Electronic Hamiltonian, use

The stationary points obtained by computational procedures on the PES are for vibrationless molecular systems. The electronic Hamiltonian used in ab initio calculations gives the total electronic energy, Eeiec. A real molecule, however, has vibrational energy even at 0 K, which is the quantum mechanical (QM) zero-point energy (ZPE), l/ihv. At absolute zero, the internal energy, Eo, is defined as the computed electronic energy plus the zero-point energy. 

Now that each Fock matrix contains the one-electron Hamiltonian, use grhfGR to add in the relevant electron-repulsion contributions. Also set up the virtual Fock matrix. 

We consider the familiar case of spin space first. In this case, the vector h in equation (2) is given by an external magnetic field B, hQ vanishes, and Hi represents the spin Zeeman term in the one-electron Hamiltonian. Using for the electron spin magnetic dipole moment... 

The wave functions iI/a and iI/b were obtained from open-shell SCF calculations based on the 11-valence-electron Hamiltonian, using the face-to-face geometry and imposing Th symmetry on the atomic configuration of each inner-sphere complex and a common point characteristic of both reactants in the activated complex. 

What remains is the calculation of the two radial functions Fn,K,(t) and Qn,)c, ( )- For this task the coupled first-order differential SCF equations have to be solved. The explicit form of these equations depends on the many-electron Hamiltonian used to calculate the total electronic energy. For the Dirac-Coulomb Hamiltonian in combination with the general multi-... 

The adiabatic picture is the standard one in quantum chemistry for the reason that, not only is it mathematically well defined, but it is also that used in ab initio calculations, which solve the electronic Hamiltonian at a particular nuclear geometry. To see the effects of vibronic coupling on the potential energy surfaces one must move to what is called a diabatic representation [1,65,180, 181]. 

We write them as i / (9) to shess that now we use the space-fixed coordinate frame. We shall call this basis diabatic, because the functions (26) are not the eigenfunction of the electronic Hamiltonian. The matrix elements of are... 

For vei y small vibronic coupling, the quadratic terms in the power series expansion of the electronic Hamiltonian in normal coordinates (see Appendix E) may be considered to be negligible, and hence the potential energy surface has rotational symmetry but shows no separate minima at the bottom of the moat. In this case, the pair of vibronic levels Aj and A2 in < 3 become degenerate by accident, and the D3/, quantum numbers (vi,V2,/2) may be used to label the vibronic levels of the X3 molecule. When the coupling of the... 

The principal semi-empirical schemes usually involve one of two approaches. The first uses an effective one-electron Hamiltonian, where the Hamiltonian matrix elements are given empirical or semi-empirical values to try to correlate the results of calculations with experiment, but no specified and clear mathematical derivation of the explicit form of this one-electron Hamiltonian is available beyond that given above. The extended Hiickel calculations are of this type. 

The implementation of the method using ab initio methods for the quantum region is straightforward. The analogous equations for the electronic Hamiltonian and the corresponding energies in this case are [51]... 

Using the notation given above for the one- and two-electron operators, the electronic Hamiltonian is... 

The aja, operator tests whether orbital i exists in the wave function, if that is the case, a one-electron orbital matrix element is generated, and similarly for the two-electron terms. Using the Hamiltonian in eq. (C.6) with the wave function in eq. (C.4) generates the first quantized operator in eq. (C.3). 

For this simple Hamiltonian, which involves the sum of one-electron Hamiltonians, we can use a wave function of the form... 

The usual EVB procedure involves diagonalizing this 3x3 Hamiltonian. However, here we wish to use a very simple model for our reaction and represent the potential surface and wavefunction of the reacting system using only two electronic states. Using a two-state system will preserve most of the important features of the potential energy surface while at the same time provide a simple model that will be more amenable to discussion than the three-state system. For the two-state system we define the following states as the reactant and product wavefunctions ... 

The relevant Hamiltonian for the gas-phase solute molecules can be treated by the same three-orbitals four-electron model used in Chapter 2. Since the energy of 3 is much higher than that of , and d>2 (see Table 2.4), we represent the system by its two lowest energy resonance structures, using now the notation fa and fa as is done in eq. (2.40). The energies of these two effective configurations are now written as... 

If we except the Density Functional Theory and Coupled Clusters treatments (see, for example, reference [1] and references therein), the Configuration Interaction (Cl) and the Many-Body-Perturbation-Theory (MBPT) [2] approaches are the most widely-used methods to deal with the correlation problem in computational chemistry. The MBPT approach based on an HF-SCF (Hartree-Fock Self-Consistent Field) single reference taking RHF (Restricted Hartree-Fock) [3] or UHF (Unrestricted Hartree-Fock ) orbitals [4-6] has been particularly developed, at various order of perturbation n, leading to the widespread MPw or UMPw treatments when a Moller-Plesset (MP) partition of the electronic Hamiltonian is considered [7]. The implementation of such methods in various codes and the large distribution of some of them as black boxes make the MPn theories a common way for the non-specialist to tentatively include, with more or less relevancy, correlation effects in the calculations. 

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. 

In contrast to the useful conceptual framework provided by the approximate approach just described, the results of more detailed molecular orbital calculations have on the whole been rather disappointing. Thus, although some semi-empirical SCF treatments were attempted, most of the earlier MO calculations for metallocene systems (18, 161, 162, 163, 164,165) suffered from such deficiencies as the neglect of the a-framework, or the use of various one-electron Hamiltonians, for example the various Wolfsberg-Helmholz techniques. Of late, Drago and his coworkers have carried out further Extended Htickel type computations for a wide range of both metallocene and bis-arene species (153, 154), and similar... 

In practice, each CSF is a Slater determinant of molecular orbitals, which are divided into three types inactive (doubly occupied), virtual (unoccupied), and active (variable occupancy). The active orbitals are used to build up the various CSFs, and so introduce flexibility into the wave function by including configurations that can describe different situations. Approximate electronic-state wave functions are then provided by the eigenfunctions of the electronic Hamiltonian in the CSF basis. This contrasts to standard HF theory in which only a single determinant is used, without active orbitals. The use of CSFs, gives the MCSCF wave function a structure that can be interpreted using chemical pictures of electronic configurations [229]. An interpretation in terms of valence bond structures has also been developed, which is very useful for description of a chemical process (see the appendix in [230] and references cited therein). 

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