Rotation Matrix
Consider two vectors u, v and two reference frames A, B.
R is the rotation matrix of the axes such that
\[ \begin{align}\begin{aligned}R u_A = u_B,\\R v_A = v_B.\end{aligned}\end{align} \]
Since \(u_A^T v_A=u_B^T v_B\),
\[R^T R=I.\]
A 2-D tensor, e.g., quadrupole moment Q, in two reference frames are
associated by
\[u_A^T Q_A v_A = u_B^T Q_B v_B.\]
It is easy to prove that
\[R Q_A R^T = Q_B.\]
Two common transformations used in Tinker are:
From (A) Local Frame (in which parameters are provided)
to (B) Global Frame (in which the calculation is done);
From (A) Global Frame (for direct pairwise electrostatics)
to (B) Quasi-Internal (QI) Frame (for optimized algebra),
as shown in Fig. 2.
Multipole Interaction in QI Frame
Once the distance vector is in QI frame, many derivatives can be simplified
as shown in the following table. f(r) does not explicitly depend on
\(r_x,r_y,r_z\).
Gradients |
Global Frame |
QI Frame |
|---|
\(\partial f(r)/\partial x_2\) |
\(f'(r)r_x/r\) |
0 |
\(\partial f(r)/\partial y_2\) |
\(f'(r)r_y/r\) |
0 |
\(\partial f(r)/\partial z_2\) |
\(f'(r)r_z/r\) |
\(f'(r)\) |
For potential energy, (4) can be used without modification in QI frame.
Since \(\partial\phi_1/\partial z_1 = -E_{z1}\), the z direction gradient
can be obtained from z direction electrostatic field (Ez):
\[\frac{\partial U}{\partial z}=-E_z C -E'_z D -E''_z Q -\cdots.\]
Once the torques are computed the same way as in the previous section
\[\tau = \tau_1 + \tau_2 = \boldsymbol{r}\times\boldsymbol{F}
= (U'_x,U'_y,U'_z)\times(0,0,r) = (U'_y r, -U'_x r, 0),\]
x and y direction gradients then become
\[\begin{split}U'_x &= -\tau_y/r, \\
U'_y &= \tau_x/r.\end{split}\]
Depending on the direction of distance vector, the signs of x and y direction
gradients may flip.
Details
In the following notes, \(A : B\) stands for \(A = A + B\).
If there is no ambiguity, \(f'\) and \(f''\) may stand for
\((f'_x,f'_y,f'_z)\) and
\((f''_{xx},f''_{yy},f''_{zz},f''_{xy},f''_{xz},f''_{yz})\), respectively.
Potential Terms |
Notes |
|---|
\(\phi_1\) |
\(\phi_1\) |
\(\phi'_{1x}\) |
\(\partial\phi_1/\partial x_1\) |
\(\phi'_{1y}\) |
\(\partial\phi_1/\partial y_1\) |
\(\phi'_{1z}\) |
\(\partial\phi_1/\partial z_1\) |
\(\phi''_{1xx}\) |
\(\partial^2\phi_1/\partial x_1^2\) |
\(\phi''_{1yy}\) |
\(\partial^2\phi_1/\partial y_1^2\) |
\(\phi''_{1zz}\) |
\(\partial^2\phi_1/\partial z_1^2\) |
\(\phi''_{1xy}\) |
\(\partial^2\phi_1/\partial x_1\partial y_1\) |
\(\phi''_{1xz}\) |
\(\partial^2\phi_1/\partial x_1\partial z_1\) |
\(\phi''_{1yz}\) |
\(\partial^2\phi_1/\partial y_1\partial z_1\) |
\(\phi_2\) |
\(\phi_2\) |
\(\phi'_{2x}\) |
\(\partial\phi_2/\partial x_2\) |
\(\phi'_{2y}\) |
\(\partial\phi_2/\partial y_2\) |
\(\phi'_{2z}\) |
\(\partial\phi_2/\partial z_2\) |
\(\phi''_{2xx}\) |
\(\partial^2\phi_2/\partial x_2^2\) |
\(\phi''_{2yy}\) |
\(\partial^2\phi_2/\partial y_2^2\) |
\(\phi''_{2zz}\) |
\(\partial^2\phi_2/\partial z_2^2\) |
\(\phi''_{2xy}\) |
\(\partial^2\phi_2/\partial x_2\partial y_2\) |
\(\phi''_{2xz}\) |
\(\partial^2\phi_2/\partial x_2\partial z_2\) |
\(\phi''_{2yz}\) |
\(\partial^2\phi_2/\partial y_2\partial z_2\) |
Charge Terms
\[\begin{split}\phi_1 &: T_{12}^{(1,1)} C_2 = B_0 C_2,\ \phi'_1 : T_{12}^{(2:4,1)} C_2 = \begin{pmatrix}
0 \\
0 \\
r B_1 C_2 \end{pmatrix}, \\
\phi''_1 &: T_{12}^{(5:13,1)} C_2 = -\begin{pmatrix}
B_1 C_2 \\
B_1 C_2 \\
(B_1 - r^2 B_2) C_2 \\
0 \\
0 \\
0 \end{pmatrix}.\end{split}\]
\[\begin{split}\phi_2 &: T_{21}^{(1,1)} C_1 = B_0 C_1,\ \phi'_2 : T_{21}^{(2:4,1)} C_1 = -\begin{pmatrix}
0 \\
0 \\
r B_1 C_1 \end{pmatrix}, \\
\phi''_2 &: T_{21}^{(5:13,1)} C_1 = -\begin{pmatrix}
B_1 C_1 \\
B_1 C_1 \\
(B_1 - r^2 B_2) C_1 \\
0 \\
0 \\
0 \end{pmatrix}.\end{split}\]
\[\begin{split}-E_{z1} &: r B_1 C_2,\ -E'_{z1} : -\begin{pmatrix}
0 \\
0 \\
B_1 - r^2 B_2 \end{pmatrix}, \\
-E''_{z1} &: -\begin{pmatrix}
r B_2 C_2 \\
r B_2 C_2 \\
(3 r B_2 - r^3 B_3) C_2 \\
0 \\
0 \\
0 \end{pmatrix}.\end{split}\]
Dipole Terms
\[\begin{split}\phi_1 &: T_{12}^{(1,2:4)} D_2 = -r B_1 D_{z2},\ \phi'_1 : T_{12}^{(2:4,2:4)} D_2 = \begin{pmatrix}
B_1 D_{x2} \\
B_1 D_{y2} \\
(B_1 - r^2 B_2) D_{z2} \end{pmatrix}, \\
\phi''_1 &: T_{12}^{(5:13,2:4)} D_2 = \begin{pmatrix}
r B_2 D_{z2} \\
r B_2 D_{z2} \\
(3 r B_2 - r^3 B_3) D_{z2} \\
0 \\
2 r B_2 D_{x2} \\
2 r B_2 D_{y2} \end{pmatrix}.\end{split}\]
\[\begin{split}\phi_2 &: T_{21}^{(1,2:4)} D_1 = r B_1 D_{z1},\ \phi'_2 : T_{21}^{(2:4,2:4)} D_1 = \begin{pmatrix}
B_1 D_{x1} \\
B_1 D_{y1} \\
(B_1 - r^2 B_2) D_{z1} \end{pmatrix}, \\
\phi''_2 &: T_{21}^{(5:13,2:4)} D_1 = -\begin{pmatrix}
r B_2 D_{z1} \\
r B_2 D_{z1} \\
(3 r B_2 - r^3 B_3) D_{z1} \\
0 \\
2 r B_2 D_{x1} \\
2 r B_2 D_{y1} \end{pmatrix}.\end{split}\]
\[\begin{split}-E_{z1} &: (B_1 - r^2 B_2) D_{z2},\ -E'_{z1} : \begin{pmatrix}
r B_2 D_{x2} \\
r B_2 D_{y2} \\
(3 r B_2 - r^3 B_3) D_{z2} \end{pmatrix}, \\
-E''_{z1} &: -\begin{pmatrix}
(B_2 - r^2 B_3) D_{z2} \\
(B_2 - r^2 B_3) D_{z2} \\
(3 B_2 - 6 r^2 B_3 + r^4 B_4) D_{z2} \\
0 \\
2 (B_2 - r^2 B_3) D_{x2} \\
2 (B_2 - r^2 B_3) D_{y2} \end{pmatrix}.\end{split}\]
Quadrupole Terms
\[\begin{split}\phi_1 &: T_{12}^{(1,5:13)} Q_2 = r^2 B_2 Q_{zz2},\ \phi'_1 : T_{12}^{(2:4,5:13)} Q_2 = -\begin{pmatrix}
2 r B_2 Q_{xz2} \\
2 r B_2 Q_{yz2} \\
(2 r B_2 - r^3 B_3) Q_{zz2} \end{pmatrix}, \\
\phi''_1 &: T_{12}^{(5:13,5:13)} Q_2 = \begin{pmatrix}
2 B_2 Q_{xx2} - r^2 B_3 Q_{zz2} \\
2 B_2 Q_{yy2} - r^2 B_3 Q_{zz2} \\
(2 B_2 - 5 r^2 B_3 + r^4 B_4) Q_{zz2} \\
4 B_2 Q_{xy2} \\
4 (B_2 - r^2 B_3) Q_{xz2} \\
4 (B_2 - r^2 B_3) Q_{yz2} \end{pmatrix}.\end{split}\]
\[\begin{split}\phi_2 &: T_{21}^{(1,5:13)} Q_1 = r^2 B_2 Q_{zz1},\ \phi'_2 : T_{21}^{(2:4,5:13)} Q_1 = \begin{pmatrix}
2 r B_2 Q_{xz1} \\
2 r B_2 Q_{yz1} \\
(2 r B_2 - r^3 B_3) Q_{zz1} \end{pmatrix}, \\
\phi''_2 &: T_{21}^{(5:13,5:13)} Q_1 = \begin{pmatrix}
2 B_2 Q_{xx1} - r^2 B_3 Q_{zz1} \\
2 B_2 Q_{yy1} - r^2 B_3 Q_{zz1} \\
(2 B_2 - 5 r^2 B_3 + r^4 B_4) Q_{zz1} \\
4 B_2 Q_{xy1} \\
4 (B_2 - r^2 B_3) Q_{xz1} \\
4 (B_2 - r^2 B_3) Q_{yz1} \end{pmatrix}.\end{split}\]
\[\begin{split}-E_{z1} &: -(2 r B_2 - r^3 B_3) Q_{zz2},\ -E'_{z1} : \begin{pmatrix}
2 (B_2 - r^2 B_3) Q_{xz2} \\
2 (B_2 - r^2 B_3) Q_{yz2} \\
(2 B_2 - 5 r^2 B_3 + r^4 B_4) Q_{zz2} \end{pmatrix}, \\
-E''_{z1} &: \begin{pmatrix}
-2 r B_3 Q_{yy2} - r^3 B_4 Q_{zz2} \\
-2 r B_3 Q_{xx2} - r^3 B_4 Q_{zz2} \\
(12 r B_3 - 9 r^3 B_4 + r^5 B_5) Q_{zz2} \\
4 r B_3 Q_{xy2} \\
4 (3 r B_3 - r^3 B_4) Q_{xz2} \\
4 (3 r B_3 - r^3 B_4) Q_{yz2} \end{pmatrix}.\end{split}\]
