged absorbed energy in fluoroproprene provides a realistic application of our method.     

The magnus expansion for the damped harmonic oscillator

It is shown that he time-evolution operator for the damped harmonic oscillator can be written as the exponential of an anti-Hermitean operator provided t is close enough to zero. The range of applicability of the Magnus expansion is discussed.     

The massey parameter expansion for the semiclassical scattering matrix

The Massey parameter expansion method for the semiclassical scattering matrix is proposed, using the Magnus formalism. The first Massey parameter, a unitary member of this expansion, is found for the interaction hamiltonian, depending on time, as exp(-τ) and τ^?n. Other possible applications of the proposed method are discussed.     

超球坐标下共线三原子反应散射的一阶Magnus数值传播法计算

使用一阶Magnus数值传播积分法计算了超球坐标描述下的共线三原子反应散射问题。应用于共线H+H_2和F+H_2体系得到了较好的结果。     

Accurate time propagation for the Schrodinger equation with an explicitly time-dependent Hamiltonian

Several different numerical propagation techniques for explicitly time-dependent Hamiltonians are discussed and compared, with the focus on models of pump-probe experiments. The quality of the rotating wave approximation is analyzed analytically, and we point out under which circumstances the modeling becomes inaccurate. For calculations with the fully time-dependent Hamiltonian, we show that for multistate systems, with either time or space dependence in the interstate coupling, the fourth order truncated Magnus expansion can be reformulated so that no commutators appear. Our results show that the split-operator method should only be used when low accuracy is acceptable. For accurate and efficient time stepping, the Magnus-Lanczos approach appears to be the best choice.     

Real-time propagation of the reduced one-electron density matrix in atom-centered Gaussian orbitals: application to absorption spectra of silicon clusters

We solve the time-dependent density functional theory equation by propagating the reduced one-electron density matrix in real-time domain. The efficiency of several standard solvers such as the short-iterative Krylov-subspace propagator, the low-order Magnus integration method with the matrix polynomial (MP) or Chebyshev matrix polynomial (CMP) expansion of the evolution operator, and Runge-Kutta algorithm are assessed. Fast methods for summing MP and CMP are implemented to speed the calculation of the matrix exponential. It is found that the exponential propagators can tolerate large time step size and retain the computational accuracy whereas the Krylov-subspace algorithm is a little inferior for a larger time step size compared with the second-order Magnus integration method with the MP/CMP expansion of the evolution operator in both weak and intense fields. As an application, we calculate the absorption spectra of hydrogen-passivated silicon nanoparticles Si(29)H(x). The popular hybrid and generalized gradient approximation exchange-correlation functionals are applied. We find that the experimental spectra can be reproduced by using B3LYP and that the silicon particles with sizes of 1 nm and the optical excitations at 3.7, 4.0, and 4.6 eV may consist of 29 Si atoms surrounded by 24 hydrogen atoms.     

Two-photon two-color nuclear magnetic resonance

Two-photon excitation has recently been demonstrated to be a practical means of exciting nuclear magnetic resonance (NMR) signals by radio-frequency (rf) irradiation at half the normal resonance frequency. In this work, two-photon excitation is treated with average Hamiltonian theory and shown to be a consequence of higher order terms in the Magnus expansion. It is shown that the excitation condition may be satisfied not only with rf at half resonance, but also with two independent rf fields, where the two frequencies sum to or differ by the resonance frequency. The technique is demonstrated by observation of proton NMR signals at 400 MHz while simultaneously exciting at 30 and 370 MHz. Advantages of this so-called two-color excitation, such as a dramatic increase in nutation rate over half-frequency excitation, along with a variety potential applications are discussed.     

Binomial Green functions

A straightforward method is developed for deriving analytic Green functions based on the binomial expansion theorem. The utility of the technique is clearly demonstrated by its application to several tight‐binding models. The simplicity of the approach gives rise to the prospect of wide applicability. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 1–4, 2000     

Accurate double many-body expansion potential energy surface for triplet H3+. II. The upper adiabatic sheet (2(3)A')

We report on a global potential energy hypersurface for the upper sheet of the lowest triplet state of H3+. The analytic representation is based on the double many-body expansion theory. The ab initio data points, calculated with a large cc-pV5Z basis, are represented with a root mean square deviation of only 5.54 cm(-1) in the energy region below the H(+)+2H(2S) dissociation threshold. The quasi-bound vibronic states supported by this surface have also been calculated.     

Role of resonances in building cross sections: Comparison between the Mittag–Leffler and the T‐matrix Green function expansion approaches

Peaks in collision cross sections are often interpreted as resonances. The complex dilation method, as well as other methods relying on analytic continuation of the scattering formalism, can be used to clarify whether these structures are true resonances in the sense that they are poles of the S‐matrix and the associated Green function. The performance of the Mittag–Leffler expansion and T‐matrix Green function expansion methods are formally and computationally compared. The two methods are applied to two model potentials. Eigenenergies, s‐wave residues, and cross sections are computed with both methods. The resonance contributions to the cross sections are further analyzed by removing the residue contributions from the Mittag–Leffler and Green function expansion sums, respectively. It is suggested that the contribution of a resonance to a cross section should be defined through its S‐matrix residue. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Analytic properties of connected moments expansions

The derivation of the connected moments expansion (CMX ) is examined as well as the singularities that arise in the series expansion for the ground-state energy. Explicit analytic results are presented that show a canceling of these singularities. Also, an alternate moments expansion (AMX ) is derived that closely models the CMX but displays a varied computational range. © 1994 John Wiley & Sons, Inc.     

Conversion of Magnus salt into diamminedichloroplatinum(II) isomers in aqueous solution

The conditions inducing conversion of Magnus salt into diamminedichloroplatinum(II) isomers were studied. Syntheses of cis-diamminedichloroplatinum(II) and trans-diamminedichloroplatinum(II), which are used to prepare potassium or ammonium amminetrichloroplatinate(II), are described. The identity and structure of diamminedichloroplatinum(II) isomers were verified by elemental analysis, X-ray powder diffraction, and IR and UV spectroscopy. A workflow for preparing potassium or ammonium amminetrichloroplatinate(II) from diamminedichloroplatinum(II) isomers was developed. This workflow appreciably increases the product yield due to the return of unused Magnus salt to the main synthesis flow.     

Energy conservation in molecular dynamics simulations of classical systems

Classical Newtonian dynamics is analytic and the energy of an isolated system is conserved. The energy of such a system, obtained by the discrete "Verlet" algorithm commonly used in molecular dynamics simulations, fluctuates but is conserved in the mean. This is explained by the existence of a "shadow Hamiltonian" H [S. Toxvaerd, Phys. Rev. E 50, 2271 (1994)], i.e., a Hamiltonian close to the original H with the property that the discrete positions of the Verlet algorithm for H lie on the analytic trajectories of H. The shadow Hamiltonian can be obtained from H by an asymptotic expansion in the time step length. Here we use the first non-trivial term in this expansion to obtain an improved estimate of the discrete values of the energy. The investigation is performed for a representative system with Lennard-Jones pair interactions. The simulations show that inclusion of this term reduces the standard deviation of the energy fluctuations by a factor of 100 for typical values of the time step length. Simulations further show that the energy is conserved for at least one hundred million time steps provided the potential and its first four derivatives are continuous at the cutoff. Finally, we show analytically as well as numerically that energy conservation is not sensitive to round-off errors.     

Two-center molecular repulsion integrals over slater functions

For the general two-electron two-center integral over Slater functions, use of the Neumann expansion for the electron-electron interaction term yields the standard auxiliary functions. These are expanded and integrated explicitly by two independent methods. The resulting simple analytic formula for the total integral is completely general, requiring only the Slater function quantum numbers and exponents and the internuclear separation. Hence all two-electron hydrid, coulomb, exchange, and one-center integrals are considered. The efficiency of calculation of this expression is compared with those of other methods, indicating an order of magnitude improvement in speed over recursion for the exchange integral.     

Chronoamperometric current at finite disk electrodes

The chronoamperometric current at a stationary finite disk electrode is studied using both analytical and digital simulation techniques. The exact long-time expansion of the current is obtained and its short-time behavior is considered. Digital simulation of the current using an explicit hopscotch algorithm is presented. In contrast to the usual explicit difference method, the ‘hopscotch’ algorithm is unconditionally stable, and thus, it is particularly suited for studying electrochemical problems at intermediate and long times. A simple analytic expressions for the current, which is accurate to 0.6% for all times, is presented.     

On dispersion force correction terms in perturbed equations of state

The first-order correction term for the contribution of dispersion forces in a perturbation expansion has been analyzed using computer simulation and perturbation theory results. It turns out that simple approximations used to make an analytic evaluation of the correction integral possible may result, ironically, in more complicated (and erroneous) behavior in comparison with the exact result.     

On early events in the Pauson-Khand reaction

[reaction: see text]. The Magnus Pauson-Khand (PK) mechanism has been studied for the first time by electrospray ionization coupled to tandem mass spectrometry. It has been found that loss of CO from the PK complex precedes olefin coordination and insertion.     

Hierarchical expansion of the kinetic energy operator in curvilinear coordinates for the vibrational self-consistent field method

A new hierarchical expansion of the kinetic energy operator in curvilinear coordinates is presented and modified vibrational self-consistent field (VSCF) equations are derived including all kinematic effects within the mean field approximation. The new concept for the kinetic energy operator is based on many-body expansions for all G matrix elements and its determinant. As a test application VSCF computations were performed on the H(2)O(2) molecule using an analytic potential (PCPSDE) and different hierarchical approximations for the kinetic energy operator. The results indicate that coordinate-dependent reduced masses account for the largest part of the kinetic energy. Neither kinematic couplings nor derivatives of the G matrix nor its determinant had significant effects on the VSCF energies. Only the zero-point value of the pseudopotential yields an offset to absolute energies which, however, is irrelevant for spectroscopic problems.