Using preconditioned adaptive step size Runge-Kutta methods for solving the time-dependent Schrödinger equation

If the Hamiltonian is time dependent it is common to solve the time-dependent Schr?dinger equation by dividing the propagation interval into slices and using an (e.g., split operator, Chebyshev, Lanczos) approximate matrix exponential within each slice. We show that a preconditioned adaptive step size Runge-Kutta method can be much more efficient. For a chirped laser pulse designed to favor the dissociation of HF the preconditioned adaptive step size Runge-Kutta method is about an order of magnitude more efficient than the time sliced method.     

Particular and homogeneous solutions of time-independent wavepacket Schrödinger equations: calculations using a subset of eigenstates of undamped or damped Hamiltonians

A variety of causal, particular and homogeneous solutions to the time-independent wavepacket Schr?dinger equation have been considered as the basis for calculations using Chebychev expansions, finite-τ expansions obtained from a partial Fourier transform of the time-dependent Schr?dinger equation, and the distributed approximating functional (DAF) representation for the spectral density operator (SDO). All the approximations are made computationally robust and reliable by damping the discrete Hamiltonian matrix along the edges of the finite grid to facilitate the use of compact grids. The approximations are found to be completely well behaved at all values of the (continuous) scattering energy. It is found that the DAF–SDO provides a suitable alternative to Chebychev propagation. Received: 29 February 2000 / Accepted: 5 April 2000 / Published online: 18 August 2000     

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.     

Algebraic approach to electronic spectroscopy and dynamics

Lie algebra, Zassenhaus, and parameter differentiation techniques are utilized to break up the exponential of a bilinear Hamiltonian operator into a product of noncommuting exponential operators by the virtue of the theory of Wei and Norman [J. Math. Phys. 4, 575 (1963); Proc. Am. Math. Soc., 15, 327 (1964)]. There are about three different ways to find the Zassenhaus exponents, namely, binomial expansion, Suzuki formula, and q-exponential transformation. A fourth, and most reliable method, is provided. Since linearly displaced and distorted (curvature change upon excitation/emission) Hamiltonian and spin-boson Hamiltonian may be classified as bilinear Hamiltonians, the presented algebraic algorithm (exponential operator disentanglement exploiting six-dimensional Lie algebra case) should be useful in spin-boson problems. The linearly displaced and distorted Hamiltonian exponential is only treated here. While the spin-boson model is used here only as a demonstration of the idea, the herein approach is more general and powerful than the specific example treated. The optical linear dipole moment correlation function is algebraically derived using the above mentioned methods and coherent states. Coherent states are eigenvectors of the bosonic lowering operator a and not of the raising operator a(+). While exp(a(+)) translates coherent states, exp(a(+)a(+)) operation on coherent states has always been a challenge, as a(+) has no eigenvectors. Three approaches, and the results, of that operation are provided. Linear absorption spectra are derived, calculated, and discussed. The linear dipole moment correlation function for the pure quadratic coupling case is expressed in terms of Legendre polynomials to better show the even vibronic transitions in the absorption spectrum. Comparison of the present line shapes to those calculated by other methods is provided. Franck-Condon factors for both linear and quadratic couplings are exactly accounted for by the herein calculated linear absorption spectra. This new methodology should easily pave the way to calculating the four-point correlation function, F(tau(1),tau(2),tau(3),tau(4)), of which the optical nonlinear response function may be procured, as evaluating F(tau(1),tau(2),tau(3),tau(4)) is only evaluating the optical linear dipole moment correlation function iteratively over different time intervals, which should allow calculating various optical nonlinear temporal/spectral signals.     

Partitioning technique procedure revisited: Formalism and first application to atomic problems

Usually the partitioning technique (PT) has been studied under two aspects: (i) as a numerical tool for solving secular equations of high order, and (ii) as a theoretical method related to the infinite‐order perturbation theory and the iteration–variation methods. Here it is shown that there exists a form of the PT equations which allows us to determine explicitly the spectrum and eigenstates of the Hamiltonian operator for different forms of potentials without the utilization of perturbative expansions or iterative equations of the type E=f(E). As a first application of the new approach, we consider the hydrogen‐atom in strong magnetic fields (B ~ 10⁹ G). This revised version was published online in July 2006 with corrections to the Cover Date.     

Vibrations of H+(D+) in stoichiometric LiNbO3 single crystal

A first principles quantum mechanical calculation of the vibrational energy levels and transition frequencies associated with protons in stoichiometric LiNbO(3) single crystal has been carried out. The hydrogen contaminated crystal has been approximated by a model one obtains by translating a supercell, i.e., a cluster of LiNbO(3) unit cells containing a single H(+) and a Li(+) vacancy. Based on the supercell model an approximate Hamiltonian operator describing vibrations of the proton sublattice embedded in the host crystal has been derived. It is further simplified to a sum of uncoupled Hamiltonian operators corresponding to different wave vectors (ks) and each describing vibrations of a quasi-particle (quasi-proton). The three dimensional (3D) Hamiltonian operator of k=0 has been employed to calculate vibrational levels and transition frequencies. The potential energy surface (PES) entering this Hamiltonian operator has been calculated point wise on a large set of grid points by using density functional theory, and an analytical approximation to the PES has been constructed by non-parametric approximation. Then, the nuclear motion Schro?dinger equation has been solved by employing the method of discrete variable representation. It has been found that the (quasi-)H(+) vibrates in a strongly anharmonic PES. Its vibrations can be described approximately as a stretching, and two orthogonal bending vibrations. The theoretically calculated transition frequencies agree within 1% with those experimentally determined, and they have allowed the assignment of one of the hitherto unassigned bands as a combination of the stretching and the bending of lower fundamental frequency.     

A semiclassical initial-value representation for quantum propagator and boltzmann operator

Starting from the position-momentum integral representation, we apply the correction operator method to the derivation of a uniform semiclassical approximation for the quantum propagator and then extend it to approximate the Boltzmann operator. In this approach, the involved classical dynamics is determined by the method itself instead of given beforehand. For the approximate Boltzmann operator, the corresponding classical dynamics is governed by a complex Hamiltonian, which can be described as a pair of real Hamiltonian systems. It is demonstrated that the semiclassical Boltzmann operator is exact for linear systems. A quantum propagator in the complex time is thus proposed and preliminary numerical results show that it is a reasonable approximation for calculating thermal correlation functions of general systems. © 2018 Wiley Periodicals, Inc.     

Making four- and two-component relativistic density functional methods fully equivalent based on the idea of "from atoms to molecule"

It is shown that four- and two-component relativistic Kohn-Sham methods of density functional theory can be made fully equivalent in all the aspects of simplicity, accuracy, and efficiency. In particular, this has been achieved based solely on physical arguments rather than on mathematical tricks. The central idea can be visualized as "from atoms to molecule," reflecting that the atomic information is employed to "synthesize" the molecular no-pair relativistic Hamiltonian. That is, the molecular relativistic Hamiltonian can, without loss of accuracy, be projected onto the positive energy states of the isolated Dirac atoms with the projector approximated simply by the superposition of the atomic ones. The dimension of the four-component Hamiltonian matrix then becomes the same as that of a two-component one. Another essential ingredient is to formulate quasirelativistic theory on matrix form rather than on operator form. The resultant quasi-four-component, normalized elimination of the small component, and symmetrized elimination of the small component approaches are critically examined by taking the molecules of MH and M(2) (M=At, E117) as examples.     

Benchmark calculations for dissipative dynamics of a system coupled to an anharmonic bath with the multiconfiguration time-dependent Hartree method

In this paper, we present benchmark results for dissipative dynamics of a harmonic oscillator coupled to an anharmonic bath of Morse oscillators. The microscopic Hamiltonian has been chosen so that the anharmonicity can be adjusted as a free parameter, and its effect can be isolated. This leads to a temperature dependent spectral density of the bath, which is studied for ohmic and lorentzian cases. Also, we compare numerically exact multiconfiguration time-dependent Hartree results with approximate solutions using continuous configuration time-dependent self-consistent field and local coherent state approximation.     

Relativistic dynamics of half-spin particles in a homogeneous magnetic field: an atom with nucleus of spin 12

An investigation of the relativistic dynamics of N+1 spin-12 particles placed in an external, homogeneous magnetic field is carried out. The system can represent an atom with a fermion nucleus and N electrons. Quantum electrodynamical interactions, namely, projected Briet and magnetic interactions, are chosen to formulate the relativistic Hamiltonian. The quasi-free-particle picture is retained here. The total pseudomomentum is conserved, and its components are distinct when the total charge is zero. Therefore, the center-of-mass motion can be separated from the Hamiltonian for a neutral (N+1)-fermion system, leaving behind a unitarily transformed, effective Hamiltonian H(0) at zero total pseudomomentum. The latter operator represents the complete relativistic dynamics in relative coordinates while interaction is chosen through order alpha4mc2. Each one-particle part in the effective Hamiltonian can be brought to a separable form for positive- and negative-energy states by replacing the odd operator in it through two successive unitary transformations, one due to Tsai [Phys. Rev. D 7, 1945 (1973)] and the other due to Weaver [J. Math. Phys. 18, 306 (1977)]. Consequently, the projector changes and the interaction that involves the concerned particle also becomes free from the corresponding odd operators. When this maneuver is applied only to the nucleus, and the non-Hermitian part of the transformed interaction is removed by another unitary transformation, a familiar form of the atomic relativistic Hamiltonian H(atom) emerges. This operator is equivalent to H(0). A good Hamiltonian for relativistic quantum chemical calculations, H(Qchem), is obtained by expanding the nuclear part of the atomic Hamiltonian through order alpha4mc2 for positive-energy states. The operator H(Qchem) is obviously an approximation to H(atom). When the same technique is used for all particles, and subsequently the non-Hermitian terms are removed by suitable unitary transformations, one obtains a Hamiltonian H(T) that is equivalent to H(atom) but is in a completely separable form. As the semidiscrete eigenvalues and eigenfunctions of the one-particle parts are known, the completely separable Hamiltonian can be used in computation. A little more effort leads to the derivation of the correct atomic Hamiltonian in the nonrelativistic limit, H(nonrel). The operator H(nonrel) is an approximation to H(T). It not only retains the relativistic and radiative effects, but also directly exhibits the phenomena of electron paramagnetic resonance and nuclear magnetic resonance.     

Cluster Amplitudes and Their Interplay with Self-Consistency in Density Functional Methods

Density functional theory (DFT) provides convenient electronic structure methods for the study of molecular systems and materials. Regular Kohn-Sham DFT calculations rely on unitary transformations to determine the ground-state electronic density, ground state energy, and related properties. However, for dissociation of molecular systems into open-shell fragments, due to the self-interaction error present in a large number of density functional approximations, the self-consistent procedure based on the this type of transformation gives rise to the well-known charge delocalization problem. To avoid this issue, we showed previously that the cluster operator of coupled-cluster theory can be utilized within the context of DFT to solve in an alternative and approximate fashion the ground-state self-consistent problem. This work further examines the application of the singles cluster operator to molecular ground state calculations. Two approximations are derived and explored: i) A linearized scheme of the quadratic equation used to determine the cluster amplitudes. ii) The effect of carrying the calculations in a non-self-consistent field fashion. These approaches are found to be capable of improving the energy and density of the system and are quite stable in either case. The theoretical framework discussed in this work could be used to describe, with an added flexibility, quantum systems that display challenging features and require expanded theoretical methods.     

Some analytical results for spin 1/2 spin systems. III: The benzene ring

In two earlier papers on multiply connectedABCD spin 1/2 spin systems, it was shown that it is possible to simplify the calculation of (i) the time dependent density matrix(t), and (ii) the time evolution of high-order multiple quantum operators, evolving in the presence of differing Zeeman offsets, scalar coupling and dipolar interactions, by subdividing the Hamiltonian into , where , is a suitable linear combination of the constants of the motion. In this paper, these techniques are applied to the benzene ring with particular emphasis on high-order multiple-quantum NMR experiments, and their interpretation in terms of specific wavefunctions. In particular, it is shown that excellent agreement between theory and experiment can be obtained for the energy splittings witnessed in the m = +4,+5 MQ-NMR transitions, with minimum effort.     

Relaxation and dephasing in open quantum systems time-dependent density functional theory: Properties of exact functionals from an exactly-solvable model system

The dissipative dynamics of many-electron systems interacting with a thermal environment has remained a long-standing challenge within time-dependent density functional theory (TDDFT). Recently, the formal foundations of open quantum systems time-dependent density functional theory (OQS-TDDFT) within the master equation approach were established. It was proven that the exact time-dependent density of a many-electron open quantum system evolving under a master equation can be reproduced with a closed (unitarily evolving) and non-interacting Kohn-Sham system. This potentially offers a great advantage over previous approaches to OQS-TDDFT, since with suitable functionals one could obtain the dissipative open-systems dynamics by simply propagating a set of Kohn-Sham orbitals as in usual TDDFT. However, the properties and exact conditions of such open-systems functionals are largely unknown. In the present article, we examine a simple and exactly-solvable model open quantum system: one electron in a harmonic well evolving under the Lindblad master equation. We examine two different representitive limits of the Lindblad equation (relaxation and pure dephasing) and are able to deduce a number of properties of the exact OQS-TDDFT functional. Challenges associated with developing approximate functionals for many-electron open quantum systems are also discussed.     

Time-dependent quasirelativistic density-functional theory based on the zeroth-order regular approximation

A time-dependent quasirelativistic density-functional theory for excitation energies of systems containing heavy elements is developed, which is based on the zeroth-order regular approximation (ZORA) for the relativistic Hamiltonian and a noncollinear form for the adiabatic exchange-correlation kernel. To avoid the gauge dependence of the ZORA Hamiltonian a model atomic potential, instead of the full molecular potential, is used to construct the ZORA kinetic operator in ground-state calculations. As such, the ZORA kinetic operator no longer responds to changes in the density in response calculations. In addition, it is shown that, for closed-shell ground states, time-reversal symmetry can be employed to simplify the eigenvalue equation into an approximate form that is similar to that of time-dependent nonrelativistic density-functional theory. This is achieved by invoking an independent-particle approximation for the induced density matrix. The resulting theory is applied to investigate the global potential-energy curves of low-lying LambdaS- and omega omega-coupled electronic states of the AuH molecule. The derived spectroscopic parameters, including the adiabatic and vertical excitation energies, equilibrium bond lengths, harmonic and anharmonic vibrational constants, fundamental frequencies, and dissociation energies, are in good agreement with those of time-dependent four-component relativistic density-functional theory and ab initio multireference second-order perturbation theory. Nonetheless, this two-component relativistic version of time-dependent density-functional theory is only moderately advantageous over the four-component one as far as computational efforts are concerned.     

Density functional restricted-unrestricted approach for nonlinear properties: application to electron paramagnetic resonance parameters of square planar copper complexes

The density functional restricted-unrestricted approach for treatments of spin polarization effects in molecular properties using spin restricted Kohn-Sham theory has been extended from linear to nonlinear properties. It is shown that the spin polarization contribution to a nonlinear property has the form of a quadratic response function that includes the zero-order Kohn-Sham operator, in analogy to the lower order case where the spin polarization correction to an expectation value has the form of a linear response function. The developed approach is used to formulate new schemes for computation of electronic g-tensors and hyperfine coupling constants, which include spin polarization effects within the framework of spin restricted Kohn-Sham theory. The proposed computational schemes are in the present work employed to study the spin polarization effects on electron paramagnetic resonance spin Hamiltonian parameters of square planar copper complexes. The obtained results indicate that spin polarization gives rise to sizable contributions to the hyperfine coupling tensor of copper in all investigated complexes, while the electronic g-tensors of these complexes are only marginally affected by spin polarization and other factors, such as choice of exchange-correlation functional or molecular structures, will have more pronounced impact on the accuracy of the results.     

Heavy (or large) ions in a fluid in an electric field: The diffusion equation exactly following from the Fokker-Planck equation

The problem of the derivation of the diffusion equation exactly following from the Fokker-Planck (or Klein-Kramers) equation for heavy (or large) particles in a fluid in an external force field is solved in the case in which the particles are ions subject to a uniform (but in general time-varying) electric field. It is found that such a diffusion equation maintains memory of the initial ion velocity distribution, unless sufficiently large values of time are considered. In such temporal asymptotic limit, the diffusion equation exactly becomes (i) the Smoluchowski equation when the electric field is constant in time, and (ii) a new equation generalizing the Smoluchowski equation, when the electric field is arbitrarily time varying. Finally, it is shown that the obtained exact (or asymptotic) results make questionable the procedures and the results of approximate theories developed in the past to get a "corrected" Smoluchowski equation when the external force can also be, in general, position dependent.     

Real-time linear response for time-dependent density-functional theory

We present a linear-response approach for time-dependent density-functional theories using time-adiabatic functionals. The resulting theory can be performed both in the time and in the frequency domain. The derivation considers an impulsive perturbation after which the Kohn-Sham orbitals develop in time autonomously. The equation describing the evolution is not strictly linear in the wave function representation. Only after going into a symplectic real-spinor representation does the linearity make itself explicit. For performing the numerical integration of the resulting equations, yielding the linear response in time, we develop a modified Chebyshev expansion approach. The frequency domain is easily accessible as well by changing the coefficients of the Chebyshev polynomial, yielding the expansion of a formal symplectic Green's operator.     

Energy-gap opening and quenching in graphene under periodic external potentials

We investigated the effects of periodic external potentials on properties of charge carriers in graphene using both the first-principles method based on density functional theory (DFT) and a theoretical approach based on a generalized effective spinor Hamiltonian. DFT calculations were done in a modified Kohn-Sham procedure that includes the effects of the periodic external potential. Unexpected energy band gap opening and quenching were predicted for the graphene superlattice with two symmetrical sublattices and those with two unsymmetrical sublattices, respectively. Theoretical analysis based on the spinor Hamiltonian showed that the correlations between pseudospins of Dirac fermions in graphene and the applied external potential, and the potential-induced intervalley scattering, play important roles in energy-gap opening and quenching.     

Electric dipole (hyper)polarizabilities of spatially confined LiH molecule

In this study we report on the electronic contributions to the linear and nonlinear static electronic electric dipole properties, namely the dipole moment (μ), the polarizability (α), and the first-hyperpolarizability (β), of spatially confined LiH molecule in its ground X (1)Σ(+) state. The finite-field technique is applied to estimate the corresponding energy and dipole moment derivatives with respect to external electric field. Various forms of confining potential, of either spherical or cylindrical symmetry, are included in the Hamiltonian in the form of one-electron operator. The computations are performed at several levels of approximation including the coupled-cluster methods as well as multi-configurational (full configuration interaction) and explicitly correlated Gaussian wavefunctions. The performance of Kohn-Sham density functional theory for the selected exchange-correlation functionals is also discussed. In general, the orbital compression effects lead to a substantial reduction in all the studied properties regardless of the symmetry of confining potential, however, the rate of this reduction varies depending on the type of applied potential. Only in the case of dipole moment under a cylindrical confinement a gradual increase of its magnitude is observed.     

Time-dependent exchange-correlation current density functionals with memory

Most present applications of time-dependent density functional theory use adiabatic functionals, i.e., the effective potential at time t is determined solely by the density at the same time. This paper discusses a method that aims to go beyond this approximation, by incorporating "memory" effects: the derived exchange-correlation potential will depend not only on present densities but also on the past. In order to ensure the potentials are causal, we formulate the action on the Keldysh contour for electrons in electromagnetic fields, from which we derive suitable Kohn-Sham equations. The exchange-correlation action is now a functional of the electron density and velocity field. A specific action functional is constructed which is Galilean invariant and yields a causal exchange-correlation vector potential for the Kohn-Sham equations incorporating memory effects. We show explicitly that the net exchange-correlation Lorentz force is zero. The potential is consistent with known dynamical properties of the homogeneous electron gas (in the linear response limit).