Nuclear displacement

Since the vibrational eigenstates of the ground electronic state constitute an orthonomial basis set, tire off-diagonal matrix elements in equation (B 1.3.14) will vanish unless the ground state electronic polarizability depends on nuclear coordinates. (This is the Raman analogue of the requirement in infrared spectroscopy that, to observe a transition, the electronic dipole moment in the ground electronic state must properly vary with nuclear displacements from... 

One starts with the Hamiltonian for a molecule H r, R) written out in terms of the electronic coordinates (r) and the nuclear displacement coordinates (R, this being a vector whose dimensionality is three times the number of nuclei) and containing the interaction potential V(r, R). Then, following the BO scheme, one can write the combined wave function [ (r, R) as a sum of an infinite number of terms... 

Gaussian can compute the vibrational spectra of molecules in their ground and excited states. In addition to predicting the frequencies and intensities of spectral lines, the program can also describe the displacements a system undergoes in its normal modes. Put another way, it can predict the direction and magnitude of the nuclear displacement that occurs when a system absorbs a quantum of energy. 

NBO population analysis 195 NBO program 196 Newman projections 290, 292 NewZMat utility xxxvii Windows xlvi nice command xxxviii nitrobenzene 165 NMR properties 21,29, 53, 104 Nobes 117 Norden 218 normal modes 65 normal termination message xlv nuclear displacements 65 numeric second derivatives 61 Nusair 119... 

Consider now the case where the perturbation A is a specific nuclear displacement, A"i Xk + AX t. The derivatives of the one- and two-electron integrals are of two types, those involving derivatives of the basis functions, and those involving derivatives of the operators. The latter are given as... 

Each normal mode of vibration can be described by a normal coordinate Qi which is a linear combination of nuclear displacement coordinates of the molecule. For the symmetric stretching vibration vi of C02, the normal coordinate is of the form... 

Here the matrix V contains the effect of the nuclear displacements therefore the inhomogeneous first term to the right is a driving term the second term to the right is of second order in the driving effect, and could be dropped in calculations. Formally, the solution for the configuration density matrix correction is... 

In Fig. 1 are shown the one-center and two-center components of transition densities for some selected molecules. From these data the actual types of the most soft nuclear displacements will be deter-... 

If the initial ground-state wavefunction (/(q is nondegenerate, the first-order term (i. e., the second term) in Eq. (1) is nonzero only for the totally-symmetrical nuclear displacements (note that g, and (dH/dQi) have the same symmetry). Information about the equilibrium nuclear configuration after the symmetrical first-order deformation will be given by equating the first-order term to zero. 

Note that for large nuclear displacements, for example, distortions exceeding about 0.3-0.4 a.u., the method based on the Lowdin transform-inverse Lowdin transform technique is not recommended. However, for smaller distortions the method discussed above appears to provide a useful approximation. 

It is now fundamental to define the normal coordinates of this vihrational system - that is to say, the nuclear displacements in a polyatomic molecule. Again in the limit of small amplitudes of vibration, the normal coordinates in the form of the vector Q, are related to the internal coordinates by a linear transformation, viz. 

The classical equation of motion1 describing the coherent phonons for a small nuclear displacement Q is that of a driven harmonic oscillator [9,10,15]... 

Optical detection offers the most conventional technique to time-resolve the coherent phonons. It includes four-wave mixing [8], transient reflectivity [9,10] and transmission [7] measurements, as well as second harmonic generation (SHG) [15,32]. Coherent nuclear displacement Q induces a change in the optical properties (e.g., reflectivity R) of the crystal through the refractive index n and the susceptibility y,... 

Fig. 4.8. (a) The measured (squares) and predicted (thick lines) intensity ratio r(q) between harmonics from H2 and that from D2 molecules as functions of harmonic order q. tqm and rcM are the predictions with and without quantum effect for nuclear dynamics see text, (b) The measured (squares) and predicted (thick line) relative phases between harmonics from H2 and those from D2 molecules as functions of harmonic order q. Measuring the relative phases directly corresponds to observing the nuclear motions, i.e., the nuclear displacement of H2 Aii1 2 (right axis) as a function of excursion time r (superior axis). For (a) and (b), vertical and horizontal errors represent SEM for 800 laser shots and those from quantum mechanical uncertainty [27]... 

Another example is provided by the minimum energy coordinates (MECs) of the compliant approach in CSA (Nalewajski, 1995 Nalewajski and Korchowiec, 1997 Nalewajski and Michalak, 1995,1996,1998 Nalewajski etal., 1996), in the spirit of the related treatment of nuclear vibrations (Decius, 1963 Jones and Ryan, 1970 Swanson, 1976 Swanson and Satija, 1977). They all allow one to diagnose the molecular electronic and geometrical responses to hypothetical electronic or nuclear displacements (perturbations). The thermodynamical Legendre-transformed approach (Nalewajski, 1995, 1999, 2000, 2002b, 2006a,b Nalewajski and Korchowiec, 1997 Nalewajski and Sikora, 2000 Nalewajski et al., 1996, 2008) provides a versatile theoretical framework for describing diverse equilibrium states of molecules in different chemical environments. 

An important, implicit assumption is that the potential energy function holds only if the nuclear displacement from a position of equilibrium is not too large. Hence, highly strained molecules are not the best targets. Whereas the MM calculations give the steric energy distribution within a molecule, which no other computational method can provide, one must be aware that this distribution is correct for the model of the molecule, not necessarily for the molecule itself (110-... 

With the availability of lasers, Brillouin scattering can now be used more confidently to study electron-phonon interactions and to probe the energy, damping and relative weight of the various hydro-dynamic collective modes in anharmonic insulating crystals.The connection between the intensity and spectral distribution of scattered light and the nuclear displacement-displacement correlation function has been extensively discussed by Griffin 236). 

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