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Ideas of Quantum Chemistry P34

Ideas of Quantum Chemistry P34 shows how quantum mechanics is applied to chemistry to give it a theoretical foundation. The structure of the book (a TREE-form) emphasizes the logical relationships between various topics, facts and methods. It shows the reader which parts of the text are needed for understanding specific aspects of the subject matter. Interspersed throughout the text are short biographies of key scientists and their contributions to the development of the field. | 296 7. Motion of Nuclei R Xl X2 X3 X4 X5 X6 . X3N T dV 1 d2V dXj 0Xi Æ dXdXj 0XiXj 7 3 i - ij J where x R - Ro is the vector with the displacements of the atomic positions from their equilibria xi Xi - Xi 0 for i 1 . 3N while the derivatives are calculated at R Ro. In Ro all the first derivatives vanish. According to the harmonic approximation the higher order terms denoted as are neglected. In effect we have R x V Ro 2 xV j x x 7-4 ij x j o In matrix notation we have V Ro x V Ro 2xTV x where V is a square force constant matrix of the Cartesian force constants V ij XX- The Newton equations of motion for all the atoms of the system can be written in matrix form as x means the second derivative with respect to time t MX -V x 7.5 where M is the diagonal matrix of the atomic masses the numbers on the diagonal are M1 M1 M1 M2 M2 M2 . because we calculate the force component along the axis k as dV dxk Xj - V x k. 0 We may use the relation M 2 M 2 M M 2 M 2 x -M 2 M - 2 V M - 2 M 2 x 7.6 where M 2 is a matrix similar to M but its elements are the square roots of the atom masses instead of the masses while the matrix M- 2 contains the inverse square roots of the masses. The last equation after multiplying from the left by . _ M 2 gives y -Ay 7.7 ._ where y M 2 x and A M 2 V M 2. 7.5 Small amplitude harmonic motion - normal modes 297 Let us try to find the solution in the form46 y ci exp iwt c2exp -jwt where the vectors cj of the dimension 3N of the complex coefficients are time independent. The coefficients cj depend on the initial conditions as well as on the A matrix. If we say that at time t 0 all the atoms are at equilibrium i.e. y t 0 0 then we obtain the relation c1 -c2 leading to the formula y L sin wt 7.8 where the vector47 L and w depend on the matrix A. Vector L is determined only to the accuracy of a multiplication constant because multiplication of L by any number does not interfere with satisfying 7.7 . When we insert the proposed solution 7.8 in 7.7 we .