The Molecular Hamiltonian and State Wavefunctions

There is a distinction between two groups of methods. The first is the finite field technique.35 In this case finite perturbations representing the external fields are added to the molecular hamiltonian and the calculation of the ground state wavefunction and energy is carried out as for the unperturbed molecule. The finite field method can be applied in conjunction with any quantum mechanical method that is available for molecular calculations. There are two principal subdivisions of the finite field method. In one of these terms of the form tff-Fi... 

In (II) the reaction field of the dipole is included in the molecular Hamiltonian, so that the QM calculation, at whatever level, is modified to give a new molecular wavefunction for one molecule at the centre of a cavity. This calculation can be carried out in the absence of an applied macroscopic field and would give the unperturbed properties (dipole moment, energy states etc) of a solvated molecule. The macroscopic field has then usually been applied in a finite field calculation of the hyperpolarizability. One source of uncertainty in this procedure arises from the fact that when the reaction field is introduced into the hamiltonian it appears in a specific form,... 

Note that here bracket does not mean just any round, square, or curly bracket but specifically the symbols and > known as the angle brackets or chevrons. Then ( /l is called a bra and Ivp) is a ket, which is much more than a word play because a bra wavefunction is the complex conjugate of the ket wavefunction (i.e., obtained from the ket by replacing all f s by -i s), and Equation 7.6 implies that in order to obtain the energies of a static molecule we must first let the Hamiltonian work to the right on its ket wavefunction and then take the result to compute the product with the bra wavefunction to the left. In the practice of molecular spectroscopy l /) is commonly a collection, or set, of subwavefunctions l /,) whose subscript index i runs through the number n that is equal to the number of allowed static states of the molecule under study. Equation 7.6 also implies the Dirac function equality... 

For the electro-nuclear model, it is the charge the only homogeneous element between electron and nuclear states. The electronic part corresponds to fermion states, each one represented by a 2-spinor and a space part. Thus, it has always been natural to use the Coulomb Hamiltonian Hc(q,Q) as an entity to work with. The operator includes the electronic kinetic energy (Ke) and all electrostatic interaction operators (Vee + VeN + Vnn)- In fact this is a key operator for describing molecular physics events [1-3]. Let us consider the electronic space problem first exact solutions exist for this problem the wavefunctions are defined as /(q) do not mix up these functions with the previous electro-nuclear wavefunctions. At this level. He and S (total electronic spin operator) commute the spin operator appears in the kinematic operator V and H commute with the total angular momentum J=L+S in the I-ffame L is the total orbital angular momentum, the system is referred to a unique origin. 

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... 

Inclusion of Pauli s exclusion principle leads to the standard methods of ab initio computational chemistry. Within these methods, molecular systems containing the same nuclei and the same number of electrons, but having a different total electronic spin, can roughly speaking be said to be different systems. Thus, matrix elements of the Hamiltonian between Slater determinants corresponding to different spin states will all be zero, and they will not interact or mix at all. The wavefunctions obtained will be pure spin states. ... 

The model that is outlined above is generated from a one-electron Hamiltonian and is only an approximation to the true wavefunction for a multielectron system. As suggested earlier, other components may be added as a linear combination to the wavefunction that has Just been derived. There are many techniques used to alter the original trial wavefunction. One of these is frequently used to improve wavefunctions for many types of quantum mechanical systems. Typically a small amount of an excited-state wavefunction is included with the minimal basis trial function. This process is called configuration interaction (Cl) because the new trial function is a combination of two molecular electron configurations. For example, in the H2+ system a new trial function can take the form... 

---