Calculators, electronic

B3.1.2 WHY IS IT SO DIFFICULT TO CALCULATE ELECTRONIC ENERGIES AND WAVEFUNCTIONS WITH REASONABLE ACCURACY ... 

To use direct dynamics for the study of non-adiabatic systems it is necessary to be able to efficiently and accurately calculate electronic wave functions for excited states. In recent years, density functional theory (DFT) has been gaining ground over traditional Hartree-Fock based SCF calculations for the treatment of the ground state of large molecules. Recent advances mean that so-called time-dependent DFT methods are now also being applied to excited states. Even so, at present, the best general methods for the treatment of the photochemistry of polyatomic organic molecules are MCSCF methods, of which the CASSCF method is particularly powerful. 

Methane, CH4, for example, has a central carbon atom bonded to four hydrogen atoms and the shape is a regular tetrahedron with a H—C—H bond angle of 109°28, exactly that calculated. Electrons in a lone pair , a pair of electrons not used in bonding, occupy a larger fraction of space adjacent to their parent atom since they are under the influence of one nucleus, unlike bonding pairs of electrons which are under the influence of two nuclei. Thus, whenever a lone pair is present some distortion of the essential shape occurs. 

Electron density represents the probability of finding an electron at a poin t in space. It is calcii lated from th e elements of th e den sity matrix. The total electron density is the sum of the densities for alpha and beta electrons. In a closed-shell RUE calculation, electron densities are the same for alpha and beta electrons. 

Ifse the electronic Spectrum dialog box to display and analyze the IfV-vis spectrum produced by a singly excitetl Cl calculation. This dialog box is available only after yon do a single point Cl semi-einpirical calculation. Electronic Spectrum is then activated on the Compute menu. 

We have the makings of an iterative computer method. Start by assuming values for the matr ix elements and calculate electron densities (charge densities and bond orders). Modify the matr ix elements according to the results of the electron density calculations, rediagonalize using the new matrix elements to get new densities, and so on. When the results of one iteration are not different from those of the last by more than some specified small amount, the results are self-consistent. 

Having filled in all the elements of the F matr ix, we use an iterative diagonaliza-tion procedure to obtain the eigenvalues by the Jacobi method (Chapter 6) or its equivalent. Initially, the requisite electron densities are not known. They must be given arbitrary values at the start, usually taken from a Huckel calculation. Electron densities are improved as the iterations proceed. Note that the entire diagonalization is carried out many times in a typical problem, and that many iterative matrix multiplications are carried out in each diagonalization. Jensen (1999) refers to an iterative procedure that contains an iterative procedure within it as a macroiteration. The term is descriptive and we shall use it from time to time. 

The calculated electronic distribution leads to an evaluation of the dipole moment of thiazole. Some values are collected in Table 1-7 that can be compared to the experimental value of 1.61 D (158). 

How is electronic potential energy computed Electrons, which are more than three orders of magnitude lighter than nuclei, are too small for classical mechanics calculations. Electronic energy must... 

Secondly, you must describe the electron spin state of the system to be calculated. Electrons with their individual spins of sj=l/2 can combine in various ways to lead to a state of given total spin. The second input quantity needed is a description of the total spin S=Esj. Since spin is a vector, there are various ways of combining individual spins, but the net result is that a molecule can have spin S of 0, 1/2, 1,. These states have a multiplicity of 2S-tl = 1, 2, 3,. ..,that is, there is only one way of orienting a spin of 0, two ways of orienting a spin of 1/2, three ways of orienting a spin of 1, and so on. 

To calculate electron production must be balanced against electron depletion. Free electrons in the gas can become attached to any of a number of species in a combustion gas which have reasonably large electron affinities and which can readily capture electrons to form negative ions. In a combustion gas, such species include OH (1.83 eV), O (1.46 eV), NO2 (3.68 eV), NO (0.09 eV), and others. Because of its relatively high concentration, its abUity to capture electrons, and thus its abUity to reduce the electrical conductivity of the gas, the most important negative ion is usuaUyOH . 

Pugmire etal. have published calculated electron densities for pyrazine (68JA697), quinoxaline (69JA6381) and phenazine and the calculated total electron densities a + v) are shown in (10), (11) and (12). 

The precise numerical values of the calculated electron densities are unimportant, as the most important feature is the relative electron density thus, the electron density at the pyrazine carbon atom is similar to that at an a-position in pyridine and this is manifest in the comparable reactivities of these positions in the two rings. In the case of quinoxaline, electron densities at N-1 and C-2 are proportionately lower, with the highest electron density appearing at position 5(8), which is in line with the observation that electrophilic substitution occurs at this position. 

Pyrazine and its derivatives have been extensively studied by proton and NMR spectroscopy and conflicting reports on the reliability of additivity rules and/or correlation of chemical shifts with calculated electron densities have appeared. 

Table 2 Calculated Electron Densities of Pyrazole, its Anion and its Cation...

Fig. 5. Calculated electronic structure by the LDA method of (a) an isolated Cuo molecule and (b) fee solid Cuo where the direct band gap at the X-point is 1.5 eV [60],...

The calculated electron affinity is about 0.02 eV too low (corresponding to a difference of about 0.5 kcal-mol ), in excellent agreement with experiment. 

There are three main methods for calculating electron correlation Configuration Interaction (Cl), Many Body Perturbation Theory (MBPT) and Coupled Cluster (CC). A word of caution before we describe these methods in more details. The Slater determinants are composed of spin-MOs, but since the Hamilton operator is independent of spin, the spin dependence can be factored out. Furthermore, to facilitate notation, it is often assumed that the HF determinant is of the RHF type. Finally, many of the expressions below involve double summations over identical sets of functions. To ensure only the unique terms are included, one of the summation indices must be restricted. Alternatively, both indices can be allowed to run over all values, and the overcounting corrected by a factor of 1/2. Various combinations of these assumptions result in final expressions which differ by factors of 1 /2, 1/4 etc. from those given here. In the present book the MOs are always spin-MOs, and conversion of a restricted summation to an unrestricted is always noted explicitly. 

According to a molecular orbital calculation of Veber and Lwowski, isoindole should be favored over its tautomer, isoindolenine, by about 8 kcal/mole. However, the calculated electronic distribution is markedly different in the two oases, particularly at position 1, and it is to be expected that the nature and pattern of substituents will play an important role in determining the position of tautomeric equilibrium between these two species. 

Figure 1.2 A schematic view of an atom. The dense, positively charged nucleus contains most of the atom s mass and is surrounded by negatively charged electrons. The three-dimensional view on the right shows calculated electron-density surfaces. Electron density increases steadily toward the nucleus and is 40 times greater at the blue solid surface than at the gray mesh surface.

Table 11. X-ray reflection data for -y-brass (CvuZna) and calculated electron densities.

The compound CusSi, which with the older valences has the same calculated electron number as AgsAl, leads with the new system to a different electron density, 104 electrons per unit cell. 

Moruzzi, V. L. Janak J. F. Williams, A. R. Calculated Electronic Properties of Metals, Pergamon Press New York, 1978. 

The research was supported in part by Direction des Recherches et Etudes Techniques and the Office of Naval Research. We would like to thank the Scientific Council of the Centre de Calcul Vectoriel pour la Recherche for providing access to a CRAY-1 computer and the Centre Inter Regional de Calcul Electron-ique for providing computational resources. In addition, we would like to thank Professors S. Candel and G. Duvaut for their support and helpful discussions. 

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