Molecular Hamiltonian terms

The molecular Hamiltonian terms resulting from the relativistic approach are collected in Table 4.6. 

The Dirac equation represents a proper relativistic form of the characteristic equation for energy. It is fulfilled for state vectors in a form of a four-component spinor. Since the upper two-component spinor dominates in the positive energy solutions for an electron, a decomposition of the Dirac equation is appropriate. 

In the presence of the electromagnetic potential a new term appears in the quadratic form of the Dirac equation. This allows an introduction of the intrinsic magnetic moment of an electron, generated by its spin, which interacts with the external magnetic field. 

The decoupling of the Dirac equation to the two-component form is a rather complicated process. It manipulates the resolvant operator which is transferred from the denominator to the numerator and then exposed to consecutive commutator relations in order to be shifted to the far right. Finally, a number of one-electron terms of the order 1/c2 is obtained some of them have no classical analogy and cannot be derived from non-relativistic theories. 

The Lorenz transformation requires some additional terms in the electron-electron interaction resulting in the Breit operator. The two-electron Breit Hamiltonian consists of the Dirac Hamiltonian for the individual electrons plus the Breit operator. The decoupling of the Breit equation to the upper-upper subspace of interest results in the appearance of several new Hamiltonian terms. 

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The mixing has nothing to do with the possibility of any molecules populating the term, which is typically 12,000 cm" above the ground state term. The population of such a term is of the order -12000/200 room temperature kT 200 cm" at 300 K), which is absolutely negligible. The mixing arises because a description of the molecular Hamiltonian in terms of Eq. (5.14) is incomplete and should be replaced with Eq. (5.15). 

In this admittedly extremely brief and hermetic summary of the quantum mechanics of static molecules, the key issue for us is that Equations 7.6 and 7.10 imply that, provided we know what the static molecular Hamiltonian H looks like and provided we can write down any set l, we can always obtain all the molecular energies E (and therefore the molecular spectrum) by computing all n2 terms... 

Before investigating the qualitative concepts of the VSEPR model it is worth noting that the details of the interactions between the electron pairs have been ascribed to a size-Pauli exclusion principle result . But objects do not repel each other simply because of their sizes (i.e. interpenetrations) only if the constituents of the objects interact is any interaction possible10). If we are to use the idea of orbital size at all we must avoid the danger of contrasting a phenomenon (electron repulsion) with one of its manifestations (steric effects). The only quantitative tests which we can apply to the VSEPR model are ones based on the terms in the molecular Hamiltonian specifically, electron repulsion. 

In general, fluctuations in any electron Hamiltonian terms, due to Brownian motions, can induce relaxation. Fluctuations of anisotropic g, ZFS, or anisotropic A tensors may provide relaxation mechanisms. The g tensor is in fact introduced to describe the interaction energy between the magnetic field and the electron spin, in the presence of spin orbit coupling, which also causes static ZFS in S > 1/2 systems. The A tensor describes the hyperfine coupling of the unpaired electron(s) with the metal nuclear-spin. Stochastic fluctuations can arise from molecular reorientation (with correlation time Tji) and/or from molecular distortions, e.g., due to collisions (with correlation time t ) (18), the latter mechanism being usually dominant. The electron relaxation time is obtained (15) as a function of the squared anisotropies of the tensors and of the correlation time, with a field dependence due to the term x /(l + x ). 

The symbol V(q,Q) stands for a kinematic operator containing spin-orbit terms, electron-phonon couplings and, eventually, a coupling to external fields. The molecular Hamiltonian is given by ... 

In this chapter, we are dealing only with spin wave functions and Hamiltonians. In the complete molecular Hamiltonian, which involves electronic spatial coordinates and momenta in addition to spin coordinates, the contact term in the Hamiltonian would have to be written in a form such that the integral over the electronic spatial wave function J/,... 

It is very important, in the theory of quantum relaxation processes, to understand how an atomic or molecular excited state is prepared, and to know under what circumstances it is meaningful to consider the time development of such a compound state. It is obvious, but nevertheless important to say, that an atomic or molecular system in a stationary state cannot be induced to make transitions to other states by small terms in the molecular Hamiltonian. A stationary state will undergo transition to other stationary states only by coupling with the radiation field, so that all time-dependent transitions between stationary states are radiative in nature. However, if the system is prepared in a nonstationary state of the total Hamiltonian, nonradiative transitions will occur. Thus, for example, in the theory of molecular predissociation4 it is not justified to prepare the physical system in a pure Born-Oppenheimer bound state and to force transitions to the manifold of continuum dissociative states. If, on the other hand, the excitation process produces the system in a mixed state consisting of a superposition of eigenstates of the total Hamiltonian, a relaxation process will take place. Provided that the absorption line shape is Lorentzian, the relaxation process will follow an exponential decay. 

Rapid molecular motions in solutions average to zero the dipolar and quadrupolar Hamiltonian terms. Hence, weak interactions (chemical shift and electron-coupled spin-spin couplings) are the main contributions to the Zeeman term. The chemical shift term (Hs) arises from the shielding effect of the fields produced by surrounding electrons on the nucleus ... 

The Born-Oppenheimer approximation permits the molecular Hamiltonian H to be separated into a component H, that depends only on the coordinates of the electrons relative to the nuclei, plus a component depending upon the nuclear coordinates. This in turn can be wriuen as a sum Hr + H, of terms for vibrational and rotational motion of ihe nuclei. 

The standard quantum chemical model for the molecular hamiltonian Hm contains, besides purely electronic terms, the Coulomb repulsion among the nuclei Vnn and the kinetic energy operator K]. The electronic terms are the electron kinetic energy operator Ke and the electron-electron Coulomb repulsion interaction Vee and interactions of electrons with the nuclei, these latter acting as sources of external (to the electrons) potential designated as Ve]q. The electronic hamiltonian He includes and is defined as... 

Just like the helium Hamiltonian, the molecular Hamiltonian H in Eq. 5.15 is composed (from left to right) of electron kinetic energy terms, nucleus-electron... 

Among the alternative definitions of the dispersion-repulsion energy, we mention the quantum mechanical approach presented in ref. [12], which has the merit of including this part of the nonelectrostatic contribution in the molecular Hamiltonian (like the electrostatic term), so that the solvent affects not only the free energy but also the electronic distribution. In this approach, however, the dispersion part is highly expensive except for very small systems, and it is not routinely used in any computational package the analytical derivatives of quantum mechanical /disp rep can be derived but they have not been implemented until now. 

The use of the exact Hamiltonian for calculating matrix elements between VB determinants leads, in the general case, to complicated expressions involving numerous bielectronic integrals, owing to the 1/r,-,- terms. Thus, for practical qualitative or semiquantitative applications, one uses an effective molecular Hamiltonian in which the nuclear repulsion and the 1/r,-, terms are only implicitly taken into account, in an averaged manner. Then, one defines a Hamiltonian made of a sum of independent monoelectronic Hamiltonians, much as in simple MO theory ... 

Whereas the quantum-mechanical molecular Hamiltonian is indeed spherically symmetrical, a simplified virial theorem should apply at the molecular level. However, when applied under the Born-Oppenheimer approximation, which assumes a rigid non-spherical nuclear framework, the virial theorem has no validity at all. No amount of correction factors can overcome this problem. All efforts to analyze the stability of classically structured molecules in terms of cleverly modified virial schemes are a waste of time. This stipulation embraces the bulk of modern bonding theories. 

All the terms in the molecular Hamiltonian transform as the totally symmetric irreducible representation (irrep) of the molecular point group. Symbolically,... 

A, H2 and H2.—H+ and H2 are particularly important molecules, which have been used both for testing new methods in quantum chemistry and also for investigation of the inclusion of small terms in the molecular Hamiltonian. The extensive earlier work has been reviewed by Kolos.22 28... 

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