Polyatomic molecules equation derivation

Equation (9.38), if restricted to two particles, is identical in form to the radial component of the electronic Schrodinger equation for the hydrogen atom expressed in polar coordinates about the system s center of mass. In the case of the hydrogen atom, solution of the equation is facilitated by the simplicity of the two-particle system. In rotational spectroscopy of polyatomic molecules, the kinetic energy operator is considerably more complex in its construction. For purposes of discussion, we will confine ourselves to two examples that are relatively simple, presented without derivation, and then offer some generalizations therefrom. More advanced treatises on rotational spectroscopy are available to readers hungering for more. 

For polyatomic molecules, (7.2) and (7.6) give the wave numbers and transition moment for an electronic transition. Besides the S selection rule (7.7), there are electronic selection rules stating between which electronic symmetry species transitions are allowed these are derived using group theory (Section 9.11). Equation (7.13) applies to polyatomics, except that Pei is now a function of the 3N —6 (or 3Af —5) normal coordinates Qr... 

We now consider the nuclear motions of polyatomic molecules. We are using the Born-Oppenheimer approximation, writing the Hamiltonian HN for nuclear motion as the sum of the nuclear kinetic-energy TN and a potential-energy term V derived from solving the electronic Schrodinger equation. We then solve the nuclear Schrodinger equation... 

The vibrations of polyatomic molecules are more complicated, for the number of possible interactions rises sharply as the number of atoms increases. However, it is possible to handle the equations of motion governing even the most complicated vibrations by lineal combination (p. 48) of the equations of motions of rather simple vibrations. For example, all vibrations of the C02 molecule are said to be derived from superposition of the four modes of vibration indicated in Figure 25-4(a), whereas all vibrations of the SO2 molecule may be likewise broken down into combinations of one or more of the vibrations indicated in 25-4(b). 

As is briefly described in the Introduction, an exact equation referred to as the Ornstein-Zernike equation, which relates h(r, r ) with another correlation function called the direct correlation function c(r, r/), can be derived from the grand canonical partition function by means of the functional derivatives. Our theory to describe the molecular recognition starts from the Ornstein-Zernike equation generalized to a solution of polyatomic molecules, or the molecular Ornstein-Zernike (MOZ) equation [12],... 

The complete partition function of a polyatomic molecule may now be represented by the product Qt X Q., where Qt is the tran.slational, including the electronic, factor, as derived above, and Q is the combined rotational and vibrational, i.e., internal, factor. Since In Q is then equal to In Qt + In Qiy equation (24.12) may be written in the form... 

Fortunately, the energy and population distributions of a metastable polyatomic molecule can be described by equations familiar from ordinary one temperature statistical thermodynamics. A multiple temperature vibrational partition function 2vib be derived, which has the form ... 

Whereas the absolute values of k depend significantly on the values of Pg. F gj, normally Pg. P e makes only a small contribution to the temperature dependence of k in polyatomic molecules. For small polyatomic molecules an empirical relation for the apparent activation energy Eg (see equation (1.21)) was derived from experiments ... 

The paper reports the derivation of the equation of motion for a polyatomic molecule. As the origin of the BFCS, unlike in this chapter, the center of mass was chosen . ... 

For a linear polyatomic molecule like acetylene or cyanogen, the rotational energy levels are the same as those of diatomic molecules in Eq. (22.2-18). Equation (25.4-13) can be used for the rotational partition function with the appropriate symmetry number and moment of inertia. The rotational energy levels of nonlinear polyatomic molecules are more complicated than those of diatomic molecules. The derivation of the rotational partition function for nonlinear molecules is complicated, and we merely cite the result ... 

Classical Dynamics of Nonequilibrium Processes in Fluids Integrating the Classical Equations of Motion Control of Microworld Chemical and Physical Processes Mixed Quantum-Classical Methods Multiphoton Excitation Non-adiabatic Derivative Couplings Photochemistry Rates of Chemical Reactions Reactive Scattering of Polyatomic Molecules Spectroscopy Computational Methods State to State Reactive Scattering Statistical Adiabatic Channel Models Time-dependent Multiconfigurational Hartree Method Trajectory Simulations of Molecular Collisions Classical Treatment Transition State Theory Unimolecular Reaction Dynamics Valence Bond Curve Crossing Models Vibrational Energy Level Calculations Vibronic Dynamics in Polyatomic Molecules Wave Packets. 

It is quite straightforward to perform quasiclassical trajectory computations (QCT) on the reactions of polyatomic molecules providing a smooth global potential energy surface is available from which derivatives can be obtained with respect to the atomic coordinates. This method is described in detail in Classical Trajectory Simulations Final Conditions. Hamilton s equations are solved to follow the motion of the individual atoms as a function of time and the reactant and product vibrational and rotational states can be set or boxed to quantum mechanical energies. The method does not treat purely quantum mechanical effects such as tunneling, resonances. or interference but it can treat the full state-to-state, eneigy-resolved dynamics of a reaction and also produces rate constants. Numerous applications to polyatomic reactions have been reported. ... 

If, however, we use a set of coordinates (which are not necessarily normal coordinates) that transform under the symmetry operations of the group to which the molecule belongs in the manner indicated by the matrices of the irreducible representations, then all cross products of the type QiQj, where Qi and Qj belong to different irreducible representations, will vanish this choice of coordinates will thus greatly simplify the solution of equation 2-55. We shall return to this point briefly a little later first we wish to derive the selection rules for optical transitions between the various possible vibrational states of polyatomic molecules. 

The close interrelations among mass, momentum, and energy transport can be explained in terms of a molecular theory of monatomic gases at low density. The continuity, motion, and energy equations can all be derived from the Boltzmann equation for the velocity distribution function, from which the molecular expressions for the flows and transport properties are produced. Similar derivations are also available for polyatomic gases, monatomic liquids, and polymeric liquids. For monatomic liquids, the expressions for the momentum and heat flows include contributions associated with forces between two molecules. For polymers, additional forces within the polymer chain should be taken into account. 

The solute-solvent 3D-RISM equation complemented with the 3D-HNC closure was first obtained by Beglov and Roux [24] within the density functional method by reduction of the generalized closure of Chandler, McCoy, and Singer for nonuniform polyatomic systems [22]. In an alternative way, the 3D-RISM integral equation was derived by Kovalenko and Hirata [28] from the six-dimensional (6D), molecular OZ equation for a solute-solvent mixture of rigid molecules at infinite dilution. The latter is written as [21]... 

An ab initio effective core potential method derived from the relativistic all-electron Dirac-Fock solution of the atom, which we call the relativistic effective core potential (RECP) method, has been widely used by several investigators to study the electronic structure of polyatomics including the lanthanide- and actinide-containing molecules. This RECP method was formulated by Christiansen et al. (1979). It differs from the conventional Phillips-Kleinman method in the representation of the nodeless pseudo-orbital in the inner region. The one-electron valence equation in an effective potential of the core electron can be written as... 

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