Otdfbt

I have this simple code,

documentclass[12pt]book
usepackagemathtools 
usepackagenewtxtext
usepackage[lite]mtpro2
begindocument
$dbarp$ and $dt$.
enddocument

with this output,

I think it's a canonical space is due to the presence of the bar... command? Sometimes to bring it back I insert a negative space ! to bring the p to the left near the d.

This little adjustment must always be done or I should create a particular macro. Which one could be always using the same command bar? It gets a bit complicated for me in my book to change the original command bar.... Thank you.

Using Steven Segletes' suggestion of ooalign$dbar#1$cr$d#1$, one can see if the problem is universal across bar. If the superposition produces a "thick" second letter, the kerning is not correct. (using newtxmath as Steven does not have mtpro2)

Unfortunately, that proves not to be the case, as some cases seem kerned correctly and others not:

documentclass[12pt]book
usepackagemathtools 
usepackagenewtxtext
usepackagenewtxmath%instead of usepackage[lite]mtpro2
newcommandtestkern[1]%
 ooalign$dbar#1$cr$d#1$
textwidth 1in
begindocument
noindenttestkerna
testkernb
testkernc
testkernd
testkerne
testkernf
testkerng
testkernh
testkerni
testkernj
testkernk
testkernl
testkernm
testkernn
testkerno
testkernp
testkernq
testkernr
testkerns
testkernt
testkernu
testkernv
testkernw
testkernx
testkerny
testkernz
enddocument

Applying a universal -1mu kern to bar

newcommandxbarmkern-1mubar
newcommandtestkern[1]%
 ooalign$dxbar#1$cr$d#1$

therefore, may fix some letter combinations, but will inevitably break others:

A proposal to renew the command of bar as follows:

letsvbarbar
renewcommandbar[1]%
 #1ThisStylesetbox0=hbox$SavedStyle#1$kern-wd0svbar,phantom#1!

This will guarantee proper kerning of the letter, as it is set first thing without the bar. Then, I kern backward across the letter, and apply the bar to a variant of the letter's phantom. In this case, the variant is a shifted phantom, which is then unshifted following the bar, so as not to introduce any stray space.

Therefore, if there are residual kerning issues, they will be with the placement of the superimposed bar, relative to the underlying properly kerned letter.

The MWE uses the ooalign test given in the question to judge the quality of the bar macro, using letter boldness as an indicator for bad kerning. As you can see, there are no thick/boldened letters signifying a kern mismatch. Therefore, with this approach, one must merely judge if the placement of the bar relative to the letter is suitable or not.

While all examples are shown relative to the letter "d", the method, in fact, will perform similarly, regardless of the preceding letter.

EDITED to work across math styles.

documentclass[12pt]book
usepackagemathtools 
usepackagenewtxtext
usepackagenewtxmath%instead of usepackage[lite]mtpro2
usepackagescalerel
letsvbarbar
renewcommandbar[1]%
 #1ThisStylesetbox0=hbox$SavedStyle#1$kern-wd0svbar,phantom#1!
newcommandtestkern[2][]%
 ooalign$#1 dbar#2$cr$#1 d#2$
textwidth 1in
begindocument
noindenttestkerna testkernb testkernc testkernd
testkerne testkernf testkerng testkernh testkerni
testkernj testkernk testkernl testkernm testkernn
testkerno testkernp testkernq testkernr testkerns
testkernt testkernu testkernv testkernw testkernx
testkerny testkernz

noindenttestkern[scriptstyle]a testkern[scriptstyle]b 
testkern[scriptstyle]c testkern[scriptstyle]d
testkern[scriptstyle]e testkern[scriptstyle]f 
testkern[scriptstyle]g testkern[scriptstyle]h 
testkern[scriptstyle]i testkern[scriptstyle]j 
testkern[scriptstyle]k testkern[scriptstyle]l 
testkern[scriptstyle]m testkern[scriptstyle]n
testkern[scriptstyle]o testkern[scriptstyle]p 
testkern[scriptstyle]q testkern[scriptstyle]r 
testkern[scriptstyle]s testkern[scriptstyle]t 
testkern[scriptstyle]u testkern[scriptstyle]v 
testkern[scriptstyle]w testkern[scriptstyle]x
testkern[scriptstyle]y testkern[scriptstyle]z

noindent$2 dbarp^2ne2d p^2$
enddocument

If I compile

documentclass[12pt]book
usepackagemathtools
usepackagenewtxtext
usepackage[lite]mtpro2
begindocument

$dbarp$

$dp$

showoutput

enddocument

I get, for the two formulas

....mathon
....LMP1/mtt/m/it/12 d
....kern1.43999
....vbox(7.88399+2.568)x7.98
.....hbox(7.88399+0.0)x0.0, shifted 3.312
......LMP2/mtt/m/n/12 N
.....kern-5.484
.....hbox(5.484+2.568)x7.98
......LMP1/mtt/m/it/12 p
....mathoff

....mathon
....LMP1/mtt/m/it/12 d
....kern1.43999
....kern-2.40001
....LMP1/mtt/m/it/12 p
....kern0.48
....mathoff

After removing the call to mtpro2, I get

....mathon
....OML/cmm/m/it/12 d
....vbox(6.77774+2.33331)x5.89717
.....hbox(6.77774+0.0)x0.0, shifted 0.99028
......OT1/cmr/m/n/12 ^^V
.....kern-5.16667
.....hbox(5.16667+2.33331)x5.89717
......OML/cmm/m/it/12 p
....mathoff

....mathon
....OML/cmm/m/it/12 d
....OML/cmm/m/it/12 p
....mathoff

The obvious difference is that d in the font LMP1/mtt/m/it/12 (pointing to mt2mit at 12pt) has an italic correction, which isn't present in the OML/cmm/m/it/12 font (pointing to cmsy10 at 12pt).

Indeed, if I do tftopl mt2mit, I get

(CHARACTER C d
 (CHARWD R 0.551)
 (CHARHT R 0.717)
 (CHARIC R 0.12)
 (COMMENT
 (KRN O 0 R -0.06)
 [...]
 (KRN C p R -0.2)
 [...]
 (KRN C t R -0.04)
 [...]
 (KRN O 263 R -0.02)
 )
 )

and with tftopl cmsy10 I get

(CHARACTER O 100
 (CHARWD R 0.611113)
 (CHARHT R 0.694445)
 )

As you can see, there is also a kerning pair with p or t, which moves in the character when there is no accent, but this is not possible when the next item is an Acc atom.

You can manually compute the appropriate kerning.

documentclass[12pt]book
usepackagemathtools 
usepackagenewtxtext
usepackage[lite]mtpro2

newcommanddwithbar[1]%
 dcomputedaccentkern#1bar#1%

makeatletter
newcommandcomputedaccentkern[1]%
 mathpalettecomputedaccentkern@#1%

newcommandcomputedaccentkern@[2]%
 begingroup
 sboxz@$m@th#1d#2$
 sboxtw@$m@th#1dkern0pt#2$%
 kerndimexprwdz@-wdtw@relax
 endgroup

makeatother


begindocument

$dbarp$ $scriptstyle dbarp$

$dwithbarp$ $scriptstyle dwithbarp$

$dp$ $scriptstyle dp$

$dbart$ $scriptstyle dbart$

$dwithbart$ $scriptstyle dwithbart$

$dt$ $scriptstyle dt$

enddocument

You can define similarly for other accents, say dot:

newcommanddwithdot[1]%
 dcomputedaccentkern#1dot#1%

or a generic

newcommanddwithacc[2]%
 dcomputedaccentkern#2#1#2%

to be called like dwithaccbarp or dwithaccdotp.