I just purchased a BCD536HP scanner along with a Diamond D3000N antenna. My question is ...what connectors and cable I would need to do this install. The antenna will be mounted about 40 feet away.
A quick look shows your antenna has an N connector. Your scanner has BNC connector.
I'd use LMR-400 cable. (Especially if you'll be listening up in the 700-800 MHz range.
One thing you could do is measure the planned run exactly and then find a place that will custom install male N connectors on both ends of a run of LMR-400 cable.
LMR-400 is pretty stiff, so to take the strain off the BNC connector on the scanner, pick up a short (3 or so foot) patch cable with a female N connector on one end and a male BNC on the other. Similar to this --> BNC Male on LMR200 UltraFlex to N Female Cable Assembly
For moderate length coax runs for a scanner antenna LMR400 is the default based on cost vs performance and the advice above about handing off the a smaller cable at the scanner is very good. Others may recommend 75 ohm RG-6 TV coax to save a few $$, which will work to some extent. I prefer to do things once and do it right, so LMR400 is my best recommendation.
The impedance of the antenna change a lot over its frequency ranges and are up to 2.0 which is either 25 ohm or 100 ohm and the BCD536 also swings between 15 ohm and 110 ohm in UHF and the same in other frequency ranges. I believe that having impedance under control will help a lot. Use a splitter at the scanner that will then isolate the scanners impedance from the coax and use a pre-amplifier at the antenna that will isolate the antenna from the coax and load the antenna with a constant impedance.
Here's a sweep of my Diamond D3000N discone antenna. The antenna uses 75FT of LMR400 at approximately 30FT above ground level. In this sweep, I'm using only a RTL-SDR FM notch filter.