Various authors have different views regarding stability order of the benzyl and t-butyl carbocations.
$$\ce{PhCH2+ ; (CH3)3C+}$$
In my opinion, resonance effect dominates, so the benzylic carbocation should be more stable. But in the other case, both the inductive and hyperconjugation effects are present, which stabilize the intermediate carbocation.
What should is the correct stability order? Especially when you consider them in SN1 reactions, and what their effect on their rate is.
I am using a very simplistic quantum chemical approach of the following isodesmic reaction:
I have used Gaussian 16 Rev. A.03 and the DF-B97D3/def2-TZVPP level of theory. The summaries of the calculations are included below.
On this level of theory the depicted reaction has an energy change of $\Delta G = \pu{- 37.1 kJ mol-1}.$ Therefore one could assume that the 2-methylpropan-2-ylium cation is more stable than the phenylmethylium cation. These values were estimated at $T = \pu{298.15 K}$ and $p = \pu{1 atm}.$
One certainly can do more calculations, but it is, however, a start.
Calculation summaries
INFO : Found route section: #P B97D3/Def2TZVPP int(ultrafinegrid) freq geom=check guess=read gfinput gfoldprint iop(6/7=3) symmetry(none) ---- calculation details : RB97D3 phch2_q1/b97d3tzvpp.freq.log temperature (T): 298.150 K pressure (p): 1.00000 atm electr. en. (E): -270.5777462860 hartree zero-point corr. (ZPE): +0.114824 hartree/particle thermal corr. (U): +0.120571 hartree/particle ther. corr. enthalpy (H): +0.121515 hartree/particle ther. corr. Gibbs en. (G): +0.085614 hartree/particle entropy (total) (S tot): +75.561 cal/(mol K) heat capacity (total) (Cv t): +22.578 cal/(mol K) ==== Next file ==== INFO : Found route section: #P B97D3/Def2TZVPP int(ultrafinegrid) freq geom=check guess=read gfinput gfoldprint iop(6/7=3) symmetry(none) ---- calculation details : RB97D3 phch3_q0/b97d3tzvpp.freq.log temperature (T): 298.150 K pressure (p): 1.00000 atm electr. en. (E): -271.4854244270 hartree zero-point corr. (ZPE): +0.125049 hartree/particle thermal corr. (U): +0.131428 hartree/particle ther. corr. enthalpy (H): +0.132372 hartree/particle ther. corr. Gibbs en. (G): +0.093359 hartree/particle entropy (total) (S tot): +82.110 cal/(mol K) heat capacity (total) (Cv t): +23.831 cal/(mol K) ==== Next file ==== INFO : Found route section: #P B97D3/Def2TZVPP int(ultrafinegrid) freq geom=check guess=read gfinput gfoldprint iop(6/7=3) symmetry(none) ---- calculation details : RB97D3 t-c4h10_q0/b97d3tzvpp.freq.log temperature (T): 298.150 K pressure (p): 1.00000 atm electr. en. (E): -158.4201368210 hartree zero-point corr. (ZPE): +0.128849 hartree/particle thermal corr. (U): +0.134606 hartree/particle ther. corr. enthalpy (H): +0.135550 hartree/particle ther. corr. Gibbs en. (G): +0.101200 hartree/particle entropy (total) (S tot): +72.297 cal/(mol K) heat capacity (total) (Cv t): +20.521 cal/(mol K) ==== Next file ==== INFO : Found route section: #P B97D3/Def2TZVPP int(ultrafinegrid) freq geom=check guess=read gfinput gfoldprint iop(6/7=3) symmetry(none) ---- calculation details : RB97D3 t-c4h9_q1/b97d3tzvpp.freq.log temperature (T): 298.150 K pressure (p): 1.00000 atm electr. en. (E): -157.5167928240 hartree zero-point corr. (ZPE): +0.114033 hartree/particle thermal corr. (U): +0.120557 hartree/particle ther. corr. enthalpy (H): +0.121501 hartree/particle ther. corr. Gibbs en. (G): +0.083653 hartree/particle entropy (total) (S tot): +79.659 cal/(mol K) heat capacity (total) (Cv t): +20.965 cal/(mol K)
(Compiled with tools-for-g09.bash, which somewhat surprisingly works for g16.)
Optimised geometries
14 phch2_q1/b97d3tzvpp.freq.xyz C 0.08663 -0.19347 0.09384 C 0.05691 -0.11475 1.46208 H 1.02702 -0.19311 -0.45239 H -0.82919 -0.25883 -0.48899 C -1.20323 -0.11721 2.16356 C -1.22180 -0.03857 3.53614 C -0.00341 0.04387 4.24119 C 1.24453 0.04873 3.58467 C 1.28552 -0.02918 2.21254 H -2.12409 -0.18167 1.59151 H -2.16067 -0.03932 4.07940 H -0.02695 0.10566 5.32621 H 2.15902 0.11360 4.16439 H 2.23039 -0.02773 1.67720 15 phch3_q0/b97d3tzvpp.freq.xyz C 0.00007 0.00000 -0.04694 C 0.00682 -0.00000 1.46143 H 1.01682 -0.00000 -0.44951 H -0.51968 -0.88199 -0.43936 H -0.51968 0.88199 -0.43936 C -1.19805 -0.00000 2.17854 C -1.20438 -0.00000 3.57251 C 0.00015 0.00000 4.27958 C 1.20614 -0.00000 3.57927 C 1.20606 -0.00000 2.18275 H -2.14093 0.00000 1.63572 H -2.14998 0.00000 4.10835 H -0.00257 0.00000 5.36594 H 2.14940 0.00000 4.11910 H 2.15082 0.00000 1.64397 14 t-c4h10_q0/b97d3tzvpp.freq.xyz C 0.00369 -0.00664 0.00284 C -0.01764 0.03020 1.53527 C 1.44139 0.03012 -0.52791 C -0.74698 -1.23338 -0.52790 H -1.04360 0.03406 1.92005 H 0.49077 -0.85020 1.94859 H 0.49182 0.92075 1.92008 H 1.46219 0.03417 -1.62347 H 1.97408 0.92055 -0.17569 H 2.00042 -0.85041 -0.18649 H -0.76068 -1.24961 -1.62346 H -0.26410 -2.15775 -0.18616 H -1.78452 -1.24933 -0.17586 H -0.51485 0.89146 -0.36388 13 t-c4h9_q1/b97d3tzvpp.freq.xyz C -0.00184 -0.00056 0.03904 C 0.01681 1.45949 0.05366 H -0.46367 1.80346 -0.88071 H 1.00698 1.90257 0.15089 H -0.66770 1.83073 0.83269 C 1.25179 -0.74789 0.02785 H 1.76765 -0.48991 -0.91708 H 1.93393 -0.35706 0.79738 H 1.13840 -1.82916 0.09679 C -1.27605 -0.71190 0.03180 H -1.24998 -1.55839 -0.66820 H -1.35257 -1.20345 1.02233 H -2.15046 -0.07921 -0.11824
(g09.chk2xyz does not work for g16.)
Is the trimethyl carbocation more stable than the benzylic carbocation?
There are a number of approaches we can take to try and answer this question. We'll start by first comparing solvolysis rate data to see which carbocation is more stable in solution, and then we can look at thermochemical data to see how the carbocation stabilities compare in the gas phase.
Solution Stability
By comparing the rates at which two compounds solvolyze we can infer which compound leads to the more stable carbocation. For example, allyl chloride solvolyzes ~8.5 times faster than i-propyl chloride (1) in agreement with the idea that the allyl carbocation is more stable than the 2-propyl carbocation. Of course, reactions must be run under conditions to minimize any non-solvolytic pathways ($\mathrm{S_{N}2}$) and insure that an $\mathrm{S_{N}1}$ mechanism is operating.
Jones further reports (1) that at $\pu{45 ^\circ C}$ in 50% $\ce{EtOH}$, t-butyl chloride solvolyzes almost 20,000 times faster than i-propyl chloride. This is in accord with the expectation that a tertiary carbocation is more stable than a secondary carbocation.
\begin{array}{|c|c|c|c|} \hline \ce{R-X} & \mathrm{k_{rel}} \\ \hline \ce{iPr-Cl} & 1 \\ \hline \ce{tBu-Cl} & 1.76 \times 10^4 \\ \hline \end{array}
Later in the book (2), we see that benzyl chloride solvolyzes 145 times faster than i-propyl chloride. at first glance this seems to tell us that (using isopropyl chloride as a common reference point) t-butyl chloride solvolyzes ~120 times $\left(\frac{1.76 \times 10^4}{145}\right)$ faster than benzyl chloride.
However this solvolysis is run at slightly lower temperature ($\pu{25 ^\circ C}$) and in pure $\ce{EtOH}$.
\begin{array}{|c|c|c|c|} \hline \ce{R-X} & \mathrm{k_{rel}} \\ \hline \ce{iPr-Cl} & 1 \\ \hline \ce{PhCH2-Cl} & 145 \\ \hline \end{array}
If we were to raise the reaction temperature to $\pu{50 ^\circ C}$ (supply more thermal energy to the reaction), this would tend to decrease the difference in relative rates. Similarly, since the dielectric constant of water is greater than the dielectric constant of ethanol, and since a higher dielectric constant facilitates ionization, if we were to rerun the second set of reactions in water-ethanol, both reaction rates would be enhanced and the difference in relative rates would decrease. So both the reaction temperature and solvent dielectric effects operate in the same direction; if we were to rerun this second set of reactions under conditions identical to the first set of reactions we would expect the relative rate to be something less than 145. If the relative rate for the second set of reactions is really 100 than t-butyl chloride would solvolyze ~176 times $\left(\frac{1.76 \times 10^4}{100}\right)$ faster than benzyl chloride. If instead of 100, the relative rate for the second set of reactions is really only 10, then we would estimate that t-butyl chloride solvolyzes ~1,760 times $\left(\frac{1.76 \times 10^4}{10}\right)$ faster than benzyl chloride.
In any case, the t-butyl chloride solvolyzes faster than benzyl chloride, suggesting that the t-butyl carbocation is slightly more stable than the benzyl carbocation in solution.
Gas-Phase Stabilty
Let's examine the following gas phase reactions. $$\ce{t-Bu-H -> t-Bu^{+} + H^{-}}$$ $$\ce{PhCH2-H -> PhCH2^{+} + H^{-}}$$ We are looking for the energy difference between them, so when we subtract them the $\ce{H^{-}}$ term cancels out. NIST provides the standard heat of formation of gaseous isobutane as ~$\pu{ -32 kcal/mol}$, while that for gaseous toluene is ~$\pu{ 12 kcal/mol}$. Using this information along with the thermochemical data provided in this answer by user55119 for the corresponding ions leads to an estimated difference in stability of ~$\pu{ 6 kcal/mol}$ (~$\pu{13 kcal/mol}$ if we use $\pu{162 kcal/mol}$ as the heat of formation of the t-butyl cation; see user55119's comment below) favoring the t-butyl carbocation. The same general result as we found above in solution, now the magnitude is larger since there is no solvent to stabilize the ions in the gas-phase.
References
Gas phase measurements give:
\begin{align} \Delta_\mathrm{f} H^\circ (\ce{PhCH2+}) &= \pu{+219 kcal mol-1} \\ \Delta_\mathrm{f} H^\circ (\ce{t-C4H9+}) &= \pu{+169 kcal mol-1} \end{align}
The tertiary-butyl cation is seemingly more stable than the benzyl cation in the gas phase. These data do not speak to the condensed phase [1].
Addendum (12/14/2017): The latest data for the gas phase heats of formation of these two cations is here along with other radicals and cations. I thank G. B. Ellison, University of Colorado, for these data.
\begin{array}{llcclc} \hline & \text{Radical} & \Delta_\mathrm{f} H_{298} (\pu{kcal mol-1}) & \qquad & \text{Cation} & \Delta_\mathrm{f} H_{298} (\pu{kcal mol-1}) \\ \hline \text{methyl} & \ce{CH3} & 35.06 \pm 0.07 & & \ce{CH3+} & 261.9 \pm 0.1 \\ \text{$t$-butyl} & \ce{C(CH3)3} & 11.9 \pm 0.2 & & \color{red}{\ce{C(CH3)3+}} & \color{red}{166.4 \pm 0.7} \\ \text{benzyl} & \ce{C6H5CH2} & 50.5 \pm 0.2 & & \color{red}{\ce{C6H5CH2+}} & \color{red}{217.6 \pm 0.2} \\ \text{tropyl} & \ce{C7H7} & 66.5 \pm 0.3 & & \ce{C7H7+} & 210.0 \pm 0.3 \\ \hline \end{array}
Benzylic is more stable because of the priority given to resonance. Like when inductive effect, hyperconjugation and resonance occur together, resonance is given preference then hyperconjugation and then inductive effect. In this case it's resonance. Moreover the explanation of aromaticity also favours this answer.
From my knowledge, the benzylic carbocation should be more stable. This is due to the fact that it is aromatic as well. Aromaticity further enhances stability.
