Diagrammatic theories of anharmonic molecular vibrations

Abstract

In this review, we summarise recent developments in our laboratory in the field of many-body quantum-mechanical calculations of the anharmonic vibrational structure of molecules. Our size-extensive vibrational self-consistent field (XVSCF) and size-extensive second-order many-body perturbation (XVMP2) methods are, unlike their parent methods (VSCF and VMP2), defined in diagrammatic formulations of the energies and Dyson self-energies, leading to manifestly size-consistent expressions for zero-point energies and anharmonic vibrational frequencies calculable with much greater efficiency. The effective one-mode potentials of XVSCF are quadratic and hence the Schrödinger equation for each mode can be solved analytically, unlike VSCF, where a basis-set expansion of wave functions on more complex one-mode potentials need to be performed; VSCF potentials and their minima (anharmonic geometry) are shown to reduce to the quadratic potentials and their minima (also given analytically) of XVSCF in the thermodynamic limit. By self-consistently solving the Dyson equation with frequency-dependent self-energies, XVMP2 has the ability to calculate anharmonic frequencies of fundamentals as well as combinations and overtones in the presence of strong anharmonic resonance without a multireference or quasi-degenerate formulation, which tends to be non-size-consistent. To eliminate the computational bottleneck of XVSCF and XVMP2, which is the high-rank force-constant evaluation, we have developed alternative algorithms in which the diagrammatic equations are recast as a small number of high-dimensional integrals and then evaluated stochastically using a Metropolis Monte Carlo (MC) method. These MC-XVSCF and MC-XVMP2 methods not only remove the need for force-constant evaluation or storage, but also take into account force constants of up to infinite order according to their importance. They are a new branch of quantum Monte Carlo which can calculate frequencies (excitation energies) directly without fixed-node errors.

Keywords:

• anharmonic vibrations
• diagrams
• Monte Carlo
• Dyson equation

Acknowledgements

We thank Dr Murat Keçeli and Dr Kiyoshi Yagi for their work on some of the developments summarised in this review, and Professor Samuel B. Trickey for introducing us to the SCP method. This work is supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences [award number DE-FG02-11ER16211]. S.H. is a Camille Dreyfus Teacher-Scholar and a Scialog Fellow of the Research Corporation for Science Advancement.

Notes

1 Note that although Equation (5(5) ) must generally be solved self-consistently due to the anharmonic frequency appearing on both the left- and right-hand sides of the equation, in this case, that is unnecessary because the first-order approximation to the Dyson self-energy does not actually depend on .

2 Note that this is not related to the self-consistent solution for required in the inverse Dyson equation for a single mode [Equation (5(5) )]; the frequency, , still does not appear in Equation (13(13) ). Instead, XVSCF() requires a self-consistent solution for the effective harmonic frequencies, , across all modes.

3 With the exception of small differences associated with restrictions on indices of summation which become negligible in the thermodynamic limit [23].