Dipole moment gradient

Comparisons of the correlation functions calculated quantum mechanically and semiclassically, like those presented in Fig. 6.2, show that the correction due to the dipole moment gradient included in (6.34) sometimes improves the accuracy especially for short propagation times. This correction affects not only the amplitude of the correlation function oscillation, but also its frequency and distortions due to the presence of high harmonics in the spectrum. An analysis of the spectrum of the correlation function indicates that including this correction in the formula enables additional quantum effects to be taken into account. 

The controlling field and its spectra, calculated both semiclassically (thin and dash lines) and quantum mechanically (bold fines), are shown in Fig. 6.5. The frequencies of the main components of the optimal field spectra are the same for all three cases. However, the optimal field obtained quantum mechanically and semiclassically with formula (6.34) contains second harmonics. This means that additional quantum effects are taken into account by the correction from the dipole moment gradient included in (6.34). 

Infrared investigations of the Stark effect can afford data on the molecular properties of adsorbates (polarizability, dipole moment gradients, and polarizability gradients). However such studies are at their very beginning and require to know the actual value of the electric field at the interface. Data on the potentials of zero charge for solid electrode materials would be welcome. 

Whereas the dipole moment gradient (the atomic polar tensor) is well defined and it is the same quantity that determines conventional infrared absorption spectra (see the chapter on vibrational spectroscopy), the gradient of the magnetic dipole moment is zero within the Born-Oppenheimer approximation. This is due to the fact that the magnetic dipole moment for a closed-shell molecule is quenched (since it corresponds to an expectation value of an imaginary operator), making the rotational strength in Eq. 2.150 zero. 

Before proceeding with some more details regarding the evaluation of the magnetic dipole moment gradient (the atomic axial tensor, Mj in the context of a nonadiabatic wavefunction, let us first note that it is from a computational point of view advantageous to derive equations with respect to Cartesian displacements of a nucleus A rather than with respect to the normal modes. These two different representations of the nuclear motion are related by a linear transformation... 

Kjaer, H. and Sauer, S. P. A. (2009). On the relation between the non-adiabatic vibrational reduced mass and the electric dipole moment gradient of a diatomic molecule. Theo. Chem. Acc., 122, 137-143. 

There are higher multipole polarizabilities tiiat describe higher-order multipole moments induced by non-imifonn fields. For example, the quadnipole polarizability is a fourth-rank tensor C that characterizes the lowest-order quadnipole moment induced by an applied field gradient. There are also mixed polarizabilities such as the third-rank dipole-quadnipole polarizability tensor A that describes the lowest-order response of the dipole moment to a field gradient and of the quadnipole moment to a dipolar field. All polarizabilities of order higher tlian dipole depend on the choice of origin. Experimental values are basically restricted to the dipole polarizability and hyperpolarizability [21, 24 and 21]. Ab initio calculations are an imponant source of both dipole and higher polarizabilities [20] some recent examples include [26, 22] ... 

Pulay P, FogarasI G, Pang F and Boggs J E 1979 Systematic ab initio gradient calculation of molecular geometries, force constants and dipole moment derivatives J. Am. Chem. Soc. 101 2550... 

In addition to total energy and gradient, HyperChem can use quantum mechanical methods to calculate several other properties. The properties include the dipole moment, total electron density, total spin density, electrostatic potential, heats of formation, orbital energy levels, vibrational normal modes and frequencies, infrared spectrum intensities, and ultraviolet-visible spectrum frequencies and intensities. The HyperChem log file includes energy, gradient, and dipole values, while HIN files store atomic charge values. 

For a quantum mechanical calculation, the single point calculation leads to a wave function for the molecular system and considerably more information than just the energy and gradient are available. In principle, any expectation value might be computed. You can get plots of the individual orbitals, the total (or spin) electron density and the electrostatic field around the molecule. You can see the orbital energies in the status line when you plot an orbital. Finally, the log file contains additional information including the dipole moment of the molecule. The level of detail may be controlled by the PrintLevel entry in the chem.ini file. 

Here q is the net charge (monopole), p, is the (electric) dipole moment, Q is the quadrupole moment, and F and F are the field and field gradient d /dr), respectively. The dipole moment and electric field are vectors, and the pF term should be interpreted as the dot product (p F = + EyPy + Ez z)- "I e quadrupole moment and field... 

The calculated dipole moment is remarkably insensitive to the size of the basis set. Note that the SVWN value in this case is substantially better than BLYP and BPW91, i.e. this is a case where the theoretically poorer method provides better results than the more advanced gradient methods. Inclusion of exact exchange again improves the performance, and provides results very close to the experimental value, even with quite small basis sets. 

The second derivatives can be calculated numerically from the gradients of the energy or analytically, depending upon the methods being used and the availability of analytical formulae for the second derivative matrix elements. The energy may be calculated using quantum mechanics or molecular mechanics. Infrared intensities, Ik, can be determined for each normal mode from the square of the derivative of the dipole moment, fi, with respect to that normal mode. 

Pulay, P., G. Fogarasi, F. Pang, and J. E. Boggs. 1979b. Systematic Ab Initio Gradient Calculation of Molecular Geometries, Force Constants, and Dipole Moment Derivatives. J. Am. Chem. Soc. 101, 2550-2560. 

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