Time-dependent close-coupling

Expansion of the wavepacket in terms of time-independent basis functions, in the same way as described in Section 3.1, provides another 

Each rotational state is coupled to all other states through the potential matrix V defined in (3.22). Initial conditions Xj(I 0) are obtained by expanding — in analogy to (3.26) — the ground-state wavefunction multiplied by the transition dipole function in terms of the Yjo- The total of all one-dimensional wavepackets Xj (R t) forms an R- and i-dependent vector x whose propagation in space and time follows as described before for the two-dimensional wavepacket, with the exception that multiplication by the potential is replaced by a matrix multiplication Vx-The close-coupling equations become computationally more convenient if one makes an additional transformation to the so-called discrete variable representation (Bacic and Light 1986). The autocorrelation function is simply calculated from 

The time-dependent approach is the starting point for several useful approximations, which are particularly important for systems with more than three degrees of freedom as well as for gaining physical insight into the fragmentation process. The following discussion is rather limited the review article of Kosloff (1988), for example, contains an extensive list of appropriate references. 

One of the main assets of the time-dependent theory is the possibility of treating some degrees of freedom quantum mechanically and others classically. Such composite methods necessarily lead to time-dependent Hamiltonians which obviously exclude time-independent approaches. We briefly outline three approximations that are frequently used in molecular dynamics studies. To be consistent with the previous sections we consider the collinear triatomic molecule ABC with Jacobi coordinates R and r. 

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The set of coupled equations (15.11) represents an example of time-dependent close-coupling as described in Section 4.2.3. It is formally equivalent to (4.25), for example, and can be solved by exactly the same numerical recipes. The dependence on the two stretching coordinates R and r is treated by discretizing the two nuclear wavepackets on a two-dimensional grid and the Fourier-expansion method is employed to evaluate the second-order derivatives in R and r. If we additionally include the rotational degree of freedom, we may expand each wavepacket in terms of... 

Abstract. Cross sections for electron transfer in collisions of atomic hydrogen with fully stripped carbon ions are studied for impact energies from 0.1 to 500 keV/u. A semi-classical close-coupling approach is used within the impact parameter approximation. To solve the time-dependent Schrodinger equation the electronic wave function is expanded on a two-center atomic state basis set. The projectile states are modified by translational factors to take into account the relative motion of the two centers. For the processes C6++H(1.s) —> C5+ (nlm) + H+, we present shell-selective electron transfer cross sections, based on computations performed with an expansion spanning all states ofC5+( =l-6) shells and the H(ls) state. 

In this chapter we discuss the close relationship between the Born-Oppenheimer treatment of molecular systems and field theory as applied to elementary particles. The theory is based on the Born-Oppenheimer non-adiabatic coupling terms which are known to behave as vector potentials in electromagnetic dynamics. Treating the time-dependent Schrodinger equation for the electrons and the nuclei we show that enforcing diabatization produces for non-Abelian time-dependent systems the four-component Curl equation as obtained by Yang and Mills (Phys. Rev. 95, 631 (1954)). 

If the coupling is zero, the bound states will live forever. However, immediately after we have switched on the coupling they start to decay as a consequence of transitions to the continuum states until they are completely depopulated. Our goal is to derive explicit expressions for the depletion of the bound states l iz) and the filling of the continuum states 2(E,0)). The method we use is time-dependent perturbation theory in the same spirit as outlined in Section 2.1, with one important extension. In Section 2.1 we explicitly assumed that the perturbation is sufficiently weak and also sufficiently short to ensure that the population of the initial state remains practically unity for all times (first-order perturbation theory). In this section we want to describe the decay process until the initial state is completely depleted and therefore we must necessarily go beyond the first-order treatment. The subsequent derivation closely follows the detailed presentation of Cohen-Tannoudji, Diu, and Laloe (1977 ch.XIII). 

Figure 12.9 depicts a comparison between classical trajectory results and exact close-coupling calculations for He--Cl2 and Ne- -Cl2, respectively. In both cases, the classical procedure reproduces the overall behavior of the final state distributions satisfactorily. Subtle details such as the weak undulations particularly for He are not reproduced, however. As shown by Gray and Wozny (1991), who treated the dissociation of van der Waals molecules in the time-dependent framework, the bimodality for He CI2 is the result of a quantum mechanical interference between two branches of the evolving wavepacket and therefore cannot be obtained in purely classical calculations. 

It can be seen that the coupling of the formation and decay processes increases with the width of the flash. In an intermediate case, the time dependence of the absorbance change will have the functional form of a double exponential, A exp(—f/i ) + B exp (—tlx"). One lifetime will be close to the lifetime of the transient species and the other to the lifetime of the pump. In the most unfavorable conditions, the functional form will be a single exponential with nearly the lifetime of the pump. The determination of the lifetime of a transient species formed by the decay of transformation of an excited state offers a similar difficulty. The reduction of methylviologen, MV2 +, by the metal to ligand charge transfer (MLCT) state19 of the Re(I) complex and the reoxidation of the produced radical, MV +, are illustrated in Equations 6.57-6.59. 

Here, w = m, n, and S. V represents the membrane potential, n is the opening probability of the potassium channels, and S accounts for the presence of a slow dynamics in the system. Ic and Ik are the calcium and potassium currents, gca = 3.6 and gx = 10.0 are the associated conductances, and Vca = 25 mV and Vk = -75 mV are the respective Nernst (or reversal) potentials. The ratio r/r s defines the relation between the fast (V and n) and the slow (S) time scales. The time constant for the membrane potential is determined by the capacitance and typical conductance of the cell membrane. With r = 0.02 s and ts = 35 s, the ratio ks = r/r s is quite small, and the cell model is numerically stiff. The calcium current Ica is assumed to adjust immediately to variations in V. For fixed values of the membrane potential, the gating variables n and S relax exponentially towards the voltage-dependent steady-state values noo (V) and S00 (V). Together with the ratio ks of the fast to the slow time constant, Vs is used as the main bifurcation parameter. This parameter determines the membrane potential at which the steady-state value for the gating variable S attains one-half of its maximum value. The other parameters are assumed to take the following values gs = 4.0, Vm = -20 mV, Vn = -16 mV, 9m = 12 mV, 9n = 5.6 mV, 9s = 10 mV, and a = 0.85. These values are all adjusted to fit experimentally observed relationships. In accordance with the formulation used by Sherman et al. [53], there is no capacitance in Eq. (6), and all the conductances are dimensionless. To eliminate any dependence on the cell size, all conductances are scaled with the typical conductance. Hence, we may consider the model to represent a cluster of closely coupled / -cells that share the combined capacity and conductance of the entire membrane area. 

In this section we outline the coupled cluster-molecular mechanics response method, the CC/MM response method. Ref. [51] considers CC response functions for molecular systems in vacuum and for further details we refer to this article. The identification of response functions is closely connected to time-dependent perturbation theory [51,65,66,67,68,69,70], Our starting point is the quasienergy and we identify the response functions as simple derivatives of the quasienergy. For a molecular system in vacuum where Hqm is the vacuum Hamiltonian for the unperturbed molecule and V" is a time-dependent perturbation we have the following time-dependent Hamiltonian, H,... 

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