An approximate coupled cluster theory via nonlinear dynamics and synergetics: The adiabatic decoupling conditions

Abstract

The coupled cluster iteration scheme is analyzed as a multivariate discrete time map using nonlinear dynamics and synergetics. The nonlinearly coupled set of equations to determine the cluster amplitudes are driven by a fraction of the entire set of cluster amplitudes. These driver amplitudes enslave all other amplitudes through a synergistic inter-relationship, where the latter class of amplitudes behave as the auxiliary variables. The driver and the auxiliary variables exhibit vastly different time scales of relaxation during the iteration process to reach the fixed points. The fast varying auxiliary amplitudes are small in magnitude, while the driver amplitudes are large, and they have a much longer time scale of relaxation. Exploiting their difference in relaxation time scale, we employ an adiabatic decoupling approximation, where each of the fast relaxing auxiliary modes is expressed as a unique function of the principal amplitudes. This results in a tremendous reduction in the independent degrees of freedom. On the other hand, only the driver amplitudes are determined accurately via exact coupled cluster equations. We will demonstrate that the iteration scheme has an order of magnitude reduction in computational scaling than the conventional scheme. With a few pilot numerical examples, we would demonstrate that this scheme can achieve very high accuracy with significant savings in computational time.

A Synergistic Approach towards Optimization of Coupled Cluster Amplitudes by Exploiting Dynamical Hierarchy

“…It was observed from a number of numerical examples [20] that the amplitudes with large magnitude (at the MP2 level) take substantially more number of iterations to reach their converged values (fixed points) than the ones with smaller magnitudes. Based on the relative magnitude of these amplitudes, the entire amplitude space can be subdivided into a Large Amplitude Subset ( LAS , spanned by $${\left\{ {T_L } \right\}}$$ with dimension n L ) and Small Amplitude Subset ( SAS , spanned by $${\left\{ {T_S } \right\}}$$ , with dimension n S ).…”

“…This thus can be interpreted as the adiabatic approximation , which holds true only in the regions of the multi‐dimensional phase space where the fast modes relax very quickly such that the whole trajectory converges to the unstable manifold and the fast modes linger around it till the convergence is reached. Note that in one of our earlier publications, [20] the adiabatic approximation was introduced simply by setting the decoupling condition. Here we further corroborate the concept by introducing a discrete‐time dependent picture and timescale decoupling in the amplitude space.…”

“…(9). Noting the fact the auxiliary amplitudes t S are significantly small compared to the principal amplitudes, t L , one may neglect [20] t S from the adiabatically decoupled expression of $${t_{S_i } }$$ : $$$\vcenter{\openup.5em\halign{$\displaystyle{#}$\cr t_{S_i (ad)} = - {{P_{s_i } (\{ t_S, t_L \} )} \over {\lambda _{S_i } }}\mathop \rightarrow \limits ^{{{\rm{\mid }}t_S {\rm{\mid }} \approx 0}}_{} - {{P_{s_i } (\{ t_L \} )} \over {\lambda _{S_i } }}\hfill\cr}}$$$ …”

“…In one of our previous publications, we developed a first‐principle based CC optimization strategy, termed as the Adiabatically Decoupled Coupled Cluster (ADCC), where we have shown that it is possible to distinguish the various amplitudes at their MP2 level into principal and auxiliary modes. Based on certain assumptions which are conceptually close in spirit to the adiabatic approximation, the authors were able to reproduce very accurate results by optimizing the variables in a reduced subspace [20] …”

“…Based on certain assumptions which are conceptually close in spirit to the adiabatic approximation, the authors were able to reproduce very accurate results by optimizing the variables in a reduced subspace. [20] The main purpose of this article is to show that some of these features of nonlinear dynamics for the systems close to classical critical points can also be applied in numerically accurate manner (although not mathematically exact) to reduce the computational scaling where nonlinear iterative optimization is involved. We will also demonstrate how the "adiabatic approximation" comes out quite naturally in a more general and rigorous mathematical way, given that some of the cluster amplitudes have significantly longer timescale of relaxation to converge.…”