From d7dc2dd9e4649e953b289cd632c5d99c73e714ef Mon Sep 17 00:00:00 2001 From: bc118 Date: Sat, 30 Nov 2024 12:20:13 -0500 Subject: [PATCH] added some suggestions or clarifications that the editor suggested. --- paper/paper.bib | 26 +++++++++++++------------- paper/paper.md | 45 +++++++++++++++++++++++++++------------------ 2 files changed, 40 insertions(+), 31 deletions(-) diff --git a/paper/paper.bib b/paper/paper.bib index 1d5bb94..005199f 100644 --- a/paper/paper.bib +++ b/paper/paper.bib @@ -44,7 +44,7 @@ @article{Martin:1998 journal = {J. Phys. Chem. B}, number = {14}, pages = {2569--2577}, -title = {{Transferable potentials for phase equilibria. 1. United-atom description of n-alkanes}}, +title = {{Transferable potentials for phase equilibria. 1. United-atom description of \emph{n}-alkanes}}, url = {https://pubs.acs.org/sharingguidelines}, volume = {102}, year = {1998} @@ -174,7 +174,7 @@ @article{Good:1970 # Mixing rule arithmetic sigma @article{Lorentz:1881, author = {Lorentz, H. A.}, -title = {Ueber die Anwendung des Satzes vom Virial in der kinetischen Theorie der Gase}, +title = {{U}eber die {A}nwendung des {S}atzes vom {V}irial in der kinetischen {T}heorie der {G}ase}, journal = {Ann. d. Phys.}, volume = {12}, pages = {127--136}, @@ -213,7 +213,7 @@ @fidgit{forcefield-utilities:2022 # MoSDeF part 1 @article{Cummings:2021, author = {Cummings, P.T. and McCabe, C. and Iacovella, C.R. and Ledeczi, A. and Jankowski, E. and Jayaraman, A. and Palmer, J.C. and Maginn, E.J. and Glotzer, S.C. and Anderson, J.A. and Siepmann, J.I. and Potoff, J. and Matsumoto, R.A. and Gilmer, J.B. and DeFever, R.S. and Singh, R. and Crawford, B.}, -Title = {Open-Source Molecular Modeling Software in Chemical Engineering, with Focus on the Molecular Simulation Design Framework (MoSDeF)}, +Title = {Open-Source Molecular Modeling Software in Chemical Engineering, with Focus on the {M}olecular {S}imulation {D}esign {F}ramework ({M}o{SD}e{F})}, journal = {AICHE J.}, volume = {67(3)}, pages = {e17206}, @@ -232,7 +232,7 @@ @article{Summers:2020 pages = {1779--1793}, pmid = {32004433}, publisher = {American Chemical Society}, -title = {{MoSDeF, a Python Framework Enabling Large-Scale Computational Screening of Soft Matter: Application to Chemistry-Property Relationships in Lubricating Monolayer Films}}, +title = {{{M}o{SD}e{F}, a Python Framework Enabling Large-Scale Computational Screening of Soft Matter: Application to Chemistry-Property Relationships in Lubricating Monolayer Films}}, url = {https://pubs.acs.org/doi/full/10.1021/acs.jctc.9b01183}, volume = {16}, year = {2020} @@ -241,7 +241,7 @@ @article{Summers:2020 # MoSDeF-dihedral-fit GitHub @fidgit{Crawford:2023b, author = "Crawford, Brad and Quach, Co and Craven, Nicholas and Iacovella, Christopher R. and McCabe, Clare and Cummings, Peter T. and Potoff, Jeffrey", - title = "MoSDeF-dihedral-fit: A simple software package to fit dihedrals via the MoSDeF software.", + title = "{M}o{SD}e{F}-dihedral-fit: A simple software package to fit dihedrals via the {M}o{SD}e{F} software.", year = "2023", publisher = "Github", url = {https://github.com/GOMC-WSU/MoSDeF-dihedral-fit}, @@ -252,7 +252,7 @@ @fidgit{Crawford:2023b # MoSDeF-GOMC part 1 @article{Crawford:2023a, author = {Crawford, Brad and Timalsina, Umesh and Quach, Co D. and Craven, Nicholas C. and Gilmer, Justin B. and McCabe, Clare and Cummings, Peter T. and Potoff, Jeffrey J.}, -title = {MoSDeF-GOMC: Python Software for the Creation of Scientific Workflows for the Monte Carlo Simulation Engine GOMC}, +title = {{M}o{SD}e{F}-{GOMC}: Python Software for the Creation of Scientific Workflows for the {M}onte {C}arlo Simulation Engine {GOMC}}, journal = {Journal of Chemical Information and Modeling}, volume = {63}, number = {4}, @@ -266,7 +266,7 @@ @article{Crawford:2023a # MoSDeF-GOMC part 2 @fidgit{Crawford:2022, author = "Crawford, Brad and Timalsina, Umesh and Quach, Co D. and Craven, Nicholas and Gilmer, Justin and Cummings, Peter T. and Potoff, Jeffrey", - title = "MoSDeF-GOMC: Python software for the creation of scientific workflows for the Monte Carlo simulation engine GOMC", + title = "{M}o{SD}e{F}-{GOMC}: Python software for the creation of scientific workflows for the {M}onte {C}arlo simulation engine {GOMC}", year = "2022", publisher = "Github", url = "https://github.com/GOMC-WSU/MoSDeF-GOMC" @@ -281,7 +281,7 @@ @article{Nejahi:2019 keywords = {Adsorption,GPU,Gibbs ensemble,Molecular simulation,Monte Carlo,Phase equilibrium}, pages = {20--27}, publisher = {Elsevier B.V.}, -title = {{GOMC: GPU Optimized Monte Carlo for the simulation of phase equilibria and physical properties of complex fluids}}, +title = {{GOMC: GPU Optimized {M}onte {C}arlo for the simulation of phase equilibria and physical properties of complex fluids}}, url = {https://doi.org/10.1016/j.softx.2018.11.005}, volume = {9}, year = {2019} @@ -296,7 +296,7 @@ @article{Nejahi:2021 keywords = {Alchemical free energy,Crankshaft move,Cyclic molecules,Exp-6 potential,Molecular Exchange Monte Carlo,Multi-particle}, pages = {100627}, publisher = {Elsevier B.V.}, -title = {{Update 2.70 to “GOMC: GPU Optimized Monte Carlo for the simulation of phase equilibria and physical properties of complex fluids”}}, +title = {{Update 2.70 to “{GOMC}: {GPU} {O}ptimized {M}onte {C}arlo for the simulation of phase equilibria and physical properties of complex fluids”}}, volume = {13}, year = {2021} } @@ -304,7 +304,7 @@ @article{Nejahi:2021 # Exp6 paper @article{Errington:1999, author = {Errington, Jeffrey R. and Panagiotopoulos, Athanassios Z.}, -title = {A New Intermolecular Potential Model for the n-Alkane Homologous Series}, +title = {A New Intermolecular Potential Model for the \emph{n}-Alkane Homologous Series}, journal = {The Journal of Physical Chemistry B}, volume = {103}, number = {30}, @@ -331,7 +331,7 @@ @article{Potoff:2009 # Mie - Hemmen paper @article{Hemmen:2015, author = {Hemmen, Andrea and Gross, Joachim}, -title = {Transferable Anisotropic United-Atom Force Field Based on the Mie Potential for Phase Equilibrium Calculations: n-Alkanes and n-Olefins}, +title = {Transferable Anisotropic United-Atom Force Field Based on the Mie Potential for Phase Equilibrium Calculations: \emph{n}-Alkanes and \emph{n}-Olefins}, journal = {The Journal of Physical Chemistry B}, volume = {119}, number = {35}, @@ -527,7 +527,7 @@ @article{Vanommeslaeghe:2014 # cause issues in dihedral transferablity @article{Chen:2015, author = {Siyan Chen and Shasha Yi and Wenmei Gao and Chuncheng Zuo and Zhonghan Hu}, -title = {Force field development for organic molecules: Modifying dihedral and 1-n pair interaction parameters}, +title = {Force field development for organic molecules: Modifying dihedral and 1-\emph{n} pair interaction parameters}, journal = {J Comput Chem.}, volume = {36}, issue = {6}, @@ -554,7 +554,7 @@ @Inbook{Mielke:2019 publisher="Springer International Publishing", address="Cham", pages="39--47", -abstract="Scientific research involves the formulation of theory to explain observed phenomena and using experimentation to test and evolve these theories. Over the past two decades, computational modeling and simulation (M{\&}S) has become accepted as the third leg of scientific research because it provides additional insights that often are impractical or impossible to acquire using theoretical and experimental analysis alone. The purpose of this chapter is to explore how M{\&}S is used in system-level healthcare research and to present some practical guidelines for its use. Two modeling approaches commonly used in healthcare research, system dynamics models and agent-based models, are presented and their applications in healthcare research are described. The three simulation paradigms, Monte Carlo simulation, continuous simulation, and discrete event simulation, are defined and the conditions for their use are stated. An epidemiology case study is presented to illustrate the use of M{\&}S in the research process.", +abstract="Scientific research involves the formulation of theory to explain observed phenomena and using experimentation to test and evolve these theories. Over the past two decades, computational modeling and simulation (M{\&}S) has become accepted as the third leg of scientific research because it provides additional insights that often are impractical or impossible to acquire using theoretical and experimental analysis alone. The purpose of this chapter is to explore how M{\&}S is used in system-level healthcare research and to present some practical guidelines for its use. Two modeling approaches commonly used in healthcare research, system dynamics models and agent-based models, are presented and their applications in healthcare research are described. The three simulation paradigms, {M}onte {C}arlo simulation, continuous simulation, and discrete event simulation, are defined and the conditions for their use are stated. An epidemiology case study is presented to illustrate the use of M{\&}S in the research process.", isbn="978-3-030-26837-4", doi="10.1007/978-3-030-26837-4_6", url="https://doi.org/10.1007/978-3-030-26837-4_6" diff --git a/paper/paper.md b/paper/paper.md index 2068ff8..6583be7 100644 --- a/paper/paper.md +++ b/paper/paper.md @@ -40,36 +40,36 @@ authors: affiliation: 2 affiliations: - - name: Atomfold, PA, USA + - name: Atomfold, PA, United States of America index: 1 - - name: Department of Chemical Engineering and Materials Science, Wayne State University, Detroit, MI 48202-4050, USA + - name: Department of Chemical Engineering and Materials Science, Wayne State University, Detroit, MI 48202-4050, United States of America index: 2 - - name: Department of Chemical and Biomolecular Engineering, Vanderbilt University, Nashville, TN 37235-1604, USA + - name: Department of Chemical and Biomolecular Engineering, Vanderbilt University, Nashville, TN 37235-1604, United States of America index: 3 - - name: Multiscale Modeling and Simulation (MuMS) Center, Vanderbilt University, Nashville, TN 37212, USA + - name: Multiscale Modeling and Simulation (MuMS) Center, Vanderbilt University, Nashville, TN 37212, United States of America index: 4 - - name: Interdisciplinary Material Science Program, Vanderbilt University, Nashville, TN 37235-0106, USA + - name: Interdisciplinary Material Science Program, Vanderbilt University, Nashville, TN 37235-0106, United States of America index: 5 - - name: School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK + - name: School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom index: 6 -date: 10 October 2024 +date: 30 November 2024 bibliography: paper.bib --- # Summary -Molecular Mechanics (MM) simulations (e.g., molecular dynamics and Monte Carlo) provide a third method of scientific discovery, adding to the traditional theoretical and experimental scientific methods [@Mielke:2019; @Siegfried:2014]. Experimental methods measure the data under set conditions (e.g., temperature and pressure), whereas the traditional theoretical methods are based on analytical equations, and sometimes their constants are fitted to experimental data. The MM simulations are deterministic and stochastic, and their models, commonly known as "force fields", can be optimized to match experimental data, similar to analytical theory-based methods [@Allen:2017; @Frekel:2002; @Jorgensen:1996; @Martin:1998; @Weiner:1984; @Weiner:1986; @Potoff:2009; @Hemmen:2015; @Errington:1999]. In larger, more complex systems, the stochastic simulation's molecules can jump large energy barriers that deterministic simulations may not be able to overcome in a reasonable timeframe, even with modern computing capabilities [@Allen:2017; @Frekel:2002]. However, deterministic and stochastic systems that provide adequate sampling for calculating a given property can provide critical insights into the system's phase space, which are not obtainable via traditional theoretical and experimental methods. Additionally, molecular simulations provide critical insights from visualizations and by obtaining chemical or material properties that do not currently exist, are not easily attainable (e.g., too expensive or dangerous) by traditional theoretical and experimental methods [@Hollingsworth:2018; @Hirst:2014], or require hard-to-achieve conditions, such as very high pressures and temperatures [@Yu:2023; @Koneru:2022; @Swai:2020; @Kumar:2022; @Louie:2021]. However, the force field parameters are ideally determined from Quantum Mechanics (QM) simulations or other methods, including the vibrational spectrum and machine learning methods [@Kania:2021; @Friederich:2018; @Vermeyen:2023; @Mayne:2013; @Schmid:2011; @Vanommeslaeghe:2014]. The MM proper dihedrals (i.e., dihedrals) are challenging to obtain if they don't currently exist for the chosen force field, inaccurately scale-up in larger molecules, or misbehave with other moiety combinations, provided some were separately derived using small molecules [@Kania:2021; @Mayne:2013]. While the same QM simulations can fit the dihedrals in most force field types, these dihedrals aren't easily transferable between force fields due to the differing parameters and formulas, including the combining rules and 1-4 scaling factors. [@Huang:2013; @Vanommeslaeghe:2010; @Vanommeslaeghe:2014; @Chen:2015]. +Molecular Mechanics (MM) simulations (e.g., molecular dynamics and Monte Carlo) provide a third method of scientific discovery, adding to the traditional theoretical and experimental scientific methods [@Mielke:2019; @Siegfried:2014]. Experimental methods measure the data under set conditions (e.g., temperature and pressure), whereas the traditional theoretical methods are based on analytical equations, and sometimes their constants are fitted to experimental data. The MM simulations are deterministic and stochastic, and their models, commonly known as "force fields", can be optimized to match experimental data, similar to analytical theory-based methods [@Allen:2017; @Frekel:2002; @Jorgensen:1996; @Martin:1998; @Weiner:1984; @Weiner:1986; @Potoff:2009; @Hemmen:2015; @Errington:1999]. In larger, more complex systems, the stochastic simulation's molecules can jump large energy barriers that deterministic simulations may not be able to overcome in a reasonable timeframe, even with modern computing capabilities [@Allen:2017; @Frekel:2002]. However, deterministic and stochastic systems that provide adequate sampling for calculating a given property can provide critical insights into the system's phase space, which are not obtainable via traditional theoretical and experimental methods. Additionally, molecular simulations provide critical insights from visualizations and by obtaining chemical or material properties that do not currently exist, are not easily attainable (e.g., too expensive or dangerous) by traditional theoretical and experimental methods [@Hollingsworth:2018; @Hirst:2014], or require hard-to-achieve conditions, such as very high pressures and temperatures [@Yu:2023; @Koneru:2022; @Swai:2020; @Kumar:2022; @Louie:2021]. However, the force field parameters are ideally determined from Quantum Mechanics (QM) simulations or other methods, including the vibrational spectrum and machine learning methods [@Kania:2021; @Friederich:2018; @Vermeyen:2023; @Mayne:2013; @Schmid:2011; @Vanommeslaeghe:2014]. The MM proper dihedrals (i.e., dihedrals) are challenging to obtain if they do not currently exist for the chosen force field, inaccurately scale-up in larger molecules, or misbehave with other moiety combinations, provided some were separately derived using small molecules [@Kania:2021; @Mayne:2013]. While the same QM simulations can fit the dihedrals in most force field types, these dihedrals are not easily transferable between force fields due to the different parameters and formulas, including the combining rules and 1-4 scaling factors. [@Huang:2013; @Vanommeslaeghe:2010; @Vanommeslaeghe:2014; @Chen:2015]. -The `MoSDeF-Dihedral-Fit` [@Crawford:2023b] library lets users quickly calculate the MM dihedrals directly from the QM simulations for several force fields (OPLS, TraPPE, AMBER, Mie, and Exp6) [@Jorgensen:1996; @Martin:1998; @Weiner:1984; @Weiner:1986; @Potoff:2009; @Hemmen:2015; @Errington:1999]. The user simply has to generate or use an existing Molecular Simulation Design Framework (MoSDeF) force field XML file [@Cummings:2021; @Summers:2020; @GMSO:2019; @forcefield-utilities:2022], provide Gaussian 16 log or Gaussian-style QM simulation files that cover the dihedral rotation (typically, 0-360 degrees), and provide the molecular structure information in a mol2 format [@Gaussian16:2016]. The `MoSDeF-Dihedral-Fit` software uses the QM and MM data to produce the dihedral for the specific force field, fitting the constants for the OPLS dihedral form and then analytically converting them to the Ryckaert-Bellemans (RB)-torsions and the periodic dihedrals. The software outputs the calculated MM dihedral points, enabling users to fit unsupported dihedral forms, provided the force fields are supported by the MoSDeF, GPU Optimized Monte Carlo (GOMC), MoSDeF-GOMC [@Crawford:2023a; @Crawford:2022; @Crawford:2023b; @Nejahi:2019; @Nejahi:2021], and vmd-python [@vmd-python:2016] software (a derivative of the VMD software [@Humphrey:1996; @Stone:2001]). +The `MoSDeF-Dihedral-Fit` [@Crawford:2023b] library allows users to quickly calculate the MM dihedrals directly from the QM simulations for several force fields (OPLS, TraPPE, AMBER, Mie, and Exp6) [@Jorgensen:1996; @Martin:1998; @Weiner:1984; @Weiner:1986; @Potoff:2009; @Hemmen:2015; @Errington:1999]. The user simply has to generate or use an existing Molecular Simulation Design Framework (MoSDeF) force field `.xml` file [@Cummings:2021; @Summers:2020; @GMSO:2019; @forcefield-utilities:2022], provide Gaussian 16 `.log` or Gaussian-style QM simulation files that cover the dihedral rotation (typically between 0-360 degrees), and provide the molecular structure information in a `.mol2` format [@Gaussian16:2016]. The `MoSDeF-Dihedral-Fit` software uses the QM and MM data to produce the dihedral for the specific force field, fitting the constants for the OPLS dihedral form (equation \ref{eqn:oplseqn}) and then analytically converting them to the Ryckaert-Bellemans torsion (equation \ref{eqn:RBeqn}) and the periodic dihedral forms (equation \ref{eqn:periodiceqn}). This analytical conversion from the OPLS dihedral form requires setting the specified parameters in the Ryckaert-Bellemans torsion and periodic dihedral forms (see equations \ref{eqn:RBeqn} and \ref{eqn:periodiceqn}). The software outputs the calculated MM dihedral points, enabling users to fit unsupported dihedral forms, provided the force fields are supported by the MoSDeF, GPU Optimized Monte Carlo (GOMC), MoSDeF-GOMC [@Crawford:2023a; @Crawford:2022; @Crawford:2023b; @Nejahi:2019; @Nejahi:2021], and vmd-python [@vmd-python:2016] software (a derivative of the VMD software [@Humphrey:1996; @Stone:2001]). # Statement of need -While many of these Molecular Mechanics (MM) force field parameters can be transferred between force fields, such as bonds, angles, and improper dihedrals (impropers), the proper dihedrals (dihedrals) can not be easily transferred due to the different combining rules (arithmetic and geometric) and 1-4 scaling factors (i.e., between the 1st and 4th bonded atoms) that were used in the development of the original parameters [@Berthelot:1898; @Good:1970; @Lorentz:1881]. The accuracy of these dihedral parameters is critical in obtaining the correct molecular conformations and configurations, which are absolutely required for understanding and analyzing the system's microstructure and physical properties (e.g., free energies, viscosities, adsorption loading, diffusion constants, and many more). +While many of these MM force field parameters can be transferred between force fields, such as bonds, angles, and improper dihedrals (often referred to as "impropers"), the proper dihedrals (dihedrals) can not be easily transferred due to the different combining rules (arithmetic and geometric) and 1-4 scaling factors (i.e., between the 1st and 4th bonded atoms) that were used in the development of the original parameters [@Berthelot:1898; @Good:1970; @Lorentz:1881]. The accuracy of these dihedral parameters is critical in obtaining the correct molecular conformations and configurations, which are required for understanding and analyzing the system's microstructure and physical properties (e.g., free energies, viscosities, adsorption loading, diffusion constants, and many more). -Some integrated dihedral fitting software currently exists for AMBER [@Horton:2022] or CHARMM-style force fields [@Mayne:2013], and other software will fit the dihedral constants to the final MM and QM energies, which need to be calculated by other means [@Guvench:2008]. However, there is a need a simple, generalized software package that supports multiple potential functions, imports QM and MM files, automatically reads and organizes the QM data, calculates the MM energies, auto-corrects the dihedral fit to account for multiple instances of the dihedral, and automatically removes the unusable cosine power series combinations due to this symmetry. The `MoSDeF-dihedral-fit` software accomplishes all this and automatically accounts for any of the common combining rules and the 1-4 scaling factors specified via the MoSDeF XML (i.e., force field) files [@Cummings:2021; @Summers:2020; @GMSO:2019; @forcefield-utilities:2022]. By allowing the user to set any other dihedral in the molecule to zero, this software avoids forcing one dihedral fit to correct the inaccurate forces of another dihedral, resulting in a problematic or bad cosine series fit; thus, providing a more flexible and accurate fit by combining multiple dihedral conformational energies in a single dihedral, a strategy used in the original and modern OPLS force fields [Jorgensen:1996: @Chao:2021]. For example, a carboxylic acid with an alkyl tail has two dihedrals in the same rotation cycle; the C-C-C-O: (O: = oxygen without hydrogen) dihedral is set to zero while the C-C-O-H dihedral is fit [@Jorgensen:1996; @Chao:2021; @Ganesh:2004]. The `MoSDeF-dihedral-fit` [@Crawford:2023b] API fills the missing gap by providing a generalized and easy solution to fitting dihedrals in their commonly used forms and outputting the MM dihedral data points so users can fit other custom dihedral forms. +Some integrated dihedral fitting software currently exists for AMBER [@Horton:2022] or CHARMM-style force fields [@Mayne:2013], and other software will fit the dihedral constants to the final MM and QM energies, which need to be calculated by other means [@Guvench:2008]. However, there is a need for a simple, generalized software package that supports multiple potential functions, imports QM and MM files, automatically reads and organizes the QM data, calculates the MM energies, auto-corrects the dihedral fit to account for multiple instances of the dihedral, and automatically removes the unusable cosine power series combinations due to this symmetry. The `MoSDeF-dihedral-fit` software accomplishes all this and automatically accounts for any of the common combining rules and the 1-4 scaling factors specified via the MoSDeF `.xml` (i.e., force field) files [@Cummings:2021; @Summers:2020; @GMSO:2019; @forcefield-utilities:2022]. By allowing the user to set any other dihedral in the molecule to zero, this software avoids forcing one dihedral fit to correct the inaccurate forces of another dihedral, resulting in a problematic or bad cosine series fit; thus, providing a more flexible and accurate fit by combining multiple dihedral conformational energies in a single dihedral, a strategy used in the original and modern OPLS force fields [@Jorgensen:1996: @Chao:2021]. For example, a carboxylic acid with an alkyl tail has two dihedrals in the same rotation cycle; the C-C-C-O: (O: = oxygen without hydrogen) dihedral is set to zero while the C-C-O-H dihedral is fit [@Jorgensen:1996; @Chao:2021; @Ganesh:2004]. The `MoSDeF-dihedral-fit` [@Crawford:2023b] API fills the missing gap by providing a generalized and easy solution to fitting dihedrals in their commonly used forms and outputting the MM dihedral data points so users can fit other custom dihedral forms. # Acknowledgements @@ -78,25 +78,31 @@ This research was partially supported by the National Science Foundation (grants # Mathematics -Proper dihedral (dihedral) forms. +**Proper dihedral (dihedral) forms** OPLS-dihedral: -$$U_{OPLS} = \frac{k_0}{2}$$ +$$ U_{OPLS} = \frac{k_0}{2} $$ $$+ \frac{k_1}{2} * (1 + cos(\theta)) + \frac{k_2}{2} * (1-cos(2 * \theta))$$ -$$+ \frac{k_3}{2} * (1 + cos(3 * \theta)) + \frac{k_4}{2} *(1-cos(4 * \theta))$$ +\begin{equation} ++ \frac{k_3}{2} * (1 + cos(3 * \theta)) + \frac{k_4}{2} *(1-cos(4 * \theta)) +\label{eqn:oplseqn} +\end{equation} -Ryckaert-Bellemans (RB)-torsions: +Ryckaert-Bellemans torsions: -$$U_{RB} = C_0$$ +$$U_{Ryckaert-Bellemans} = C_0$$ $$+ C_1 * cos(\psi) + C_2 * cos(\psi)^2$$ $$+ C_3 * cos(\psi)^3 + C_4 * cos(\psi)^4$$ -$$\psi = \theta - 180^o$$ +\begin{equation} +\psi = \theta - 180^o +\label{eqn:RBeqn} +\end{equation} Periodic-dihedral: @@ -108,4 +114,7 @@ $$+ K_3 * (1 + cos(n_3*\theta - d_3)) + K_4 * (1 + cos(n_4*\theta) - d_4)$$ $$where: n_0 = 0 ; n_1 = 1 ; n_2 = 2 ; n_3 = 3 ; n_4 = 4 $$ -$$d_0 = 90^o ; d_1 = 180^o ; d_2 = 0^o ; d_3 = 180^o ; d_4 = 0^o $$ +\begin{equation} +d_0 = 90^o ; d_1 = 180^o ; d_2 = 0^o ; d_3 = 180^o ; d_4 = 0^o +\label{eqn:periodiceqn} +\end{equation} \ No newline at end of file