A single ground rod, unless in sea water, has a resistance to mother earth
between 50 and 200 ohms. Let's take it to be 100 ohms.

Efficiency of a 1/4-wave vertical, feedpoint resistance = 37 ohms, is 27
percent.

Efficiency of a 5/8-wave vertical, feedpoint resistance = 50 ohms, is 33
percent.

Efficiency of a 1/2-wave vertical, feedpoint resistance = 2500 ohms, is 96
percent.

The difference in radiation pattern in a typical back yard, in the vertical
plane, is neither here nor there.

The 1/2-wave antenna also needs the most simple L and C matching network.

But I'd never recommend a ground rod anyway. Not worth the time, trouble
and expense unless extremely short of real estate at ground level.

Roy, the problem of choice lies in over-complication by too 'clever',
'knowledgeable' old-wives and gurus rather than under-complication.
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Reg, G4FGQ