Tinker9 User Manual
Fig. 2 Global frame g and QI frame i of atoms 1 and 2. The z direction of this QI frame is chosen along the distance vector.¶
Consider two vectors u, v and two reference frames A, B. R is the rotation matrix of the axes such that
Since \(u_A^T v_A=u_B^T v_B\),
A 2-D tensor, e.g., quadrupole moment Q, in two reference frames are associated by
It is easy to prove that
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.
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
\(\partial f(r)/\partial y_2\)
\(f'(r)r_y/r\)
\(\partial f(r)/\partial z_2\)
\(f'(r)r_z/r\)
\(f'(r)\)
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):
Once the torques are computed the same way as in the previous section
x and y direction gradients then become
Depending on the direction of distance vector, the signs of x and y direction gradients may flip.
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
\(\phi_1\)
\(\phi'_{1x}\)
\(\partial\phi_1/\partial x_1\)
\(\phi'_{1y}\)
\(\partial\phi_1/\partial y_1\)
\(\phi'_{1z}\)
\(\partial\phi_1/\partial z_1\)
\(\phi''_{1xx}\)
\(\partial^2\phi_1/\partial x_1^2\)
\(\phi''_{1yy}\)
\(\partial^2\phi_1/\partial y_1^2\)
\(\phi''_{1zz}\)
\(\partial^2\phi_1/\partial z_1^2\)
\(\phi''_{1xy}\)
\(\partial^2\phi_1/\partial x_1\partial y_1\)
\(\phi''_{1xz}\)
\(\partial^2\phi_1/\partial x_1\partial z_1\)
\(\phi''_{1yz}\)
\(\partial^2\phi_1/\partial y_1\partial z_1\)
\(\phi_2\)
\(\phi'_{2x}\)
\(\partial\phi_2/\partial x_2\)
\(\phi'_{2y}\)
\(\partial\phi_2/\partial y_2\)
\(\phi'_{2z}\)
\(\partial\phi_2/\partial z_2\)
\(\phi''_{2xx}\)
\(\partial^2\phi_2/\partial x_2^2\)
\(\phi''_{2yy}\)
\(\partial^2\phi_2/\partial y_2^2\)
\(\phi''_{2zz}\)
\(\partial^2\phi_2/\partial z_2^2\)
\(\phi''_{2xy}\)
\(\partial^2\phi_2/\partial x_2\partial y_2\)
\(\phi''_{2xz}\)
\(\partial^2\phi_2/\partial x_2\partial z_2\)
\(\phi''_{2yz}\)
\(\partial^2\phi_2/\partial y_2\partial z_2\)