Vibrational probability density

The nature of the intemuclear distance, r, is the object of interest in this chapter. In Eq. (5.1) it has the meaning of an instantaneous distance i.e., at the instant when a single electron is scattered by a particular molecule, r is the value that is evoked by the measurement in accordance with the probability density of the molecular state. Thus, when electrons are scattered by an ensemble of molecules in a given vibrational state v, characterized by the wave function r /v(r), the molecular intensities, Iv(s), are obtained by averaging the electron diffraction operator over the vibrational probability density. 

We performed a series of theoretical studies on pump-probe diffraction patterns with a twofold objective the first aim is to evaluate the effect of electronic and vibrational excitation on electron diffraction patterns, compared to that of structural rearrangements that are the primary goal for observation in structural dynamics measurements. Secondly, we wish to explore to what extent electronic and vibrational probability density distributions are observable using the pump-probe electron diffraction methodology. Previously we have discussed the effect of electronic excitation in atomic systems,[3] and the observability of vibrational excitation in diatomic and triatomic systems.[4,5] We have now extended this work to the 8-atomic molecule s-tetrazine (C2H2N4). 

The probability density plots for the first three vibrational states ( o, < i and ( 2 of the IBr molecule are plotted in Fig. 3 and the probability density profile... 

Fig. 3 Probability density for the Fig. 4 Probability density for the vibrational eigenstates (/>o, and (j)2 optimal superpositions and tp ...

The probability density for the ground vibrational level peaks at 4.66 a.u., that for at 4.8 a.u. and for at 4.58 a.u. The optimal wave-function seems to maximize flux out of channel 2 (I -I- Br ) by localizing the probability density to the left of that given by the (po wavefiinction and... 

Figure 12. Potential energy contour plots for He + I Cl(B,v = 3) and the corresponding probability densities for the n = 0, 2, and 4 intermolecular vibrational levels, (a), (c), and (e) plotted as a function of intermolecular angle, 0 and distance, R. Modified with permission from Ref. 40. The I Cl(B,v = 2/) rotational product state distributions measured following excitation to n = 0, 2, and 4 within the He + I Cl(B,v = 3) potential are plotted as black squares in (b), (d), and (f), respectively. The populations are normalized so that their sum is unity. The l Cl(B,v = 2/) rotational product state distributions calculated by Gray and Wozny [101] for the vibrational predissociation of He I Cl(B,v = 3,n = 0,/ = 0) complexes are shown as open circles in panel (b). Modified with permission from Ref. [51].

From the individual contributions of the modes to the msd along the c-axis ( 6 pm ) and along the a-axis ( 8 pm ), the corresponding calculated molecular Lamb-Mossbauer factors for the c-cut crystal (/Lm,c = 0.90) and for the a-cut crystal = 0.87) were derived. Comparison with the experimental /-factor, i.e., / P = 0.20(1) and/ N> = 0.12(1) [45], indicates that by far the largest part of the iron msd must be due to intermolecular vibrations (acoustic modes) of the nitroprusside anions and its counter ions. This behavior is reflected in the NIS spectrum of GNP by the considerable onset of absorption probability density below 30 meV in Fig. 9.36a. 

A molecule is composed of positively charged nuclei surrounded by electrons. The stability of a molecule is due to a balance among the mutual repulsions of nuclear pairs, attractions of nuclear-electron pairs, and repulsions of electron pairs as modified by the interactions of their spins. Both the nuclei and the electrons are in constant motion relative to the center of mass of the molecule. However, the nuclear masses are much greater than the electronic mass and, as a result, the nuclei move much more slowly than the electrons. Thus, the basic molecular structure is a stable framework of nuclei undergoing rotational and vibrational motions surrounded by a cloud of electrons described by the electronic probability density. 

As has been mentioned above, the inclusion of basis functions (49) with high power values, nik, is very essential for the calculations of molecular systems. It is especially important for highly vibrationally excited states where there are many highly localized peaks in the nuclear correlation function. To illustrate this point, we calculated this correlation function (it corresponds to the internuclear distance, r -p = r ), which is the same as the probability density of pseudoparticle 1. The definition of this quantity is as follows ... 

Figure 3.1 Energy levels and wave functions of harmonic oscillator. Heavy line bounding potential (3.2). Light solid lines quantum-mechanic probability density distributions for various quantum vibrational numbers see section 1.16.1). Dashed lines classical probability distribution maximum classical probability is observed in the zone of inversion of motion where velocity is zero. From McMillan (1985). Reprinted with permission of The Mineralogical Society of America.

Scheme 4.2 Bond energy as a function of hydrogen position (black solid line), assuming identical pff, values for the donor and acceptor, relative to the lowest vibrational energy level of the hydrogen atom (highlighted by a dotted line), (a) A standard, symmetric hydrogen bond (b) the corresponding low-barrier hydrogen bond (LBHB). The red line represents the probability density function [27, 28].

Fig. 21.3 Probability density function for different vibrational levels of normal vibrations. Dependence of energy change on the value of torsion angle corresponding to the out-of-plane deformation of pyrimidine ring in uracil. MP2/6-31G(d,p) level calculations...

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