Molecular orbital theory Hamiltonian operator

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.. 

In the molecular orbital theory and electronic spectroscopy we are interested in the electronic wave functions of the molecules. Since each of the symmetry operations of the point group carries the molecule into a physically equivalent configuration, any physically observable property of the molecule must remain unchanged by the symmetry operation. Energy of the molecule is one such property and the Hamiltonian must be unchanged by any symmetry operation of the point group. This is only possible if the symmetry operator has values 1. Hence, the only possible wave functions of the molecules are those which are either symmetric or antisymmetric towards the symmetry operations of the... 

As it is well known (see, for example, [16,17]), the Hiickel molecular orbital theory is based on a Hamiltonian operator, ff defined by means of the matrix elements... 

However, there also exists a third possibility. By using a famous relation due to Dirac, the relativistic effects can be (in a nonunique way) divided into spin-independent and spin-dependent terms. The former are collectively called scalar relativistic effects and the latter are subsumed under the name spin-orbit coupling (SOC). The scalar relativistic effects can be straightforwardly included in the one-electron Hamiltonian operator h. Unless the investigated elements are very heavy, this recovers the major part of the distortion of the orbitals due to relativity. The SOC terms may be treated in a second step by perturbation theory. This is the preferred way of approaching molecular properties and only breaks down in the presence of very heavy elements or near degeneracy of the investigated electronic state. 

Hartree-Fock-Roothaan Closed-Shell Theory. Here [7], the molecular spin-orbitals it where the subscript labels the different MOs, are functions of (af, 2/", z") (where /z stands for the coordinate of the /zth electron) and a spin function. The configurational wave function is represented by a single determinantal antisymmetrized product wave function. The total Hamiltonian operator 2/F is defined by... 

In 1969, Thomas published two papers [11,12] in which a molecular structure theory was developed without invoking the Bom-Oppenheimer approximation. In these publications and two further papers published in 1970 [13,14], Thomas studied methane, ammonia, water and hydrogen fluoride adding the kinetic energy operators of the protons to the electronic hamiltonian and using Slater-type orbitals centered on the heavier nuclei for the protonic wave functions. Over the years, a number of authors [15-23] have attempted the development of a non-Bom-Oppenheimer theory of molecular structure, but problems of accuracy and/or feasibility remain for applications to arbitrary molecular systems. 

Extended Hiickel Theory (EHT) uses the highest degree of approximation of any of the approaches we have already considered. The Hamiltonian operator is the least complex and the basis set of orbitals includes only pure outer atomic orbitals for each atom in the molecule. Many of the interactions that would be considered in semi-empirical MO theory are ignored in EHT. EHT calculations are the least computationally expensive of all, which means that the method is often used as a quick and dirty means of obtaining electronic information about a molecule. EHT is suitable for all elements in the periodic table, so it may be applied to organometallic chemistry. Although molecular orbital energy values and thermodynamic information about a molecule are not accessible from EHT calculations, the method does provide useful information about the shape and contour of molecular orbitals. 

M0ller and Plesset proposed an alternative way to tackle the problem of electron correlation [Moller and Plesset 1934] Their method is based upon Rayleigh-Schrodinger perturbation theory, in which the true Hamiltonian operator X is expressed as the sum of a zeroth-order Hamiltonian Xq (for which a set of molecular orbitals can be obtained) and a perturbation, f " ... 

The question as to whether or not orbital angular momentum is a good quantum number in electronic states of molecules, as well as in atomic states, is one that is extraordinarily important in molecular theory. A good quantum number means physically that the dynamical variable is a good constant of the electronic motion. In quantum mechanics a sufficient condition for the conservation of a dynamical variable is that the operator for the variable commutes with the Hamiltonian operator. 

The outline of the review is as follows in the next section (Sect. 2) we introduce the basic ideas of effective Hamiltonian theory based on the use of projection operators. The effective Hamiltonian (1-5) for the ligand field problem is constructed in several steps first by analogy with r-electron theory we use the group product function method of Lykos and Parr to define a set of n-electron wavefimctions which define a subspace of the full -particle Hilbert space in which we can give a detailed analysis of the Schrodinger equation for the full molecular Hamiltonian H (Sect. 3 and 4). This subspace consists of fully antisymmetrized product wavefimctions composed of a fixed ground state wavefunction, for the electrons in the molecule other than the electrons which are placed in states, constructed out of pure d-orbitals on the... 

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