We know that there are those who light chanukah candles at sunset and other at tzeis but there are some poskim lighting between at an intermediate time, for example :
- Rav Moshe Feinstein, Rav Elyashiv => 10 minutes after sunset
- Rav Yaakov Kamenetsky, Chazon Ish => 20 minutes after sunset
- Rav Aaron Kotler => 15 minutes before tzeis
I read that for the 10 minutes of Rav Moshe Feinstein it is based on the following reasoning :
- tzeis = 50 minutes after sunset = Rabbenu Tam 72 minutes adjusted downwards for NY = 4 mil
- so 3/4 mil = 10 minutes
- so geonim tzeis is 10 minutes past sunset
Thereby lighting 10 minutes after shkia we are yotzei both opinions : we light just after sunset & we light at (geonim) tzeis.
For Chazon Ish reasoning here what I've found : The Chazon Ish zt'l lit the Chanukah lights on weekdays twenty minutes after sunset. On Motzei Shabbos, however, he lit them only after ma'ariv (which he davened approximately three quarters of an hour after sunset) and making havdoloh. [The report in Tohoras Habayis (part 2, p. 273) that the Chazon Ish lit Chanukah lights twenty minutes after sunset even on Motzei Shabbos--from which the author deduces that melochos may be done this soon after sunset when there is a pressing necessity--is erroneous].
The source of the minhag of the Chazon Ish of when to light Chanukah lights on weekdays is found in the Bei'ur HaGra in Yoreh Dei'ah, Hilchos Milah chap. 266:17, who explicitly mentions Chanukah lights!
The Chazon Ish based his reasoning on the fact that there must be some darkness for Chanukah lighting, since "what good is a candle during the day?" In central Eretz Yisroel we light 20-21 minutes after sunset (depending when Chanukah falls according to the solar calendar). The sun is at that time 4.8 degrees below the horizon.
Using the sun's position below the horizon as our criterion, these are the times to light Chanukah lights in selected cities worldwide according to the Chazon Ish: Zurich 27-28 minutes after sunset; Vienna 28-29 minutes; Sydney 22 minutes; Strasbourg 28 minutes; Stockholm 43-44 minutes; Rome 24-25 minutes; Paris 28-29 minutes; New York 23-24 minutes; Gateshead 35 minutes; Melbourne 23-24 minutes; Manchester 33 minutes; Los Angeles 21-22 minutes; London 31 minutes; Johannesburg 20 minutes; Cleveland 24 minutes; Chicago 24-25 minutes; Buenos Aires 22-23 minutes; Amsterdam 32 minutes.
The custom in Yerushalayim, based on the traditions of the talmidei Hagra, is to light at sunset.
See Birur Halocho Teliso'oh, Orach Chaim 772
Do you have any idea for the other rabbonim calculations ?