Initio Wavefunctions

The text by Hehre et al. explains the procedures used to obtain molecular wavefunctions. Let the molecular orbitals Tj be defined in terms of N normalized basis functions by 

When calculating the wavefunction it is important to make a choice of basis set J. t that is suitable for the available computing power and the accuracy desired. A straightforward early approach to basis set construction was to fit an accurate Slater-type atomic orbital (STO) with n gaussians,called STO- G. The quality of STO- G wavefunctions increases as n increases. It was determined that = 3 was a good starting point, and the STO-3G basis set has been widely used, particularly where computing resources were limited or for lai er molecules. 

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The electron capture processes are driven by non-adiabatic couplings between molecular states. All the non-zero radial and rotational eoupling matrix elements have therefore been evaluated from ab initio wavefunctions. 

In this section, we present a unified picture of the different electronic effects that combine to determine methyl rotor potentials in the S0, Sp and D0 electronic states of different substituted toluenes. Our approach is based on analysis of ab initio wavefunctions using the natural bond orbitals (NBOs)33 of Weinhold and cowork-ers. We will attempt to decompose the methyl torsional potential into two dominant contributions. The first is repulsive steric interactions, which are important only when an ortho substituent is present. The second is attractive donor-acceptor interactions between CH bond pairs and empty antibonding orbitals vicinal to the CH bonds. In the NBO basis, these attractive interactions dominate the barrier in ethane (1025 cm-1) and in 2-methylpropene (1010 cm-1) see Figure 3. By analogy, donor-acceptor attractions are important in toluenes whenever there is a substantial difference in bond order between the two ring CC bonds adjacent to the C-CH3 bond. Viewed the other way around, we can use the measured methyl rotor potential as a sensitive probe of local ring geometry. 

However, despite their proven explanatory and predictive capabilities, all well-known MO models for the mechanisms of pericyclic reactions, including the Woodward-Hoffmann rules [1,2], Fukui s frontier orbital theory [3] and the Dewar-Zimmerman treatment [4-6] share an inherent limitation They are based on nothing more than the simplest MO wavefunction, in the form of a single Slater determinant, often under the additional oversimplifying assumptions characteristic of the Hiickel molecular orbital (HMO) approach. It is now well established that the accurate description of the potential surface for a pericyclic reaction requires a much more complicated ab initio wavefunction, of a quality comparable to, or even better than, that of an appropriate complete-active-space self-consistent field (CASSCF) expansion. A wavefunction of this type typically involves a large number of configurations built from orthogonal orbitals, the most important of which i.e. those in the active space) have fractional occupation numbers. Its complexity renders the re-introduction of qualitative ideas similar to the Woodward-Hoffmann rules virtually impossible. 

The electronic overlap populations in all three cases were calculated from the one electron Extended Hiickel MO s. For the photocyclizations of 1,2-difuryl ethylenes very similar results were obtained also from minimal basis set ab-initio wavefunctions l The possibility of obtaining useful reactivity analyses from wave-functions which are easily available even for large systems could prove to be an important practical consideration for further applications of this method. The dependence on Sri sj ill (5) ensures that electronic overlap populations show the desirable physical characteristics for their use as reactivity measures strong falling-off with increasing interatomic distance and proper directional dependence. This last point is of particular significance for bond formation in polyenes. Thus for two C 2 p atomic... 

Let us compare the SCF dipole moments obtained by MNDO type methods either from the NDDO or the quasi ab initio wavefunctions to the experimental dipole moments. Since experimental results were taken into account in the parametrization of MNDO type methods [32-34], it is more appropriate to... 

Currently, two different lines of approach for such calculations may be distinguished. They use either very good quality ab initio wavefunctions or semiempiri-cal methods to describe the ground state.71 The obvious advantage of the former approach as compared with the second can hardly be stressed enough. However, precise ab initio calculations are computationally so demanding that systematic calculations for medium-sized molecules have not yet been carried out, despite striking advances in computational facilities. 

Both of these methods essentially use ab initio wavefunctions to deduce the composition of the MOs. An alternative approach is to use semi-empirical functions, obtained from the MINDO or CNDO methods, to calculate the constants directly from the hamiltonian. A major problem in such work is that in many cases only valence shell electrons are included and consequently spin-other orbit effects cannot be calculated directly. Hinkley, Walker, and Richards122 have shown from ab initio calculations that this shielding for a given atom often bears a constant ratio to theZ/r3 terms. Such a ratio could be... 

Unfortunately the calculation of transition probabilities has not been attempted in many instances and the results are not encouraging. Apart from a valiant early attempt by Davies173 for Na2 in a calculation where not all electrons were specifically considered, the first serious attempt to compute transition probabilities from ab initio wavefunctions was due to La Paglia.174 In this important paper La Paglia considered the dipole strengths of various 1 -iE transitions for some first-row diatomic molecules. 

Tel. 518-276-2678, e-mail breneman quant.chem.rpi.edu Computes electrostatic potential-derived charges from ab initio wavefunctions generated by one of the Gaussian 86/88/90 packages. This program is a modification of CHELP by Dr. Lisa E. Chirlian and Dr. Michelle M. Francl. UNIX and VMS machines. 

F. J. Olivares del Valle, M. A. Aguilar, and S. Tolosa, /. Mol. Struct. (THEOCHEM), 279, 223 (1993). Polarizable Continuum Model Calculation Including Electron Correlation in the Ab Initio Wavefunction. 

The method that is used in most of the work described in this chapter is the distributed multipole analysis (DMA) of Stone,which is implemented in the CADPAC ab initio suite. DMA is based on the density matrix p,y of the ab initio wavefunction of the molecule, expressed in terms of the Gaussian primitives q that comprise the atomic orbital basis set ... 

The MEDLA method is based on the following electron density fragment additivity principle. A natural scheme for the implementation of this principle has been d cribed in [12]. Consider an LCAO ab initio wavefunction of a small molecule of some fixed conformation K. If n is the number of atomic orbitals three-dimensional position vector variable, and P is the n X n density matrix, then the electronic density p(r) of the molecule is given by... 

Momentum-space concepts are not, in general, familiar to the chemist and so we outline first the calculation of momentum-space electron densities, p p), from ab initio wavefunctions. The form of pip) for different molecules is discussed, using as examples (i) the ground state of H2, (ii) bond formation in BH", and (iii) the n-orbitals in large conjugated polyenes. 

The calculation of ab initio wavefunctions for the larger molecules associated with applications such as those discussed later in this paper is computationally demanding. Accordingly, we have also examined these model systems using semi-empirical wavefunctions taken from the well-established MOPAC program [16] such wavefunctions can be obtained very cheaply. Table 3 collects together values of Rab, Sab< Tab and Dab for the model series, evaluated using... 

In ab initio wavefunction pictures using Koopmans theorem, t] becomes half of the energy gap between HOMO and LUMO as [51] ... 

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