Molecular orbitals ground state energy

The simplest molecular orbital method to use, and the one involving the most drastic approximations and assumptions, is the Huckel method. One str ength of the Huckel method is that it provides a semiquantitative theoretical treatment of ground-state energies, bond orders, electron densities, and free valences that appeals to the pictorial sense of molecular structure and reactive affinity that most chemists use in their everyday work. Although one rarely sees Huckel calculations in the resear ch literature anymore, they introduce the reader to many of the concepts and much of the nomenclature used in more rigorous molecular orbital calculations. 

To begin a more general approach to molecular orbital theory, we shall describe a variational solution of the prototypical problem found in most elementary physical chemistry textbooks the ground-state energy of a particle in a box (McQuanie, 1983) The particle in a one-dimensional box has an exact solution... 

In using the concept of molecular orbital theory to discuss the absorption of light by organic molecules, we concentrate on two molecular orbitals in particular. The highest occupied molecular orbital (HOMO) is the ground-state molecular orbital of highest energy with electrons in it and the lowest unoccupied molecular orbital (LUMO) is the... 

When one of the two electrons of opposite spins (belonging to a molecular orbital of a molecule in the ground state) is promoted to a molecular orbital of higher energy, its spin is in principle unchanged (Section 2.3) so that the total spin quantum number (S = Es , with s — I y or 1) remains equal to zero. Because the multi-... 

The variational principle is employed to find the "best set of molecular orbitals which can be represented by a LCAO and lead to a minimum value of the ground-state energy ... 

In the final exploration of the quantum chemistry unit students use a computational chemistry package (eg. Spartan, Gaussian, CaChe, etc.) to calculate the ground state energies, molecular orbitals, and in some cases the excited state energies, of two proton transfer tautomers. Calculations are performed at several different levels of theory, and use both semi-empirical and ab initio methods. Several different basis sets are compared in the ab initio calculations. The students use the results of these calculations to estimate the likelihood of excited state proton transfer. The calculations require CPU time ranging from a couple of minutes to a couple of hours on the PCs available to the students in the laboratory. 

Thus, let us assume that two different external potentials can each be consistent with the same nondegenerate ground-state density po- We will call these two potentials Va and wj, and the different Hamiltonian operators in which they appear and Ht,. With each Hamiltonian will be associated a ground-state wave function Pq and its associated eigenvalue Eq. The variational theorem of molecular orbital theory dictates that the expectation value of the Hamiltonian a over the wave function b must be higher than the ground-state energy of a, i.e.. 

Fig. 4.15 The n molecular orbitals and n energy levels for a two-p-orbital system in the simple Hiickel method. The MOs are composed of the basis functions (two p AOs) and the eigenvectors, while the energies of the MOs follow from the eigenvalues (Eq. 4.66). The paired arrows represent a pair of electrons of opposite spin (in the electronic ground state of the neutral ethene molecule i[/ is occupied and i//2 is empty)...

Photoelectron spectral measurements have prompted high-accuracy near-Hartree-Fock calculations on the Is hole states of 02. 261 Calculations were reported at Re for molecular O2. The frozen-orbital approximation evaluated the energy of Oj from the RHF calculations of Schaefer250 reported above. Then the IP are the difference between the O2 ground-state energy and the Ot energy. The IP obtained was 563.5 eV. Direct hole-state calculations for the relevant states of OJ, with the MO constrained to be of g or a symmetry, were also carried out. For the orbital occupancy (16), the computed IP was 554.4 eV. Finally, the restriction to g and u... 

Fig. 37. Molecular orbitals of the nitro group (a) with the lowest energy level, (b) with the highest energy level, (c) with the ground state energy level [88].

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... 

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