Schrodinger equation for a molecule

As was also previously noted in Section 9.3.1, the completely general rigid-rotor Schrodinger equation for a molecule characterized by three unique axes and associated moments of inertia does not lend itself to easy solution. However, by pursuing a generalization of the classical mechanical rigid-rotor problem, one can derive a quantum mechanical approximation that is typically quite good. Within that approximation, the rotational partition function becomes... 

Beyond the Molecular Orbital Approach Introduction.—In principle an exact solution to the non-relativistic Schrodinger equations for a molecule can be achieved by the configuration interaction technique. A complete set of one-electron spin orbitals i is used to form a complete set of Slater determinants by choosing all possible ordered sets of n elements of the set of 4u s. A linear combination of these determinants is then used ... 

The solution of the Schrodinger equation for a molecule follows a certain approximation scheme. It is assumed that (Table 1.6) ... 

The reality, of course, is that any complete set of square integrable functions, such as all the atomic orbitals of any atom, provide for an exact description of the eigenfunction solutions to the Schrodinger equation for a molecule. However, since such an approach is not practicable even on the largest modem computers, we settle for an approximation, which leads to good results when comparisons with experimental data are made. This is the essence of the LCAO-MO approximation and it would be normal to extend equation 6.2 and take linear combinations of valence atomic orbitals chosen from all... 

FF Finite Field Method of solving the Schrodinger equation for a molecule... 

On the basis of Equation 11.40, we recognize the importance of obtaining analytical expressions, wherever possible, for molecular partition functions. We will use certain approximations, the first being that the Schrodinger equation for a molecule is separable into parts such that there are additive contributions to the energy of the molecule in a given state. 

The time-independent Schrodinger equation for a molecule with N nuclei... 

The advent of inexpensive, fast computers has allowed chemists to develop methods for displaying the electron distribution within molecules. This distribution is obtained, in principle, by solving the Schrodinger equation for a molecule. Although the solution can be obtained only by using approximate methods, these methods provide an electrostatic potential map, a way to visualize the charge distribution within a molecule. 

To describe the orientations of a diatomic or linear polyatomic molecule requires only two angles (usually termed 0 and ([)). For any non-linear molecule, three angles (usually a, P, and y) are needed. Hence the rotational Schrodinger equation for a nonlinear molecule is a differential equation in three-dimensions. 

The Schrodinger equation for a collection of particles like a molecule is very similar. In this case, P would be a function of the coordinates of all the particles in the system as well as t. 

Electronic structure methods are aimed at solving the Schrodinger equation for a single or a few molecules, infinitely removed from all other molecules. Physically this corresponds to the situation occurring in the gas phase under low pressure (vacuum). Experimentally, however, the majority of chemical reactions are carried out in solution. Biologically relevant processes also occur in solution, aqueous systems with rather specific pH and ionic conditions. Most reactions are both qualitatively and quantitatively different under gas and solution phase conditions, especially those involving ions or polar species. Molecular properties are also sensitive to the environment. 

The ab initio method begins by solving the Schrodinger equation for the orbitals of electrons around a molecule, using as little simplification and approximation as is practical. This exact method is available only for small molecules with few atoms. The Schrodinger equation for a system with only one nucleus and a single electron... 

We know that not all solids conduct electricity, and the simple free electron model discussed previously does not explain this. To understand semiconductors and insulators, we turn to another description of solids, molecular orbital theory. In the molecular orbital approach to bonding in solids, we regard solids as a very large collection of atoms bonded together and try to solve the Schrodinger equation for a periodically repeating system. For chemists, this has the advantage that solids are not treated as very different species from small molecules. 

In order to calculate the absorption spectrum in the time-independent approach one solves the time-independent Schrodinger equation for a series of total energies and evaluates the overlap of the total continuum wavefunction, defined in (2.70), with the bound wavefunction of the parent molecule, ( tot(E) Pio I o(Ei)). Any structures in the spectrum are thus related to the energy dependence of the stationary wavefunction "Jftot(E). As illustrated schematically in Figure 7.4 for the one-... 

The function ij/(r, 9, p) (clearly ij/ could also be expressed in Cartesians), depends functionally on r, 6, p and parametrically on n, l and inm for each particular set (n. I, mm ) of these numbers there is a particular function with the spatial coordinates variables r, 6, p (or x, y, z). A function like /rsiiir is a function of x and depends only parametrically on k. This ij/ function is an orbital ( quasi-orbit the term was invented by Mulliken, Section 4.3.4), and you are doubtless familiar with plots of its variation with the spatial coordinates. Plots of the variation of ij/2 with spatial coordinates indicate variation of the electron density (recall the Bom interpretation of the wavefunction) in space due to an electron with quantum numbers n, l and inm. We can think of an orbital as a region of space occupied by an electron with a particular set of quantum numbers, or as a mathematical function ij/ describing the energy and the shape of the spatial domain of an electron. For an atom or molecule with more than one electron, the assignment of electrons to orbitals is an (albeit very useful) approximation, since orbitals follow from solution of the Schrodinger equation for a hydrogen atom. 

In an exact representation of the interaction between a solute and a solvent, i.e., solvation, the solvent molecules must be explicitly taken into account. That is, the solvent is described on a microscopic level, where the individual solvent molecules are considered explicitly. The interaction potential between solvent molecules and between solvent molecules and the solute can, in principle, be found by solving the electronic Schrodinger equation for a system consisting of all the involved molecules. Typically, in practice, a more empirical approach is followed where the interaction potential is described by parameterized energy functions. These potential energy functions (often referred to as force fields) are typically parameterized as pairwise atom-atom interactions. 

It is not possible to obtain a direct solution of a Schrodinger equation for a structure containing more than two particles. Solutions are normally obtained by simplifying H by using the Hartree-Fock approximation. This approximation uses the concept of an effective field V to represent the interactions of an electron with all the other electrons in the structure. For example, the Hartree-Fock approximation converts the Hamiltonian operator (5.7) for each electron in the hydrogen molecule to the simpler form ... 

Electronic structure methods are aimed at solving the Schrodinger equation for a single or a tew molecules, infinitely removed front all other molecules. Physically... 

The models for the control processes start with the Schrodinger equation for the molecule in interaction with a laser field that is treated either as a classical or as a quantized electromagnetic field. In Section II we describe the Floquet formalism, and we show how it can be used to establish the relation between the semiclassical model and a quantized representation that allows us to describe explicitly the exchange of photons. The molecule in interaction with the photon field is described by a time-independent Floquet Hamiltonian, which is essentially equivalent to the time-dependent semiclassical Hamiltonian. The analysis of the effect of the coupling with the field can thus be done by methods of stationary perturbation theory, instead of the time-dependent one used in the semiclassical description. In Section III we describe an approach to perturbation theory that is based on applying unitary transformations that simplify the problem. The method is an iterative construction of unitary transformations that reduce the size of the coupling terms. This procedure allows us to detect in a simple way dynamical or field induced resonances—that is, resonances that... 

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