Nuclear “position

Here each < ) (0 is a vibrational wavefiinction, a fiinction of the nuclear coordinates Q, in first approximation usually a product of hamionic oscillator wavefimctions for the various nomial coordinates. Each j (x,Q) is the electronic wavefimctioii describing how the electrons are distributed in the molecule. However, it has the nuclear coordinates within it as parameters because the electrons are always distributed around the nuclei and follow those nuclei whatever their position during a vibration. The integration of equation (Bl.1.1) can be carried out in two steps—first an integration over the electronic coordinates v, and then integration over the nuclear coordinates 0. We define an electronic transition moment integral which is a fimctioii of nuclear position ... 

This assumption breaks down in many molecules, especially upon photo-excitation, since excited states are often close to each other or even cross one another (i.e. have the same electronic energy at a given nuclear position). Thus, the fiill Scluodinger wavefiinction needs to be considered ... 

The time dependence of the molecular wave function is carried by the wave function parameters, which assume the role of dynamical variables [19,20]. Therefore the choice of parameterization of the wave functions for electronic and nuclear degrees of freedom becomes important. Parameter sets that exhibit continuity and nonredundancy are sought and in this connection the theory of generalized coherent states has proven useful [21]. Typical parameters include molecular orbital coefficients, expansion coefficients of a multiconfigurational wave function, and average nuclear positions and momenta. We write... 

In this minimal END approximation, the electronic basis functions are centered on the average nuclear positions, which are dynamical variables. In the limit of classical nuclei, these are conventional basis functions used in moleculai electronic structure theoiy, and they follow the dynamically changing nuclear positions. As can be seen from the equations of motion discussed above the evolution of the nuclear positions and momenta is governed by Newton-like equations with Hellman-Feynman forces, while the electronic dynamical variables are complex molecular orbital coefficients that follow equations that look like those of the time-dependent Hartree-Fock (TDHF) approximation [24]. The coupling terms in the dynamical metric are the well-known nonadiabatic terms due to the fact that the basis moves with the dynamically changing nuclear positions. 

Vhen calculating the total energy of the system, we should not forget the Coulomb inter-ction between the nuclei this is constant within the Born-Oppenheimer approximation Dr a given spatial arrangement of nuclei. When it is desired to change the nuclear positions,... 

The wave function T is a function of the electron and nuclear positions. As the name implies, this is the description of an electron as a wave. This is a probabilistic description of electron behavior. As such, it can describe the probability of electrons being in certain locations, but it cannot predict exactly where electrons are located. The wave function is also called a probability amplitude because it is the square of the wave function that yields probabilities. This is the only rigorously correct meaning of a wave function. In order to obtain a physically relevant solution of the Schrodinger equation, the wave function must be continuous, single-valued, normalizable, and antisymmetric with respect to the interchange of electrons. 

The reaction coordinate is one specific path along the complete potential energy surface associated with the nuclear positions. It is possible to do a series of calculations representing a grid of points on the potential energy surface. The saddle point can then be found by inspection or more accurately by using mathematical techniques to interpolate between the grid points. 

An alkyl group activates all nuclear positions, the 0- and p-positions more than the r-position. The activation is not very strong. 

It will be noticed that this account makes no allowance for the electrostatic interaction of the positive pole with the electrophile, the nitro-nium ion. This should generally work for deactivation, and its influence at nuclear positions should be in the order ortho > meta > para. This point is resumed below. 

The electrostatic interaction of the charge on the orienting substituent, and those at the nuclear positions, with that of the approaching electrophile. 

HyperChem computes the Hessian using numerical second derivative of the total energy with respect to the nuclear positions based on the analytically calculated first derivatives in ab initio methods and any of the semi-empirical methods, except the Extended Hiickel. Vibration calculations in HyperChem using an ab initio method may take much longer than calculations using the semi-empirical methods. 

By using an aromatic aldehyde carrying an electron-releasing group the intermediate cation can be stabilized. This is the basis of the widely-used Ehrlich colour reaction for pyrroles, indoles and furans which have a free reactive nuclear position (Scheme 21). 

Mercury(II) acetate tends to mercurate all the free nuclear positions in pyrrole, furan and thiophene to give derivatives of type (74). The acetoxymercuration of thiophene has been estimated to proceed ca. 10 times faster than that of benzene. Mercuration of rings with deactivating substituents such as ethoxycarbonyl and nitro is still possible with this reagent, as shown by the formation of compounds (75) and (76). Mercury(II) chloride is a milder mercurating agent, as illustrated by the chloromercuration of thiophene to give either the 2- or 2,5-disubstituted product (Scheme 25). 

Thus, in the very short time (10 s) required for excitation, the molecule does not undergo changes in nuclear position or in the spin state of the promoted electron. After the excitation, however, these changes can occur very rapidly. The new minimum-energy... 

Energy calculations and geometry optimizations ignore the vibrations in molecular systems. In this way, these computations use an idealized view of nuclear position. In reality, the nuclei in molecules are constantly in motion. In equilibrium states, these vibrations are regular and predictable, and molecules can be identified by their characteristic spectra. 

Molecular frequencies depend on the second derivative of the energy with respect to the nuclear positions. Analytic second derivatives are available for the Hartree-Fock (HF keyword). Density Functional Theory (primarily the B3LYP keyword in this book), second-order Moller-Plesset (MP2 keyword) and CASSCF (CASSCF keyword) theoretical procedures. Numeric second derivatives—which are much more time consuming—are available for other methods. 

The equilibrium geometries produced by electronic structure theory correspond to the spectroscopic geometry R, which assumes that there is no nuclear motion. Contrast this to the Rg geometry, defined via the vibrationally-averaged nuclear positions. 

The Born-Oppenheimer approximation is the first of several approximations used to simplify the solution of the Schradinger equation. It simplifies the general molecular problem by separating nuclear and electronic motions. This approximation is reasonable since the mass of a typical nucleus is thousands of times greater than that of an electron. The nuclei move very slowly with respect to the electrons, and the electrons react essentially instantaneously to changes in nuclear position. Thus, the electron distribution within a molecular system depends on the positions of the nuclei, and not on their velocities. Put another way, the nuclei look fixed to the electrons, and electronic motion can be described as occurring in a field of fixed nuclei. 

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