Converting and dealing with inverse centimeters(cm-1),nanometer(nm-1),seconds(s-1)?

I assume you are referring to interconverting between ##lambda##, ##nu##, and ##tildenu##, or something like that? I say this because ##lambda## is typically wavelength in ##”nm”##, ##nu## is typically frequency in ##”s”^-1##, and ##tildenu## means energy in wavenumbers (##”cm”^-1##). If you want ##”nm”^(-1)##, just reciprocate ##”nm”##.
There are four possibilities for conversions that I could cover:
##lambda->nu##
##nu->tildenu##
##nu->lambda##
##tildenu->nu##
However, recognize that if you can do 1 and 2, you have done 3 and 4 backwards, and if you can do 1 and 2 consecutively, you can go straight from ##lambda## to ##tildenu## (same with 3 and 4 but ##tildenu## to ##lambda##). So, I will only show 1 and 2.
##lambda->nu##
Suppose we have ##lambda = 600nm## for yellow light and we want its frequency in ##s^-1##.
What we want is to convert from a unit of length to a unit of reciprocal time, which requires something that has ##”length”/”time”## units… The speed of light works great here, and it’s about ##3xx10^8 “m/s”##. Therefore:
Reciprocate the wavelength
Convert to ##”m”##
Multiply by the speed of light
##overbrace((1/(600 cancel(“nm”))))^(lambda)xx((10^9 cancel(“nm”))/(1 cancel(“m”)))xx(3xx10^8 cancel(“m”)/”s”) = underbrace(color(blue)(5.bar55xx10^(-3) “s”^-1))_(nu)##
##nu->tildenu##
This is fairly straightforward. We have ##1/”s”## and want ##1/”cm”##. Suppose we have a frequency of ##6xx10^(-3) “s”^-1##.
Divide by the speed of light
Convert to ##”cm”##
##overbrace((6xx10^(-3) 1/cancel(“s”)))^(nu)xx(cancel(“s”)/(3xx10^8 cancel(“m”)))xx((1 cancel(“m”))/(100 “cm”)) = underbrace(color(blue)(2xx10^(-13) “cm”^(-1)))_(tildenu)##

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