Linear molecules energy

The linear molecule energy levels are given by E = BJ(J + 1) where J is the rotational quantum number for rotation perpendicular to the bond, and B is the rotational constant. The energy level structure is... 

The simplest case is a transition in a linear molecule. In this case there is no orbital or spin angular momentum. The total angular momentum, represented by tire quantum number J, is entirely rotational angular momentum. The rotational energy levels of each state approximately fit a simple fomuila ... 

For chemically bound molecules, it is usual to analyse tlie vibrational energy levels in teniis of normal modes, a non-linear (or linear) molecule witli V atoms has 3 V - 6 (or 3 V - 5) vibrational degrees of freedom. There is a... 

The expressions for the rotational energy levels (i.e., also involving the end-over-end rotations, not considered in the previous works) of linear triatomic molecules in doublet and triplet II electronic states that take into account a spin orbit interaction and a vibronic coupling were derived in two milestone studies by Hougen [72,32]. In them, the isomorfic Hamiltonian was inboduced, which has later been widely used in treating linear molecules (see, e.g., [55]). 

Isolated Linear Molecule Figure 6 shows the error in total energy for an isolated linear molecule H-(-C=C-)5-H. It is obvious that for the same level of accuracy, the time step in the SISM can be ten times or more larger as in the LFV. Furthermore, the LFV method is stable for only very short time steps, up to 5 fs, while the SISM is stable even for a time step up to 200 fs. However, such large time steps no longer represent physical reality and arc a particular property identified with linear molecules without bending or torsional intramolecular interactions. 

An N-atom molecular system may he described by dX Cartesian coordinates. Six independent coordinates (five for linear molecules, three fora single atom) describe translation and rotation of the system as a whole. The remaining coordinates describe the nioleciiUir configuration and the internal structure. Whether you use molecular mechanics, quantum mechanics, or a specific computational method (AMBER, CXDO. etc.), yon can ask for the energy of the system at a specified configuration. This is called a single poin t calculation. ... 

So, for any atom, the orbitals can be labeled by both 1 and m quantum numbers, which play the role that point group labels did for non-linear molecules and X did for linear molecules. Because (i) the kinetic energy operator in the electronic Hamiltonian explicitly contains L2/2mer2, (ii) the Hamiltonian does not contain additional Lz, Lx, or Ly factors. 

The rotational energy levels for a prolate and an oblate symmetric rotor are shown schematically in Figure 5.6. Although these present a much more complex picture than those for a linear molecule the fact that the selection mles... 

For a symmetric rotor the modification Eg to the rotational energy levels in an electric field S is larger than in a linear molecule and is given, approximately, by... 

If we use a contour map to represent a three-dimensional surface, with each contour line representing constant potential energy, two vibrational coordinates can be illustrated. Figure 6.35 shows such a map for the linear molecule CO2. The coordinates used here are not normal coordinates but the two CO bond lengths rj and r2 shown in Figure 6.36(a). It is assumed that the molecule does not bend. 

Many of the reactions in which acetylene participates, as well as many properties of acetylene, can be understood in terms of the stmcture and bonding of acetylene. Acetylene is a linear molecule in which two of the atomic orbitals on the carbon are sp hybridized and two are involved in 7T bonds. The lengths and energies of the C—H O bonds and C=C<7 + 27t bonds are as follows ... 

The development of linear ffee-energy relationships in aliphatic molecules is complicated because steric and conformational factors come into pl along with... 

It turns out that certain electronic states of atoms and linear molecules, even those requiring many-determinant wavefunctions, may have an energy expression... 

The second aspect is more fundamental. It is related to the very nature of chemistry (quantum chemistry is physics). Chemistry deals with fuzzy objects, like solvent or substituent effects, that are of paramount importance in tautomerism. These effects can be modeled using LFER (Linear Free Energy Relationships), like the famous Hammett and Taft equations, with considerable success. Quantum calculations apply to individual molecules and perturbations remain relatively difficult to consider (an exception is general solvation using an Onsager-type approach). However, preliminary attempts have been made to treat families of compounds in a variational way [81AQ(C)105]. 

Free energy of solution formation containing linear molecules after degradation equals ... 

Rotational Energy Levels The rotational energy of a molecule depends upon the molecular geometry. For a linear molecule that behaves as a rigid rotator,3... 

For a polyatomic molecule, the complex vibrational motion of the atoms can be resolved into a set of fundamental vibrations. Each fundamental vibration, called a normal mode, describes how the atoms move relative to each other. Every normal mode has its own set of energy levels that can be represented by equation (10.11). A linear molecule has (hr) - 5) such fundamental vibrations, where r) is the number of atoms in the molecule. For a nonlinear molecule, the number of fundamental vibrations is (3-q — 6). 

Lewis. G. N. 1. 248. 264, 265 line integrals 605-8 linear molecules electronic energy levels 506 fundamental frequencies 645 inertia, moments of 643 vibrational energy levels 504 Linhart. G.A. 481-3 liquid crystals 4... 

Hamiltonian operator, 2,4 for many-electron systems, 27 for many valence electron molecules, 8 semi-empirical parametrization of, 18-22 for Sn2 reactions, 61-62 for solution reactions, 57, 83-86 for transition states, 92 Hammond, and linear free energy relationships, 95... 

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