The present paper deals with classifying the real quadratic number fields k = Q($\sqrt{d}$) having specific continued fraction expansion of the integral basis element where d ≡ 2, 3(mod4) is a square free positive integer. Certain parametric representations are determined to calculate fundamental unit $\epsilon_d = (t_d + u_d\sqrt{d})/2$ of such real quadratic number fields as well as the parametrized forms of d. Moreover, Yokoi's d-invariants $n_d$ and $m_d$ in the relation to continued fraction expansion of $w_d$ are mentioned by using coefficients of fundamental unit for such real quadratic fields. All results are concluded in the tables.