Nicht & helper verb word order

In Lesson 6.9, I'm having trouble understanding the proper word order, especially with regard to "nicht" or the specification of a time or place.  I'm not sure when a word order is incorrect, vs just one of several acceptable alternatives.
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In the sentences

Er darf die Tür nicht öffnen.
Er darf das Telefon nicht benutzen.
Du musst das nicht essen.
Sie dürfen hier nicht rauchen.

"nicht" comes near the end, before the infinitive.

In the sentences

Du musst nicht zu Hause bleiben.
Es darf nicht mehr als fünf Seiten haben.
Er muss nicht den Zug nehmen.

"nicht" comes after the helper verb instead of before the infinitive.

Is there a rule governing the placement of "nicht" here, or is this a something that can be said both ways.  For example:

Er darf die Tür nicht öffnen.
Er darf nicht die Tür öffnen.
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Regarding time or place, the translation of "I have to work this evening" is given as

Heute Abend muss ich arbeiten.

Would

Ich muss heute Abend arbeiten.

be equally correct?
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It gets even more complicated when you combine "nicht" with time or place.  For example, are both of these correct:

Sie dürfen hier nicht rauchen.
Hier dürfen Sie nicht rauchen.

Or perhaps even:
Sie dürfen nicht hier rauchen.

This presents a similar issue:

Morgen muss er nicht arbeiten.

Are these alternatives also correct:

Er muss Morgen nicht arbeiten.
Er muss nicht Morgen arbeiten.
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Thanks.

Hallo RobertD50,

Thank you for these questions!

There are a few placement rules for nicht which should help answer your questions.

1.
"Er darf die Tür nicht öffnen." - "He is not allowed to open the door."
"Er darf das Telefon nicht benutzen." - "He is not allowed to use the telephone."
"Du musst das nicht essen." - "You don't have to eat that."
"Sie dürfen hier nicht rauchen." - "You (formal) are not allowed to smoke here."

This is the standard placement of nicht when infinitives are present so nicht goes in front of them which means that the infinitive is what is being negated.

2.
"Er muss nicht den Zug nehmen." - "He doesn't have to take the train."

Here nicht is not in front of the infinitive which actually changes the meaning of the sentence. So in this case "the train" is what is being negated which implies that the person doesn't have to take that specific train but maybe the next one.

3.
"Du musst nicht zu Hause bleiben." - "You don't have to stay at home."
For sentences with prepositional phrases, nicht will go in front of the preposition.

"Es darf nicht mehr als fünf Seiten haben." - "It cannot have more than five pages."
In this example, nicht comes before the adverb which means it is negating "mehr als fünf Seiten" specifically.

4.
"Er darf die Tür nicht öffnen." and "Er darf nicht die Tür öffnen." are both correct. As explained above, "the train" is what is being negated.

5.
"Heute Abend muss ich arbeiten." and  "Ich muss heute Abend arbeiten." are both correct.

6.
"Sie dürfen hier nicht rauchen.",  "Hier dürfen Sie nicht rauchen." and "Sie dürfen nicht hier rauchen." are all correct. However, the third phrase wouldn't be as common as the other two and it is also negating and emphasising the word "hier" like saying "You are not allowed to smoke here but over there is fine."

7.
This is similar to the above.
"Morgen muss er nicht arbeiten." and "Er muss morgen nicht arbeiten." would be the most common phrases to say but "Er muss nicht morgen arbeiten." is also correct when you want to negate and put emphasis on the word "morgen".

I hope this clears things up! There is also a lesson on nicht which will give you more information on this topic: Lesson 5.6 'Making Things Negative: NICHT and KEIN'
You can also have a look at Lesson 7.6 'Word Order Rules'.

Viele Grüße
Julia
 

Julia,

Thank you for taking the time w/ these detailed explanations.  It’s helping me understand when I’m saying something wrong vs an acceptable, or at least understandable, variant.  At the risk of sounding like I’m trying to split hairs, re:

"Er muss nicht den Zug nehmen." - "He doesn't have to take the train."

How would you translate "Er muss den Zug nicht nehmen."?

In my mind, in English saying "He doesn't have to take the train."
means/implies
“He doesn’t have to go by train; he could take the bus, plane, or any other mode of transportation he prefers.”
—not really
“He doesn’t have to take this train, he could take that train.”  But I guess this starts to get into using definitive articles to mean this or that, a topic of another lesson.

Again, thanks for your patience.

Hallo RobertD50,

In the sentence "Er muss den Zug nicht nehmen." nicht goes in front of the infinitive and since nehmen is the main verb of the sentence, the entire sentence is basically being negated.
I would still translate it the same while also keeping in mind that the meaning is slightly different as that person might be able to go on the bus or in a taxi instead of taking the train or he might not need any form of transportation at all. 

For now I would suggest not to get stuck on the finer points and focus on learning to put nicht in front of infinitives as this would be most common but also keep in mind that when nicht is placed in front of something else then that part of the sentence is being negated and the emphasis is being put on that part of the sentence specifically. 

I know this seems very complex and maybe a bit overwhelming but it does get easier and will eventually all make sense.

Grüße
Julia