Can hydride shift and methyl shift happen one after another in a carbocation?

Question

Can a hydride shift be followed by a methyl shift for stabilizing a carbocation? In case both are possible, which will occur first?

Answer 1

Multiple rearrangements can happen if the resulting species is stabilized by it. In general hydride shift is more favourable over methyl shift because of its smaller size.

The example proposed by James was not known to me but its truly an amazing example.

James:
It’s possible for multiple hydride/alkyl shifts to occur. One amazing example is in the biosynthesis of lanosterol.

Multiple shifts are certainly possible, and they could happen, but generally will only happen if each shift generates a successively more stable carbocation. For example, you probably wouldn’t see a shift if it involved turning a tertiary carbocation into a secondary carbocation.

Source

From Wikipedia:

Answer 2

Yes it is possible. But it must fulfil some basic condition for that shift.

The empty p-orbital of the carbocation must be in plane with the $\ce{C-H}$ or the $\ce{C-C}$ bond. When the alignment of a carbon–hydrogen bond with a vacant p orbital takes place allowing for hyperconjugation, a "pseudo-double-bond" develops. As illustrated in Figure 5.13, this can be envisioned as a double bond with a closely associated hydrogen ion.

If, as shown in Figure 5.13, hyperconjugation results in the formation of species possessing both double-bond character and associated hydrogen ions, equilibrium-controlled migration of the associated hydrogen ion can be expected. This transformation, shown in Scheme 5.8, is known as a 1,2-hydride shift and results in the migration of a proton from carbon 1 to carbon 2.

While the example illustrated in Scheme 5.8 shows equilibrium between two chemically identical carbocations, there are factors influencing the direction of these transformations when applied to more complex systems. If we consider Scheme 5.9, we notice that the positive charge migrates exclusively to the tertiary center, reflecting the increased stability of tertiary carbocations over primary carbocations. In general, where 1,2-hydride shifts are possible, rearrangement of less stable carbocations to more stable carbocations is expected.

It is important to remember that hydride shifts occur much more readily than the corresponding alkyl shifts. In fact,as a general rule, alkyl shifts will not occur unless a hydride shift cannot take place.

Alkyl migrations occur when the resulting carbocation is more stable than the starting carbocation. But for hydride shift its not such a hard and fast rule.

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Source: "Arrow Pushing in Organic Chemistry, An Easy Approach to Understanding Reaction Mechanisms" by Daniel E. Levy; Copyright 2008 by John Wiley & Sons, Inc.; page 110 in chapter 6