Dear Karsten and co-authors,

Here are some comments on the paper.  The first set are things which probably
should be changed, the second set are more a matter of opinion or style
and can be ignored if you like.  I have not finished going through some
of the sections assigned to other folks, but will do so and send a second note
if I see anything.  I'll make sure you get it by noon.

You did a great job!  

I am sorry but I did not cross-check my corrections against others.  
I'm hoping you'll quickly see/recall which ones you already made.

Best Regards,
Richard

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p2-3: the input parameters are not accurately known and hence need to be
      adjusted.
      Is this the key point or is it the gross simplification, eg reduction
      in number of bands, nature of interaction, etc that is the restriction?

p6:    to leading order in $Z$ ($Z$: number of nearest-neighbor sites),  
       immediately below you use Z for the component of Hund's rule coupling,
       and later for the partition function.

p9:    Define Z in denominator of equation 20?

p9:     will be discussed in more *details* in
        details--->detail

p10:    atomic limit for the self-energy *off* half-filling\cite{Kajueter},
        I have not had a chance to look up the citation, but I thought
        somehow it was in the case of exactly two bands and at half filling
        (not off) that perturbation theory accidently worked also at
        strong coupling.

p11:     onto a sum of non-interacting problems by 
       would be better to say noninteracting problems where the single
       particle moves in a flcutuating, time-dependent field.

p11:     by a sum of $\Lambda M (2M-1)$ *non-interacting* auxiliary
        I am not sure what 'non-interacting' means here.  It is true that
        if one had a phonon field coupled to electrons, the phonon
        field would have its own dynamics apart from the electrons,
        whereas the HS field only has the fermion determinant.  But still
        the HS fields do interact when the fermions are integrated out.
        I'd eliminate the word 'noninteracting'.  (There are in fact 
        formulations of HS transformations where the HS fields do interact,
        eg see Fahy and Hamann.)

p11:    In Eq 25 the sum over m',s',m'',s'' seems confusing.  Where are
        these variables in the object being summed?

p12:    Since the sum in Eq. (\ref{hidimsum}) consists of $2^{\lambda M(2M-1)}$
        should \lambda be \Lambda?

p14:    band-width and also offers an in other situations.
        something missing here?

p14:    Again, $H_{LDA}^0(\vec{k})$, $\Sigma(z)$ and hence $G^0_{\sigma}(\zeta)$
        \vec{k}  ---> {\bf k}

p16:    *conception-al* simplifications arise: (i) the subtraction of
        conceptual

p31:   M. Born und R. Oppenheimer, *1927*, Ann. Phys. (Leibzig) {\bf 84},~457.
       In other citations, the year s at the end.

p32:   An introduction *into* LDA+DMFT less complete than the present Pr
       into --> to

p32:   Phys. Rev. B {\bf 58}, 12749 (1998)).
       extra paranthesis at end

p33:   Universit\"at Hamburg 1998, *Shaker* Verlag, Aachen 1999.
       Is it really 'Shaker' and not 'Springer'?

p34:   Citations 57 and 33 are the same.

p34:   2082 (1995); J. Schlipf, M. Jarrell, P. G. J. van Dongen, N. *Bl~Aümer*,

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The following are points which do not need to be addressed, but struck
me as possible places where a different style might be considered:

p1:   Is it a bit unbalanced that Chapter 2 has a detailed breakdown
      into sections but not the other chapters?  One could consider
      breaking chapter 5, for example, into spectral function and energy,
      or whatever.

p2:   Because of this, one *(generally)* either needs to make substantial
      remove parantheses?

p4:   it. Instead of minimizing $E[\rho ]$ *w.r.t.* $\rho $ one minimizes
      suggest writing out 'with respect to'

p8:   Hence, it does not contain true many-body physics. While LDA+U is
      I thought that what the word 'it' was referring to was not completely
      clear.  Maybe it ---> LDA+U?

p8:   make the unit matrix '1' in equation (19) and othe locations boldface?

p9:   Figure 2 does not carry a lot of content.  Eliminate?

p12:  The review of Monte Carlo seems unnecessary to me.  

p13:    Figure 5:   Did you want to write something in the 'no' box at the 
        bottom like 'keep s_old'?