The Electronic Energy

This energy region is that which corresponds to the electron energy states of the molecules, which are quantified and so can have only well-defined discrete values. 

XPS X-ray photoelectron spectroscopy [131-137] Monoenergetic x-rays eject electrons from various atomic levels the electron energy spectrum is measured Surface composition, oxidation state... 

The electronic energy, as detennined from must be added to tire ion-ion interactions to obtain the structural energies. This is a straightforward calculation for confined systems. For extended systems such as crystals, the calculations can be done using Madelimg summation techniques [2]. 

A quantum mechanical treatment of molecular systems usually starts with the Bom-Oppenlieimer approximation, i.e., the separation of the electronic and nuclear degrees of freedom. This is a very good approximation for well separated electronic states. The expectation value of the total energy in this case is a fiinction of the nuclear coordinates and the parameters in the electronic wavefunction, e.g., orbital coefficients. The wavefiinction parameters are most often detennined by tire variation theorem the electronic energy is made stationary (in the most important ground-state case it is minimized) with respect to them. The... 

The electronic energy W in the Bom-Oppenlieimer approxunation can be written as W= fV(q, p), where q is the vector of nuclear coordinates and the vector p contains the parameters of the electronic wavefimction. The latter are usually orbital coefficients, configuration amplitudes and occasionally nonlinear basis fiinction parameters, e.g., atomic orbital positions and exponents. The electronic coordinates have been integrated out and do not appear in W. Optimizing the electronic parameters leaves a function depending on the nuclear coordinates only, E = (q). We will assume that both W q, p) and (q) and their first derivatives are continuous fimctions of the variables q- and py... 

Gorse C and Capitelli M 1996 Non-equilibrium vibrational, electronic and dissociation kinetics in molecular plasmas and their coupling with the electron energy distribution function NATO ASI Series C 482 437-49... 

Strict degeneracy between the electronic energy surfaces therefore requires the existence of points Qq at which = H b Q) and //ab (Q) = 0. These two... 

B. A. Hess and C. M. Marian, Relativistic Effects in the Calculation of the Electronic Energy, in Computational Molecular Spectroscopy, P. Jensen and P. Bunker, eds., John Wiley Sc Sons, Inc., Chichester, UK, 2000, pp. 169-220. 

Generating the potential energy surface (PCS) using this equation requires solutions for many configurations ofnnclei. In molecular mechanics, the electronic energy is not evaluated explicitly. 

The quaniity, (R). the sum of the electronic energy computed 111 a wave funciion calculation and the nuclear-nuclear coulomb interaciion .(R.R), constitutes a potential energy surface having 15X independent variables (the coordinates R j. The independent variables are the coordinates of the nuclei but having made the Born-Oppenheimer approximation, we can think of them as the coordinates of the atoms in a molecule. 

Tire total energy equals the sum of the nuclear energy (the electrostatic repulsion between the positively charged nuclei) and the electronic energy. The electronic energy comprises... 

VVe can now start to separate the integral in Equation (2.74) into individual terms and identify the various contributions to the electronic energy ... 

N() -e that the summations are over the N/2 occupied orbitals. Other properties can be cali ulated from the density matrix for example, the electronic energy is ... 

The iotal energy of a system is equal to the sum of the electronic energy and the Coulombic nuclear repulsion energy ... 

After diagonalization of the EHT matrix, the lowest 4 orbitals have an energy sum of about —70 eV. The electronic energy for these doubly occupied orbitals is 2(—70) = — 140 eV. The energy gain of the molecule relative to its atoms is —140 — ( — 110) = —30eV = —690 kcal mol (1 eV = 23 kcal mol ) therefore, the molecule is stable relative to its atoms. We can envision an energy cycle with three steps (Eig. 7-5) ... 

Many physical properties of a molecule can be calculated as expectation values of a corresponding quantum mechanical operator. The evaluation of other properties can be formulated in terms of the "response" (i.e., derivative) of the electronic energy with respect to the application of an external field perturbation. 

The electronic partition function of the transition state is expressed in terms of the activation energy (the energy of the transition state relative to the electronic energy of the reactants) E as ... 

Molecular enthalpies and entropies can be broken down into the contributions from translational, vibrational, and rotational motions as well as the electronic energies. These values are often printed out along with the results of vibrational frequency calculations. Once the vibrational frequencies are known, a relatively trivial amount of computer time is needed to compute these. The values that are printed out are usually based on ideal gas assumptions. 

This last equation is the nuclear Schrodinger equation describing the motion of nuclei. The electronic energy computed from solving the electronic Schrodinger equation (3) on page 163 plus the nuclear-nuclear interactions Vjjjj(R,R) provide a potential for nuclear motion, a Potential Energy Surface (PES). 

The potential energy of vibration is a function of the coordinates, xj,. .., z hence it is a function of the mass-weighted coordinates, qj,. .., q3N. For a molecule, the vibrational potential energy, U, is given by the sum of the electronic energy and the nuclear repulsion energy ... 

In absorption spectroscopy a beam of electromagnetic radiation passes through a sample. Much of the radiation is transmitted without a loss in intensity. At selected frequencies, however, the radiation s intensity is attenuated. This process of attenuation is called absorption. Two general requirements must be met if an analyte is to absorb electromagnetic radiation. The first requirement is that there must be a mechanism by which the radiation s electric field or magnetic field interacts with the analyte. For ultraviolet and visible radiation, this interaction involves the electronic energy of valence electrons. A chemical bond s vibrational energy is altered by the absorbance of infrared radiation. A more detailed treatment of this interaction, and its importance in deter-... 

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