Coefficients of atomic orbitals in molecular

Observables calculated from approximate wavefunctions as in Equation 1.13 are called expectation values, an expression used in probability theory. In practice, we will always have to be satisfied with approximate wavefunctions. How can we choose between different approximations And if our trial wavefunction has adjustable parameters (such as the coefficients of atomic orbitals in molecular orbitals see Section 4.1), how can we choose the adjustable parameters best values Here, Rayleigh s variation theorem is of great value. It tells us that the expectation value for the ground state energy E, (E ), calculated from an approximate wavefunction (P is always larger than the true energy E (Equation 1.14). Proof of the variation theorem is given in textbooks on quantum mechanics.18... 

Table 1. Coefficients of atomic orbitals in molecular orbitals of H2OI...

Thus, a rule of thumb says that valid normal coordinates must be either entirely symmetric or entirely antisymmetric over the most common symmetry operations if m and n label any two internal coordinates related by symmetry, then either Ukm = cikn or akm = —akn- This rule of thumb holds also in the construction of the coefficients of atomic orbitals in molecular orbitals (see Section 3.5). The bending mode of water straddles the mirror plane and is already symmetry-adapted if the displacements of the two hydrogen atoms are equal and in opposite directions (if the motions were in the same direction, that would be a molecular rotation, not a vibration). 

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