About OOPSE OOPSE News Features Download CVS Mailing Lists Documentation Authors Funding Results	DUFF Energy Functions The total potential energy function in DUFF is $$V = \sum^{N}_{I=1} V^{I}_{\text{Internal}} + \sum^{N-1}_{I=1} \sum_{J>I} V^{IJ}_{\text{Cross}},$$ (3.7) where $V^{I}_{\text{Internal}}$ is the internal potential of molecule $I$ : $$V^{I}_{\text{Internal}} = \sum_{\theta_{ijk} \in I} V_{\text{bend... ...le}} (\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) \biggr].$$ (3.8) Here $V_{\text{bend}}$ is the bend potential for all 1, 3 bonded pairs within the molecule $I$ , and $V_{\text{torsion}}$ is the torsion potential for all 1, 4 bonded pairs. The pairwise portions of the non-bonded interactions are excluded for atom pairs that are involved in the smae bond, bend, or torsion. All other atom pairs within a molecule are subject to the LJ pair potential. The bend potential of a molecule is represented by the following function: $$V_{\text{bend}}(\theta_{ijk}) = k_{\theta}( \theta_{ijk} - \theta_0 )^2,$$ (3.9) where $\theta_{ijk}$ is the angle defined by atoms $i$ , $j$ , and $k$ (see Fig. 3.1), $\theta_0$ is the equilibrium bond angle, and $k_{\theta}$ is the force constant which determines the strength of the harmonic bend. The parameters for $k_{\theta}$ and $\theta_0$ are borrowed from those in TraPPE.[18] The torsion potential and parameters are also borrowed from TraPPE. It is of the form: $$V_{\text{torsion}}(\phi) = c_1[1 + \cos \phi] + c_2[1 + \cos(2\phi)] + c_3[1 + \cos(3\phi)],$$ (3.10) where: $$\cos\phi = (\hat{\mathbf{r}}_{ij} \times \hat{\mathbf{r}}_{jk}) \cdot (\hat{\mathbf{r}}_{jk} \times \hat{\mathbf{r}}_{kl}).$$ (3.11) Here, $\hat{\mathbf{r}}_{\alpha\beta}$ are the set of unit bond vectors between atoms $i$ , $j$ , $k$ , and $l$ . For computational efficiency, the torsion potential has been recast after the method of CHARMM,[1] in which the angle series is converted to a power series of the form: $$V_{\text{torsion}}(\phi) = k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0,$$ (3.12) where: $$k_0 = c_1 + c_3,$$ $$k_1 = c_1 - 3c_3,$$ $$k_2 = 2 c_2,$$ $$k_3 = 4c_3.$$ By recasting the potential as a power series, repeated trigonometric evaluations are avoided during the calculation of the potential energy. The cross potential between molecules $I$ and $J$ , $V^{IJ}_{\text{Cross}}$ , is as follows: $$V^{IJ}_{\text{Cross}} = \sum_{i \in I} \sum_{j \in J} \biggl[ V_{... ...ky}} (\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) \biggr],$$ (3.13) where $V_{\text{LJ}}$ is the Lennard Jones potential, $V_{\text{dipole}}$ is the dipole dipole potential, and $V_{\text{sticky}}$ is the sticky potential defined by the SSD model (Sec. 3.2.2). Note that not all atom types include all interactions. The dipole-dipole potential has the following form: $$V_{\text{dipole}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, \bolds... ...l_comment_mark>25 (\boldsymbol{\hat{u}}_j \cdot \hat{\mathbf{r}}_{ij}) \biggr].$$ (3.14) Here $\mathbf{r}_{ij}$ is the vector starting at atom $i$ pointing towards $j$ , and $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ are the orientational degrees of freedom for atoms $i$ and $j$ respectively. The magnitude of the dipole moment of atom $i$ is $\vert\mu_i\vert$ , $\boldsymbol{\hat{u}}_i$ is the standard unit orientation vector of $\boldsymbol{\Omega}_i$ , and $\boldsymbol{\hat{r}}_{ij}$ is the unit vector pointing along $\mathbf{r}_{ij}$ ( $\boldsymbol{\hat{r}}_{ij}=\mathbf{r}_{ij}/\vert\mathbf{r}_{ij}\vert$ ).
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	Updated on January 16, 2006