Coefficients, orbital

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... 

The basic self-consistent field (SCF) procedure, i.e., repeated diagonalization of the Fock matrix [26], can be viewed, if sufficiently converged, as local optimization with a fixed, approximate Hessian, i.e., as simple relaxation. To show this, let us consider the closed-shell case and restrict ourselves to real orbitals. The SCF orbital coefficients are not the... 

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. 

Molecular orbitals were one of the first molecular features that could be visualized with simple graphical hardware. The reason for this early representation is found in the complex theory of quantum chemistry. Basically, a structure is more attractive and easier to understand when orbitals are displayed, rather than numerical orbital coefficients. The molecular orbitals, calculated by semi-empirical or ab initio quantum mechanical methods, are represented by isosurfaces, corresponding to the electron density surfeces Figure 2-125a). 

In view of this, early quantum mechanical approximations still merit interest, as they can provide quantitative data that can be correlated with observations on chemical reactivity. One of the most successful methods for explaining the course of chemical reactions is frontier molecular orbital (FMO) theory [5]. The course of a chemical reaction is rationali2ed on the basis of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), the frontier orbitals. Both the energy and the orbital coefficients of the HOMO and LUMO of the reactants are taken into account. 

As an example, we shall discuss the Diels-Alder reaction of 2-methoxybuta-l,3-diene with acrylonitrile. Figure 3-7 gives the reaction equation, the correlation diagram of the HOMOs and LUMOs, and the orbital coefficients of the correlated HOMO and LUMO. 

FMO theory requires that a HOMO of one reactant has to be correlated with the LUMO of the other reactant. The decision between the two alternatives - i.e., from which reactant the HOMO should be taken - is made on the basis of which is the smaller energy difference in our case the HOMO of the electron rich diene, 3.1, has to be correlated with the LUMO of the electron-poor dienophile, 3.2. The smaller this HOMO-LUMO gap, the higher the reactivity will be. With the HOMO and LUMO fixed, the orbital coefficients of these two orbitals can explain the regios-electivity of the reaction, which strongly favors the formation of 3.3 over 3.4. 

The importance of FMO theory hes in the fact that good results may be obtained even if the frontier molecular orbitals are calculated by rather simple, approximate quantum mechanical methods such as perturbation theory. Even simple additivity schemes have been developed for estimating the energies and the orbital coefficients of frontier molecular orbitals [6]. 

Figure 3-7. FMO trealirent of a) a Dlels-Alder reaction equation, b) correlation diagram, c) orbital coefficients,...

The initial values, a, , are derived by correlations with dipole moments of a series of conjugated systems. The exchange integrals are taken from Abraham and Hudson [38] and are considered as being independent of charge. The r-charges are then calculated from the orbital coefficients, c,j, of the HMO theory according to Eq. (14). 

In the Lowdin approach to population analysis [Ldwdin 1970 Cusachs and Politzer 1968] the atomic orbitals are transformed to an orthogonal set, along with the molecular orbital coefficients. The transformed orbitals in the orthogonal set are given by ... 

The FMO coefficients also allow cpralitative prediction of the kinetically controlled regioselectivity, which needs to be considered for asymmetric dienes in combination with asymmetric dienophiles (A and B in Scheme 1.1). There is a preference for formation of a o-bond between the termini with the most extreme orbital coefficients ... 

The regioselectivity benefits from the increased polarisation of the alkene moiety, reflected in the increased difference in the orbital coefficients on carbon 1 and 2. The increase in endo-exo selectivity is a result of an increased secondary orbital interaction that can be attributed to the increased orbital coefficient on the carbonyl carbon ". Also increased dipolar interactions, as a result of an increased polarisation, will contribute. Interestingly, Yamamoto has demonstrated that by usirg a very bulky catalyst the endo-pathway can be blocked and an excess of exo product can be obtained The increased di as tereo facial selectivity has been attributed to a more compact transition state for the catalysed reaction as a result of more efficient primary and secondary orbital interactions as well as conformational changes in the complexed dienophile" . Calculations show that, with the polarisation of the dienophile, the extent of asynchronicity in the activated complex increases . Some authors even report a zwitteriorric character of the activated complex of the Lewis-acid catalysed reaction " . Currently, Lewis-acid catalysis of Diels-Alder reactions is everyday practice in synthetic organic chemistry. 

The Huckel method and is one of the earliest and simplest semiempirical methods. A Huckel calculation models only the 7t valence electrons in a planar conjugated hydrocarbon. A parameter is used to describe the interaction between bonded atoms. There are no second atom affects. Huckel calculations do reflect orbital symmetry and qualitatively predict orbital coefficients. Huckel calculations can give crude quantitative information or qualitative insight into conjugated compounds, but are seldom used today. The primary use of Huckel calculations now is as a class exercise because it is a calculation that can be done by hand. 

The Lowdin population analysis scheme was created to circumvent some of the unreasonable orbital populations predicted by the Mulliken scheme, which it does. It is different in that the atomic orbitals are first transformed into an orthogonal set, and the molecular orbital coefficients are transformed to give the representation of the wave function in this new basis. This is less often used since it requires more computational work to complete the orthogonalization and has been incorporated into fewer software packages. The results are still basis-set-dependent. 

For systems with unpaired electrons, it is not possible to use the RHF method as is. Often, an unrestricted SCF calculation (UHF) is performed. In an unrestricted calculation, there are two complete sets of orbitals one for the alpha electrons and one for the beta electrons. These two sets of orbitals use the same set of basis functions but different molecular orbital coefficients. 

Wave functions can be visualized as the total electron density, orbital densities, electrostatic potential, atomic densities, or the Laplacian of the electron density. The program computes the data from the basis functions and molecular orbital coefficients. Thus, it does not need a large amount of disk space to store data, but the computation can be time-consuming. Molden can also compute electrostatic charges from the wave function. Several visualization modes are available, including contour plots, three-dimensional isosurfaces, and data slices. 

The second step determines the LCAO coefficients by standard methods for matrix diagonalization. In an Extended Hiickel calculation, this results in molecular orbital coefficients and orbital energies. Ab initio and NDO calculations repeat these two steps iteratively because, in addition to the integrals over atomic orbitals, the elements of the energy matrix depend upon the coefficients of the occupied orbitals. HyperChem ends the iterations when the coefficients or the computed energy no longer change the solution is then self-consistent. The method is known as Self-Consistent Field (SCF) calculation. 

You can interpret the stereochemistry and rates of many reactions involving soft electrophiles and nucleophiles—in particular pericyclic reactions—in terms of the properties of Frontier orbitals. This applies in particular to pericyclic reactions. Overlap between the HOMO and the LUMO is a governing factor in many reactions. HyperChem can show the forms of orbitals such as HOMO and LUMO in two ways a plot at a slice through the molecule and as values in a log file of the orbital coefficients for each atom. 

The two equations couple because the alpha Fock matrix depends on both the alpha and the beta solutions, C and cP (and sim ilarly for the beta Fock matrix). The self-consistent dependence of the Fock matrix on molecular orbital coefficients is best represen ted, as before, via the den sity matrices an d pP, wh ich essen -tially state the probability of describing an electron of alpha spin, and the probability of finding one of beta spin ... 

The orbital coefficients for the MO of energy a for the phenalenyl system described in Fig. 9.7 are as shown below. Predict the general appearance of the NMR spectra of the anion and cation derived from phenalene. 

Fig. 10.3. Orbital coefficients for HOMO and next highest n orbital for some substituted benzenes. (From CNDO/2 ealculations. Ortho and meta eoefficients have been averaged in the case of the unsymmetrical methoxy and formyl substituents. Orbital energies are given in atomic units.)...

Orbital energies are given in electron volts. The size of the circles give relative indication of orbital coefficients at each carbon. 

Fig. 11.14. Orbital coefficients for HOMO and LUMO n MOs of some common 1,3-dipoles. [From K. N. Houk, J. Sims, R. E. Duke, Jr., R. W. Strozier, and J. K. George, J. Am. Chem. Soc. 95 7287 (1973).]...

In contrast, when ot,P-unsaturated multiple bond systems act as dienophiles in concerted [4+2] cycloaddition reactions, they react across the C=C double bond Periselectivity as well as regiochemistry are explained on the basis of the size of the orbital coefficients and the resonance integrals [25S]... 

The atomic orbital contributions for each atom in the molecule are given for each molecular orbital, numbered in order of increasing energy (the MO s energy is given in the row labeled EIGENVALUES preceding the orbital coefficients). The symmetry of the orbital and whether it is an occupied orbital or a virtual (unoccupied) orbital appears immediately under the orbital number. 

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