Wavefunctions amplitude

This factor is responsible for an increase in the correlated wavefunction amplitude and hence in the charge distribution of both electrons when these electrons are on opposite sides of the nucleus ( 12 = 180° see Fig. 1.2(h)). 

All the alternative variants of the MPn may be implemented using a relaxed density matrix or a unrelaxed density matrix, in analogy with the Cl solvation methods. In the first case the correlated electronic density is computed as a first derivatives of the free energy, while in the second case only the MPn perturbative wavefunction amplitudes are necessary. 

Fig. 7. Contour plot of the a2n MO. The values of the wavefunction amplitudes are not given611...

It is perhaps not immediately clear how one may go about solving the Tj and T2 amplitude equations given in Eqs. [152] and [153] for the individual amplitudes, tf and Iff- . A simple rearrangement of the equations, however, provides a more palatable form of these expressions that leads to a simple iterative approach for determining the coupled cluster wavefunction amplitudes. For example, the first few terms of Eq. [152] may be written as... 

The coupled cluster and configuration interaction equations presented thus far in this chapter have implicitly used spin-dependent molecular orbitals for their definitions of determinants, integrals, and wavefunction amplitudes. 

The fundamental idea is that the pump and probe pulses create wavepackets, which evolve on the excited state potential surface. Interference between the excited state wavefunction amplitudes created by the two pulses affects the population transferred to the excited state. The population that is measureable in a typical incoherent experiment (spontaneous fluorescence, field ionization, excitation to a different excited state by a nanosecond pulsed laser) is proportional... 

In the periodic cluster approach, the size of the supercell influences the number of orbitals and thus the discrete representation of the band structure. Using this model to describe localized defects such as vacancies introduces an additional difficulty because now the spatial extent of the vacancy wavefunction must be related to the size of the supercell. A 64-atom supercell was found to be too small to adequately describe the vacancy wavefunction, the amplitude of the wave function of a monovacancy at the cell boundary is still V4 of the maximum value. This means that the wavefunction has a considerable overlap with the vacancy wavefunctions of the neighboring cells. This artificial defect-defect interaction leads to a dispersion of about 0.8 eV for the vacancy states. When a 216-atom supercell is considered, the wavefunction amplitude at the cell boundary is only Vs of the maximum value and the dispersion is less than 0.2 A supercell of at... 

In the case of the mixture He + He, Q corresponds to the wavefunction amplitude of the hyperfluid component n to the concentration of He(x), and g = ps — P4 with being beyond experiment in this case. 

Fig. 2.8 Contour plots of the wavefunction amplitudes for the highest occupied molecular orbital HOMO) and lowest unoccupied molecular orbital (LUMO) of 3-methylindole. Positive amplitudes are indicated by solid lines, negative amplitudes by dotted lines, and zero by dot-dashed lines. The plane of the map is parallel to the plane of the indole ring and is above the ring by o as in Fig. 2.7, panels D and E. The contour intervals for the amplitude are 0.05ao. Small contributions from the carbon and hydrogen atoms of the methyl group are neglected. The straight black lines indicate the carbon and nitrogen skeleton of the molecule. The atomic coefficients for the molecular orbitals were obtained as described by Callis [37-39]. Slater-type atomic orbitals (Eq. 2.40) with with f = 3.071/A (1.625/ao) and 3.685/A (1.949/ o) were used to represent C and N, respectively...

Fig. 4.3 Wavefunction amplitudes (A, B) and probability densities (C, D) for two pure states and a superposition state. The dotted and dashed curves are for the first two eigenstates of an electron in a one-dimensional box of unit length, as given by Eq. (2.23) with n= or 2 and respectively). The solid curves are for the superposition 2 Ta + l b)- (A, C) The signed amplitudes of the wavefunctions and the probability densities at time t = 0, when all the wavefunctions are real. (B, D) The corresponding functions at time t = (ll2)h/ Ei, — Ea), where Ea and Ef, are the energies of the pure states...

The idea of the superposition of waves is useful to bear in mind when we deal with chemical bonding the individual waves are akin to the AOs and their superposition gives the MOs of sets of atoms. We should expect regions where the AOs are in phase to result in a relatively large wavefunction amplitude, while regions where opposite phases come together will give a low, or even zero, amplitude. 

Fig, 5.13 Results of a linear-combination-of-atomic-orbitals calculation of the effect of photoexcitation on TT-electron distribution in 1,4,5,8-tetramino anthraquinone. The size of the black gray) circles gives the positive (negative) wavefunction amplitude for the electrons localized in the Pj-orbitals [31]... 

A graph of the wavefunction s amplitudes is given in Figure 7.11. In area I, the value of the square of the wavefunction amplitude is depicted. As was shown earlier for free particles, this value does not depend on coordinates and, within the given area, is constant. In area II the solution is not periodic but exponential (as was obtained when solving a potential... 

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