| 838 |
|
\caption{\label{tab:tableenergy}Radial functions used in the energy |
| 839 |
|
and torque equations. The $f, g, h, s, t, \mathrm{and} u$ |
| 840 |
|
functions used in this table are defined in Appendices B and C.} |
| 841 |
< |
\begin{tabular}{|c|c|l|l|} \hline |
| 842 |
< |
Generic&Bare Coulomb&Taylor-Shifted&Gradient-Shifted |
| 841 |
> |
\begin{tabular}{|c|c|l|l|l|} \hline |
| 842 |
> |
Generic&Bare Coulomb&Taylor-Shifted (TSF)&Shifted Potential (SP)&Gradient-Shifted (GSF) |
| 843 |
|
\\ \hline |
| 844 |
|
% |
| 845 |
|
% |
| 848 |
|
$v_{01}(r)$ & |
| 849 |
|
$\frac{1}{r}$ & |
| 850 |
|
$f_0(r)$ & |
| 851 |
< |
$f(r)-f(r_c)-(r-r_c)g(r_c)$ |
| 851 |
> |
$f(r)-f(r_c)$ & |
| 852 |
> |
SP $-(r-r_c)g(r_c)$ |
| 853 |
|
\\ |
| 854 |
|
% |
| 855 |
|
% |
| 858 |
|
$v_{11}(r)$ & |
| 859 |
|
$-\frac{1}{r^2}$ & |
| 860 |
|
$g_1(r)$ & |
| 861 |
< |
$g(r)-g(r_c)-(r-r_c)h(r_c)$ \\ |
| 861 |
> |
$g(r)-g(r_c)$ & |
| 862 |
> |
SP $-(r-r_c)h(r_c)$ \\ |
| 863 |
|
% |
| 864 |
|
% |
| 865 |
|
% |
| 867 |
|
$v_{21}(r)$ & |
| 868 |
|
$-\frac{1}{r^3} $ & |
| 869 |
|
$\frac{g_2(r)}{r} $ & |
| 870 |
< |
$\frac{g(r)}{r}-\frac{g(r_c)}{r_c} -(r-r_c) |
| 870 |
> |
$\frac{g(r)}{r}-\frac{g(r_c)}{r_c}$ & |
| 871 |
> |
SP $-(r-r_c) |
| 872 |
|
\left( -\frac{g(r_c)}{r_c^2} + \frac{h(r_c)}{r_c} \right)$ \\ |
| 873 |
|
$v_{22}(r)$ & |
| 874 |
|
$\frac{3}{r^3} $ & |
| 875 |
|
$\left(-\frac{g_2(r)}{r} + h_2(r) \right)$ & |
| 876 |
< |
$\left(-\frac{g(r)}{r}+h(r) \right) |
| 877 |
< |
-\left(-\frac{g(r_c)}{r_c}+h(r_c) \right)$ \\ |
| 875 |
< |
&&& $ ~~~-(r-r_c) \left( \frac{g(r_c)}{r_c^2}-\frac{h(r_c)}{r_c}+s(r_c) \right)$ |
| 876 |
< |
\\ |
| 876 |
> |
$\left(-\frac{g(r)}{r}+h(r) \right)$ & SP \\ |
| 877 |
> |
&&& $~~~-\left(-\frac{g(r_c)}{r_c}+h(r_c) \right)$ & $~~~-(r-r_c) \left( \frac{g(r_c)}{r_c^2}-\frac{h(r_c)}{r_c}+s(r_c) \right)$\\ |
| 878 |
|
% |
| 879 |
|
% |
| 880 |
|
% |
| 882 |
|
$v_{31}(r)$ & |
| 883 |
|
$\frac{3}{r^4} $ & |
| 884 |
|
$\left(-\frac{g_3(r)}{r^2} + \frac{h_3(r)}{r} \right)$ & |
| 885 |
< |
$\left( -\frac{g(r)}{r^2}+\frac{h(r)}{r} \right) |
| 886 |
< |
-\left(-\frac{g(r_c)}{r_c^2}+\frac{h(r_c)}{r_c} \right) $\\ |
| 886 |
< |
&&&$ ~~~ -(r-r_c) \left(\frac{2g(r_c)}{r_c^3}-\frac{2h(r_c)}{r_c^2}+\frac{s(r_c)}{r_c} \right)$ |
| 887 |
< |
\\ |
| 885 |
> |
$\left( -\frac{g(r)}{r^2}+\frac{h(r)}{r} \right)$ & SP \\ |
| 886 |
> |
&&& $-\left(-\frac{g(r_c)}{r_c^2}+\frac{h(r_c)}{r_c} \right) $ & $~~~-(r-r_c) \left(\frac{2g(r_c)}{r_c^3}-\frac{2h(r_c)}{r_c^2}+\frac{s(r_c)}{r_c} \right)$ \\ |
| 887 |
|
% |
| 888 |
|
$v_{32}(r)$ & |
| 889 |
|
$-\frac{15}{r^4} $ & |
| 890 |
|
$\left( \frac{3g_3(r)}{r^2} - \frac{3h_3(r)}{r} + s_3(r) \right)$ & |
| 891 |
< |
$\left( \frac{3g(r)}{r^2} - \frac{3h(r)}{r} + s(r) \right) |
| 892 |
< |
- \left( \frac{3g(r_c)}{r_c^2} - \frac{3h(r_c)}{r_c} + s(r_c) \right)$ \\ |
| 893 |
< |
&&&$ ~~~ -(r-r_c) \left( \frac{-6g(r_c)}{r_c^3}+\frac{6h(r_c)}{r_c^2}-\frac{3s(r_c)}{r_c}+t(r_c) \right)$ |
| 895 |
< |
\\ |
| 891 |
> |
$\left( \frac{3g(r)}{r^2} - \frac{3h(r)}{r} + s(r) \right)$ & SP \\ |
| 892 |
> |
&&& $~~~- \left( \frac{3g(r_c)}{r_c^2} - \frac{3h(r_c)}{r_c} + s(r_c) |
| 893 |
> |
\right)$ & $~~~-(r-r_c) \left( \frac{-6g(r_c)}{r_c^3}+\frac{6h(r_c)}{r_c^2}-\frac{3s(r_c)}{r_c}+t(r_c) \right)$ \\ |
| 894 |
|
% |
| 895 |
|
% |
| 896 |
|
% |
| 898 |
|
$v_{41}(r)$ & |
| 899 |
|
$\frac{3}{r^5} $ & |
| 900 |
|
$\left(-\frac{g_4(r)}{r^3} +\frac{h_4(r)}{r^2} \right) $ & |
| 901 |
< |
$\left( -\frac{g(r)}{r^3} + \frac{h(r)}{r^2} \right) |
| 902 |
< |
- \left( -\frac{g(r_c)}{r_c^3} + \frac{h(r_c)}{r_c^2} \right)$ \\ |
| 905 |
< |
&&&$ ~~~ -(r-r_c) \left( \frac{3g(r_c)}{r_c^4}-\frac{3h(r_c)}{r_c^3}+\frac{s(r_c)}{r_c^2} \right)$ |
| 901 |
> |
$\left( -\frac{g(r)}{r^3} + \frac{h(r)}{r^2} \right)$ & SP \\ |
| 902 |
> |
&&& $~~~- \left( -\frac{g(r_c)}{r_c^3} + \frac{h(r_c)}{r_c^2} \right)$& $~~~-(r-r_c) \left( \frac{3g(r_c)}{r_c^4}-\frac{3h(r_c)}{r_c^3}+\frac{s(r_c)}{r_c^2} \right)$ |
| 903 |
|
\\ |
| 904 |
|
% 2 |
| 905 |
|
$v_{42}(r)$ & |
| 906 |
|
$- \frac{15}{r^5} $ & |
| 907 |
|
$\left( \frac{3g_4(r)}{r^3} - \frac{3h_4(r)}{r^2}+\frac{s_4(r)}{r} \right)$ & |
| 908 |
< |
$\left( \frac{3g(r)}{r^3} - \frac{3h(r)}{r^2}+\frac{s(r)}{r} \right) |
| 909 |
< |
-\left( \frac{3g(r_c)}{r_c^3} - \frac{3h(r_c)}{r_c^2}+\frac{s(r_c)}{r_c} \right)$ \\ |
| 910 |
< |
&&&$ ~~~ -(r-r_c) \left(- \frac{9g(r_c)}{r_c^4}+\frac{9h(r_c)}{r_c^3} |
| 911 |
< |
-\frac{4s(r_c)}{r_c^2} + \frac{t(r_c)}{r_c}\right)$ |
| 912 |
< |
\\ |
| 908 |
> |
$\left( \frac{3g(r)}{r^3} - \frac{3h(r)}{r^2}+\frac{s(r)}{r} |
| 909 |
> |
\right)$ & SP \\ |
| 910 |
> |
&&& $~~~-\left( \frac{3g(r_c)}{r_c^3} - |
| 911 |
> |
\frac{3h(r_c)}{r_c^2}+\frac{s(r_c)}{r_c} \right)$ & $~~~-(r-r_c) \left(- \frac{9g(r_c)}{r_c^4}+\frac{9h(r_c)}{r_c^3} |
| 912 |
> |
-\frac{4s(r_c)}{r_c^2} + \frac{t(r_c)}{r_c}\right)$ \\ |
| 913 |
|
% 3 |
| 914 |
|
$v_{43}(r)$ & |
| 915 |
|
$ \frac{105}{r^5} $ & |
| 916 |
|
$\left(-\frac{15g_4(r)}{r^3}+\frac{15h_4(r)}{r^2}-\frac{6s_4(r)}{r} + t_4(r)\right) $ & |
| 917 |
< |
$\left(-\frac{15g(r)}{r^3}+\frac{15h(r)}{r^2}-\frac{6s(r)}{r} + t(r)\right)$ \\ |
| 918 |
< |
&&&$~~~ -\left(-\frac{15g(r_c)}{r_c^3}+\frac{15h(r_c)}{r_c^2}-\frac{6s(r_c)}{r_c} + t(r_c)\right)$ \\ |
| 919 |
< |
&&&$~~~ -(r-r_c)\left(\frac{45g(r_c)}{r_c^4}-\frac{45h(r_c)}{r_c^3}+\frac{21s(r_c)}{r_c^2} |
| 920 |
< |
-\frac{6t(r_c)}{r_c}+u(r_c) \right)$ \\ \hline |
| 917 |
> |
$\left(-\frac{15g(r)}{r^3}+\frac{15h(r)}{r^2}-\frac{6s(r)}{r} + |
| 918 |
> |
t(r)\right)$ & SP $-(r-r_c)\left(\frac{45g(r_c)}{r_c^4}-\frac{45h(r_c)}{r_c^3}\right.$\\ |
| 919 |
> |
&&& |
| 920 |
> |
$~~~-\left(-\frac{15g(r_c)}{r_c^3}+\frac{15h(r_c)}{r_c^2}-\frac{6s(r_c)}{r_c} |
| 921 |
> |
+ t(r_c)\right)$ & $~~~~~~~\left.+\frac{21s(r_c)}{r_c^2} |
| 922 |
> |
-\frac{6t(r_c)}{r_c}+u(r_c) \right)$ \\ |
| 923 |
> |
\hline |
| 924 |
|
\end{tabular} |
| 925 |
|
\end{sidewaystable} |
| 926 |
|
% |
