Effective nuclear Hamiltonian

One of the most interesting aspects of the Interacting Boson Model concerns its connections with the underlying fermion space. The understanding of the mechanism through which bosonlike features arise from effective nuclear hamiltonians provides, in fact, a way to relate collective spectra to the fermion motion. 

The right member of this equation contains an operator which depends only on the nucler coordinates, X, and on the quantum number n. As a result, the following approximation may be written for an effective nuclear Hamiltonian operator ... 

The Group Theory for Non-Rigid Molecules considers isoenergetic isomers, and the interconversion motions between them. Because of the discernability between the identical nuclei, each isomer possesses a different electronic Hamiltonian operator in (3), with different eigenfunctions, but the same eigenvalue. In contrast, a non-rigid molecule has an unique effective nuclear Hamiltonian operator (7). 

The Born-Oppenheimer regime is attained when the electronic eigenenergies ( (0) =i are well separated so that the nuclear motion takes place on a single electronic potential energy surface, let us say E (0). This may be described by the effective nuclear Hamiltonian... 

Accordingly, E is also used as the effective potential for the nuclear Hamiltonian ... 

In this review, we have mainly studied the correlation energy connected with the standard unrelativistic Hamiltonian (Eq. II.4). This Hamiltonian may, of course, be refined to include relativistic effects, nuclear motion, etc., which leads not only to improvements in the Hartree-Fock scheme, but also to new correlation effects. The relativistic correlation and the correlation connected with the nuclear motion are probably rather small but may one day become significant. 

A simple example of an effective operator with which the reader will be familiar is the use of Zeff e r as the effective nuclear potential experienced by an electron outside of a closed inner shell. Thus, we may compute the energies and wavefunctions for a 2s or 2p electron outside a shell, using the hydrogen-like Hamiltonian,... 

Moreover, for the observables depending on external electric field, its specific effect has to be investigated the electric field induces new terms in the nuclear Hamiltonian, due to the change of equilibrium geometry and the nuclear motion perturbation. Pandey and Santry (14) has brought to the fore this effect and calculated the correction which only concerns the parallel component. It is represented by the following expression ... 

If the electric quadrupole splitting of the 7 = 3/2 nuclear state of Fe is larger than the magnetic perturbation, as shown in Fig. 4.13, the nij = l/2) and 3/2) states can be treated as independent doublets and their Zeeman splitting can be described independently by effective nuclear g factors and two effective spins 7 = 1/2, one for each doublet [67]. The approach corresponds exactly to the spin-Hamiltonian concept for electronic spins (see Sect. 4.7.1). The nuclear spin Hamiltonian for each of the two Kramers doublets of the Fe nucleus is ... 

Technically, the time-independent Schrodinger equation (2) is solved for clamped nuclei. The Hamiltonian is broken into its electronic part, He, including the nuclear Coulomb repulsion energy, and the nuclear Hamiltonian HN. At this level, mass polarization effects are usually neglected. The wave function is therefore factorized as usual (r,X)= vP(r X)g(X). Formally, the electronic wave function d lnX) and total electronic energy, E(X), are obtained after solving the equation for each value of X ... 

In our discussion the usual Born-Oppenheimer (BO) approximation will be employed. This means that we assume a standard partition of the effective Hamiltonian into an electronic and a nuclear part, as well as the factorization of the solute wavefunction into an electronic and a nuclear component. As will be clear soon, the corresponding electronic problem is the main source of specificities of QM continuum models, due to the nonlinearity of the effective electronic Hamiltonian of the solute. The QM nuclear problem, whose solution gives information on solvent effects on the nuclear structure (geometry) and properties, has less specific aspects, with respect the case of the isolated molecules. In fact, once the proper potential energy surfaces are obtained from the solution of the electronic problem, such a problem can be solved using the standard methods and approximations (mechanical harmonicity, and anharmonicity of various order) used for isolated molecules. The QM nuclear problem is mainly connected with the vibrational properties of the nuclei and the corresponding spectroscopic observables, and it will be considered in more detail in the contributions in the book dedicated to the vibrational spectroscopies (IR/Raman). This contribution will be focused on the QM electronic problem. 

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 ... 

In more physical language, the effective rotational hamiltonian in each vibrational state is obtained by averaging the original hamiltonian over the vibrational co-ordinates using the true vibrational wavefunctions, obtained by an appropriate perturbation of the harmonic oscillator basis functions. It is an extension of the Bom-Oppenheimer separation of the electronic from the nuclear motion, to achieve a separation of the vibrational from the rotational motion. 

We introduced the field-free nuclear Hamiltonian in section 3.10. Again by analogy with the electronic Hamiltonian, we include the effects of external magnetic fields by replacing P, by P, — Z,eA l in equation (III.248) and the effects of an external electric field by addition of the term Y,a Zae(pa, this treatment is only really justified if the nuclei behave as Dirac particles. The nuclear Zeeman Hamiltonian is thus ... 

The most important terms in the effective hyperftne Hamiltonian are those which describe the nuclear quadrupole and nuclear spin-rotation interactions ... 

We therefore adapt the locally quadratic Hamiltonian treatment of Gaussian wave packets, pioneered by Heller [18], to a system with an induced adiabatic vector potential. The locally quadratic theory replaces the anharmonic time-independent nuclear Hamiltonian by a time-dependent Hamiltonian which is taken to be of second order about the instantaneous center of the wave packet. Since the nuclear wave packet continually evolves under an effective harmonic Hamiltonian, an initially Gaussian wave form remains Gaussian. The treatment yields equations of motion for the wave function parameters that can be solved numerically [36-38]. The locally quadratic Hamiltonian includes a second order expansion of the scalar potential, consisting of the last three terms in Eq. (2.18), which we write as... 

In this connection, it is of interest to note that when nonadiabatic effects are small, as in strongly coupled systems at low energy or substantial vibronic angular momentum, a semi-classical quantization scheme beginning with the adiabatic nuclear Hamiltonian yields fairly... 

The effective nuclear motion Hamiltonian, depending only on the q, is expressed in terms of matrix elements of the Hamiltonian just as before, between pairs of functions like (21) but with internal coordinate parts like (24) integrated over the z as well as the angular factors. Doing this yields an equation rather like (23) but with coupling between different electronic states, labelled by p. The term analogous to (23) become... 

In this Section we will derive the effective rotational Hamiltonian within the rigid nuclear frame approximation, i.e., under the simplifying assumption that the nuclei may be considered as frozen at their equilibrium positions within the molecule, (For a discussion of vibrational effects compare Appendix III.) Furthermore all intramolecular magnetic interactions are neglected since they lead only to comparatively small shifts and splittings of the rotational absorption lines, which in most cases cannot be observed with the standard resolution of a micro-wave spectrograph. 

Table 3.1 Effective Nuclear Charge (Scaling Parameter) Z ff for Approximate Spin-Orbit Interaction Calculations Using the One-Electron Term in the Breit-Pauli Hamiltonian (Equation 3.8, developed by Koseki et...

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