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The Journal of Chemical Physics

Generalized CC-TDSCF and LCSA: The system-energy representation

Sergio López-López, Mathias Nest, Rocco Martinazzo

Published Jan 5, 2011• 10 citations

Abstract

Typical (sub)system-bath quantum dynamical problems are often investigated by means of (approximate) reduced equations of motion. Wavepacket approaches to the dynamics of the whole system have gained momentum in recent years and there is hope that properly designed approximations to the wavefunction will allow one to correctly describe the subsystem evolution. The continuous-configuration time-dependent self-consistent field (CC-TDSCF) and local coherent-state approximation (LCSA) methods, for instance, use a simple Hartree product of bath single-particle-functions for each discrete variable representation (DVR) state introduced in the Hilbert space of the subsystem. Here we focus on the above two methods and replace the DVR states with the eigenstates of the subsystem Hamiltonian, i.e., we adopt an energy-local representation for the subsystem. We find that stable and semiquantitative results are obtained for a number of dissipative problems, at the same (small) computational cost of the original methods. Furthermore, we find that both methods give very similar results, thus suggesting that coherent-states are well suited to describe (local) bath states. As a whole, present results highlight the importance of the system basis-set in the selected-multiconfiguration expansion of the wavefunction. They suggest that accurate and yet computationally cheap methods may be simply obtained from CC-TDSCF/LCSA by letting the subsystem states be variationally optimized.

DOI: 10.1063/1.3518418

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Typical (sub)system-bath quantum dynamical problems are often investigated by means of (approximate) reduced equations of motion. Wavepacket approaches to the dynamics of the whole system have gained momentum in recent years and there is hope that properly designed approximations to the wavefunction will allow one to correctly describe the subsystem evolution. The continuous-configuration time-dependent self-consistent field (CC-TDSCF) and local coherent-state approximation (LCSA) methods, for instance, use a simple Hartree product of bath single-particle-functions for each discrete variable representation (DVR) state introduced in the Hilbert space of the subsystem. Here we focus on the above two methods and replace the DVR states with the eigenstates of the subsystem Hamiltonian, i.e., we adopt an energy-local representation for the subsystem. We find that stable and semiquantitative results are obtained for a number of dissipative problems, at the same (small) computational cost of the original methods. Furthermore, we find that both methods give very similar results, thus suggesting that coherent-states are well suited to describe (local) bath states. As a whole, present results highlight the importance of the system basis-set in the selected-multiconfiguration expansion of the wavefunction. They suggest that accurate and yet computationally cheap methods may be simply obtained from CC-TDSCF/LCSA by letting the subsystem states be variationally optimized.

A novel quantum method to deal with typical system-bath dynamical problems is introduced. Subsystem discrete variable representation and bath coherent-state sets are used to write down a multiconfigurational expansion of the wave function of the whole system. With the help of the Dirac-Frenkel variational principle, simple equations of motion--a kind of Schrodinger-Langevin equation for the subsystem coupled to (pseudo) classical equations for the bath--are derived. True dissipative dynamics at all times is obtained by coupling the bath to a secondary, classical Ohmic bath, which is modeled by adding a friction coefficient in the derived pseudoclassical bath equations. The resulting equations are then solved for a number of model problems, ranging from tunneling to vibrational relaxation dynamics. Comparison of the results with those of exact, multiconfiguration time-dependent Hartree calculations in systems with up to 80 bath oscillators shows that the proposed method can be very accurate and might be of help in studying realistic problems with very large baths. To this end, its linear scaling behavior with respect to the number of bath degrees of freedom is shown in practice with model calculations using tens of thousands of bath oscillators.

The feasibility of using a basis set approach to the study of quantum dissipative dynamics is investigated for the spin-boson model, a system of two discrete states linearly coupled to a harmonic bath. The infinite Hamiltonian is discretized to a finite number of degrees of freedom. Traditional basis set approach, in a multiconfiguration time-dependent Hartree context, is used to solve the time-dependent Schrödinger equations by explicitly including all the degrees of freedom (“system”+“bath”). Quantities such as the reduced density matrix are then evaluated via a quadrature summation/Monte Carlo procedure over a certain number of time-dependent wave functions. Numerically exact results are obtained by systematically increasing the number of bath modes used to represent the condensed phase environment, as well as other variational parameters (number of basis functions, configurations, etc.). The potential of the current method is briefly discussed.

We discuss a generalized dynamic mean-field method combining the advantages of explicit pair correlations and of configuration interaction. The approximate dynamical method, which we call time-dependent self-consistent-field configuration interaction (TDSCF2-CI), is constructed by including N(N−1)/2 TDSCF2 configurations. In each configuration a given pair of N coupled modes is directly (not in the mean-field approach) correlated; the N(N−1)/2 configurations include all such choices of pairs. As such, it has both the usual advantages of TDSCF and improvements due to some inclusion of correlations (exact results for any two-mode problem, improved descriptions of dynamical corrections, and greater accuracy). A three-mode model Hamiltonian is analyzed using five approximate treatments, i.e., the usual TDSCF, the three TDSCF2 forms, and the TDSCF2-CI one. The quantities for comparison with the exact results include the decay P(t) of the initial state, the time dependencies of the energies e(i) of individual modes, and the overlap S(t) of the corresponding approximate wave function with the exact one. We find, indeed, that explicit inclusion of pair correlations improves the description of the quantum dynamics of the system.

Accurate descriptions of many-particle quantum systems subject to laser interactions can be found using real-time ab initio methods. Of these the arguably most popular and exact is the multi-configuration time-dependent Hartree-Fock (MCTDHF) method. However, MCTDHF suffers from computational limitations in that it quickly becomes too time consuming. The orbital-adaptive time-dependent coupled-cluster (OATDCC) method represents a hierarchy of approximations to MCTDHF that are less computationally expensive while retaining as much accuracy as possible. Building on an existing codebase we have in this thesis generalized the OATDCC method to include Q-space orbital equations. A novel ground state solver is implemented, employing adiabatic switching, since imaginary time propagation is not feasible "out of the box". Furthermore, we implement a sinc-discrete variable representation basis for one-dimensional model systems. We demonstrate that ionization and high-harmonic processes can be described using the OATDCCD method. Comparison with the more accurate, yet more expensive, multiconfigurational time-dependent Hartree-Fock method indicates that the OATDCCD method is an excellent approximation.

An efficient method, the self-consistent hybrid method, is proposed for accurately simulating time-dependent quantum dynamics in complex systems. The method is based on an iterative convergence procedure for a dynamical hybrid approach. In this approach, the overall system is first partitioned into a “core” and a “reservoir” (an initial guess). The former is treated via an accurate quantum mechanical method, namely, the time-dependent multiconfiguration self-consistent field or multiconfiguration time-dependent Hartree approach, and the latter is treated via a more approximate method, e.g., classical mechanics, semiclassical initial value representations, quantum perturbation theories, etc. Next, the number of “core” degrees of freedom, as well as other variational parameters, is systematically increased to achieve numerical convergence for the overall quantum dynamics. The method is applied to two examples of quantum dissipative dynamics in the condensed phase: the spin-boson problem and the electronic resonance decay in the presence of a vibrational bath. It is demonstrated that the method provides a practical way of obtaining accurate quantum dynamical results for complex systems.