Conformational Energy Terms

The term

$$E_{BOND} = \sum_{bonds} k_b ( r - r_{0} )^{2}$$ (4.4)

describes the covalent bond energy where the sum is carried out over all covalent bonds in the molecular structure selected by the constraints interaction statement; $r$ is the actual bond length, $k_b$ is energy constants, and $r_0$ is equilibrium constants specified by parameter statements (Section 3.2.1).

The term

$$E_{ANGL} = \sum_{angles} ( k_{\theta} (\theta - \theta_{0} )^{2} + k_{ub} ( r_{13} - r_{ub} )^{2} )$$ (4.5)

describes the bond angle energy where the sum is carried out over all bond angles in the molecular structure selected by the constraints interaction statement; $k_\theta$ and $k_{ub}$ are energy constants and $\theta _0$ and $r_{ub}$ are equilibrium constants specified by parameter statements (Section 3.2.1). $\theta$ is the actual value of the angle, and $r_{13}$ is the distance between the first and the third atom defining the angle. The second term in Eq. 4.5 is the Urey-Bradley term, which is used by certain force fields (Burkert and Allinger, 1982). The default value for $k_{ub}$ is zero.

The angle between two planes, the first being defined through atoms i,j,k and the second through atoms j,k,l, is defined as a torsion angle where the atoms i,j,k,l are specified by the dihedral and improper statements (Section 3.1.1). The terms

$$\begin{eqnarray*} E_{DIHE} & = & \sum_{dihedrals} \sum_{i=1,m} \left\{ \begin{a... ...\delta_i)^{2}$ if $n_i=0$} \\ \end{array}\right. \nonumber \\ \end{eqnarray*}$$

describe the dihedral and improper energy terms. The sums are carried out over all dihedral or improper angles in the molecular structure selected by the constraints interaction statement; $\phi_i$ is the actual torsion angle, $k_{\phi_i}$ are energy constants, $n_i$ are periodicities, $m_i$ are multiplicities, and $\delta_i$ are phase shifts (Section 3.2.1). Note that the definition of dihedral and improper angles is identical. However, X-PLOR maintains two separate topology and parameter lists for dihedral and improper angles. Historically, improper angles are mostly used with $n=0$ to maintain planarity or chirality, whereas dihedral angles are used with $n>0$ to describe multi-minimum torsion potentials.

The specification of multiple dihedral or torsion angles with $m>1$ allows one to carry out a cosine expansion of a torsion potential for a particular instance involving four atoms or four atom types. Multiple dihedral angles and improper angles have to be indicated by using the MULTiple option both in the definition of the molecule's topology (Section 3.1.1) and in the specification of the corresponding parameter (Section 3.2.1). Internally, X-PLOR stores multiple dihedral or improper angles as multiple instances of the same combination of atoms or atom types.