(675d) Virial Coefficients Of Hydrogen and Nitrogen Including Quantum Effects Using Path Integral Monte Carlo

Accurate virial coefficients are important for predicting properties of light gases such as hydrogen, helium etc., especially at low temperatures where the quantum effects aren't negligible. In addition to predicting properties of gases, virial coefficients (particularly the second virial coefficient) can be utilized as a tool to check the accuracy of various potential models against experimental data. Hence, the accuracy of such coefficients are of vital importance to many scientists and fellow engineers. The Mayer-sampling Monte Carlo method [1] provides an efficient route to calculating accurate virial coefficients. To incorporate quantum effects of light gases in which the positions of the atoms/molecules fluctuate by a significant amount at low temperatures, one needs to use path integral techniques. Path Integral Monte Carlo (PIMC) methods involve discretizing the quantum fluctuations of the atoms/molecules into “beads” that form closed paths or rings. Neighboring beads act as if they are harmonically connected to each other and the strength of the harmonic spring is directly proportional to temperature. The interaction energy is then computed as the average of the potential evaluated between corresponding beads of different rings. At very low temperatures, a large number of beads are required to accurately represent the path of the atoms/molecules.

PIMC methods have been applied by us and others to study quantum virial coefficients of He [2 - 5] and H₂[6]. We now extend this approach to compute higher order virial coefficients using flexible potentials (if available) of small diatomic molecules such as H₂ and N₂. For diatomic molecules, the complexity in terms of the orientational sampling under the approximation of quantum rigid rotors involved in the PIMC methods can be greatly reduced by a new algorithm we introduce. We studied the probability distribution of placing each successive bead on the surface of a sphere as a function of the angle φ between the previous bead and the current bead position vectors. We noticed that for reasonably low φ, (φ  ̴< 15 – 20 degrees) we could analytically express the probability distribution functions for all the beads as one universal probability distribution function with a parameter k, that was different for each bead. The analytical nature of this universal probability distribution function is the key feature that makes the algorithm fast, as it can be computed on the fly. Also, since we are regrowing the entire ring from scratch for each Monte-Carlo move, the possibility of different starting points of the these rings is accounted for. The assumption of low φ is reasonable even at low temperatures where the harmonic springs become weak. We present results for the H₂ and N₂ systems demonstrating the importance of the quantum effects, and providing a comparison with experimentally-obtained virial coefficients, where available.

[1] – Jayant K. Singh, David A. Kofke., Phys. Rev. Lett. 92 (22), 220601 (2004),DOI:10.1103/PhysRevLett.92.220601

[2] – Giovanni Garberoglio, Allan H. Harvey, J. Res. Natl. Inst. Stand. Technol. 114, 249 (2009), DOI:10.6028/jres.114.018

[3] – Giovanni Garberoglio, Allan H. Harvey, J. Chem. Phys. 134, 134106 (2011), DOI:10.1063/1.3573564

[4] – Giovanni Garberoglio, Micheal R. Moldover, Allan H. Harvey, J. Res. Natl. Inst. Stand. Technol. 116, 729 (2011), DOI:10.6028/jres.116.016

[5] – Katherine R.S. Shaul, Andrew J. Schultz, David A. Kofke, J. Chem. Phys. 137, 184101 (2012), DOI:10.1063/1.4764857

[6] – Giovanni Garberoglio, Chem. Phys. Lett. 557, 26 (2013), DOI:10.1016/j.cplett.2012.11.090