Time-dependent equation wave function propagation

The preceding remarks all apply to a steadily propagating one-dimensional detonation. It is a relatively simple task to solve the algebraic equation for the over-all steady-state motion. The differential equations for the structure may be solved in the steady state, but the task is tedious and, in addition, detailed knowledge of reaction rates needed in the equation is not available. It is not too difficult to solve the time-dependent over-all equation if a burning velocity for the flame as a function of temperature and pressure is assumed (J4). It is not practicable to solve the time-dependent equations which govern the structure of the wave with any certainty because of the lack of kinetic information, in addition to the mathematical difficulty. The acceleration of the slowly moving flame front as it sends forward pressure waves which coalesce into shock waves that eventually are coupled to a zone of reaction to form a detonation wave has been observed experimentally (LI, L2). 

Perhaps the most straightforward method of solving the time-dependent Schrodinger equation and of propagating the wave function forward in time is to expand the wave function in the set of eigenfunctions of the unperturbed Hamiltonian [41], Hq, which is the Hamiltonian in the absence of the interaction with the laser field. 

One way in which we can solve the problem of propagating the wave function forward in time in the presence of the laser field is to utilize the above knowledge. In order to solve the time-dependent Schrodinger equation, we normally divide the time period into small time intervals. Within each of these intervals we assume that the electric field and the time-dependent interaction potential is constant. The matrix elements of the interaction potential in the basis of the zeroth-order eigenfunctions y i Vij = (t t T(e(t)) / ) are then evaluated and we can use an eigenvector routine to compute the eigenvectors, = S) ... 

Selloni et al. [48] were the first to simulate adiabatic ground state quantum dynamics of a solvated electron. The system consisted of the electron, 32 K+ ions, and 31 Cl ions, with electron-ion interactions given by a pseudopotential. These simulations were unusual in that what has become the standard simulating annealing molecular dynamics scheme, described in the previous section, was not used. Rather, the wave function of the solvated electron was propagated forward in time with the time-dependent Schrodinger equation,... 

A fundamental theoretical issue for computational photochemistry is the treatment of the hop (nonadiabatic) event. One needs to add the time propagation of the solutions of the time-dependent Schrodinger equation for electronic motion to the classical propagation of the nuclei, thus obtaining the populations of each adiabatic state. The time-dependent wave function for electronic motion is just a time-dependent configuration interaction vector ... 

The initial condition for the dynamics at / = 0 was again chosen to be a v = 3 overtone excited CH oscillator. The subsequent dynamics was then generated by propagating the time-dependent wave function in the active space. If we let C(/) denote the column vector of the expansion coefficients of the various basis functions at time t, then this state vector evolves according to the equation C(t) = U(r)C(r), where U(r) = exp[—iH/] is the time propagator associated with the vibrational Hamiltonian matrix H. In order to evaluate the propagator, we used the Chebyshev expansion (Sec. II.C.3). 

Nuclear motion can be described by quantum mechanical propagation of the vibrational wave function or classical motion on the time-dependent adiabatic potentials. In our approach, the classical equations of motion are solved by the velocity Verlet algorithm. The typical time increment for integration. At, was 0.5 fs. 

The simplest polarization propagator corresponds to choosing an HF reference and including only the h2 operator, known as the Random Phase Approximation (RPA), which is identical to Time-Dependent Hartree-Fock (TDHF), with the corresponding density functional version called Time-Dependent Density Functional Theory (TDDFT). For the static case co= 0) the resulting equations are identical to those obtained from a coupled Hartree-Fock approach (Section 10.5). When used in conjunction with coupled cluster wave functions, the approach is usually called Equation Of Motion (EOM) methods. ... 

The basic technique used to propagate the wave packet in the spatial domain is the fast Fourier transform method [287, 288, 299, 300]. The time-dependent Schrodinger equation is solved numerically, employing the second-order differencing approach [299, 301]. In this approach the wave function Sit t = t St is constructed recursively from the wave functions at t and t" = t — St. The operator including the potential energy is applied in phase space and that of the kinetic energy in momentum space. Therefore, for each... 

The Hamiltonian and the coordinates are discretized by means of the generalized pseudospectral (GPS) method in prolate spheroidal coordinates [44-47], allowing optimal and nonuniform spatial grid distribution and accurate solution of the wave functions. The time-dependent Kohn-Sham Equation 3.5 can be solved accurately and efficiently by means of the split-operator method in the energy representation with spectral expansion of the propagator matrices [44-46,48]. We employ the following split operator, second-order short-time propagation formula [40] ... 

The formulation of approximate response theory based on an exponential parame-trization of the time-dependent wave function, Eq. (11.36), and the Ehrenfest theorem, Eq. (11.40), can also be used to derive SOPPA and higher-order Mpller-Plesset perturbation theory polarization propagator approximations (Olsen et al., 2005). Contrary to the approach employed in Chapter 10, which is based on the superoperator formalism from Section 3.12 and that could not yet be extended to higher than linear response functions, the Ehrenfest-theorem-based approach can be used to derive expressions also for quadratic and higher-order response functions. In the following, it will briefly be shown how the SOPPA linear response equations, Eq. (10.29), can be derived with this approach. 

Let us assume that at r = 0 the wave function If is given in MCTDH form, i.e., given by equation (16). (The question of how to define and generate an initial-state wave function is addressed below.) We want to propagate If while preserving its MCTDH form. As was done above, we derive first-order differential equations (equations of motion) for A and by employing the time-dependent variational principle equation (5). But before doing so, we partition the Hamiltonian H into a separable and residual part ... 

The frequency dependence of the propagation constant appears as wave deformation in time domain. This is measured as a voltage waveform at distance X when a step (or impulse) function voltage is applied to the sending end of a semi-infinite line. The voltage waveform, which is distorted from the original waveform, is called "step (impulse) response of wave deformation," and is defined in Equation 1.201. 

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