Bounds Matrix

The bounds matrix encodes the geometric constraints between all atom pairs. It is the central data structure of distance geometry — all subsequent steps derive from it.

Matrix Layout

For $N$ atoms, the bounds matrix $B$ is an $N\times N$ matrix:

• Upper triangle ($i<j$): upper bounds $u_{ij}$
• Lower triangle ($j<i$): lower bounds $l_{ij}$ (stored as $B[j][i]$)
• Diagonal: zero

1-2 Bounds (Bonded Pairs)

For directly bonded atoms, the distance is the bond length computed from UFF (Universal Force Field) parameters:

$$r_{ij}=r_i+r_j+r_{\text{BO}}-r_{\text{EN}}$$

where:

Bond order correction:

$$r_{\text{BO}}=-0.1332(r_i+r_j)\ln (\text{BO})$$

Electronegativity correction:

$$r_{\text{EN}}=\frac{r_i{r}_j{(\sqrt{{\chi }_i}-\sqrt{{\chi }_j})}^2}{{\chi }_i{r}_i+{\chi }_j{r}_j}$$

These terms come from the UFF parameterization. $r_i$ is the covalent radius, ${\chi }_i$ is the GMP electronegativity, and BO is the bond order.

The bounds are very tight: $[r_{ij}-0.01,r_{ij}+0.01]$ Å.

Special Case: Conjugated Heterocycles

For conjugated bonds in 5-membered rings involving heteroatoms (like pyrrole, furan), an additional squish factor of 0.2 Å is applied to the lower bound to accommodate aromatic bond length variation:

$$l_{ij}=r_{ij}-0.01-0.2$$

UFF Parameters

Selected covalent radii and electronegativities:

Element	$r_i$ (Å)	${\chi }_i$
H	0.354	2.20
C (SP3)	0.757	2.55
C (SP2)	0.732	2.55
C (SP)	0.706	2.55
N (SP3)	0.700	3.04
O (SP3)	0.658	3.44
F	0.668	3.98
S	1.047	2.58
Cl	1.044	2.87
Br	1.214	2.74

1-3 Bounds (Angle Paths)

For atoms separated by two bonds (sharing a common neighbor), the distance is computed via the law of cosines:

$$d_{13}=\sqrt{d_1^2+d_2^2-2d_1{d}_2\cos \theta }$$

The equilibrium angle $\theta$ depends on hybridization:

Hybridization	Angle $\theta$
SP	180°
SP2	120°
SP3	109.47°

The tolerance for 1-3 bounds is ±0.04 Å, doubled for each "large SP2 atom" in the path (atomic number > 13 that is SP2 and in a ring).

When multiple paths connect the same atom pair, union semantics apply: the lower bound is the minimum lower bound and the upper bound is the maximum upper bound across all paths.

1-4 Bounds (Torsion Paths)

For atoms separated by three bonds, the distance depends on the torsion angle between the two outer atoms. We compute the extreme distances at cis ($\varphi =0°$) and trans ($\varphi =180°$) configurations.

The cis and trans distances are computed from planar geometry:

$$d_{\text{cis}}=\sqrt{(x_3-x_0{)}^2+(y_3-y_0{)}^2}$$

where:

• $x_0=r_{12}\cos \alpha$, $y_0=r_{12}\sin \alpha$
• $x_3=r_{23}-r_{34}\cos \beta$, $y_3=r_{34}\sin \beta$

For the trans case, $y_3=-r_{34}\sin \beta$.

Stereochemistry Constraints

Bond Type	Lower Bound	Upper Bound
E/Z stereo	forced cis or trans	same
SP2–SP2 (not in ring)	$d_{\text{cis}}$	$d_{\text{trans}}$
SP2–SP2 (in same ring)	$d_{\text{cis}}$	$d_{\text{cis}}$ (planar)
Amide / Ester	$d_{\text{cis}}$	$d_{\text{trans}}$ with bias
Default	min($d_{\text{cis}},d_{\text{trans}}$)	max($d_{\text{cis}},d_{\text{trans}}$)

VDW Bounds (Non-bonded)

For atom pairs with topological distance ≥ 4, the lower bound comes from van der Waals radii:

$$l_{ij}=f\cdot (r_{\text{vdw}}^i+r_{\text{vdw}}^j)$$

The scaling factor $f$ depends on topological distance:

Topological Distance	Factor $f$
4	0.70
5	0.85
≥6	1.00

The upper bound for non-bonded pairs defaults to ${10}^6$ Å (effectively unconstrained) until tightened by triangle smoothing.

Triangle Inequality Smoothing

After populating bounds from all sources, Floyd-Warshall tightens them:

for k in 0..N:
    for i in 0..N:
        for j in (i+1)..N:
            u[i][j] = min(u[i][j], u[i][k] + u[k][j])
            l[i][j] = max(l[i][j], l[i][k] - u[k][j], l[k][j] - u[i][k])
            if l[i][j] > u[i][j]:
                INFEASIBLE

This is $O(N^3)$ and typically tightens many VDW upper bounds from ${10}^6$ to reasonable values.

Fallback Strategy

If triangle smoothing detects infeasibility ($l_{ij}>u_{ij}$), sci-form tries a sequence of fallbacks:

1. Strict bounds + set15 (1-5 bounds enabled) — most constrained
2. No set15 — relax 1-5 constraints
3. Soft smoothing — allow $l_{ij}>u_{ij}$ with tolerance 0.05 Å

This ensures even strained molecules can produce initial coordinates, with the force field correcting violations later.