Wave function, viii

Bernal, M. J. M., and S. F. Boys. 1952a. Electronic wave functions. VIII. A calculation for the ground states Na, Ne, F . Philosophical Transactions of the Royal Society of London 245 139-154. 

Benchmark calculations with correlated molecular wave functions. VIII. Bond energies and equilibrium geometries of the CH , and C2Hn n = 1-4) series ,... 

In Chapter VIII, Haas and Zilberg propose to follow the phase of the total electronic wave function as a function of the nuclear coordinates with the aim of locating conical intersections. For this purpose, they present the theoretical basis for this approach and apply it for conical intersections connecting the two lowest singlet states (Si and So). The analysis starts with the Pauli principle and is assisted by the permutational symmetry of the electronic wave function. In particular, this approach allows the selection of two coordinates along which the conical intersections are to be found. 

VIII. Permutational Symmetry of Rotational Wave Function... 

As was shown in the preceding discussion (see also Sections VIII and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in + and Xu electronic states antisymmetric for odd J values in + and electronic states symmetric for odd J values in Xj and X+ electronic states and antisymmetric for even J values in X and X+ electronic states. Note that the vibrational ground state is symmetric under permutation of the two nuclei. The most restrictive result arises therefore when the nuclear spin quantum number of the individual nuclei is 0. In this case, the nuclear spin function is always symmetric with respect to interchange of the identical nuclei, and hence only totally symmetric rovibronic states are allowed since the total wave function must be symmetric for bosonic systems. For example, the 12C nucleus has zero nuclear spin, and hence the rotational levels with odd values of J do not exist for the ground electronic state ( X+) of 12C2. 

VIII. PERMUTATIONAL SYMMETRY OF ROTATIONAL WAVE FUNCTION... 

TABLE VIII The Symmetry Properties of Wave Functions of 6Li3 Electronically First-Excited State in S3 Permutation Group ... 

The perturbation corrections to the wave function for Z = Z2 are given in Table VII, Appendix C, up to fifth order and the corrections to the energy are given in rational form to fifth order in Table VIII, Appendix C. More extensive tabulations in floating point form have been given by Clay (1979). 

In Section VII we describe how expressions for geometrical derivatives of molecular properties may be derived using the formalism developed for energy derivatives. We also discuss alternative definitions that may be used to determine geometrical derivatives of molecular properties for wave functions which do not satisfy the Hellmann-Feynman relationship for the property in question. Finally, in Section VIII we describe how translational and rotational symmetries may be used to reduce the cost of derivative calculations. 

Theoretical work is concerned with the relative contributions of (5-VII) and (5-VIII) and leads to the conclusion that the resonance represented by (5-VII) is of importance only for the strongest, shortest bonds. It has been estimated, for example, that in an O—H-0 bond with the O—O distance 2.78 A, and the O—H distance 1.0 A (fairly typical parameters), structure (5-VIIb) appears in the over-all wave function to the extent of only about 4%. Thus, it is believed that most hydrogen bonds are basically electrostatic but this then raises another question. If unshared electron pairs are concentrated along the direction of hybrid orbitals, will the proton approach the atom Y preferentially along these directions In other words, does the proton see the atom Y as a structureless concentration of negative charge or as an atomic dipole The answer to this question is not entirely clear cut, because in most cases where the angle 9 in (5-IX) is in accord with the latter idea it is possible to... 

Figure 5 Computational results. The average survival probability, P, and the average wave-function extent, L, as a function of time, on a log/log scale, for different coupling strength 4> -Note that the scale for log(L) is offset by 0.5. The labels on the plot indicate the following values of log(7 ) (i) 3.8 (ii) 3.3 (iii) 2.8 (iv) 2.3 (v) 1.8 (vi) 1.3 (vii) 0.8 (viii) 0.3 (ix) -0.2. For each curve the time has been multiplied by the harmonic frequency v.

As was shown in the preceding discussion (see also Sections VIII and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in and electronic states antisymmetric for odd J values in and electronic states symmetric for odd J values in S7 and electronic... 

As we have indicated, the CASVB procedures may also be used for the fully-variational optimization of modem VB wave functions. This allows us to investigate for the ground states of these model polyene systems the extent to which the modem VB representations obtained from the CASSCF wave functions by means of the overlap criterion resemble the corresponding flilly-variational spin-coupled results. The energies from such flilly-variational calculations, which include full optimization of the core orbitals, are labelled SC in Table VIII. All of the CASVB wave functions were found to be symmetry pure, without the need to impose any symmetry relationships amongst the orbitals. 

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