beunruhigen
to worry
To cause someone unease, concern, or fear.
Die Nachricht von dem Unfall hat mich tief beunruhigt.
The news of the accident deeply worried me.

to worry / unsettle
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to worry
To cause someone unease, concern, or fear.
Die Nachricht von dem Unfall hat mich tief beunruhigt.
The news of the accident deeply worried me.

The verb comes from the noun 'die Unruhe' (unease/restlessness). The prefix 'be-' and the ending '-igen' turn it into a verb meaning 'to cause unease.'
Seine späte Ankunft verursachte große Unruhe.
His late arrival caused great unease.
Pay attention to the difference between the transitive and reflexive uses. In the transitive use, something 'beunruhigt' (worries/unsettles) a person in the accusative; in the reflexive use, a person in the nominative worries themselves.
Sein seltsames Verhalten beunruhigt mich. (transitiv)
His strange behavior worries me. (transitive)
'Beunruhigen' (to worry/unsettle) is an inseparable verb. The prefix 'be-' stays firmly attached to the verb stem in all tenses and sentence constructions.
Sie hat sich wegen der schlechten Nachrichten sehr beunruhigt.
She was very worried because of the bad news.
When something worries a person, the subject is often an abstract concept or a piece of information. Typical examples are: 'eine Nachricht' (a message/news), 'ein Geräusch' (a noise), 'die Stille' (the silence), 'eine Entwicklung' (a development), or 'das Verhalten' (the behavior).
Die plötzliche Stille im Haus beunruhigte ihn mehr als der Lärm zuvor.
The sudden silence in the house worried him more than the noise before.
to occupy / employ
'beschäftigen' and 'beunruhigen' can both have to do with thoughts and worries. 'beschäftigen' means that a topic stays on someone's mind; this can be neutral or slightly worrying. 'beunruhigen' is more clearly negative: something makes someone uneasy, anxious, or worried. If you only want to say that you are thinking about something, 'beschäftigen' fits. If you want to show that something causes worry or fear, 'beunruhigen' is more precise.

Es beunruhigt mich, dass der Zug schon wieder ausfällt.
It worries me that the train has been canceled yet again.
Der lange Stau beschäftigt sie noch im Büro.
The long traffic jam is still on her mind at the office.