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The UFF implementation

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

UFF = $\displaystyle \sum^{{N_B}}_{}$$\displaystyle {\frac{{1}}{{2}}}$KIJ$\displaystyle \left(\vphantom{ r-r_{IJ} }\right.$r - rIJ$\displaystyle \left.\vphantom{ r-r_{IJ} }\right)^{2}_{}$ (5.1)
+ $\displaystyle \sum^{{N_A}}_{}$$\displaystyle \left\{\vphantom{ \begin{array}{r@{ : }l} \frac{K_{IJK}}{4} \left...
...heta + C^A_2 \cos(2\theta) \right) & \text{general case}\ \end{array} }\right.$$\displaystyle \begin{array}{r@{ : }l} \frac{K_{IJK}}{4} \left( 1 - \cos (2\thet...
...1 \cos \theta + C^A_2 \cos(2\theta) \right) & \text{general case}\ \end{array}$    
+ $\displaystyle \sum^{{N_T}}_{}$$\displaystyle {\frac{{1}}{{2}}}$Vφ$\displaystyle \left(\vphantom{1-\cos \left(n\phi_0\right) \cos(n\phi) }\right.$1 - cos$\displaystyle \left(\vphantom{n\phi_0}\right.$0$\displaystyle \left.\vphantom{n\phi_0}\right)$cos()$\displaystyle \left.\vphantom{1-\cos \left(n\phi_0\right) \cos(n\phi) }\right)$    
+ $\displaystyle \sum^{{N_I}}_{}$Vω$\displaystyle \left(\vphantom{C^I_0 + C^I_1\cos \omega + C^I_2 \cos 2 \omega }\right.$CI0 + CI1cosω + CI2cos 2ω$\displaystyle \left.\vphantom{C^I_0 + C^I_1\cos \omega + C^I_2 \cos 2 \omega }\right)$    
+ $\displaystyle \sum^{{N_{nb}}}_{}$DIJ$\displaystyle \left(\vphantom{ -2 \left(\frac{x_{IJ}}{x}\right)^6 + \left(\frac{x_{IJ}}{x}\right)^{12} }\right.$ -2$\displaystyle \left(\vphantom{\frac{x_{IJ}}{x}}\right.$$\displaystyle {\frac{{x_{IJ}}}{{x}}}$$\displaystyle \left.\vphantom{\frac{x_{IJ}}{x}}\right)^{6}_{}$ + $\displaystyle \left(\vphantom{\frac{x_{IJ}}{x}}\right.$$\displaystyle {\frac{{x_{IJ}}}{{x}}}$$\displaystyle \left.\vphantom{\frac{x_{IJ}}{x}}\right)^{{12}}_{}$$\displaystyle \left.\vphantom{ -2 \left(\frac{x_{IJ}}{x}\right)^6 + \left(\frac{x_{IJ}}{x}\right)^{12} }\right)$    
+ $\displaystyle \sum^{{N_{nb}}}_{}$$\displaystyle {\frac{{q_I \cdot q_J}}{{\epsilon \cdot x}}}$    

The Fourier coefficients CA0, CA1, CA2 of the general angle terms are evaluated as a function of the natural angle θ0:

CA2 = $\displaystyle {\frac{{1}}{{4 \sin^2 {\theta_0}}}}$ (5.2)
CA1 = - 4⋅CA2cosθ0 (5.3)
CA0 = CA2$\displaystyle \left(\vphantom{ 2 \cos^2{\theta_0}+1 }\right.$2 cos2θ0 + 1$\displaystyle \left.\vphantom{ 2 \cos^2{\theta_0}+1 }\right)$ (5.4)

The expressions in the engery term are:
NB, NA, NT, NI, Nnb
the numbers of the bond-, angle-, torsion-, inversion- and the non bonded-terms.
forceconstants of the bond- and angle-terms.
r, rIJ
bond distance and natural bond distance of the two atoms I and J.
θ, θ0
angle and natural angle for three atoms I - J - K.
CA0, CA1, CA2
Fourier coefficients of the general angle terms.
φ, φ0
torsion angle and natural torison angle of the atoms I - J - K - L.
height of the torsion barrier.
periodicity of the torsion potential.
inversion- or out-of-plane-angle at atom I.
height of the inversion barrier.
CI0, CI1, CI2
Fourier coefficients of the inversions terms.
x, xIJ
distance and natural distance of two non bonded atoms I and J.
depth of the Lennard-Jones potential.
qI, ε
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 sp3 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 [38]. 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 [39,40,33,41]. Pulay's DIIS procedure is implemented for big molecule to accelarate the optimization [42,32].

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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