Wave functions methane

Electron Nuclear Dynamics (48) departs from a variational form where the state vector is both explicitly and implicitly time-dependent. A coherent state formulation for electron and nuclear motion is given and the relevant parameters are determined as functions of time from the Euler equations that define the stationary point of the functional. Yngve and his group have currently implemented the method for a determinantal electronic wave function and products of wave packets for the nuclei in the limit of zero width, a "classical" limit. Results are coming forth protons on methane (49), diatoms in laser fields (50), protons on water (51), and charge transfer (52) between oxygen and protons. 

In order to calculate the orbitals for a methane molecule, the four Lv functions of the four hydrogen atoms and the functions 2s, 2px, 2py and 2pz of the carbon atom are combined to give eight wave functions, four of which are bonding and four of which are antibonding. The four bonding wave functions are ... 

Jahn, H. A. (1938), A New Coriolis Perturbation in the Methane Spectrum. I. Vibrational-Rotational Hamiltonian and Wave Functions, Proc. Roy. Soc. A 168,469. 

Use of this wave function with Eq. (19) then yields a theoretical value for Ahh in CH4 of 12.5 cps which is to be compared with the experimental value of 12.3 0.6 cps. Valence bond calculations of this nature have successfully accounted for the variation with H—C—H angle of the proton-proton coupling constants in substituted methanes (45) (Fig. 3), for the difference between AHwcls and AHwran across double bonds in ethylenes, and for the difference between AHH [Pg.241]

Mathematically, the formation of sp3 or tetrahedral orbitals for methane is more complicated but not basically different. The results are four equivalent hybrid orbitals, each containing one part s to three parts p in each wave function, directed to the corners of a tetrahedron. As in the case of sp hybrids, the hybridization of s and p has... 

It is known that, in the MO framework, the nondynamical electron correlation is accounted for by means of a so-called CASSCF calculation, which is nothing else than a full Cl in a given space of orbitals and electrons, in which the orbitals and the coefficients of the configurations are optimized simultaneously. If the active space includes all the valence orbitals and electrons, then the totality of the nondynamical correlation of the valence electrons is accounted for. In the VB framework, an equivalent VB calculation, defined with pure AOs or purely localized hybrid atomic orbitals (HAOs), would involve all the covalent and ionic structures that may possibly be generated for the molecule at hand. Note that the resulting covalent—ionic VB wave function would have the same dimension as the valence—CASSCF one (e.g., 1764 VB structures for methane, and 1764 MO SCF configurations in the CASSCF framework). 

The GVB and SC methods provide wave functions that are, of course, much more compact than the corresponding valence—CASSCF one (e.g., only 14 spin-coupling modes for methane with the SC method, and a single one with the GVB method). Owing to this difference in size, the GVB and SC methods cannot be expected to include the totality of the nondynamical correlation, even if these two methods treat well, by definition, the left—right correlation for each bond of the molecule. Physically, this is because the various local ionic... 

Molecular orbital calculations of the DME steam reforming and the methane production increased by C02 injection to coal seam was performed by the DV-Xa method. The equation for the wave function i, (r) at a space point r in atomic units is denoted as... 

Central Field Model of Tetrahedral and Octahedral Molecules.—The idea is very simple, and has long been exploited in the sense of one-centre expansions of molecular wave functions in a molecule like CH4. However, to exemplify the way the density description can afford answers to questions (i)—(iii) above, we take the model literally in which, in methane for example, we smear the protons uniformly over the surface of a sphere of radius R, equal, in the methane example to the C—H bond length. 

This picture has the advantage over that in Fig. 1.12 that the C H bonds do have a direct relationship with the lines drawn on the conventional structure (Fig. 1.18b). The two descriptions of the overall wave function for methane lead to identical electron distributions hybridisation involves the same approximations, and the taking of s and p orbitals in various proportions and various combinations, as those used to arrive at Fig. 1.12. 

Ethylene has the well-known classical >2/1 structure with a barrier to rotation. The next in complexity of the simple hydrides is the methyl radical CH3. The obvious (sp2) planar arrangement can only accommodate six of the seven valence electrons. The electronic configuration of this molecule can therefore not be described in terms of either atomic wave functions or hybrid orbitals. An alternative approach is to view the structure of the methyl radical as a reduced-symmetry form, derived from the structure of methane, to be considered next. 

The main effect is already taken into account if symmetry numbers are included in the densities of states. The symmetry number is a correction to the density of states that allows for the fact that indistinguishable atoms occupy symmetry-related positions and these atoms have to obey the constraints of the Pauli principle (i.e. the wave function must have a definite symmetry with respect to any permutation), whereas the classical density of states contains no such constraint. The density of states is reduced by a factor that is equal to the dimension of the rotational subgroup of the molecule. When a molecule is distorted, its symmetry is reduced, and so its symmetry number changes by a proportion that is equivalent to the number of indistinguishable ways in which the distortion may be produced. For example, the rotational subgroup of the methane molecule is T, whose dimension is 12, whereas the rotational subgroup of a distorted molecule in which one bond is stretched is C3, whose dimension is 3. The ratio of these symmetry numbers, 4, is the number of ways in which the distortion can occur, i.e. the reaction path degeneracy. 

In 1969, Thomas published two papers [11,12] in which a molecular structure theory was developed without invoking the Bom-Oppenheimer approximation. In these publications and two further papers published in 1970 [13,14], Thomas studied methane, ammonia, water and hydrogen fluoride adding the kinetic energy operators of the protons to the electronic hamiltonian and using Slater-type orbitals centered on the heavier nuclei for the protonic wave functions. Over the years, a number of authors [15-23] have attempted the development of a non-Bom-Oppenheimer theory of molecular structure, but problems of accuracy and/or feasibility remain for applications to arbitrary molecular systems. 

Such a transformation can be used for relocalizing a given set of delocalized molecular orbitals in conformity with the chemical formula. For instance, the occupied orbitals of methane can be transformed into orbitals very close to simple two-center MO s constructed from tetrahedral sp3 hybrid orbitals and Is hydrogen orbitals 24,25,26) a. unitary transformation can hardly modify the wave function, except for an immaterial phase factor therefore, it leads to a description which is as valid as that in terms of the canonical delocalized Hartree-Fock orbitals. Of course, the localization obtained in this way is not perfect, but it is usually much better than is often believed. In the case of methane, the best localized orbitals are uniquely determined by symmetry 27> for less symmetric molecules one needs a criterion for best localization 28 29>, a problem on which we shall not insist here. A careful inspection reveals that there are three classes of compounds ... 

---