A time-dependent multi-reference unified cluster cumulant formulation to study the subdynamics of quantum system coupled both to thermal and stochastic bath
Department of Chemistry, Rishi Bankim Chandra College, Naihati, North 24-Parganas-743 165, West Bengal, India E-mail: hanrkm1@yahoo.co.in Manuscript received online 05 March 2019, accepted 25 April 2019 This is to introduce a time-local effective hamiltonian unified cluster cumulant method based on the time-dependent multireference cluster cumulant (TDMRCC) and thermal cluster cumulant (TCC) strategy developed by Mukherjee and his coworkers. A factorized ansatz for UIP = [UI] ex [UI ]MP with P = |j i > Ä |0b> Ä|0f>, where |ji >,|0b > and |0f > denote the purely quantum mechanical vacuum state, the thermal vacuum state, the base state for the solvent degrees of freedom respectively, and [UI]ex consists of operators admixing the thermally as well as the stochastically projected model space functions with that of the virtual space outside the model space whose functions govern the time-evolution of [UI ]M via the time-local effective operator, Veff [b,x,t ](=<< vl[ul]ex>b>f ), where < ... >b and < ... >f denote thermal and stochastic averaging respectively. The exponential ansatz is introduced both for [UI]ex and [UI ]M and these ansatze are normal ordered with respect to
Department of Chemistry, Rishi Bankim Chandra College, Naihati, North 24-Parganas-743 165, West Bengal, India E-mail: hanrkm1@yahoo.co.in Manuscript received online 05 March 2019, accepted 25 April 2019 This is to introduce a time-local effective hamiltonian unified cluster cumulant method based on the time-dependent multireference cluster cumulant (TDMRCC) and thermal cluster cumulant (TCC) strategy developed by Mukherjee and his coworkers. A factorized ansatz for UIP = [UI] ex [UI ]MP with P = |j i > Ä |0b> Ä|0f>, where |ji >,|0b > and |0f > denote the purely quantum mechanical vacuum state, the thermal vacuum state, the base state for the solvent degrees of freedom respectively, and [UI]ex consists of operators admixing the thermally as well as the stochastically projected model space functions with that of the virtual space outside the model space whose functions govern the time-evolution of [UI ]M via the time-local effective operator, Veff [b,x,t ](=<< vl[ul]ex>b>f ), where < ... >b and < ... >f denote thermal and stochastic averaging respectively. The exponential ansatz is introduced both for [UI]ex and [UI ]M and these ansatze are normal ordered with respect to
Department of Physical Chemistry, Indian Association for the Cultivation of Science, Kolkata- 700 032, India E-Mail : pcdm@mahendra.iacs.res.in Present address: Department of Physical Science, R. K. M. Sikshanamandira, Belurmath, Howrah-711 202, India Present address : Department of Chemistry, R. K. M. Residential College, Narendrapur, Kolkata-700 103, India Also at : Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore-560 064, India Manuscript received 4 October 2002 We present here a non-perturbative cumulant expansion method of studying stochastically averaged dynamics of a TLS coupled to a gaussian colored noise bath. It uses a factorized cluster Ansatz for the evolution operator, U(t) suggested in an earlier formalism (Ref. 22). The factorization separates the operators involving only the system variables from those containing the intractions involving the bath variables. The stochastically averaged U(t) is obtained in a compact way by way of introducing bath-normal ordered products, which have zero bath averages for the bath-variables satisfying gaussian factorization property. We use a Wick-like operator reordering theorem (Ref. 25) to have a bath-normal ordered cluster expansion representation of U(t) and subsequently Novikov's theorem (Ref. 27) to get equations for the cluster amplitudes without free bath variables.The subdynamics of the system where the bath-variables are averaged out is governed in our method by a time-local effective hamiltonian involving system variables only. The method is efficacious when the bath is strongly colored and its coupling with the system is strong, e.g. when nonperturbative effects are prominent. The efficacy of the method is illustrated by computing power spectra from a model system which mimics nonadiabatic vibronic coupling simulated by colored noise.
We have developed in this paper a nonperturbative cluster-expansion strategy for generating the stochastically averaged time-evolution operator for quantum systems driven by Ornstein-Uhlenbeck (OU) colored noise. The method induces a boson mapping of the (real or complex) stochastic variable f, and interprets the stochastic average of a pair of variables f at two different times as the expectation value of the time-ordered product of the associated bosons with respect to the boson vacuum \ensuremath{\Vert}0B〉. The stochastic evolution is thus mapped onto a deterministic evolution in an expanded Fock space. The evolution of the system from the groups of state of interest is monitored using our recently developed nonperturbative time-dependent multireference coupled-cluster (TDMRCC) method. In this, the evolution operator U is written in a factorized form UexUM, where UM evolves in the space of the starting functions (model space) and Uex brings in the virtual functions. Uex and UM are both written as normal ordered exponentials involving cluster operators. In the present context, the TDMRCC method translates into one for generating the evolution operator UB for the Hamiltonian containing the additional boson variables, and the stochastic averaging of UB is realized as the expectation value 〈0B\ensuremath{\Vert}UB\ensuremath{\Vert}0B〉.We call the TDMRCC method involving the expanded Fock space a cluster cumulant method. We have analyzed the relation of the cluster cumulant approach with the Fox-Kubo operator cumulant expansion and the method of marginal averages involving the Fokker-Planck operator of the OU process. It has been shown that an order by order expansion of our equations in the power of stochastic coupling generates the perturbative cumulant results of Fox and Kubo. It is also demonstrated that the traditional Fokker-Planck method of marginal averages uses a Kubo-Schr\"odinger operator which is related to our boson mapped Hamiltonian by a transformation which converts the Fokker-Planck operator \ensuremath{\Gamma} to a manifestly Hermitian form. However, the method of marginal averages involves a linear expansion of UB in terms of eigenfunctions of \ensuremath{\Gamma}, while the cluster cumulant method uses a cluster expansion, which is nonlinear in the expansion parameters. The former is thus analogous to a configuration-interaction expansion, while the latter is analogous to a coupled-cluster strategy. The method is illustrated by applying it to a simple yet nontrivial problem, viz., a harmonic oscillator linearly perturbed by a (real or complex) stochastic variable. Survivalities of both the ground and the first excited states have been considered for a wide range of coupling strengths and noise correlation times. Our method provides exact results in this case; these have been used as benchmarks for assessing performance of the second- and fourth-order cumulant results.
We present in this paper a systematic nonperturbative cluster-cumulant method for deriving thermal averages of operators in quantum many-body systems. The method combines the advantages of the cumulant expansion scheme of thermodynamic perturbation theory, the approach of thermofield dynamics as a finite-temperature field theory, and the time-dependent coupled-cluster theory extended to ``imaginary time.'' We have generalized the concepts of cumulants in a nonperturbative manner and have posited on the statistical operator an exponential-like ansatz containing connected, size-extensive operators in the exponent. These latter cumulantlike operators have been termed ``cluster cumulants'' by us. For a compact treatment, we have derived an alternative thermal field theory in which a time-ordered product is expanded in terms of ``thermal normal products'' of operators and thermal contractions---leading to a ``thermal Wick's theorem.'' The thermal normal products are the finite-temperature analogs of the ordinary normal products and have zero thermal averages. Operators in these products commute (anticommute) under permutations for bosons (fermions). This thermal representation is shown to be unitarily related to the traditional thermofield dynamics formulation, but has the advantage of using only the physical variables. The imaginary-time evolution of the statistical operator is treated by our recently formulated time-dependent cluster-cumulant theory. The partition function is evaluated as an exponential of a connected quantity. As an illustrative example, we have computed the partition function of an anharmonic oscillator with equally weighted cubic and quartic perturbation for a wide range of coupling, extending to the strongly nonperturbative regime. We study the behavior of free energy in the low-temperature limit and verify numerically the validity of the Kohn-Luttinger theorem [Phys. Rev. 118, 41 (1960)] for this system. We also show that our formalism is a natural nonperturbative analog of the thermodynamic perturbative theory by showing that a perturbative solution of the thermal-cluster-cumulant equations generates a variation of the Bloch--Balian--de Dominicis theory.
The traditional state universal multi-reference coupled cluster (SUMRCC) theory uses the Jeziorski-Monkhorst (JM) based Ansatz of the wave operator: Ω = Σ(μ)Ω(μ)|φ(μ)><φ(μ)|, where Ω(μ) = exp(T(μ)) is the cluster representation of the component of Ω inducing virtual excitations from the model function φ(μ). In the first formulations, φ(μ)s were chosen to be single determinants and T(μ)s were defined in terms of spinorbitals. This leads to spin-contamination for the non-singlet cases. In this paper, we propose and implement an explicitly spin-free realization of the SUMRCC theory. This method uses spin-free unitary generators in defining the cluster operators, {T(μ)}, which even at singles-doubles truncation, generates non-commuting cluster operators. We propose the use of normal-ordered exponential parameterization for Ω:Σ(μ){exp(T(μ))}|φ(μ)><φ(μ)|, where {} denotes the normal ordering with respect to a common closed shell vacuum which makes the "direct term" of the SUMRCC equations terminate at the quartic power. We choose our model functions {φ(μ)} as unitary group adapted (UGA) Gel'fand states which is why we call our theory UGA-SUMRCC. In the spirit of the original SUMRCC, we choose exactly the right number of linearly independent cluster operators in {T(μ)} such that no redundancies in the virtual functions {χ(μ)(l)} are involved. Using example applications for electron detached/attached and h-p excited states relative to a closed shell ground state we discuss how to choose the most compact and non-redundant cluster operators. Although there exists a more elaborate spin-adapted JM-like ansatz of Datta and Mukherjee (known as combinatoric open-shell CC (COS-CC), its working equations are more complex. Results are compared with those from COS-CC, equation of motion coupled cluster methods, restricted open-shell Hartree-Fock coupled cluster, and full configuration interaction. We observe that our results are more accurate with respect to most other theories as a result of the use of the cluster expansion structure for our wave operator. Our results are comparable to those from the more involved COS-CC, indicating that our theory captures the most important aspects of physics with a considerably simpler scheme.
We present a time-dependent formulation of coupled cluster theory. This theory allows for direct computation of the free energy of quantum systems at finite temperature by imaginary time integration and is closely related to the thermal cluster cumulant theory of Mukherjee and co-workers [ Chem. Phys. Lett. 1992 , 192 , 55 -61 ; Phys. Rev. E 1993 , 48 , 3373 -3389 ; Chem. Phys. Lett. 2001 , 335 , 281 -288 ; Chem. Phys. Lett. 2002 , 352 , 63 -69 ; Int. J. Mod. Phys. B 2003 , 17 , 5367 -5377 ]. Our derivation of the finite-temperature theory highlights connections to perturbation theory and to zero-temperature coupled cluster theory. We show explicitly how the finite-temperature coupled cluster singles and doubles amplitude equations can be derived in analogy with the zero-temperature theory and how response properties can be efficiently computed using a variational Lagrangian. We discuss the implementation for realistic systems and showcase the potential utility of the method with calculations of the exchange correlation energy of the uniform electron gas under warm dense matter conditions.
Department of Chemistry, Visva-Bharati, Shantiniketan-731 235, India indian Association for the Cultivation of Science, Jadavpur, Kolkata-700 032, India E-mail : pcdsr@mahendra.iacs.res.in Fax: 91-33-24722805 Max-Pianck-Institut fur Physik Komplexer Systeme, Nothnitzer Strabe 38, 01187 Dresden, Germany Manuscript received 12 November 2002 Arrhenius rate law is now more than hundred years old. The dynamical basis of this phenomenological law was provided by Kramers in early nineteenforties. Since late seventies, the extensive experimental investigation in chemical dynamics and allied areas centered around Kramers' dynamical rate theory. The present paper is an outline of further development of rate theory in quantum mechanical context which takes into account of thermal activation and tunneling within an unified description. The theory is based on a new scheme for quantum Brownian motion recently proposed by us to establish quantum analogues of Kramers', Smoluchowski and Fokker-Pianck equations with true probability distribution functions in c-number phase space.