How does Bohr's model fail to account for atomic spectra of multi-electron atoms?

Essentially, the presence of a second electron changes the energy levels, which makes it so that for each #n#, there is more than one energy level.

Bohr's model doesn't account for a single #n# having more than one possible energy; it assumes only one energy level for each #n#.

For #bb"H"# atom only, the first few energy levels are written like this, where bottom to top is increasing in energy:

#ul(" ") " " ul(" ") " " ul(" ") " " ul(" ") " " ul(" ") " " ul(" ") " " ul(" ") " " ul(" ") " " ul(" ")#
#" "" "underbrace(" "" "" "" "" ") " " underbrace(" "" "" "" "" "" "" "" "" ")#
#3s" "" "" "3p" "" "" "" "" "" "" "3d#
#" "#
#" "#
#" "#
#" "#
#ul(" ") " " ul(" ") " " ul(" ") " " ul(" ")#
#" "" "underbrace(" "" "" "" "" ")#
#2s" "" "" "2p#
#" "#
#" "#
#" "#
#" "#
#" "#
#" "#
#" "#
#" "#
#" "#
#ul(" ")#
#1s#

However, when we include a second electron, we include the effect of electron-electron repulsion.

Since electrons generally move very quickly, the repulsions are fast enough to influence the difference in energy levels for the above electron "shells".

Based on the type of shell label (#s,p,d,f,g,...#), these split like so (for a multi-electron atom):

#" "" "" "" "" "" "" "ul(" ") " " ul(" ") " " ul(" ") " " ul(" ") " " ul(" ")#
#" "" "" "" "" "" "" "underbrace(" "" "" "" "" "" "" "" "" ")#
#" "" "" "" "" "" "" "" "" "" "" "3d#
#" "" "ul(" ") " " ul(" ") " " ul(" ")#
#" "" "underbrace(" "" "" "" "" ")#
#" "" "" "" "3p#
#" "#
#ul(" ")#
#3s#
#" "#
#" "#
#" "#
#" "#
#" "" "ul(" ") " " ul(" ") " " ul(" ")#
#" "" "underbrace(" "" "" "" "" ")#
#" "" "" "" "2p#
#" "#
#" "#
#" "#
#" "#
#" "#
#" "#
#" "#
#ul(" ")#
#2s#
#" "#
#" "#
#" "#
#" "#
#" "#
#" "#
#" "#
#" "#
#" "#
#" "#
#" "#
#" "#
#ul(" ")#
#1s#

(for this diagram, these are just relative energy differences.)

As a result, the simple #n = 1, 2, 3, . . . , oo# labels for each energy level no longer work for multi-electron atoms, because there is no longer just one energy level for a single #bb(n)#.

Since there are now energy "sublevels", the electromagnetic spectra of atoms is not so easily predicted unless one knows exactly how the energy levels have split.