The problem of small oscillations can be solved through the study of molecular vibrations which, further, can be introduced by considering the elementary dynamic principles. The solution to the problem of small oscillations can be found classically, since it is much easier to find it in classical mechanics than in quantum mechanics. One of the most powerful tools for simplifying the treatment of molecular vibrations is the use of symmetry coordinates. Symmetry coordinates are the linear combination of internal coordinates and will be discussed in detail later in this chapter. When a molecule vibrates, the atoms are displaced from their equilibrium positions. We consider a set of generalized coordinates q_1,q_2,q_3……… q_n (the displacements of the N atoms from their equilibrium positions) to formulate the theory of small vibrations. Since these generalized coordinates do not explicitly involve time, then classically the kinetic energy (T) is given by2T = ∑_(i,j)▒〖k_ij (q_i ) ̇ 〗 (q_j ) ̇ (2.01) where k_ij = (∂ ^2 T)/(∂(q_i ) ̇∂(q_j ) ̇ ) (2.02) and potential energy, V is given by2V (q_1,q_2,q_3…q_n )=2V_0+2∑_i▒(∂V/〖∂ q〗_i ) q_i+∑_(i,j)▒((∂^2 V)/(〖∂q〗_i 〖∂q〗_j )) q_i q_j+ higher order terms (2.03)In the expression of potential energy (V ) given by equation (2.03), higher order terms can be neglected for sufficiently small vibration amplitudes. To match the equilibrium position, the arbitrary zero of the potential must be moved to eliminate V_0. Consequently the term (∂V/〖∂q〗_i ) becomes zero for the minimum equilibrium energy. Therefore, the expression for V will be reduced to2V = ∑_(i,j)▒((∂^2 V)/(〖∂q〗_i 〖∂q〗_j )) q_i q_j (2.04)2V = ∑_( i,j)▒f_ij q_i ...... middle of the paper ......k and the vanishing determinant, a fixed value of λ = λk, is chosen accordingly. Therefore, for λ = λk, the coefficients of the unknown amplitude Aj in equation (2.12) will become fixed and then it will be possible to obtain the solution Ajk (the additional subscript k will serve to indicate the correspondence with the particular values ​​of λk). Such a system of equations does not uniquely determine the Ajk but provides their relationships. A convenient mathematical solution designated by the quantities mjk is defined in terms of an arbitrary solution A_jk^' by the formula m_jk = (A_jk^')/[∑_j▒(A_jk^' )^2 ]^(1⁄2) ( 2.14) These amplitudes are normalized∑_j▒(m_jk )^2 =1 (2.15)So the solution of the real physical problem can be obtained by taking A_jk= N_k m_jk (2.16) where N_k are the constants and can be determined from the initial values ​​of q_j eq ̇_j.