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Here's an incorrect derivation of the conversion from rows of a Snellen chart to diopters -- this derivation results in the correct sign but too small a value:

Interpret "20/40 correction" to mean that when a lens of power d_lens is placed in front of the eye, objects at 12 m form an image at the same location (retina) as objects at 6 m would for the naked eye. Assume the ciliary muscles do not change the power of the naked eye, d_eye, when viewing objects at 6 m or 12 m. So we have the two equations:

1/6 + 1/v = d_eye; and 
1/12 + 1/v = d_lens + d_eye; 

Solving them yields:

d_lens = -1/12 = -0.08

The results of converting between rows of a snellen chart and diopters is indicated in the tables at these two links: How to Convert Diopters to 20/20 Vision and Conversion Table for Distance Vision

Question: What is the derivation of the relationship between rows of a snellen chart and diopters?

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