  Despite being a staple of synthetic plastics and biomolecules, helical
polymers are scarcely studied with Gaussian-basis-set {\it ab initio}
electron-correlated methods on an equal footing with molecules. This article
introduces an {\it ab initio} second-order many-body Green's-function [MBGF(2)]
method with nondiagonal, frequency-dependent Dyson self-energy for infinite
helical polymers using screw-axis-symmetry-adapted Gaussian-spherical-harmonics
basis functions. Together with the Gaussian-basis-set density-functional theory
for energies, analytical atomic forces, translational-period force, and
helical-angle force, it can compute correlated energy, quasiparticle energy
bands, structures, and vibrational frequencies of an infinite helical polymer,
which smoothly converge at the corresponding oligomer results. These methods
can handle incommensurable structures, which have an infinite translational
period and are hard to characterize by any other method, just as efficiently as
commensurable structures. We apply these methods to polyethylene ($2/1$ helix),
polyacetylene (Peierls' system), and polytetrafluoroethylene ($13/6$ helix) to
establish the quantitative accuracy of MBGF(2)/cc-pVDZ in simulating their
(angle-resolved) ultraviolet photoelectron spectra, and of B3LYP/cc-pVDZ or
6-31G** in reproducing their structures, infrared and Raman band positions,
phonon dispersions, and (coherent and incoherent) inelastic neutron scattering
spectra. We then predict the same properties for infinitely catenated chains of
nitrogen or oxygen and discuss their possible metastable existence under
ambient conditions. They include planar zigzag polyazene (N$_2$)$_x$ (Peierls'
system), $11/3$-helical isotactic polyazane (NH)$_x$, $9/4$-helical isotactic
polyfluoroazane (NF)$_x$, and $7/2$-helical polyoxane (O)$_x$ as potential
high-energy-density materials.
