TOWARD UNDERSTANDING VIBRATIONS OF POLYATOMIC MOLECULES

For a polyatomic molecule in the Born-Oppenheimer approximation, the quantum-mechanical virial theorem takes the form V = 2W+∑α<β Rαβ(∂W/∂Rαβ) or T=-W- +∑α<βRαβ(∂W/∂Rαβ), where W is the vibrational potential-energy function, expressed as a function of the internuclear distances Rap, and V and T are the electronic potential energy and electronic kinetic energy, as functions of Rαβ. If W for a particular case has the form W = W(0)+W(-1)+W(-2) + ⋯, (A) where W(P) is a function of the Rαβ homogeneous of degree P, then V=2W(0)+W(-1) +⋯ (B) and T=-W(0) +W(-2)+⋯. (C) Also, (B) would imply (A) and (C), and (C) would imply (A) and (B). These facts suggest use of empirical vibrational potential functions of tie form (A), for then the terms W (-1) will represent the Coulomb-like conformation-dependent part of the electronic potential energy, and the terms W(-2) will be the particle-in-a-box-like conformation-dependent part of the electronic kinetic energy. To illustrate the method, the vibrations of the CO2 molecule are treated. With R1 and R1 the two CO distances, R2 the CO distance, and γ the OCO angle, the vibrational potential is written as W=W0+W1(R 1-1+R2-1)+W11(R 1-2+R2-2)+W111(R 1-3+R1-3)+W3(R 3-1) +W12(R1+R2) -2+Wγ(R1R1)-1 tan 2[1/2(π-γ)]. (D) Values of the six (five independent) parameters W1 to Wγ are found which approximate 12 constants in the Overend-Suzuki valence-force potential function. Values of force constants determined from (D) are as follows (empirical values in parentheses): quadratic constants, 10.3 (10.3), 1.7 (1.7), 0.4 (0.4); cubic constants, -29.6 (-29.6), -2.9 (-2.9), -1.0 (-0.7); quartic constants, 57.2 (47.3), 4.6 (4.3), 3.1 (6.0), 1.6 (1.1), 0.4 (3.5), 0.1 (0.0).