Here is one approach. I split the columns using = into two columns. To get the spacing right (and save typing) I used @{$\null=\null$} between the columns. Unfortunately, this meant having to use \multicolumn every time I didn't want a = to appear.
The only remaining problem was that when the label is too large, the extra space is added to the right half. To fix that, I calculated the difference and added extra space to the left half using \makebox. Now the equal signs are centered.
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{multirow, adjustbox, mathtools}
\begin{document}
\begin{table}[htbp]
\begin{adjustbox}{width=\columnwidth,center}
\sbox0{Calculated Difference}%
\sbox1{Experimental Difference}%
\sbox2{$\null=\null$}%
\dimen0=\dimexpr \wd0 - \wd2\relax
\dimen1=\dimexpr \wd1 - \wd2\relax
\begin{tabular}{|c|r@{$\null=\null$}l|r@{$\null=\null$}l|r@{$\null=\null$}l|c|}
\hline
Force Field & \multicolumn{2}{c|}{Calculated Energy} & \multicolumn{2}{c|}{Calculated Difference} & \multicolumn{2}{c|}{Experimental Difference} & Error \\
\hline
\multirow{4}{*}{OPLS-AA}
& tt & 2.55447 & \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \\ \cline{2-8}
& $\text{g}^+\text{g}^+$ & 4.19785 & \makebox[0.5\dimen0][r]{$\text{g}^+\text{g}^+$-tt} & 1.64338
& \makebox[0.5\dimen1][r]{$\text{g}^+\text{g}^+$-tt} & 1.62 & 0.02338 \\ \cline{2-8}% note \makebox
& $\text{g}^+\text{g}^-$ & 5.56306 & $\text{g}^+\text{g}^-$-tt & 3.00859 & $\text{g}^+\text{g}^-$-tt & 3.51 & -0.50141 \\ \cline{2-8}
& tg & 3.36156 & tg-tt & 0.80709 & tg-tt & 0.86 & -0.05291 \\ \cline{2-8}
\hline
\multirow{4}{*}{AM1}
& tt & -37.97281 & \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \\ \cline{2-8}
& $\text{g}^+\text{g}^+$ & -36.52659 & $\text{g}^+\text{g}^+$-tt & 1.44622 & $\text{g}^+\text{g}^+$-tt & 1.62 & -0.17378 \\ \cline{2-8}
& $\text{g}^+\text{g}^-$ & -35.85939 & $\text{g}^+\text{g}^-$-tt & 2.11342 & $\text{g}^+\text{g}^-$-tt & 3.51 & -1.39658 \\ \cline{2-8}
& tg & -37.25846 & tg-tt & 0.71435 & tg-tt & 0.86 & -0.14565 \\ \cline{2-8}
\hline
\multirow{4}{*}{PDDG/PM3}
& tt & -34.61227 & \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \\ \cline{2-8}
& $\text{g}^+\text{g}^+$ & -33.94462 & $\text{g}^+\text{g}^+$-tt & 0.66765 & $\text{g}^+\text{g}^+$-tt & 1.62 & -0.95235 \\ \cline{2-8}
& $\text{g}^+\text{g}^-$ & -33.29225 & $\text{g}^+\text{g}^-$-tt & 1.32002 & $\text{g}^+\text{g}^-$-tt & 3.51 & -2.18998 \\ \cline{2-8}
& tg & -34.38819 & tg-tt & 0.22408 & tg-tt & 0.86 & -0.63592 \\ \cline{2-8}
\hline
\end{tabular}
\end{adjustbox}
\end{table}
\end{document}

If you really want to center the fields...
Note: I used a tabular in \sbox3 mostly to show what you have to do it you are not sure which entry is the largest. Actually, 2.22222 and 2.22 would be close enough (2 is usually the widest digit).
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{multirow, adjustbox, mathtools}
\begin{document}
\begin{table}[htbp]
\begin{adjustbox}{width=\columnwidth,center}
\sbox0{Calculated Difference}%
\sbox1{$\null=\null$}%
\sbox2{$\text{g}^+\text{g}^-$-tt}% left
\sbox3{\begin{tabular}{@{}l@{}}
1.64338\\
3.00859\\
0.80709\\
1.44622\\
2.11342\\
0.71435\\
0.66765\\
1.32002\\
0.22408
\end{tabular}}% right
\dimen0=\dimexpr \wd2 + \wd0 - \wd1 - \wd3\relax
\sbox0{Experimental Difference}%
\sbox3{\begin{tabular}{@{}l@{}}
1.62\\
3.51\\
0.86
\end{tabular}}% right
\dimen1=\dimexpr \wd2 + \wd0 - \wd1 - \wd3\relax
\begin{tabular}{|c|r@{$\null=\null$}l|r@{$\null=\null$}l|r@{$\null=\null$}l|c|}
\hline
Force Field & \multicolumn{2}{c|}{Calculated Energy} & \multicolumn{2}{c|}{Calculated Difference} & \multicolumn{2}{c|}{Experimental Difference} & Error \\
\hline
\multirow{4}{*}{OPLS-AA}
& tt & 2.55447 & \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \\ \cline{2-8}
& $\text{g}^+\text{g}^+$ & 4.19785 & \makebox[0.5\dimen0][r]{$\text{g}^+\text{g}^+$-tt} & 1.64338
& \makebox[0.5\dimen1][r]{$\text{g}^+\text{g}^+$-tt} & 1.62 & 0.02338 \\ \cline{2-8}% note \makebox
& $\text{g}^+\text{g}^-$ & 5.56306 & $\text{g}^+\text{g}^-$-tt & 3.00859 & $\text{g}^+\text{g}^-$-tt & 3.51 & -0.50141 \\ \cline{2-8}
& tg & 3.36156 & tg-tt & 0.80709 & tg-tt & 0.86 & -0.05291 \\ \cline{2-8}
\hline
\multirow{4}{*}{AM1}
& tt & -37.97281 & \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \\ \cline{2-8}
& $\text{g}^+\text{g}^+$ & -36.52659 & $\text{g}^+\text{g}^+$-tt & 1.44622 & $\text{g}^+\text{g}^+$-tt & 1.62 & -0.17378 \\ \cline{2-8}
& $\text{g}^+\text{g}^-$ & -35.85939 & $\text{g}^+\text{g}^-$-tt & 2.11342 & $\text{g}^+\text{g}^-$-tt & 3.51 & -1.39658 \\ \cline{2-8}
& tg & -37.25846 & tg-tt & 0.71435 & tg-tt & 0.86 & -0.14565 \\ \cline{2-8}
\hline
\multirow{4}{*}{PDDG/PM3}
& tt & -34.61227 & \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \\ \cline{2-8}
& $\text{g}^+\text{g}^+$ & -33.94462 & $\text{g}^+\text{g}^+$-tt & 0.66765 & $\text{g}^+\text{g}^+$-tt & 1.62 & -0.95235 \\ \cline{2-8}
& $\text{g}^+\text{g}^-$ & -33.29225 & $\text{g}^+\text{g}^-$-tt & 1.32002 & $\text{g}^+\text{g}^-$-tt & 3.51 & -2.18998 \\ \cline{2-8}
& tg & -34.38819 & tg-tt & 0.22408 & tg-tt & 0.86 & -0.63592 \\ \cline{2-8}
\hline
\end{tabular}
\end{adjustbox}
\end{table}
\end{document}

-and$-$are significantly different symbols. Alas, I tend to forget that myself. – John Kormylo Mar 22 '19 at 16:41