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To make my question more clear, I want to ask two questions:

  1. Is it possible for molecule to be achiral even though molecule has chirality center?

  2. Is it possible for molecule to be chiral even though molecule doesn't have chirality center?

I thought of these questions because the definitions of chiral center and chiral do not have a one to one correspondence.

A chiral center is defined by carbon having four different ligands around it and chiral is defined by a compound being non-superimposable with its mirror image. They are quite related, but I think there will be counterexamples that satisfy one condition but not satisfy the other one.

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The answer to both questions is yes.

In regards to the first question, meso compounds like tartaric acid have multiple chiral centers, but are not chiral overall because they have a mirror plane.

For the second question, a compound can be geometrically chiral without having a chiral center. A cool example class of compounds is helicenes, which are helical arrangements of aromatic rings. These compounds have axial chirality, where the stereoisomers are the left and right handed versions of the helix.

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    $\begingroup$ A simpler example for (2) is penta-2,3-diene. Because the methyl groups are strung out along the cumulative double bond axis, not on a single central atom, the molecular symmetry is lowered and one of the symmetry elements that goes missing is a mirror plane. $\endgroup$
    – Oscar Lanzi
    Commented Mar 3, 2019 at 1:31

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