A/-Schrodinger equation

This establishes our assertion that the former roots are overwhelmingly more numerous than those of the latter kind. Before embarking on a formal proof, let us illustrate the theorem with respect to a representative, though specific example. We consider the time development of a doublet subject to a Schrodinger equation whose Hamiltonian in a doublet representation is [13,29]... 

Although Eq. (139) looks like a Schrodinger equation that contains a vector potential x, it cannot be interpreted as such because t is an antisymmetric matrix (thus, having diagonal terms that are equal to zero). This inconvenience can be repaired by employing the following unitary bansformation ... 

Various kinds of mixed quantum-classical models have been introduced in the literature. We will concentrate on the so-called quantum-classical molecular dynamics (QCMD) model, which consists of a Schrodinger equation coupled to classical Newtonian equations (cf. Sec. 2). 

Quantum mechanics is cast in a language that is not familiar to most students of chemistry who are examining the subject for the first time. Its mathematical content and how it relates to experimental measurements both require a great deal of effort to master. With these thoughts in mind, the authors have organized this introductory section in a manner that first provides the student with a brief introduction to the two primary constructs of quantum mechanics, operators and wavefunctions that obey a Schrodinger equation, then demonstrates the application of these constructs to several chemically relevant model problems, and finally returns to examine in more detail the conceptual structure of quantum mechanics. 

Solutions to a Schrodinger equation for this last Hamiltonian (7) describe the vibrational, rotational, and translational states of a molecular system. This release of HyperChem does not specifically explore solutions to the nuclear Schrodinger equation, although future releases may. Instead, as is often the case, a classical approximation is made replacing the Hamiltonian by the classical energy ... 

Molecular quantum mechanics finds the solution to a Schrodinger equation for an electronic Hamiltonian, Hgjg., that gives a total energy, Egjg(-(R) + V (R,R). Repeated solutions at different nuclear configurations, R, lead to some approximate potential energy sur-... 

Rather than solve a Schrodinger equation with the Nuclear Hamiltonian (above), a common approximation is to assume that atoms are heavy enough so that classical mechanics is a good enough approximation. Motion of the particles on the potential surface, according to the laws of classical mechanics, is then the subject of classical trajectory analysis or molecular dynamics. These come about by replacing Equation (7) on page 164 with its classical equivalent ... 

The first term on the left-hand side of equation (10.18) has the form of a Schrodinger equation for nuclear motion, so that we may identify the expansion coefficient Xk Q) as a nuclear wave function for the electronic state k. The second term couples the influence of all the other electronic states to the nuclear motion for a molecule in the electronic state k. 

Solutions to the Schrodinger equation Hcj) = E(f> are the molecular wave functions 0, that describe the entangled motion of the three particles such that (j) 4> represents the density of protons and electron as a joint probability without any suggestion of structure. Any other molecular problem, irrespective of complexity can also be developed to this point. No further progress is possible unless electronic and nuclear variables are separated via the adiabatic simplification. In the case of Hj that means clamping the nuclei at a distance R apart to generate a Schrodinger equation for electronic motion only, in atomic units,... 

Marcus, R. A. Schrodinger equation for strongly interacting electron-transfer systems, J.Phys.Chem., 96 (1992), 1753-1757... 

The trial wave functions of a Schrodinger equation are expressed as determinant of the HF orbitals. This will give coupled nonlinear equations. The amplitudes were solved usually by some iteration techniques so the cc energy is computed as... 

The Hamiltonian (2.23) represents the general expansion in terms of the elements Gap, and it corresponds to a Schrodinger equation with a generic potential. In some special cases, one does not have in Eq. (2.23) generic coefficients e ap, apY8 but only those combinations that can be written as invariant Casimir operators of G and its subalgebras, GdG dG"D ", This situation... 

There is another manner in which perturbation theory is used in quantum chemistry that does not involve an externally applied perturbation. Quite often one is faced with solving a Schrodinger equation to which no exact solution has been (yet) or can be found. In such cases, one often develops a model Schrodinger equation which in some sense is designed to represent the system whose full Schrodinger equation can not be solved. The... 

The problem with the calculation is that s and p functions centered on the atoms do not provide sufficient mathematical flexibility to adequately describe the wave function for the pyramidal geometry. This is true even though the atoms nitrogen and hydrogen can individually be reasonably well described entirely by s and p functions. The molecular orbitals, which are eigenfunctions of a Schrodinger equation involving multiple nuclei at various positions in space, require more mathematical flexibility than do the atoms. 

Exercise. Similarly (2.10) is a Schrodinger equation with the potential W,... 

Our Schrodinger-Langevin equation (5.2) involves no T or equivalent parameter it is therefore not possible to obtain a Schrodinger equation with damping. 

In the present section we are concerned with genuine internal noise. We consider a closed, isolated many-body system, whose evolution is given by a Schrodinger equation. Remember that in the classical case in III.2 we gave a macroscopic description in terms of a reduced set of macroscopic variables, which obey an autonomous set of differential equations. These equations are approximate and deviations appear in the form of fluctuations, which are a vestige of the large number of eliminated microscopic variables. Our task is to carry out this program in the framework of quantum mechanics. 

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 ... 

The only model available for direct quantum-mechanical study of interatomic interaction is the hydrogen molecular ion Hj. If the two protons are considered clamped in position at a fixed distance apart, the single electron is represented by a Schrodinger equation, which can be separated in confo-cal elliptic coordinates. On varying the interproton distance for a series of calculations a complete mapping of the interaction for all possible configurations is presumably achieved. This is not the case. Despite its reasonable appearance the model is by no means unbiased. 

This must now be walked into a quantum-mechanical formalism. What we have learned above permits us to write a Schrodinger equation similar to Eq. (3.39.14), whose solutions will be of the harmonic oscillator type ... 

We shall proceed as follows. We shall first diagonalize the Schrbdinger problem [Eq. (3.46)] with respect to the vibrational and rotational quantum numbers (Section 5.1). We arrive in this way at a Schrodinger equation in the variable p with an effective potential function for each vibration—rotation state. A least squares procedure that includes the numerical integration of the Schrodinger equation for this effective Hamiltonian will be used to determine the harmonic force field and the doubleminimum inversion potential function for ( NHa, NHs), ( ND3, NTa) and NH2D, ND2H (Section 5.2). 

We now specialize the discussion to the ligand field theory situation and define the orthonormal set of spin-orbitals we shall use in the determinantal expansion of the many-electron functions Vyy for the groups M and L. First we suppose that we have a set of k orbitals describing the one-electron states in the metal atom these will be orthonormal solutions of a Schrodinger equation for a spherically symmetric potential, V<,(r), which may be thought of as the average potential about the metal atom which an electron experiences ... 

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