A Stochastic Algorithm for the Isobaric-Isothermal Ensemble with Ewald Summations for all Long Range Forces

. Author manuscript; available in PMC: 2016 Dec 8.

Published in final edited form as: J Chem Theory Comput. 2015 Nov 25;11(12):5624–5637. doi: 10.1021/acs.jctc.5b00648

e^{L_{4} \frac{Δ t}{2}}

η \leftarrow η + \frac{P_{η}}{M_{η}} \frac{Δ t}{2}

e^{L_{5} \frac{Δ t}{2}}

ξ \leftarrow ξ + \frac{P_{ξ}}{M_{ξ}} \frac{Δ t}{2}

e^{L_{6} \frac{Δ t}{2}}

P_{ξ} \to P_{ξ} + (\frac{P_{V}^{2}}{M_{V}} - k_{B} T_{ext}) \frac{Δ t}{2}

e^{L_{7} \frac{Δ t}{2}}

{\vec{P}}_{μ i} \leftarrow {\vec{p}}_{μ i} exp (- \frac{P_{η}}{M_{η}} \frac{Δ t}{2})

e^{L_{8} \frac{Δ t}{2}}

{\vec{P}}_{μ i} \leftarrow {\vec{p}}_{μ i} + ({\vec{F}}_{μ i} - \frac{m_{μ i}}{M_{μ}} {\vec{F}}_{μ}) \frac{Δ t}{2}

e^{L_{9} \frac{Δ t}{2}}

{\vec{P}}_{μ} \leftarrow {\vec{P}}_{μ} exp [- \frac{Δ t}{2} (\frac{P_{η}}{M_{η}} + \frac{P_{V}}{3 V M_{V}})]

e^{L_{10} \frac{Δ t}{2}}

{\vec{P}}_{μ} \leftarrow {\vec{P}}_{μ} + \frac{Δ t}{2} {\vec{F}}_{μ}

{\vec{p}}_{μ}

← SHAKE COORDINATES

T^{*} \leftarrow \frac{1}{g k_{B}} [\sum_{i, μ} \frac{{\vec{p}}_{μ i}^{2}}{m_{μ i}} + \sum_{μ} \frac{{\vec{P}}_{μ}^{2}}{M_{μ}}]

e^{L_{11} \frac{Δ t}{2}}

P_{η} \leftarrow P_{η} + g k_{B} (T^{*} - T_{ext}) \frac{Δ t}{2}

e^{L_{12} \frac{Δ t}{2}}

{\vec{R}}_{μ} \leftarrow {\vec{R}}_{μ} exp (\frac{Δ t}{2} \frac{P_{V}}{3 V M_{V}})

e^ℒ₁₃Δ^t

{\vec{R}}_{μ} \leftarrow {\vec{R}}_{μ} + Δ t \frac{{\vec{P}}_{μ}}{M_{μ}}

e^ℒ₁₄Δ^t

{\vec{r}}_{μ i} \leftarrow {\vec{r}}_{μ i} + Δ t \frac{{\vec{p}}_{μ i}}{m_{μ i}}

{\vec{F}}_{μ i}

{\vec{F}}_{μ}

← FORCES CALCULATION

e^{L_{12} \frac{Δ t}{2}}

{\vec{R}}_{μ} \leftarrow {\vec{R}}_{μ} exp (\frac{Δ t}{2} \frac{P_{V}}{3 V M_{V}})

e^{L_{11} \frac{Δ t}{2}}

P_{η} \leftarrow P_{η} + g k_{B} (T^{*} - T_{ext}) \frac{Δ t}{2}

e^{L_{10} \frac{Δ t}{2}}

{\vec{P}}_{μ} \leftarrow {\vec{P}}_{μ} + \frac{Δ t}{2} {\vec{F}}_{μ}

e^{L_{9} \frac{Δ t}{2}}

{\vec{P}}_{μ} \leftarrow {\vec{P}}_{μ} exp [- \frac{Δ t}{2} (\frac{P_{η}}{M_{η}} + \frac{P_{V}}{3 V M_{V}})]

e^{L_{8} \frac{Δ t}{2}}

{\vec{p}}_{μ i} \leftarrow {\vec{p}}_{μ i} + ({\vec{F}}_{μ i} - \frac{m_{μ i}}{M_{μ}} {\vec{F}}_{μ}) \frac{Δ t}{2}

e^{L_{7} \frac{Δ t}{2}}

{\vec{p}}_{μ i} \leftarrow {\vec{p}}_{μ i} exp (- \frac{P_{η}}{M_{η}} \frac{Δ t}{2})

{\vec{p}}_{μ}

← SHAKE MOMENTA

e^{L_{6} \frac{Δ t}{2}}

P_{ξ} \leftarrow P_{ξ} + (\frac{P_{V}^{2}}{M_{V}} - k_{B} T_{ext}) \frac{Δ t}{2}

e^{L_{5} \frac{Δ t}{2}}

ξ \leftarrow ξ + \frac{P_{ξ}}{M_{ξ}} \frac{Δ t}{2}

e^{L_{4} \frac{Δ t}{2}}

η \leftarrow η + \frac{P_{η}}{M_{η}} \frac{Δ t}{2}

e^{L_{3} ν \frac{Δ t}{2}}

V \leftarrow V + \frac{P_{V}}{M_{V}} \frac{ν Δ t}{2}

Π^{*} \leftarrow \frac{1}{3 V} \sum_{μ} M_{μ} {\dot{\vec{R}}}_{μ}^{2} + \frac{1}{6 V} \sum_{μ} \sum_{μ^{'}} \sum_{i \in μ} \sum_{j \in μ^{'}} ({\vec{R}}_{μ} - {\vec{R}}_{μ^{'}}) \cdot ({\vec{f}}_{μ i} - {\vec{f}}_{μ^{'} j})

e^{δ L_{1} ν \frac{Δ t}{2}}

P_{V} \leftarrow e^{- γ \frac{ν Δ t}{2}} P_{V} + \frac{σ M_{V}^{1 / 2}}{\sqrt{2 γ}} \sqrt{1 - e^{- γ ν Δ t}} R (t)

e^{L_{2} ν \frac{Δ t}{2}}

P_{V} \leftarrow P_{V} + (Π^{*} - Π_{ext}) \frac{ν Δ t}{2}

e^{L_{1} ν \frac{Δ t}{2}}

P_{V} \leftarrow P_{V} exp (- \frac{P_{ξ}}{M_{ξ}} \frac{ν Δ t}{2})