http://milesmathis.com/tide.html
The solar tide on the Moon should be 171 times smaller than the terrestrial effect. More importantly, the visible tide on the Moon should be symmetrical front and back. Is this what we find? Not at all. The Moon rotates relative to the Sun, so we would not expect to find a solar effect on the Moon, beyond a tiny constant shift in the crust opposite the direction of this rotation. The rotation of the Moon on its axis relative to the Sun does not cause a further tide from the Sun, or add to the tidal effect, but it acts to shift the tides we have already calculated, just as the rotation of the Earth shifts the ocean tides, causing them to travel.