Normal ordered exponential approach to thermal properties and time-correlation functions: general theory and simple examples

Abstract

A normal ordered exponential parametrisation is used to obtain equations for thermal one-and two-particle reduced density matrices, as well as free energies, partition functions and entropy for both Fermionic (electronic) and Bosonic (vibrational) Hamiltonians. A first principles derivation of the equations, relying only on a simple Wick's theorem and starting from the differential equation $\frac{\mathrm{d}\hat{D}}{\mathrm{d}\beta }=-(\hat{H}-\mu \hat{N})\hat{D}$, is presented that yields a differential equation for the amplitudes representing density cumulants, as well as the grand potential. In contrast to other approaches reported in the literature, the theory does not use the interaction picture and an integral formulation as a starting point but rather requires a propagation of the resulting differential equation for the amplitudes in the Schrödinger representation. While the theory is applicable to general classes of many-body problems in principle, here, the theory is illustrated using simple model systems. As an application of the related time-dependent formulation numerically exact time-autocorrelation functions and absorption spectra are obtained for harmonic Franck–Condon problems. These examples illustrate the basic soundness of the scheme and are used for pedagogical purposes. Other approaches in the literature are only discussed briefly and no detailed comparative discussion is attempted.

GRAPHICAL ABSTRACT

Keywords:

• Thermal properties
• normal ordered exponential
• coupled cluster
• electronic absorption spectra

Acknowledgments

It is our pleasure to contribute to the festschrift for Prof. John F. Stanton. M. N. has had many enjoyable and fruitful interactions with JFS in the past and the authors think it is fitting to contribute a pedagogically oriented paper to commemorate the joyful occasion of John's 60th birthday. The authors would like to thank Neil Raymond for his help with the MCTDH calculation of the FC spectrum for formaldehyde.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research has been supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).