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We calculate harmonic spectra and shapes of attosecond-pulse trains using numerical solutions of Non-Born-Oppenheimer time-dependent Shrödinger equation for 1D H2 molecules in an intense laser pulse. A very strong signature of nuclear motion is seen in the time profiles of high-order harmonics. In general the nuclear motion shortens the part of the attosecond-pulse train originating from the first electron contribution, but it may enhance the second electron contribution for longer pulses. The shape of time profiles of harmonics can thus be used for monitoring the nuclear motion.

This article evaluates the impact of partial or total covariate inclusion or exclusion on the class enumeration performance of growth mixture models (GMMs). Study 1 examines the effect of including an inactive covariate when the population model is specified without covariates. Study 2 examines the case in which the population model is specified with 2 covariates influencing only the class membership. Study 3 examines a population model including 2 covariates influencing the class membership and the growth factors. In all studies, we contrast the accuracy of various indicators to correctly identify the number of latent classes as a function of different design conditions (sample size, mixing ratio, invariance or noninvariance of the variance-covariance matrix, class separation, and correlations between the covariates in Studies 2 and 3) and covariate specification (exclusion, partial or total inclusion as influencing class membership, partial or total inclusion as influencing class membership, and the growth factors in a class-invariant or class-varying manner). The accuracy of the indicators shows important variation across studies, indicators, design conditions, and specification of the covariates effects. However, the results suggest that the GMM class enumeration process should be conducted without covariates, and should rely mostly on the Bayesian information criterion (BIC) and consistent Akaike information criterion (CAIC) as the most reliable indicators under conditions of high class separation (as indicated by higher entropy), versus the sample size adjusted BIC or CAIC (SBIC, SCAIC) and bootstrapped likelihood ratio test (BLRT) under conditions of low class separation (indicated by lower entropy). (PsycINFO Database Record

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Numerical solutions of the time-dependent Schrödinger equation, using a new unitary numerical method, are used to obtain gauge-independent photoionization angular distributions for two-dimensional (planar) models of aligned H 2 and H 2 + molecules in different electronic states. Simulations are performed with few-cycle attosecond (as) extreme-ultraviolet (xuv) laser pulses at wavelengths λ = 10 nm [ω = 4.56 a.u. (atomic units), τ = 2π/ω = 1.38 a.u. = 33.4 as], and λ = 5 nm (ω = 9.11 a.u., τ = 16.7 as) for linear and circular polarization ionizations at R e = 1.675 a.u., where molecular orbital configurations dominate for which the electron wavelengths λ e R e and at large internuclear distance R = 10 a.u. (λ e < R) where Heitler-London atomic configurations including ionic states H + H − are more appropriate. The simulations allow one to investigate the effects of electron correlation and entanglement in H 2 at different R due to electron spin exchange as compared to the delocalized one-electron H 2 + system. An ultrashort (delta function) pulse model is used to interpret interferences in momentum distributions.

The time-dependent Schrödinger Equation (TDSE) is a parabolic partial differential equation (PDE) comparable to a diffusion equation but with imaginary time. Due to its first order time derivative, exponential integrators or propagators are natural methods to describe evolution in time of the TDSE, both for time-independent and time-dependent potentials. Two splitting methods based on Fer and/or Magnus expansions allow for developing unitary factorizations of exponentials with different accuracies in the time step △t. The unitary factorization of exponentials to high order accuracy depends on commutators of kinetic energy operators with potentials. Fourth-order accuracy propagators can involve negative or complex time steps, or real time steps only but with gradients of potentials, i.e. forces. Extending the propagators of TDSE's to imaginary time allows to also apply these methods to classical many-body dynamics, and quantum statistical mechanics of molecular systems.

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