Coefficients of atomic orbitals

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... 

The SCF LCAO ab initio representation of the molecular electronic density p(r) is a function of the 3D position variable r, and is defined in terms of a set of h atomic orbitals (p,(r), i= 1,2,..., n. The n x n dimensional density matrix P can be computed from the set of self-consistent coefficients of atomic orbitals in the occupied molecular orbitals. In terms of this density matrix P, the electronic density p(r) of the molecule can be written as... 

The first term in Equation 11.1 is the Coulomb integral, 3b apj apj gj f e total charges on atoms k and I e is the local dielectric constant is the distance between atoms k and 1 Cf-a is the coefficient of atomic orbitals (a) in molecular orbital (r), where r refers to the molecular orbitals on one molecule and s refers to those on the other and Ef and Eg are the energies of the molecular orbitals. 

Here the indices A and B refer to atoms, k and I label atomic orbitals centered on atoms A and B, respectively, and i labels the molecular orbitals in the SCF approximation - which may be either doubly or singly occupied depending on the RHF or UHF case, respectively c,vt represents the coefficient of atomic orbital k in the molecular orbital i and Su is the overlap integral between atomic orbitals k and 1. The charge of the atom appears by comparison between the gross population Pa and the atomic number Z ... 

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

As an observation, the coefficients of atomic orbitals in (2.126) are the real ones, i.e., those that fix the size of the lobes of atomic orbitals in associated molecular orbital, therefore having the role of describing the population of those atomic orbitals in molecule. 

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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