The problem of small oscillations can be solved through the study of molecular vibrations which further, can be introduced by considering the elementary dynamical principles. The solution for the problem of small oscillations can be found out classically, as it is much easier to find its solution in classical mechanics than that in quantum mechanics. One of the most powerful tools to simplify the treatment of molecular vibrations is by use of symmetry coordinates. Symmetry coordinates are the linear combination of internal coordinates and will be discussed later in detail in this chapter.
When a molecule vibrates, the atoms get displaced from their respective equilibrium positions. Consider a set of generalized co-ordinates q_1,q_2,q_3……… q_n (the displacements of the N atoms from their equilibrium positions) in order to formulate the theory of small vibration. As these generalized coordinates do not involve the time explicitly, so classically, kinetic energy (T) is given by
2T = ∑_(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 by
2V (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), the higher order terms can be neglected for sufficiently small amplitudes of vibration. To make coinciding with the equilibrium position, the arbitrary zero of potential must be shifted to eliminate V_0. Consequently the term (∂V/〖∂q〗_i ) becomes zero for the minimum energy in equilibrium. Therefore, the expression of V will be reduced to
2V = ∑_(i,j)▒((∂^2 V)/(〖∂q〗_i 〖∂q〗_j )) q_i q_j (2.04)
2V = ∑_(i,j)▒f_ij q_i ...
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...ke the vanishing determinant, a fixed value of λ = λk, is chosen accordingly. Therefore, at λ = λk, the coefficients of the unknown amplitude Aj in equation (2.12) will become fixed and then it will be possible in obtaining the solution Ajk (the additional subscript k will be used to indicate the correspondence with the particular values of λk). Such a system of equations does not determine the Ajk uniquely but gives their ratios. A convenient mathematical solution designated by the quantities mjk are 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)
Thus the solution of the actual 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 by the initial values of q_j and q ̇_j.
5. Collected Papers, Charles Hartshorne and Paul Weiss, (edd.) (Cambridge: The Belknap Press of Harvard University Press, 1960). Volume and page number, respectively, noted in the text.
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... the vibrations should be equal to the frequency of the IR in order for the radiation to be absorbed. That would modify the amplitude of the molecular vibrations.
2) Fundamentals of Physics Extended: Fifth Edition. David Hanley, Robert Resnick, Jearl Walker. Published by John Wiley & Sons, Inc, New York, Chichester, Brisbane, Toronto, Singapore. 1997.
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