Appendix A: conservation of energy in CONDORR
Two point masses
Two “structureless” particles with masses m1 and m2 are located at coordinates r1 and r2. Define a particle separation vector s≡r1−r2 and a separation velocity \( {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{s}}} \equiv {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}_{1} - {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}_{2} \). The interaction force F1 on particle 1 is equal and opposite to the force F2 on particle 2, so the subscripts for the force vectors can be dropped if a force F≡F1=−F2 is defined. For a pair of structureless particles, the potential energy due to their interaction depends solely on the interaction distance, so F is parallel to s. To demonstrate conservation of energy, it is necessary and sufficient to show that dH/dT=0, where H=T+U is the classical Hamiltonian (T is the kinetic energy and U is the potential energy).
The kinetic energy of the system is
$$ T = {\sum\limits_{i = 1,2} {T_{i} } } = {\sum\limits_{i = 1,2} {\frac{{m_{i} {\mathbf{\dot{r}}}^{2}_{i} }} {2}} } $$
According to Newton’s second law, F=ma, where F is the force and a is the acceleration. The acceleration of each particle is
$$ {\mathbf{a}}_{i} = {\mathbf{\ifmmode\expandafter\ddot\else\expandafter\"\fi{r}}}_{i} = \frac{{\mathbf{F}}} {{m_{i} }} $$
The time-derivative of the kinetic energy is
$$ \begin{array}{*{20}l} {{\frac{{\operatorname{d} T}} {{\operatorname{d} t}}} \hfill} & { = \hfill} & {{{\sum\limits_{i = 1,2} {\frac{\operatorname{d} } {{\operatorname{d} t}}\frac{{m_{i} {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}^{2}_{i} }} {2}} } = {\sum\limits_{i = 1,2} {\frac{{m_{i} }} {2}\frac{{\operatorname{d} {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}^{2}_{i} }} {{\operatorname{d} {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}_{i} }}\frac{{\operatorname{d} {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}_{i} }} {{\operatorname{d} t}}} } = {\sum\limits_{i = 1,2} {\frac{{m_{i} }} {2}2{\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}_{i} \cdot \frac{{\operatorname{d} {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}_{i} }} {{\operatorname{d} t}}} } = {\sum\limits_{i = 1,2} {m_{i} {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}_{i} \frac{{{\mathbf{F}}_{i} }} {{m_{i} }}} }} \hfill} \\ {{} \hfill} & { = \hfill} & {{{\sum\limits_{i = 1,2} {{\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}_{i} \cdot {\mathbf{F}}_{i} } } = {\left( {{\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}_{1} - {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}_{2} } \right)} \cdot {\mathbf{F}} = {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{s}}} \cdot {\mathbf{F}}} \hfill} \\ \end{array} $$
The time-derivative of the potential energy is
$$ \frac{{\operatorname{d} U}} {{\operatorname{d} t}} = \frac{{\operatorname{d} U}} {{\operatorname{d} {\mathbf{s}}}}\frac{{\operatorname{d} {\mathbf{s}}}} {{\operatorname{d} t}} = \frac{{\operatorname{d} U}} {{\operatorname{d} {\mathbf{s}}}}{\mathbf{\dot{s}}} $$
Conservation of energy requires that
$$ \frac{{\operatorname{d} T}} {{\operatorname{d} t}} = - \frac{{\operatorname{d} U}} {{\operatorname{d} t}} $$
Then,
$$ \begin{aligned} {\mathbf{\dot{s}}} \cdot {\mathbf{F}} = - \frac{{\operatorname{d} U}} {{\operatorname{d} {\mathbf{s}}}} \cdot {\mathbf{\dot{s}}} & \\ {\mathbf{F}} = - \frac{{\operatorname{d} U}} {{\operatorname{d} {\mathbf{s}}}} = - \nabla _{s} U \equiv - {\left[ {{\mathbf{i}}\frac{{\partial U}} {{\partial s_{x} }} + {\mathbf{j}}\frac{{\partial U}} {{\partial s_{y} }} + {\mathbf{k}}\frac{{\partial U}} {{\partial s_{z} }}} \right]} & \\ \end{aligned} $$
where ∇s is the gradient with respect to s, and i, j, and k are unit vectors parallel to the x, y and z-axes. This simple, well established relationship is the basis for molecular dynamics algorithms that are limited to the interaction of “structureless” particles (i.e., point masses with no angular momentum).
The interaction of an atom within a rigid residue with an atom outside the residue
The residue is idealized as a sphere, with a mass m1, a moment of intertia I, and a center of mass r1 (see Fig. 7). Here, the external atom is treated as a point mass m2 at coordinates r2. The kinetic energy of the system, including that due to rotation of the residue is
$$ T = {\sum\limits_{i = 1,2} {T_{i} } } = \frac{{I\omega ^{2} }} {2} + {\sum\limits_{i = 1,2} {\frac{{m_{i} {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}^{2}_{i} }} {2}} } $$
The angular acceleration of the residue is \( \dot{\omega } = \Gamma /I \), where Γ is a torque. The time derivative of the kinetic energy of the system can be expressed in a form analogous to that given above for two point masses. That is, the angular velocity ω can be treated the same way that the linear velocity ṙ was treated in the above derivation.
$$ \frac{{\operatorname{d} T}} {{\operatorname{d} t}} = \frac{\operatorname{d} } {{\operatorname{d} t}}\frac{{I\omega ^{2} }} {2} + {\sum\limits_{i = 1,2} {\frac{\operatorname{d} } {{\operatorname{d} t}}\frac{{m_{i} {\mathbf{\dot{r}}}^{2}_{i} }} {2}} } = \omega \cdot \Gamma + {\left( {{\mathbf{\dot{r}}}_{1} - {\mathbf{\dot{r}}}_{2} } \right)} \cdot {\mathbf{F}} $$
It is important to note that this equation defines F as the force on the residue at its center of mass. (Conservation of linear momentum requires that the force on the external atom is -F.) Furthermore, the separation vector s≡r3−r2 is defined with reference to the two atoms rather than the residue’s center of mass. Accounting for both translation and rotation, the separation velocity of the atoms is
$$ {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{s}}} = {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}_{3} - {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}_{2} = {\left[ {{\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}_{1} + {\left( {\omega \times {\mathbf{R}}} \right)}} \right]} - {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}_{2} $$
where R is the vector from the center of mass of the residue to the interacting atom within the residue. The time derivative of the potential energy is thus
$$ \frac{{\operatorname{d} U}} {{\operatorname{d} t}} = \frac{{\operatorname{d} U}} {{\operatorname{d} {\mathbf{s}}}}\frac{{\operatorname{d} {\mathbf{s}}}} {{\operatorname{d} t}} = \frac{{\operatorname{d} U}} {{\operatorname{d} {\mathbf{s}}}} \cdot {\mathbf{\dot{s}}} = \frac{{\operatorname{d} U}} {{\operatorname{d} {\mathbf{s}}}} \cdot {\left[ {{\mathbf{\dot{r}}}_{1} - {\mathbf{\dot{r}}}_{2} + {\left( {\omega \times {\mathbf{R}}} \right)}} \right]} $$
Conservation of energy requires that
$$ \begin{aligned} \frac{{\operatorname{d} T}} {{\operatorname{d} t}} = - \frac{{\operatorname{d} U}} {{\operatorname{d} t}} & \\ \dot{\omega } \cdot \Gamma + {\left( {{\mathbf{\dot{r}}}_{1} - {\mathbf{\dot{r}}}_{2} } \right)} \cdot {\mathbf{F}} = - {\left[ {{\mathbf{\dot{r}}}_{1} - {\mathbf{\dot{r}}}_{2} + \omega \times {\mathbf{R}}} \right]} \cdot \frac{{\operatorname{d} U}} {{\operatorname{d} {\mathbf{s}}}} & \\ \dot{\omega } \cdot \Gamma + {\left( {{\mathbf{\dot{r}}}_{1} - {\mathbf{\dot{r}}}_{2} } \right)} \cdot {\mathbf{F}} = {\left[ {{\mathbf{\dot{r}}}_{1} - {\mathbf{\dot{r}}}_{2} + \omega \times {\mathbf{R}}} \right]} \cdot {\mathbf{F}} & \\ \omega \cdot \Gamma = \omega \times {\mathbf{R}} \cdot {\mathbf{F}} & \\ \omega \cdot \Gamma = \omega \cdot {\mathbf{R}} \times {\mathbf{F}} & \\ \Gamma = {\mathbf{R}} \times {\mathbf{F}} & \\ \end{aligned} $$
where a substitution was made based on the assertion (derived above) that \( - \frac{{{\text{d}}U}} {{{\text{d}}{\mathbf{s}}}} = {\mathbf{F}} \) and where the triple scalar product on the right hand side of the fourth equation was replaced with an equivalent expression.
These equations show that conservation of energy for the interaction of an external object with an atom within a rigid residue requires two applications of the interaction force F to the residue: (1) F must be applied to the residue at its center of mass, affecting its linear momentum; (2) a torque Γ=R×F must be applied to the residue, affecting its angular momentum. The opposite force -F must also be applied to the external object. It is not necessary to apply any torque to the external object if it is a point mass. However, if the external object is an atom within another rigid residue, the second residue must be treated in exactly the same way as the first, except that a force equal to −F must be used.
This recipe also conserves both linear and angular momentum. Conservation of linear momentum P is explicitly invoked in the derivation, as F≡F1=−F2. Accordingly,
$$ \frac{{\operatorname{d} {\mathbf{P}}}} {{\operatorname{d} t}} = \frac{\operatorname{d} } {{\operatorname{d} t}}{\left( {m_{1} {\mathbf{r}}_{1} + m_{2} {\mathbf{r}}_{2} } \right)} = m_{1} \frac{{{\mathbf{F}}_{1} }} {{m_{1} }} + m_{2} \frac{{{\mathbf{F}}_{2} }} {{m_{2} }} = {\mathbf{F}}_{1} + {\mathbf{F}}_{2} = 0 $$
The angular momentum L of the system depends on the rotation of the residue (Lrot) and the relative motion of the objects (Ltrans). That is,
$$ {\mathbf{L}} = {\mathbf{L}}_{{{\text{rot}}}} + {\mathbf{L}}_{{{\text{trans}}}} $$
Conservation of angular momentum requires that
$$ \frac{{\operatorname{d} {\mathbf{L}}}} {{\operatorname{d} t}} = \frac{{\operatorname{d} {\mathbf{L}}_{{{\text{rot}}}} }} {{\operatorname{d} t}} + \frac{{\operatorname{d} {\mathbf{L}}_{{{\text{trans}}}} }} {{\operatorname{d} t}} = 0 $$
By definition, the time derivative of Lrot is dLrot/dt=Γ. The time derivative of Ltrans can be defined relative to the center of mass c of the system, as illustrated in Fig. 8. The center of mass c is
$$ {\mathbf{c}} = \frac{{{\sum\limits_{\alpha = x,y,z} {{\mathbf{u}}_{\alpha } {\left( {m_{1} r_{{1,\alpha }} + m_{2} r_{{2,\alpha }} } \right)}} }}} {{m_{1} + m_{2} }} $$
where uα are unit vectors along the x, y, and z-axes. Define two vectors A1≡r1−c and A2≡r2−c, as shown in Fig. 8. Substituting the expression for c into these definitions, expanding, and recollecting terms yields the following expressions.
$$ {\mathbf{A}}_{1} = \frac{{m_{2} }} {{m_{1} + m_{2} }}{\left( {{\mathbf{r}}_{1} - {\mathbf{r}}_{2} } \right)} $$
$$ {\mathbf{A}}_{2} = \frac{{m_{1} }} {{m_{1} + m_{2} }}{\left( {{\mathbf{r}}_{2} - {\mathbf{r}}_{1} } \right)} $$
Then, Ltrans can be expressed
$$ \begin{array}{*{20}l} {{{\mathbf{L}}_{{{\text{trans}}}} } \hfill} & {{ = m_{1} {\left( {{\mathbf{A}}_{1} \times {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}_{1} } \right)} + m_{2} {\left( {{\mathbf{A}}_{2} \times {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}_{2} } \right)}} \hfill} \\ {{} \hfill} & {{ = \frac{{m_{1} m_{2} }} {{m_{1} + m_{2} }}{\left[ {{\left( {{\mathbf{r}}_{1} - {\mathbf{r}}_{2} } \right)} \times {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}_{1} + {\left( {{\mathbf{r}}_{2} - {\mathbf{r}}_{1} } \right)} \times {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}_{2} } \right]}} \hfill} \\ {{} \hfill} & {{ = \mu {\left[ {{\mathbf{A}}_{{\text{r}}} \times {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}_{2} - {\mathbf{A}}_{{\text{r}}} \times {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}_{1} } \right]}} \hfill} \\ {{} \hfill} & {{ = \mu {\mathbf{A}}_{{\text{r}}} \times {\left( {{\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}_{2} - {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{r}}}_{1} } \right)}} \hfill} \\ {{} \hfill} & {{ = \mu {\mathbf{A}}_{{\text{r}}} \times {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{A}}}_{{\text{r}}} } \hfill} \\ \end{array} $$
where a new vector Ar≡r2−r1 has been defined and the reduced mass μ≡m1m2/(m1+m2) have been substituted into the expressions. The time-derivative of Ltrans is then
$$ \begin{array}{*{20}l} {{\frac{\operatorname{d} } {{\operatorname{d} t}}{\mathbf{L}}_{{{\text{trans}}}} } \hfill} & {{ = \frac{\operatorname{d} } {{\operatorname{d} t}}\mu {\left[ {{\mathbf{A}}_{{\text{r}}} \times {\mathbf{\dot{A}}}_{{\text{r}}} } \right]}} \hfill} \\ {{} \hfill} & {{ = \mu {\left[ {{\left( {\frac{{\operatorname{d} {\mathbf{A}}_{{\text{r}}} }} {{\operatorname{d} t}} \times {\mathbf{\dot{A}}}_{{\text{r}}} } \right)} + {\left( {{\mathbf{A}}_{{\text{r}}} \times \frac{{\operatorname{d} {\mathbf{\dot{A}}}_{{\text{r}}} }} {{\operatorname{d} t}}} \right)}} \right]}} \hfill} \\ {{} \hfill} & {{ = \mu {\left[ {{\left( {{\mathbf{\dot{A}}}_{{\text{r}}} \times {\mathbf{\dot{A}}}_{{\text{r}}} } \right)} + {\left( {{\mathbf{A}}_{{\text{r}}} \times \frac{{\operatorname{d} {\mathbf{\dot{A}}}_{{\text{r}}} }} {{\operatorname{d} t}}} \right)}} \right]}} \hfill} \\ {{} \hfill} & {{ = \mu {\left[ {{\mathbf{A}}_{{\text{r}}} \times \frac{{\operatorname{d} {\mathbf{\dot{A}}}_{{\text{r}}} }} {{\operatorname{d} t}}} \right]}} \hfill} \\ {{} \hfill} & {{ = \mu {\left[ {{\mathbf{A}}_{{\text{r}}} \times \frac{\operatorname{d} } {{\operatorname{d} t}}{\left( {{\mathbf{\dot{r}}}_{2} - {\mathbf{\dot{r}}}_{1} } \right)}} \right]}} \hfill} \\ {{} \hfill} & {{ = \mu {\left[ {{\mathbf{A}}_{{\text{r}}} \times {\left( {\frac{{{\mathbf{F}}_{2} }} {{m_{2} }} - \frac{{{\mathbf{F}}_{1} }} {{m_{1} }}} \right)}} \right]}} \hfill} \\ {{} \hfill} & {{ = - \mu {\left[ {{\mathbf{A}}_{{\text{r}}} \times {\left( {\frac{{m_{1} {\mathbf{F}}}} {{m_{1} m_{2} }} + \frac{{m_{2} {\mathbf{F}}}} {{m_{1} m_{2} }}} \right)}} \right]}} \hfill} \\ {{} \hfill} & {{ = - {\mathbf{A}}_{{\text{r}}} \times {\mathbf{F}}} \hfill} \\ \end{array} $$
Here, the product rule was invoked, the cross product of vector \( {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{A}}}_{{\text{r}}} \) with itself was identified as zero, the vector \( {\mathbf{\ifmmode\expandafter\dot\else\expandafter\.\fi{A}}}_{{\text{r}}} \) was expanded, the acceleration of each object was expressed in terms of F=F1=–F2, and the reduced mass μ was cancelled. Appropriately, the expression for the time derivative of angular momentum is independent of the masses of the objects. Conservation of momentum requires that
$$ \frac{{\operatorname{d} {\mathbf{L}}}} {{\operatorname{d} t}} = \frac{{\operatorname{d} {\mathbf{L}}_{{{\text{trans}}}} }} {{\operatorname{d} t}} + \frac{{\operatorname{d} {\mathbf{L}}_{{{\text{rot}}}} }} {{\operatorname{d} t}} = 0 $$
Substituting and recalling that Γ=R×F,
$$ \begin{aligned} \frac{{\operatorname{d} {\mathbf{L}}_{{{\text{trans}}}} }} {{\operatorname{d} t}} = - \frac{{\operatorname{d} {\mathbf{L}}_{{{\text{rot}}}} }} {{\operatorname{d} t}} & \\ - {\mathbf{A}}_{{\text{r}}} \times {\mathbf{F}} = - \Gamma & \\ {\mathbf{A}}_{{\text{r}}} \times {\mathbf{F}} = {\mathbf{R}} \times {\mathbf{F}} & \\ \end{aligned} $$
The angular momentum L is conserved only if F is defined such that the last equation is true.
This can be shown geometrically with reference to Fig. 9. The force vector F is always parallel to the line (defined by s) passing through the two atoms. Another line parallel to the line defined by s is drawn through the center of mass of the spherical residue. The vectors F, R and Ar are all in the plane defined by these two lines, so the two cross products of interest are both normal to this plane and thereby parallel. The length of each of these cross products is equal to the distance d between the two lines. That is,
$$ \begin{aligned} & {\left| {{\mathbf{R}} \times {\mathbf{F}}} \right|} = {\left| {\mathbf{R}} \right|}{\left| {\mathbf{F}} \right|}\sin \theta _{1} = d \\ & {\left| {{\mathbf{A}}_{{\text{r}}} \times {\mathbf{F}}} \right|} = {\left| {{\mathbf{A}}_{{\text{r}}} } \right|}{\left| {\mathbf{F}} \right|}\sin \theta _{2} = d \\ \end{aligned} $$
As the two vectors of interest are parallel and have the same length, they are equal, demonstrating that angular momentum is conserved.
Evaluation of torsional potentials for the interaction of two residues
The torsional angle ϕ is specified by four atoms, located at coordinates a1, a2, a3, and a4 (see Fig. 10). Atoms 1, 2, and 3 are constituents of residue 1, whose center of mass is at r1, and atom 4 is a constituent of residue 2, whose center of mass is at r2. (Two copies of the third atom exist, as this atom is shared by the two residues. However, coordinates of the atom 3 copy associated with residue 1 are used for calculating torsional angles and potentials.) As illustrated in Fig. 10, it is possible to find a coordinate transformation that results in a new system in which atom 2 is at the origin, atom 3 is on the z’-axis, and atom 4 is in the x’,z’-plane. The coordinate system is chosen such that the z’-coordinate of atom 3 and the x’-coordinate of atom 4 are both positive. Unit vectors parallel to the primed axes can be calculated as follows. The vector A3=a3−a2 is parallel to the z’-axis, so the unit vector \( {\mathbf{u}}_{z} = {\mathbf{A}}_{3} /{\left| {{\mathbf{A}}_{3} } \right|} \) defines the z’-axis. The vector A4=a4−a3 is in x’,z’-plane, so the vector A
y
=u
z
×A4 is parallel to the y’-axis and the unit vector \( {\mathbf{u}}_{y} = {\mathbf{A}}_{y} /{\left| {{\mathbf{A}}_{y} } \right|} \) defines the y’-axis. The unit vector u
x
=u
y
×u
z
defines the x’-axis. The vector Ap is defined as the projection of A4 onto the x’-axis. Vectors Ap and Ay have the same length l, which is
$$ \begin{array}{*{20}l} {{\ell } \hfill} & {{ = {\left| {{\mathbf{A}}_{p} } \right|} = {\mathbf{u}}_{x} \cdot {\mathbf{A}}_{4} = {\left| {{\mathbf{u}}_{x} } \right|}{\left| {{\mathbf{A}}_{4} } \right|}\cos {\left( \theta \right)}} \hfill} \\ {{} \hfill} & {{ = {\left| {{\mathbf{A}}_{y} } \right|} = {\left| {{\mathbf{u}}_{z} \times {\mathbf{A}}_{4} } \right|} = {\left| {{\mathbf{u}}_{z} } \right|}{\left| {{\mathbf{A}}_{4} } \right|}\sin {\left( {\frac{\pi } {2} - \theta } \right)}} \hfill} \\ {{} \hfill} & {{ = {\left| {{\mathbf{u}}_{z} } \right|}{\left| {{\mathbf{A}}_{4} } \right|}\cos {\left( \theta \right)}} \hfill} \\ \end{array} $$
where θ is the angle between A4 and the x’-axis.
Define a fictitious atom f with coordinates a
f
=a4. A fictitious force F parallel to the y’-axis acting on atom f (rigidly attached to residue 1) and the equal and opposite force −F acting on atom 4 (rigidly attached to residue 2) for a time dt will cause atom f and atom 4 (initially coincident) to become separated by a distance ds, changing the dihedral angle ϕ by an increment \( {\text{d}}\phi = \frac{{{\text{d}}s}} {l} \). The potential energy gradient dU/ds associated with the fictitious force can be expressed in terms of the torsional potential gradient dU/dφ:
$$ \frac{{\operatorname{d} U}} {{\operatorname{d} s}} = \frac{{\operatorname{d} U}} {{\operatorname{d} \phi }}\frac{{\operatorname{d} \phi }} {{\operatorname{d} s}} = \frac{{\operatorname{d} U}} {{\operatorname{d} \phi }}\frac{1} {l} $$
The force F is parallel to u
y
, and has a magnitude of −dU/ds. That is,
$$ {\mathbf{F}} = - \frac{{\operatorname{d} U}} {{\operatorname{d} s}}{\mathbf{u}}_{y} = - \frac{{\operatorname{d} U}} {{\operatorname{d} \phi }}\frac{1} {l}{\mathbf{u}}_{y} = - \frac{{\operatorname{d} U}} {{\operatorname{d} \phi }}\frac{1} {{{\left| {{\mathbf{A}}_{y} } \right|}}}\frac{{{\mathbf{A}}_{y} }} {{{\left| {{\mathbf{A}}_{y} } \right|}}} = - \frac{{\operatorname{d} U}} {{\operatorname{d} \phi }}\frac{{{\mathbf{A}}_{y} }} {{{\mathbf{A}}_{y} \cdot {\mathbf{A}}_{y} }} $$
where A
y
=u
z
×A4, as defined above.
Fictitious forces calculated in this way can be treated as interaction forces (described above) involving pairs of atoms on different residues. (Fictitious atom f is associated with residue 1 and atom 4 is associated with residue 2.) Doing so preserves the total energy, linear momentum, and angular momentum of the system.
Atomic interaction potentials
Atomic interaction potentials have the form:
$$ V = Q\varepsilon {\left[ {{\left( {\frac{\sigma } {r}} \right)}^{m} - {\left( {\frac{\sigma } {r}} \right)}^{n} } \right]} $$
where Q, σ, m, and n are constants, −ε is the potential at its minimum value, and r is the distance between atoms. Define a distance rmin where the potential is minimized. That is,V=Vmin=−ε when r=rmin. Then,
$$ \begin{array}{*{20}l} {{\frac{{\operatorname{d} V}} {{\operatorname{d} r}}} \hfill} & {{ = Q\varepsilon \frac{\operatorname{d} } {{\operatorname{d} r}}{\left[ {{\left( {\frac{\sigma } {r}} \right)}^{m} - {\left( {\frac{\sigma } {r}} \right)}^{n} } \right]}} \hfill} \\ {{} \hfill} & {{ = Q\varepsilon {\left[ { - m\sigma ^{m} {\left( {\frac{1} {r}} \right)}^{{m + 1}} + n\sigma ^{n} {\left( {\frac{1} {r}} \right)}^{{n + 1}} } \right]}} \hfill} \\ {{} \hfill} & {{ = \frac{{Q\varepsilon }} {r}{\left[ { - m{\left( {\frac{\sigma } {r}} \right)}^{m} + n{\left( {\frac{\sigma } {r}} \right)}^{n} } \right]}} \hfill} \\ \end{array} $$
Define the ratio ρ=m/n and note that the derivative is zero at the minimum,
$$ \begin{array}{*{20}l} {{\frac{{\operatorname{d} V_{{\min }} }} {{\operatorname{d} r}}} \hfill} & {{ = \frac{{Q\varepsilon }} {{r_{{\min }} }}{\left[ { - \rho n{\left( {\frac{\sigma } {{r_{{\min }} }}} \right)}^{{\rho n}} + n{\left( {\frac{\sigma } {{r_{{\min }} }}} \right)}^{n} } \right]} = 0} \hfill} \\ {{\rho n{\left( {\frac{\sigma } {{r_{{\min }} }}} \right)}^{{\rho n}} } \hfill} & {{ = n{\left( {\frac{\sigma } {{r_{{\min }} }}} \right)}^{n} } \hfill} \\ {{{\left( {\frac{\sigma } {{r_{{\min }} }}} \right)}^{{(\rho - 1)n}} } \hfill} & {{ = \frac{1} {\rho }} \hfill} \\ {{\sigma ^{{(\rho - 1)n}} } \hfill} & {{ = r^{{(\rho - 1)n}}_{{\min }} \rho ^{{ - 1}} } \hfill} \\ {\sigma \hfill} & {{ = r_{{\min }} \rho ^{{\frac{{ - 1}} {{(\rho - 1)n}}}} } \hfill} \\ \end{array} $$
Then,
$$ \begin{array}{*{20}l} {V \hfill} & {{ = Q\varepsilon {\left[ {{\left( {\frac{\sigma } {r}} \right)}^{{\rho n}} - {\left( {\frac{\sigma } {r}} \right)}^{n} } \right]}} \hfill} \\ {{} \hfill} & {{ = Q\varepsilon {\left[ {{\left( {\rho ^{{\frac{{ - 1}} {{(\rho - 1)n}}}} \frac{{r_{{\min }} }} {r}} \right)}^{{\rho n}} - {\left( {\rho ^{{\frac{{ - 1}} {{(\rho - 1)n}}}} \frac{{r_{{\min }} }} {r}} \right)}^{n} } \right]}} \hfill} \\ {{} \hfill} & {{ = Q\varepsilon {\left[ {{\left( {\rho ^{{\frac{{ - \rho n}} {{(\rho - 1)n}}}} } \right)}{\left( {\frac{{r_{{\min }} }} {r}} \right)}^{{\rho n}} - {\left( {\rho ^{{\frac{{ - n}} {{(\rho - 1)n}}}} } \right)}{\left( {\frac{{r_{{\min }} }} {r}} \right)}^{n} } \right]}} \hfill} \\ {{} \hfill} & {{ = Q\varepsilon {\left[ {{\left( {\rho ^{{\frac{{ - \rho }} {{\rho - 1}}}} } \right)}{\left( {\frac{{r_{{\min }} }} {r}} \right)}^{{\rho n}} - {\left( {\rho ^{{\frac{{ - 1}} {{\rho - 1}}}} } \right)}{\left( {\frac{{r_{{\min }} }} {r}} \right)}^{n} } \right]}} \hfill} \\ \end{array} $$
Define ε such that when r=rmin , then V=−ε, so
$$ \begin{aligned} & - \varepsilon = Q\varepsilon {\left[ {{\left( {\rho ^{{\frac{{ - \rho }} {{\rho - 1}}}} } \right)}{\left( {\frac{{r_{{\min }} }} {{r_{{\min }} }}} \right)}^{{\rho n}} - {\left( {\rho ^{{\frac{{ - 1}} {{\rho - 1}}}} } \right)}{\left( {\frac{{r_{{\min }} }} {{r_{{\min }} }}} \right)}^{n} } \right]} \\ & \frac{{ - 1}} {Q} = {\left[ {{\left( {\rho ^{{\frac{{ - \rho }} {{\rho - 1}}}} } \right)} - {\left( {\rho ^{{\frac{{ - 1}} {{\rho - 1}}}} } \right)}} \right]} \\ & Q = - {\left[ {{\left( {\rho ^{{\frac{{ - \rho }} {{\rho - 1}}}} } \right)} - {\left( {\rho ^{{\frac{{ - 1}} {{\rho - 1}}}} } \right)}} \right]}^{{ - 1}} \\ \end{aligned} $$
The atomic interaction energies can be calculated by substituting this value of Q into the general potential expression derived above:
$$ \begin{array}{*{20}l} {V \hfill} & {{ = Q\varepsilon {\left[ {{\left( {\rho ^{{\frac{{ - \rho }} {{\rho - 1}}}} } \right)}{\left( {\frac{{r_{{\min }} }} {r}} \right)}^{{\rho n}} - {\left( {\rho ^{{\frac{{ - 1}} {{\rho - 1}}}} } \right)}{\left( {\frac{{r_{{\min }} }} {r}} \right)}^{n} } \right]}} \hfill} \\ {{} \hfill} & {{ = \varepsilon {\left[ {{\left( {\rho ^{{\frac{{ - \rho }} {{\rho - 1}}}} } \right)} - {\left( {\rho ^{{\frac{{ - 1}} {{\rho - 1}}}} } \right)}} \right]}^{{ - 1}} {\left[ {{\left( {\rho ^{{\frac{{ - \rho }} {{\rho - 1}}}} } \right)}{\left( {\frac{{r_{{\min }} }} {r}} \right)}^{{\rho n}} - {\left( {\rho ^{{\frac{{ - 1}} {{\rho - 1}}}} } \right)}{\left( {\frac{{r_{{\min }} }} {r}} \right)}^{n} } \right]}} \hfill} \\ \end{array} $$
For example, for the familiar Lennard-Jones 6–12 potential, m=12, n=6, ρ=2, and Q=4. That is,
$$ \begin{array}{*{20}l} {{V_{{6 - 12}} } \hfill} & {{ = 4\varepsilon {\left[ {{\left( {2^{{\frac{{ - 2}} {{2 - 1}}}} } \right)}{\left( {\frac{{r_{{\min }} }} {r}} \right)}^{{12}} - {\left( {2^{{\frac{{ - 1}} {{2 - 1}}}} } \right)}{\left( {\frac{{r_{{\min }} }} {r}} \right)}^{6} } \right]}} \hfill} \\ {{} \hfill} & {{ = 4\varepsilon {\left[ {\frac{1} {4}{\left( {\frac{{r_{{\min }} }} {r}} \right)}^{{12}} - \frac{1} {2}{\left( {\frac{{r_{{\min }} }} {r}} \right)}^{6} } \right]}} \hfill} \\ {{} \hfill} & {{ = \varepsilon {\left[ {{\left( {\frac{{r_{{\min }} }} {r}} \right)}^{{12}} - 2{\left( {\frac{{r_{{\min }} }} {r}} \right)}^{6} } \right]}} \hfill} \\ \end{array} $$
The force F due to a potential of this general type is
$$ \begin{array}{*{20}l} {F \hfill} & {{ = \frac{{ - \operatorname{d} V}} {{\operatorname{d} r}}} \hfill} \\ {{} \hfill} & {{ = - Q\varepsilon \frac{\operatorname{d} } {{\operatorname{d} r}}{\left[ {{\left( {\rho ^{{\frac{{ - \rho }} {{\rho - 1}}}} } \right)}{\left( {\frac{{r_{{\min }} }} {r}} \right)}^{{\rho n}} - {\left( {\rho ^{{\frac{{ - 1}} {{\rho - 1}}}} } \right)}{\left( {\frac{{r_{{\min }} }} {r}} \right)}^{n} } \right]}} \hfill} \\ {{} \hfill} & {{ = - Q\varepsilon {\left( {\rho ^{{\frac{{ - \rho }} {{\rho - 1}}}} } \right)}r^{{\rho n}}_{{\min }} \frac{{\operatorname{d} r^{{ - \rho n}} }} {{\operatorname{d} r}} + Q\varepsilon {\left( {\rho ^{{\frac{{ - 1}} {{\rho - 1}}}} } \right)}r^{n}_{{\min }} \frac{{\operatorname{d} r^{{ - n}} }} {{\operatorname{d} r}}} \hfill} \\ {{} \hfill} & {{ = - Q\varepsilon {\left( {\rho ^{{\frac{{ - \rho }} {{\rho - 1}}}} } \right)}r^{{\rho n}}_{{\min }} {\left( { - \rho nr^{{ - \rho n - 1}} } \right)} + Q\varepsilon {\left( {\rho ^{{\frac{{ - 1}} {{\rho - 1}}}} } \right)}r^{n}_{{\min }} {\left( { - nr^{{ - n - 1}} } \right)}} \hfill} \\ {{} \hfill} & {{ = - Q\varepsilon {\left( {\rho ^{{\frac{{ - \rho }} {{\rho - 1}}}} } \right)}r^{m}_{{\min }} {\left( { - mr^{{ - m - 1}} } \right)} + Q\varepsilon {\left( {\rho ^{{\frac{{ - 1}} {{\rho - 1}}}} } \right)}r^{n}_{{\min }} {\left( { - nr^{{ - n - 1}} } \right)}} \hfill} \\ {{} \hfill} & {{ = Q\varepsilon m{\left( {\rho ^{{\frac{{ - \rho }} {{\rho - 1}}}} } \right)}r^{m}_{{\min }} r^{{ - m - 1}} - Q\varepsilon n{\left( {\rho ^{{\frac{{ - 1}} {{\rho - 1}}}} } \right)}r^{n}_{{\min }} r^{{ - n - 1}} } \hfill} \\ \end{array} $$
Defining the constants
$$ \begin{aligned} & A_{F} = Q\varepsilon m{\left( {\rho ^{{\frac{{ - \rho }} {{\rho - 1}}}} } \right)}r^{m}_{{\min }} \\ & B_{F} = Q\varepsilon n{\left( {\rho ^{{\frac{{ - 1}} {{\rho - 1}}}} } \right)}r^{n}_{{\min }} \\ \end{aligned} $$
makes it possible to write a simple expression for the force F:
$$ F = A_{F} r^{{ - m - 1}} - B_{F} r^{{ - n - 1}} $$
Interpolation of potential surfaces and surface gradients
The energy surface \( {\left( {U_{{\Omega _{1} ,\Omega _{2} }} } \right)} \) is approximated as a function of (Ω1, Ω2) at discrete points (ω
i
,ω
j
), arranged in a square pattern, allowing the coordinates at each vertex to be specified using only one index (i or j). Define an inverse distance η between the vertices,
$$ \eta \equiv \frac{1} {{\Delta \omega }} = \frac{1} {{\omega _{{i + 1}} - \omega _{i} }} = \frac{1} {{\omega _{{j + 1}} - \omega _{j} }} $$
Define “edge gradients” σa,i,j along the lines connecting adjacent vertices:
$$ \sigma _{{1,i,j}} \equiv {\left( {\frac{{\Delta U}} {{\Delta \omega _{i} }}} \right)}_{{i,j}} \equiv \frac{{U_{{i + 1,j}} - U_{{i,j}} }} {{\omega _{{i + 1}} - \omega _{i} }} = {\left( {U_{{i + 1,j}} - U_{{i,j}} } \right)}\eta $$
$$ \sigma _{{2,i,j}} \equiv {\left( {\frac{{\Delta U}} {{\Delta \omega _{j} }}} \right)}_{{i,j}} \equiv \frac{{U_{{i,j + 1}} - U_{{i,j}} }} {{\omega _{{j + 1}} - \omega _{j} }} = {\left( {U_{{i,j + 1}} - U_{{i,j}} } \right)}\eta $$
For a specific point at coordinates (Ω1,Ω2) within the square area bounded by points (ω
i
,ω
j
) and (ωi+1,ωj+1), the gradient component \( {\left( {\partial U/\partial \Omega _{1} } \right)} \) can be approximated by calculating the values \( U^{{rel}}_{{\omega _{i} ,\Omega _{2} }} \) and \( U^{{rel}}_{{\omega _{{i + 1}} ,\Omega _{2} }} \) at coordinates (ω
i
,Ω2) and (ωi+1,Ω2) relative to the value Ui,j and making a linear extrapolation of U between these points. That is,
$$ \begin{array}{*{20}l} {{\frac{{\partial U}} {{\partial \Omega _{1} }}} \hfill} & {{ \approx \frac{{U^{{rel}}_{{\omega _{{i + 1}} ,\Omega _{2} }} - U^{{rel}}_{{\omega _{i} ,\Omega _{2} }} }} {{\Delta \omega _{i} }}} \hfill} \\ {{} \hfill} & {{ = \frac{{\sigma _{{1,i,j}} \cdot \Delta \omega _{i} + (\Omega _{2} - \omega _{j} )\sigma _{{2,i + 1,j}} - (\Omega _{2} - \omega _{j} )\sigma _{{2,i,j}} }} {{\Delta \omega _{i} }}} \hfill} \\ {{} \hfill} & {{ = {\left( {\frac{{\sigma _{{1,i,j}} }} {\eta } + (\Omega _{2} - \omega _{j} )\sigma _{{2,i + 1,j}} - (\Omega _{2} - \omega _{j} )\sigma _{{2,i,j}} } \right)}\eta } \hfill} \\ {{} \hfill} & {{ = \sigma _{{1,i,j}} + (\Omega _{2} - \omega _{j} )(\sigma _{{2,i + 1,j}} - \sigma _{{2,i,j}} )\eta } \hfill} \\ \end{array} $$
Similarly,
$$ \begin{aligned} & \frac{{\partial U}} {{\partial \Omega _{2} }} \quad \approx \frac{{U^{{rel}}_{{\Omega _{1} ,\omega _{{j + 1}} }} - U^{{rel}}_{{\Omega _{1} ,\omega _{j} }} }} {{\Delta \omega _{j} }}{\left( {\frac{{\sigma _{{2,i,j}} }} {\eta } + (\Omega _{1} - \omega _{i} )\sigma _{{1,i,j + 1}} - (\Omega _{1} - \omega _{i} )\sigma _{{2,i,j}} } \right)}\eta \\ & \quad = \sigma _{{2,i,j}} + (\Omega _{1} - \omega _{i} )(\sigma _{{1,i,j + 1}} - \sigma _{{1,i,j}} )\eta \\ \end{aligned} $$
It is possible to define another parameter αi,j based on the definition of σa,i,j:
$$ \alpha _{{i,j}} \equiv {\left( {\sigma _{{1,i,j + 1}} - \sigma _{{1,i,j}} } \right)} = {\left( {\sigma _{{2,i + 1,j}} - \sigma _{{2,i,j}} } \right)} = {\left( {U_{{i + 1,j + 1}} - U_{{i,j + 1}} - U_{{i + 1,j}} + U_{{i,j}} } \right)}\eta $$
So that, in general,
$$ \frac{{\partial U}} {{\partial \Omega _{a} }} \approx \sigma _{{a,i,j}} + {\left( {\Omega _{b} - \omega _{{b,c}} } \right)}\alpha _{{i,j}} \eta $$
where: if a=1, then b=2 and c=j; if a=2, then b=1 and c=i.
This is efficiently implemented by calculating the values of variables in the following order:
$$ i{\text{,}}\,\,\,j{\text{,}}\;\;\;\sigma _{{1,i,j}} ,\;\;\;\sigma _{{2,i,j}} $$
and finally
$$ \alpha _{{i,j}} = - \sigma _{{1,i,j + 1}} - \sigma _{{1,i,j}} = U_{{i + 1,j + 1}} - U_{{i,j + 1}} - \sigma _{{1,i,j}} $$
This can be extended to a hypersurface describing the potential U as a function of three variables. Define
$$ \sigma _{{1,i,j,k}} \equiv {\left( {\frac{{\Delta U}} {{\Delta \omega }}} \right)}_{{i,j,k}} \equiv \frac{{U_{{i + 1,j,k}} - U_{{i,j,k}} }} {{\omega _{{i + 1}} - \omega _{i} }} = {\left( {U_{{i + 1,j,k}} - U_{{i,j,k}} } \right)}\eta $$
The gradient components \( {\left( {\partial U/\partial \Omega _{1} } \right)} \) and \( {\left( {\partial U/\partial \Omega _{2} } \right)} \) can be calculated in any plane defined by (Ω3=ω
k
=constant) as described above. Then, for the point at coordinates (Ω1, Ω2, Ω3) within the cubic solid bounded by points (ω
i
,ω
j
,ω
k
) and (ωi+1,ωj+1,ωk+1), one can trace the changes in U from the point (ω
i
,ω
j
,ω
k
) to the points (Ω1, Ω2, Ω
k
) and (Ω1, Ω2, Ωk+1), making it possible to define the interpolated values \( {\left( {U^{{rel}}_{{\Omega _{1} ,\Omega _{2} ,\omega _{k} }} } \right)} \) and \( {\left( {U^{{rel}}_{{\Omega _{1} ,\Omega _{2} ,\omega _{{k + 1}} }} } \right)} \) relative to the value (Ui,j,k). This makes it possible to find a linear approximation for \( {\left( {\partial U/\partial \Omega _{3} } \right)} \) solely in terms of other, easily calculated gradient values. That is,
$$ \begin{aligned} & U^{{rel}}_{{\Omega _{1} ,\Omega _{2} ,\omega _{{3,k}} }} = {\left( {\Omega _{2} - \omega _{j} } \right)}\sigma _{{2,i,j,k}} + {\left( {\Omega _{1} - \omega _{i} } \right)}{\left( {\sigma _{{1,i,j,k}} + {\left( {\Omega _{2} - \omega _{j} } \right)}{\left( {\sigma _{{2,i + 1,j,k}} - \sigma _{{2,i,j,k}} } \right)}\eta } \right)} \\ & U^{{rel}}_{{\Omega _{1} ,\Omega _{2} ,\omega _{{3,k + 1}} }} = \frac{{\sigma _{{3,i,j,k}} }} {\eta } + {\left( {\Omega _{2} - \omega _{j} } \right)}\sigma _{{2,i,j,k + 1}} + {\left( {\Omega _{1} - \omega _{i} } \right)}{\left( {\sigma _{{1,i,j,k + 1}} + {\left( {\Omega _{2} - \omega _{j} } \right)}{\left( {\sigma _{{2,i + 1,j,k + 1}} - \sigma _{{2,i,j,k + 1}} } \right)}\eta } \right)} \\ \end{aligned} $$
Alternatively,
$$ \begin{aligned} & U^{{rel}}_{{\Phi ,\Psi ,\omega _{k} }} = {\left( {\Omega _{1} - \omega _{i} } \right)}\sigma _{{1,i,j,k}} + {\left( {\Omega _{2} - \omega _{j} } \right)}{\left( {\sigma _{{2,i,j,k}} + {\left( {\Omega _{1} - \omega _{i} } \right)}{\left( {\sigma _{{1,i,j + 1,k}} - \sigma _{{1,i,j,k}} } \right)}\eta } \right)} \\ & U^{{rel}}_{{\Phi ,\Psi ,\omega _{{k + 1}} }} = \frac{{\sigma _{{3,i,j,k}} }} {\eta } + {\left( {\Omega _{1} - \omega _{i} } \right)}\sigma _{{1,i,j,k + 1}} + {\left( {\Omega _{2} - \omega _{j} } \right)}{\left( {\sigma _{{2,i,j,k + 1}} + {\left( {\Omega _{1} - \omega _{i} } \right)}{\left( {\sigma _{{1,i,j + 1,k + 1}} - \sigma _{{1,i,j,k + 1}} } \right)}\eta } \right)} \\ \end{aligned} $$
Then,
$$ \begin{array}{*{20}l} {{\frac{{\partial U}} {{\partial \Omega _{3} }}} \hfill} & {{ = {\left( {U^{{rel}}_{{\Omega _{1} ,\Omega _{2} ,\omega _{{k + 1}} }} - U^{{rel}}_{{\Omega _{1} ,\Omega _{2} ,\omega _{k} }} } \right)}\eta } \hfill} \\ {{} \hfill} & {{ = {\left( {\frac{{\sigma _{{3,i,j,k}} }} {\eta } + {\left( {\Omega _{2} - \omega _{j} } \right)}\sigma _{{2,i,j,k + 1}} + {\left( {\Omega _{1} - \omega _{i} } \right)}{\left( {\sigma _{{1,i,j,k + 1}} + {\left( {\Omega _{2} - \omega _{j} } \right)}{\left( {\sigma _{{2,i + 1,j,k + 1}} - \sigma _{{2,i,j,k + 1}} } \right)}\eta } \right)}} \right)}\eta } \hfill} \\ {{} \hfill} & {{ - {\left( {{\left( {\Omega _{2} - \omega _{j} } \right)}\sigma _{{2,i,j,k}} + {\left( {\Omega _{1} - \omega _{i} } \right)}{\left( {\sigma _{{1,i,j,k}} + {\left( {\Omega _{2} - \omega _{j} } \right)}{\left( {\sigma _{{2,i + 1,j,k}} - \sigma _{{2,i,j,k}} } \right)}\eta } \right)}} \right)}\eta } \hfill} \\ {{} \hfill} & {{ = {\left( {\Omega _{1} - \omega _{{1,i}} } \right)}{\left( {\Omega _{2} - \omega _{j} } \right)}{\left( {\sigma _{{2,i + 1,j,k + 1}} - \sigma _{{2,i,j,k + 1}} - \sigma _{{2,i + 1,j,k}} + \sigma _{{2,i,j,k}} } \right)}\eta ^{2} } \hfill} \\ {{} \hfill} & {{ + {\left[ {{\left( {\Omega _{2} - \omega _{j} } \right)}{\left( {\sigma _{{2,i,j,k + 1}} - \sigma _{{2,i,j,k}} } \right)} + {\left( {\Omega _{1} - \omega _{i} } \right)}{\left( {\sigma _{{1,i,j,k + 1}} - \sigma _{{1,i,j,k}} } \right)}} \right]}\eta + \sigma _{{3,i,j,k}} } \hfill} \\ \end{array} $$
Alternatively,
$$ \begin{array}{*{20}l} {{\frac{{\partial U}} {{\partial \Omega _{3} }}} \hfill} & {{ = {\left( {U^{{rel}}_{{\Omega _{1} ,\Omega _{2} ,\omega _{{k + 1}} }} - U^{{rel}}_{{\Omega _{1} ,\Omega _{2} ,\omega _{k} }} } \right)}\eta } \hfill} \\ {{} \hfill} & {{ = {\left( {\frac{{\sigma _{{3,i,j,k}} }} {\eta } + {\left( {\Omega _{1} - \omega _{i} } \right)}\sigma _{{1,i,j,k + 1}} + {\left( {\Omega _{2} - \omega _{j} } \right)}{\left( {\sigma _{{2,i,j,k + 1}} + {\left( {\Omega _{1} - \omega _{i} } \right)}{\left( {\sigma _{{1,i,j + 1,k + 1}} - \sigma _{{1,i,j,k + 1}} } \right)}\eta } \right)}} \right)}\eta } \hfill} \\ {{} \hfill} & {{ - {\left( {{\left( {\Omega _{1} - \omega _{i} } \right)}\sigma _{{2,i,j,k}} + {\left( {\Omega _{2} - \omega _{j} } \right)}{\left( {\sigma _{{2,i,j,k}} + {\left( {\Omega _{1} - \omega _{i} } \right)}{\left( {\sigma _{{1,i,j + 1,k}} - \sigma _{{1,i,j,k}} } \right)}\eta } \right)}} \right)}\eta } \hfill} \\ {{} \hfill} & {{ = {\left( {\Omega _{1} - \omega _{i} } \right)}{\left( {\Omega _{2} - \omega _{j} } \right)}{\left( {\sigma _{{1,i,j + 1,k + 1}} - \sigma _{{1,i,j,k + 1}} - \sigma _{{1,i,j + 1,k}} + \sigma _{{1,i,j,k}} } \right)}\eta ^{2} } \hfill} \\ {{} \hfill} & {{ + {\left[ {{\left( {\Omega _{2} - \omega _{j} } \right)}{\left( {\sigma _{{2,i,j,k + 1}} - \sigma _{{2,i,j,k}} } \right)} + {\left( {\Omega _{1} - \omega _{i} } \right)}{\left( {\sigma _{{1,i,j,k + 1}} - \sigma _{{1,i,j,k}} } \right)}} \right]}\eta + \sigma _{{3,i,j,k}} } \hfill} \\ \end{array} $$
The two above expressions for \( {\left( {\partial U/\partial \Omega _{3} } \right)} \) are equivalent as
$$ \begin{aligned} & {\left( {U_{{i + 1,j + 1,k + 1}} - U_{{i,j + 1,k + 1}} } \right)}\eta - {\left( {U_{{i + 1,j,k + 1}} - U_{{i,j,k + 1}} } \right)}\eta - {\left( {U_{{i + 1,j + 1,k}} - U_{{i,j + 1,k}} } \right)}\eta + {\left( {U_{{i + 1,j,k}} - U_{{i,j,k}} } \right)}\eta \\ & = {\left( {\sigma _{{1,i,j + 1,k + 1}} - \sigma _{{1,i,j,k + 1}} - \sigma _{{1,i,j + 1,k}} + \sigma _{{1,i,j,k}} } \right)} \\ \end{aligned} $$
and
$$ \begin{aligned} & {\left( {U_{{i + 1,j + 1,k + 1}} - U_{{i + 1,j,k + 1}} } \right)}\eta - {\left( {U_{{i,j + 1,k + 1}} - U_{{i,j,k + 1}} } \right)}\eta - {\left( {U_{{i + 1,j + 1,k}} - U_{{i + 1,j,k}} } \right)}\eta + {\left( {U_{{i,j + 1,k}} - U_{{i,j,k}} } \right)}\eta \\ & = {\left( {\sigma _{{2,i + 1,j,k + 1}} - \sigma _{{2,i,j,k + 1}} - \sigma _{{2,i + 1,j,k}} + \sigma _{{2,i,j,k}} } \right)} \\ \end{aligned} $$
and it is clear that the last two expressions are identical when stated in terms of U.
Permutation of the indices gives values for \( {\left( {\partial U/\partial \Omega _{1} } \right)} \) and \( {\left( {\partial U/\partial \Omega _{2} } \right)} \)
$$ \begin{aligned} & \frac{{\partial U}} {{\partial \Omega _{1} }} \quad = {\left( {\Omega _{2} - \omega _{j} } \right)}{\left( {\Omega _{3} - \omega _{k} } \right)}{\left( {\sigma _{{2,i + 1,j,k + 1}} - \sigma _{{2,i + 1,j,k}} - \sigma _{{2,i,j,k + 1}} + \sigma _{{2,i,j,k}} } \right)}\eta ^{2} \\ & \quad + {\left[ {{\left( {\Omega _{2} - \omega _{j} } \right)}{\left( {\sigma _{{2,i + 1,j,k}} - \sigma _{{2,i,j,k}} } \right)} + {\left( {\Omega _{3} - \omega _{k} } \right)}{\left( {\sigma _{{3,i + 1,j,k}} - \sigma _{{3,i,j,k}} } \right)}} \right]}\eta + \sigma _{{1,i,j,k}} \\ \end{aligned} $$
$$ \begin{aligned} & \frac{{\partial U}} {{\partial \Omega _{2} }} \quad = {\left( {\Omega _{1} - \omega _{i} } \right)}{\left( {\Omega _{3} - \omega _{k} } \right)}{\left( {\sigma _{{3,i + 1,j + 1,k}} - \sigma _{{3,i,j + 1,k}} - \sigma _{{3,i + 1,j,k}} + \sigma _{{3,i,j,k}} } \right)}\eta ^{2} \\ & \quad + {\left[ {{\left( {\Omega _{1} - \omega _{i} } \right)}{\left( {\sigma _{{1,i,j + 1,k}} - \sigma _{{1,i,j,k}} } \right)} + {\left( {\Omega _{3} - \omega _{k} } \right)}{\left( {\sigma _{{3,i,j + 1,k}} - \sigma _{{3,i,j,k}} } \right)}} \right]}\eta + \sigma _{{2,i,j,k}} \\ \end{aligned} $$
Three of the six faces of the cubic solid bounded by points (ω
i
,ω
j
,ω
k
) and (ωi+1,ωj+1,ωk+1) can each be associated with a parameter αa,i,j,k, which is analogous to the parameter αi,j defined for the two-dimensional case above. That is,
$$ \alpha _{{1,i,j,k}} \equiv {\left( {\sigma _{{2,i,j,k + 1}} - \sigma _{{2,i,j,k}} } \right)} = {\left( {\sigma _{{3,i,j + 1,k}} - \sigma _{{3,i,j,k}} } \right)} = {\left( {U_{{i,j + 1,k + 1}} - U_{{i,j,k + 1}} - U_{{i,j + 1,k}} + U_{{i,j,k}} } \right)}\eta $$
$$ \alpha _{{2,i,j,k}} \equiv {\left( {\sigma _{{1,i,j,k + 1}} - \sigma _{{1,i,j,k}} } \right)} = {\left( {\sigma _{{3,i + 1,j,k}} - \sigma _{{3,i,j,k}} } \right)} = {\left( {U_{{i + 1,j,k + 1}} - U_{{i,j,k + 1}} - U_{{i + 1,j,k}} + U_{{i,j,k}} } \right)}\eta $$
$$ \alpha _{{3,i,j,k}} \equiv {\left( {\sigma _{{1,i,j + 1,k}} - \sigma _{{1,i,j,k}} } \right)} = {\left( {\sigma _{{2,i + 1,j,k}} - \sigma _{{2,i,j,k}} } \right)} = {\left( {U_{{i + 1,j + 1,k}} - U_{{i,j + 1,k}} - U_{{i + 1,j,k}} + U_{{i,j,k}} } \right)}\eta $$
Another parameter βi,j,k can be defined by extending the definition of αa,i,j,k to include, for example, α1,i+1,j,k , which is associated with a cubic solid adjacent to the one being considered. βi,j,k can be expressed in many different ways, some of which may be more computationally convenient:
$$ \begin{aligned} & \beta _{{i,j,k}} \quad = \sigma _{{3,i + 1,j + 1,k}} - \sigma _{{3,i,j + 1,k}} - \sigma _{{3,i + 1,j,k}} + \sigma _{{3,i,j,k}} = \alpha _{{2,i,j + 1,k}} - \alpha _{{2,i,j,k}} = \alpha _{{1,i + 1,j,k}} - \alpha _{{1,i,j,k}} \\ & \quad = \sigma _{{2,i + 1,j,k + 1}} - \sigma _{{2,i,j,k + 1}} - \sigma _{{2,i + 1,j,k}} + \sigma _{{2,i,j,k}} = \alpha _{{3,i,j,k + 1}} - \alpha _{{3,i,j,k}} = \alpha _{{1,i + 1,j,k}} - \alpha _{{1,i,j,k}} \\ & \quad = \sigma _{{1,i,j + 1,k + 1}} - \sigma _{{1,i,j + 1,k}} - \sigma _{{1,i,j,k + 1}} + \sigma _{{1,i,j,k}} = \alpha _{{2,i,j + 1,k}} - \alpha _{{2,i,j,k}} = \alpha _{{3,i,j,k + 1}} - \alpha _{{3,i,j,k}} \\ & \quad = {\left( {U_{{i + 1,j + 1,k + 1}} - U_{{i,j + 1,k + 1}} } \right)} - {\left( {U_{{i + 1,j + 1,k}} - U_{{i,j + 1,k}} } \right)} - {\left( {U_{{i + 1,j,k + 1}} - U_{{i,j,k + 1}} } \right)} + {\left( {U_{{i + 1,j,k}} - U_{{i,j,k}} } \right)} \\ & \quad = {\left( {U_{{i + 1,j + 1,k + 1}} - U_{{i,j + 1,k + 1}} } \right)} - {\left( {U_{{i + 1,j + 1,k}} - U_{{i,j + 1,k}} } \right)} - \alpha _{{2,i,j,k}} \\ & \quad = U_{{i + 1,j + 1,k + 1}} - U_{{i,j + 1,k + 1}} - U_{{i + 1,j,k + 1}} - U_{{i + 1,j + 1,k}} + U_{{i + 1,j,k}} + U_{{i,j + 1,k}} + U_{{i,j,k + 1}} - U_{{i,j,k}} \\ \end{aligned} $$
Then, the components of the gradient at coordinates (Ω1, Ω2, Ω3) can be simply expressed as
$$ \frac{{\partial U}} {{\partial \Omega _{1} }} = {\left( {\Omega _{2} - \omega _{j} } \right)}{\left( {\Omega _{3} - \omega _{k} } \right)}\beta _{{i,j,k}} \eta ^{2} + {\left[ {{\left( {\Omega _{2} - \omega _{j} } \right)}\alpha _{{3,i,j,k}} + {\left( {\Omega _{3} - \omega _{k} } \right)}\alpha _{{2,i,j,k}} } \right]}\eta + \sigma _{{1,i,j,k}} $$
$$ \frac{{\partial U}} {{\partial \Omega _{2} }} = {\left( {\Omega _{1} - \omega _{i} } \right)}{\left( {\Omega _{3} - \omega _{k} } \right)}\beta _{{i,j,k}} \eta ^{2} + {\left[ {{\left( {\Omega _{1} - \omega _{i} } \right)}\alpha _{{3,i,j,k}} + {\left( {\Omega _{3} - \omega _{k} } \right)}\alpha _{{1,i,j,k}} } \right]}\eta + \sigma _{{2,i,j,k}} $$
$$ \frac{{\partial U}} {{\partial \Omega _{3} }} = {\left( {\Omega _{1} - \omega _{i} } \right)}{\left( {\Omega _{2} - \omega _{j} } \right)}\beta _{{i,j,k}} \eta ^{2} + {\left[ {{\left( {\Omega _{2} - \omega _{j} } \right)}\alpha _{{1,i,j,k}} + {\left( {\Omega _{1} - \omega _{i} } \right)}\alpha _{{2,i,j,k}} } \right]}\eta + \sigma _{{3,i,j,k}} $$