Diagrammatic many-body methods for anharmonic molecular vibrational properties

Diagrammatic many-body methods for computing the energies and other properties of anharmonic vibrations have been developed based on the Dyson equation formalism for the single-particle vibrational Green's function and the many-body perturbation theory for the total zero-point energy. Unlike similar methods based on the vibrational self-consistent field (VSCF) approximation, these XVSCF and XVMP2 methods are guaranteed to be size-consistent at the formalism level, meaning that they are applicable not only to small molecules but also to larger systems including condensed phases. The XVSCF method, initially developed by Keçeli and Hirata, is extended to calculate anharmonic corrections to geometries as well as vibrational frequencies and energies, and rendered identical to the VSCF method in the thermodynamic limit despite orders of magnitude lower computational cost. When XVSCF is formulated in terms of the Dyson equation, it is additionally revealed to be an approximation to the self-consistent phonon (SCP) method which is commonly used in solid-state physics. Furthermore, the development of XVSCF in terms of Green's functions enables the formulation of the concept of Dyson coordinates and Dyson geometries, conceived as anharmonic generalizations of the normal coordinates and equilibrium geometries of the harmonic approximation, which represent a formally exact effectively harmonic treatment of molecular and crystal vibrations, similar to the concept of Dyson orbitals from the field of electronic structure theory. Many-body perturbation theory based on XVSCF is referred to as XVMP2 and is showed to be both more efficient and more powerful than standard VMP2 methods. XVMP2 inherits the computational efficiency and manifest size-consistency of XVSCF, and additionally, through the Dyson-equation formalism, it is able to directly compute vibrational fundamental, overtone, and combination frequencies directly even in the presence of anharmonic resonance. This makes XVMP2 a rare example of a perturbative method which can defeat strong correlation. The XVSCF and XVMP2 methods are formulated in both deterministic algorithms which rely on the computation of a large number of anharmonic force constants, and stochastic algorithms which require no stored representation of the PES. This is a significant advance because the computation and storage of the PES is a significant bottleneck in terms of accuracy and computational cost. The Monte Carlo XVSCF and Monte Carlo XVMP2 methods, as they are called, are uncommon among stochastic methods in that they can compute anharmonic frequencies directly, without noisy, small differences between large total vibrational energies and without sign problems that plague other forms of quantum Monte Carlo such as DMC