Hamiltonian function description

In this regard, we should notice that the time evolution of a quantum system is ruled by two different types of eigenvalues corresponding to the wave function and the statistical descriptions. On the one hand, we have the eigenenergies of the Hamiltonian within the wave function description. On the other hand, we have the eigenvalues of the Landau-von Neumann superoperator in the Liouville formulation of quantum mechanics. These quantum Liouvillian eigenvalues j are related to the Bohr frequencies according to... 

The traditional a—ir description of multiple bonding and the expectation that conformational effects could somehow be derived from a molecular Hamiltonian function can both safely be discounted. Looking for a fresh approach in terms of quantum potential, returns the argument to the general form of polar wave functions... 

Here, attention will be drawn to the important work of Rosina [79] and to the subsequent discussion of his study by Mazziotti [80], As summarized in Ref. [80], Rosina showed that the ground-state 2 DM for a quantum system completely determines the exact N-electron ground-state wave function without any specific knowledge of the exact Hamiltonian except that it has no more than two-particle interactions. Mazziotti [80] points out that a consequence of this theorem is that any ground-state electronic 2 DM precisely determines within the ensemble N-representable space a unique series of higher p-DMs where 2 < p < N. He asserts that these results provide important justification for the functional description of the higher DMs in terms of the 2 DM. 

As before, we need a functional description of the wavefunction at f = 0,4 (x, 0). We will then expand that in terms of eigenfunctions of the free-particle hamiltonian. As we have seen in Section 2-5, the free-particle eigenfunctions may be written exp( i-Jim Ex/h), where E is any noimegative number. These are also eigenfunctions for the momentum operator, with eigenvalues -JlmEh. 

Example 9.1 illustrates that the three different equations of motion produce the same description for the motion of a system, only by different routes. Why present three different ways of doing the same thing Because all three ways are not equally easy to apply to all situations Newton s laws are most popular for straight-line motion. However, for other systems (like systems involving revolution about a center) or when knowledge of the total energy of a system is important, the other forms are more appropriate to use. We will find later that for atomic and molecular systems, the Hamiltonian function is used almost exclusively. 

By use of the Hamiltonian description established in this subsection, it can be shown that the Hamiltonian equations are equivalent to the more familiar Newton s second law of motion in Newtonian mechanics, adopting a transformation procedure similar to the one used assessing the Lagrangian equations. In this case we set qi = Ti and substitute both the Hamiltonian function H (2.24) and subsequently the Lagrangian function L (2.6) into one of Hamilton s equations of motion (2.29). The preliminary results can be expressed as ... 

The most commonly used semiempirical for describing PES s is the diatomics-in-molecules (DIM) method. This method uses a Hamiltonian with parameters for describing atomic and diatomic fragments within a molecule. The functional form, which is covered in detail by Tully, allows it to be parameterized from either ah initio calculations or spectroscopic results. The parameters must be fitted carefully in order for the method to give a reasonable description of the entire PES. Most cases where DIM yielded completely unreasonable results can be attributed to a poor fitting of parameters. Other semiempirical methods for describing the PES, which are discussed in the reviews below, are LEPS, hyperbolic map functions, the method of Agmon and Levine, and the mole-cules-in-molecules (MIM) method. 

During the evolution of the current semi-empirical methods (AMI, PM3) a number of refinements were made to the core-repulsion function in order to improve, for example, the description of hydrogen-bonding [20, 21]. Such changes have in general led to increased flexibility within the modified semi-empirical Hamiltonians resulting in quite marked improvements in the accuracy of the parent methods. 

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). 

The solute-solvent system, from the physical point of view, is nothing but a system that can be decomposed in a determined collection of electrons and nuclei. In the many-body representation, in principle, solving the global time-dependent Schrodinger equation with appropriate boundary conditions would yield a complete description for all measurable properties [47], This equation requires a definition of the total Hamiltonian in coordinate representation H(r,X), where r is the position vector operator for all electrons in the sample, and X is the position vector operator of the nuclei. In molecular quantum mechanics, as it is used in this section, H(r,X) is the Coulomb Hamiltonian[46]. The global wave function A(r,X,t) is obtained as a solution of the equation ... 

The hfs (or quadrupole) tensors of geometrically (chemically) equivalent nuclei can be transformed into each other by symmetry operations of the point group of the paramagnetic metal complex. For an arbitrary orientation of B0 these nuclei may be considered as nonequivalent and the ENDOR spectra are described by the simple expressions in (B 4). If B0 is oriented in such a way that the corresponding symmetry group of the spin Hamiltonian is not the trivial one (Q symmetry), symmetry adapted base functions have to be used in the second order treatment for an accurate description of ENDOR spectra. We discuss the C2v and D4h covering symmetry in more detail. 

The model of a reacting molecular crystal proposed by Luty and Eckhardt [315] is centered on the description of the collective response of the crystal to a local strain expressed by means of an elastic stress tensor. The local strain of mechanical origin is, for our purposes, produced by the pressure or by the chemical transformation of a molecule at site n. The mechanical perturbation field couples to the internal and external (translational and rotational) coordinates Q n) generating a non local response. The dynamical variable Q can include any set of coordinates of interest for the process under consideration. In the model the system Hamiltonian includes a single molecule term, the coupling between the molecular variables at different sites through a force constants matrix W, and a third term that takes into account the coupling to the dynamical variables of the operator of the local stress. In the linear approximation, the response of the system is expressed by a response function X to a local field that can be approximated by a mean field V ... 

A semiclassical description is well established when both the Hamilton operator of the system and the quantity to be calculated have a well-defined classical analog. For example, there exist several semiclassical methods for calculating the vibrational autocorrelation function on a single excited electronic surface, the Fourier transform of which yields the Franck-Condon spectmm [108, 109, 150, 244]. In particular, semiclassical methods based on the initial-value representation of the semiclassical propagator [104-111, 245-248], which circumvent the cumbersome root-search problem in boundary-value-based semiclassical methods, have been successfully applied to a variety of systems (see, for example, Refs. 110, 111, 161, and 249 and references therein). The mapping procedure introduced in Section VI results in a quantum-mechanical Hamiltonian with a well-defined classical limit, and therefore it... 

---