How do the findings of quantum physics affect the theories of classical chemistry?

to a large degree, they explain them. Chemists have long known that certain elements acted similarly to certain other elements, but it took the idea of quantized energy levels to explain why. Similarly, many techniques that chemist now use rely on their understanding of quantum mechanics at some level. NMR, IR, UV-Vis, Fluorescence, AE, Raman, Auger, and other spectroscopies all rely on the concept of quantized energy levels. In the case of NMR this is the quantization of nuclear spin; in IR and Raman it is vibrational states; in UV-Vis, Fluorescence, and AE it is outer electron energy levels; in Auger it is inner electron energy levels.

FYI: for anyone who doesn't know, "quantized" means that only certain states are allowed (e.g. 1, 2, 3, ... but not 1.3485). This is the main principle of quantum mechanics.