The UFF implementation

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

$$\sum^{{N_B}}_{} {\frac{{1}}{{2}}} \left(\vphantom{ r-r_{IJ} }\right. \left.\vphantom{ r-r_{IJ} }\right)^{2}_{}$$ E _UFF = (5.1)

$$\sum^{{N_A}}_{} \left\{\vphantom{ \begin{array}{r@{~:~}l} \frac{K_{IJK}}{4} \left... ...heta + C^A_2 \cos(2\theta) \right) & \text{general case}\\ \end{array} }\right. \begin{array}{r@{~:~}l} \frac{K_{IJK}}{4} \left( 1 - \cos (2\thet... ...1 \cos \theta + C^A_2 \cos(2\theta) \right) & \text{general case}\\ \end{array}$$ +

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

$$\sum^{{N_I}}_{} \left(\vphantom{C^I_0 + C^I_1\cos \omega + C^I_2 \cos 2 \omega }\right. \left.\vphantom{C^I_0 + C^I_1\cos \omega + C^I_2 \cos 2 \omega }\right)$$ +

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

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

The Fourier coefficients C^A₀, C^A₁, C^A₂ of the general angle terms are evaluated as a function of the natural angle θ₀:

$${\frac{{1}}{{4 \sin^2 {\theta_0}}}}$$ C^A₂ (5.2)

C^A₁ = - 4⋅C^A₂cosθ₀ (5.3)

$$\left(\vphantom{ 2 \cos^2{\theta_0}+1 }\right. \left.\vphantom{ 2 \cos^2{\theta_0}+1 }\right)$$ C^A₀ (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.

θ, θ₀

angle and natural angle for three atoms I - J - K.

C^A₀, C^A₁, C^A₂

Fourier coefficients of the general angle terms.

φ, φ₀

torsion angle and natural torison angle of the atoms I - J - K - L.

V_φ

height of the torsion barrier.

n

periodicity of the torsion potential.

ω

inversion- or out-of-plane-angle at atom I.

V_ω

height of the inversion barrier.

C^I₀, C^I₁, C^I₂

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, 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 [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:

$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