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The UFF implementation

The uff implementation follows the paper by Rappé [7]. The energy expression in uff is as follows:

$\displaystyle \mathrm{E} _{UFF} =$ $\displaystyle \sum^{N_B} \frac{1}{2}\cdot K_{IJ} \cdot \left( r-r_{IJ} \right)^2$ (5.1)
$\displaystyle +$ $\displaystyle \sum^{N_A} \left\{ \begin{array}{r@{ : }l} \frac{K_{IJK}}{4} \lef...
...theta + C^A_2 \cos(2\theta) \right) & \text{general case}\ \end{array} \right.$    
$\displaystyle +$ $\displaystyle \sum^{N_T}\frac{1}{2}\cdot V_{\phi} \cdot \left(1-\cos \left(n\phi_0\right) \cos(n\phi) \right)$    
$\displaystyle +$ $\displaystyle \sum^{N_I} V_\omega \cdot \left(C^I_0 + C^I_1\cos \omega + C^I_2 \cos 2 \omega \right)$    
$\displaystyle +$ $\displaystyle \sum^{N_{nb}} D_{IJ} \cdot \left( -2 \left(\frac{x_{IJ}}{x}\right)^6 + \left(\frac{x_{IJ}}{x}\right)^{12} \right)$    
$\displaystyle +$ $\displaystyle \sum^{N_{nb}} \frac{q_I \cdot q_J}{\epsilon \cdot x}$    

The Fourier coefficients $ C^A_0,C^A_1,C^A_2$ of the general angle terms are evaluated as a function of the natural angle $ \theta_0$:

$\displaystyle C^A_2$ $\displaystyle = \frac{1}{4 \sin^2 {\theta_0}}$ (5.2)
$\displaystyle C^A_1$ $\displaystyle = -4 \cdot C^A_2 \cos{\theta_0}$ (5.3)
$\displaystyle C^A_0$ $\displaystyle = C^A_2 \left( 2 \cos^2{\theta_0}+1 \right)$ (5.4)

The expressions in the engery term are:
$ N_B, N_A, N_T, N_I, N_{nb}$
the numbers of the bond-, angle-, torsion-, inversion- and the non bonded-terms.
$ K_{IJ}, K_{IJK}$
forceconstants of the bond- and angle-terms.
$ r,r_{IJ}$
bond distance and natural bond distance of the two atoms $ I$ and $ J$.
$ \theta,\theta_0$
angle and natural angle for three atoms $ I-J-K$.
$ C^A_0,C^A_1,C^A_2$
Fourier coefficients of the general angle terms.
$ \phi,\phi_0$
torsion angle and natural torison angle of the atoms $ I-J-K-L$.
$ V_\phi$
height of the torsion barrier.
$ n$
periodicity of the torsion potential.
$ \omega$
inversion- or out-of-plane-angle at atom $ I$.
$ V_\omega$
height of the inversion barrier.
$ C^I_0,C^I_1,C^I_2$
Fourier coefficients of the inversions terms.
$ x,x_{IJ}$
distance and natural distance of two non bonded atoms $ I$ and $ J$.
$ D_{IJ}$
depth of the Lennard-Jones potential.
$ q_I,\epsilon$
partial charge of atoms $ I$ and dielectric constant.
One major difference in this implementation concerns the atom types. The atom types in Rappé's paper have an underscore "_". In the present implementation an sp$ ^3$ C atom has the name "C 3" instead of "C_3". Particularly the bond terms are described with the harmonic potential and the non-bonded van der Waals terms with the Lennard-Jones potential. The partial charges needed for electrostatic nonbond terms are calculated with the Charge Equilibration Modell (QEq) from Rappé [34]. There is no cutoff for the non-bonded terms.

The relaxation procedure distinguishes between molecules wih more than 90 atoms and molecules with less atoms. For small molecules it consists of a Newton step followed by a linesearch step. For big molecules a quasi-Newton relaxation is done. The BFGS update of the force-constant matric is done [35,36,29,37]. Pulay's DIIS procedure is implemented for big molecule to accelarate the optimization [38,28].

The coordinates for any single atom can be fixed by placing an 'f' in the third to eighth column of the chemical symbol/flag group. As an example, the following coordinates specify acetone with a fixed carbonyl group:

    2.02693271108611      2.03672551266230      0.00000000000000      c
    1.08247228252865     -0.68857387733323      0.00000000000000      c f
    2.53154870318830     -2.48171472134488      0.00000000000000      o     f
   -1.78063790034738     -1.04586399389434      0.00000000000000      c
   -2.64348282517094     -0.13141435997713      1.68855816889786      h
   -2.23779643042546     -3.09026673535431      0.00000000000000      h
   -2.64348282517094     -0.13141435997713     -1.68855816889786      h
    1.31008893646566      3.07002878668872      1.68840815751978      h
    1.31008893646566      3.07002878668872     -1.68840815751978      h
    4.12184425921830      2.06288409251899      0.00000000000000      h

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