The UFF implementation

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

$$\mathrm{E}\,_{UFF} = \sum^{N_B} \frac{1}{2}\cdot K_{IJ} \cdot \left( r-r_{IJ} \right)^2$$ (3.1)

$$+ \sum^{N_A} \left\{ \begin{array}{r@{~:~}l} \frac{K_{IJK}}{4} \lef... ...theta + C^A_2 \cos(2\theta) \right) & \text{general case}\\ \end{array} \right.$$

$$+ \sum^{N_T}\frac{1}{2}\cdot V_{\phi} \cdot \left(1-\cos \left(n\phi_0\right) \cos(n\phi) \right)$$

$$+ \sum^{N_I} V_\omega \cdot \left(C^I_0 + C^I_1\cos \omega + C^I_2 \cos 2 \omega \right)$$

$$+ \sum^{N_{nb}} D_{IJ} \cdot \left( -2 \left(\frac{x_{IJ}}{x}\right)^6 + \left(\frac{x_{IJ}}{x}\right)^{12} \right)$$

$$+ \sum^{N_{nb}} \frac{q_I \cdot q_J}{\epsilon \cdot x}$$

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 $\theta_0$:

$$C^A_2 = \frac{1}{4 \sin^2 {\theta_0}}$$ (3.2)

$$C^A_1 = -4 \cdot C^A_2 \cos{\theta_0}$$ (3.3)

$$C^A_0 = C^A_2 \left( 2 \cos^2{\theta_0}+1 \right)$$ (3.4)

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 [35]. 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 [36,37,30,38]. Pulay's DIIS procedure is implemented for big molecule to accelarate the optimization [39,29].

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:

$coord
    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
$end