In the Modern Steinitz Defense to the Ruy Lopez there is a line that runs 1. e4 e5 2. Nf3 Nc6 3. Bb5 a6 4. Ba4 d6 5. Bxc6.

What is the motivation for this unprovoked exchange?

I don't claim to be an expert in this line but I am quite sure that the idea is once ...d6 is played by exchanging Black is forced to capture with the b pawn.

Note in the regular exchange line after 4 B:c6 d:c6 is usually played.

---

You may be right. I just don't know why recapturing with the b-pawn is so undesireable for black as to warrent giving up the two bishops.

In the normal exchange variation, black recaptures with the d-pawn for tactical reasons I believe. I may be wrong about this, but if black could recapture with the b-pawn and still regain his center pawn that would probably be preferable (since he would not be giving white a k-side pawn majority), but doing so only allows him to get back a wing pawn for his center pawn.

Correction - I looked up the Exchange Variation w/ 4...bxc6, and it looks as though the reason that is not played is because it transposes to this very line a tempo down for black after 1. e4 e5 2. Nf3 Nc6 3. Bb5 a6 4. Bxc6 bxc6 5. Nf3 d6

What is white's move after bxc6?

---

6. d4 is usual.

After black captures with the b pawn he will have a weak a pawn as well as the doubled pawns on the c file. He now has  pawn islands instead of 2 so his pawn structure is much worse than it is in the normal exchange variation. As to whether this compensates for the loss of tempo and the 2 bishops advantage? Ha - well its not the most popular line is it????

White generally exchanges here because Black has defended his e5 pawn with ...d6, making the threat of ...Nf6xe4 a real one.  White must either then defend the e4 pawn with 5 d3, which spoils his basic plan to play d2-d4, or 5 Nc3, which is not challenging for Black (actually transposing into the Four Knights Game), or exchange at c6.

Bxc6 is the only serious attempt at advantage:  in exchange for giving up the Bishop pair, White doubles Black's c-pawns and makes his d2-d4 both easier and more effective by removing the Nc6's control of the d4 and e5 squares.

The resulting position offers imbalances with chances for both sides.  White has more space and freedom to move in the center, and will swiftly get his King to safety.  Black is cramped, but his only weakness is the doubled c-pawn, which actually helps in central control, and he has the two Bishops which may eventually dominate if the game opens up.  He is a bit behind on development, so White has a slight initiative to pursue.

Errrrm there is no knight at f6. The above mentioned line was:

 

Actually it does make it a lot easier to see what is going on with the diagram...

Quite so, I was thinking of the Berlin move order.  But the basic reasoning is unchanged anyway, and Black will surely play ...Nf6 in the near future.

Now actually 5. Bxc6+ is not the most popular after black playes 4...d6 but it's definitely the most annoying to meet and also the reason why I don't quite like this line for black.

The basic idea behind playing this move is to really fix blacks pawnstructure while he can because black is very likely to play Bd7 next move in order to stop white from doing this. And of course it is also possible to allow black to play Bd7 but then black can choose between some very interesting plans.

Some examples: